Vector Bundles VOLUME1 Foundarions and Stiejel- Whitney Classes
This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUEL EILENBERG AND HYMAN BASS A list of recent titles in this series appears at the end of this volume.
Vector Bundles VOLUME1 Foundations and Stiejel- Whitney Classes
Howard Osborn Depurtment of Muthemutics tin iv ers ity of IIlino is u t tirhunu - Chumpuign tirbunu, lllino is
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1982
ACADEMIC PRESS A
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u/ Huitouir Biute J o ~ u n o ~ r t Publishus h.
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7DX
Linrary o f Corgress Cataloqiy in Fualication Data Osborn, Howard. Vector bundles. (Pure and applied mathematics;
)
Incluaes oiDliograpnica1 references and index. Contents: v. 1. Foundations and Stiefel-wnitney classes. I. Vector ounales. 2. Cnaracteristic classes. I. Title. 11. Series: Pure and applied mathematics (kaoemic Press) ; PA3.P8 LQA612.631 ISBN 0-12-529301-1
510s [514'.224] ( v . I)
82-163Y
PRINTED IN THE UNITED STATES OF AMERICA 82 83 84 85
9 8 7 6 5 4 3 2 1
To Jean, and lo our children, Mark, Stephen, Adrienne, and Emily
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Contents
xi
Prejuce
i
Introduction Chapter 1 Base Spaces 0. I. 2. 3. 4. 5. 6. 7. 8. 9.
Introduction The Category of Base Spaces Some Simplicial Spaces More Simplicial Spaces Weak Simplicial Spaces CW Spaces Smooth Manifolds Grassmann Manifolds Some More Coverings The Mayer-Vietoris Technique 10. Remarks and Exercises
Chapter 11 0. 1, 2. 3.
5
5 8 II 15 19 23 34 39 41 41
Fiber Bundles
Introduction Fibre Bundles and Fiber Bundles Coordinate Bundles Bundles over Contractible Spaces 4. Pullbacks along Homotopic Maps 5. Reduction of Structure Groups
57 60 66 13 15 80
vii
...
Contents
Vlll
86 93 91
6 . Polar Decompositions 7. The Leray-Hirsch Theorem 8. Remarks and Exercises
Chapter III Vector Bundles Introduction Real Vector Bundles Whitney Sums and Products Riemannian Metrics Sections of Vector Bundles Smooth Vector Bundles Vector Fields and Tangent Bundles Canonical Vector Bundles The Homotopy Classification Theorem More Smooth Vector Bundles 10. Orientable Vector Bundles I I . Complex Vector Bundles 12. Realifications and Complexifications 13. Remarks and Exercises 0. I. 2. 3. 4. 5. 6. 7. 8. 9.
Chapter IV 0. I. 2. 3. 4. 5. 6. 7. 8. 9.
105 105 108 112
I I6 1 I9 126 140 147 158 160 169 172 178
2/2 Euler Classes
Introduction The 2/2 Thom Class Ute H"(E, Ec) Properties of 212 Thom Classes E / 2 Euler Classes Gysin Sequences and H*(RPa ; 212) The Splitting Principle Axioms for hi2 Euler Classes Z/2 Euler Classes of Product Line Bundles The 212 Thom Class VcE H"(P,, , Ps) Remarks and Exercises
,
191
192 194 196 199 201 205 206 209 21 1
Chapter V Stiefel-Whitney Classes 0. I. 2. 3. 4. 5. 6. 7.
Introduction Multiplicative 212 Classes Whitney Product Formulas Orientability The Rings H*(Cm(Wm); Z/2) Axioms for Stiefel-Whitney Classes Dual Classes Remarks and Exercises
215 216 219 226 227 233 237 238
ix
Contents
Chapter VI
Unoriented Manifolds
0. Introduction I . H / 2 Fundamental Classes 2. h / 2 Poincare-Lefschetz Duality 3. Multiplicative 212 Classes of RP" 4. Nonimmersions and Nonembeddings 5. Stiefel-Whitney Numbers 6. Stiefel-Whitney Genera 1. HI2 Thorn Forms 8. The Thorn-Wu Theorem 9. Remarks and Exercises
249 25 1 256 258 260 265 274 216 285 281
References
303
Glossary of Notation Index
351 359
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Preface
Vector bundles provide the background for the proper formulation of many classical and modem problems of differential topology, whose solutions via characteristic classes and K-theory are among the mathematical triumphs of the past few decades. These volumes are an introduction to vector bundles, characteristic classes, and K-theory and to some of their applications : Volume 1 : Foundations and Stiejel- Whitney Classes Volume 2 : Euler, Chern, and Pontrjagin Classes Volume.3 : K - Theory and Integrality Theorems The exposition is based on various courses the author has presented at the University of Illinois, with a one-semester course in singular homology and cohomology as the only prerequisite; no further background is necessary. Appropriate portions of differential topology, Lie groups, and homotopy theory, for example, are explicitly introduced as needed, with complete proofs in most cases; in a few exceptional cases, specific references are substituted for proofs. Each chapter ends with illustrative remarks and exercises. The remarks constitute a guide to much of the literature on vector bundles, characteristic classes, and K-theory from 1935 to 1981. This first volume is designed both for self-study and as a classroom text about real vector bundles and their 2/2 characteristic classes. For classroom use, one should perhaps not dwell on the background details of Chapter I, whose results are often taken for granted in any event; only some scattered definitions and the Mayer-Vietoris technique itself need any special emphasis. The remaining chapters can then easily be adapted for presentation xi
xii
Preface
either in a one-quarter course or in a one-semester course, with very few omissions in the latter case. The author is indebted to many mathematical friends for their direct and indirect contributions to this work: to Professor S . S. Chern for an exciting and informative 1953 graduate course based on the newly published Steenrod [4]; to Professor John Milnor, who has been an inspiration since shared undergraduate days at Princeton University, and who provided Milnor [3] ; to Professor Rene Thorn, whose pithy and patient explanations turned mathematical abstractions into virtually tactile objects ; to Professor Emery Thomas, whose clearly delivered lectures (Thomas [2]) rekindled the author’s interest in vector bundles; to Professor Peter Hilton and Professor Raoul Bott, who provided much encouragement and moral support during early stages of the project; to Professor Felix Albrecht and Professor Philippe Tondeur, who read large portions of the completed typescript and provided many constructive suggestions; to Professor Wu Wen-Tsun, who addressed himself to the details of Chapter VI with the astonishing enthusiasm one expects only of someone discovering a new world (which he had in fact helped to create); to Professor Hassler Whitney, who created vector bundles in the first place, and who warmly shared several days of companionship and chamber music; and to many other mathematical friends and colleagues who helped clarify questions as they arose. The author is equally grateful to Mrs. Doris Bartle for her assistance in proofreading the bibliography and to the staff of Academic Press for their cooperation during the production of the book. I thank my wife, Jean, and our children, Mark, Stephen, Adrienne, and Emily, for their patience, understanding, and love throughout the entire project. University ?f Illinois at Urbana-Champaign June, I982
HOWARDOSRORN
Introduction
In the first half of this volume both real and complex vector bundles are introduced and classified, and many examples are given which will play a role in later applications. The second half of the volume concentrates exclusively on certain 2/2 cohomology classes assigned to real vector bundles; there are several applications to classical problems in differential topology. (Complex vector bundles reappear in the next volume, where further cohomology classes are assigned to both real and complex vector bundles.) Let E X be a continuous map from a topological space E (the total space) onto a topological space X (the base space). Suppose that there is an open covering .( Uil i E I } of X such that each inverse image 7c-'(Ui) c E is homeomorphic to the product U ix R". The homeomorphisms n-'( U , )5 U ix R" need not be unique; however, they will belong to a certain family of homeomorphisms whose composition with the first projection U i x R" + U iis the restriction of 7c to 7c-'(Ui).If an intersection U i n U j of two sets in the covering U iI i E I } is nonempty, then for each x E U i n U j one has a composition ( x } x R" n- ' ( { x } ) (x) x R", hence a homeomorphism R" -+ R". If all the latter homeomorphisms result from the usual action GL(m,R) x R" R"' of the general linear group GL(m,R), then the projection E 3 X is a coordinate bundle representing a real vector bundle 5 of' rank m over the base space X . The bundle 5 is also called a real m-plane bundle over X , itsjber being R". For example, any smooth m-dimensional manifold X is covered by open coordinate patches U i , each of which has coordinate functions x', . . . , x" -+
1
2
Introduction
used to describe a homeomorphism from U ito an open set in R"; the partial differentiations a / ? x ' , . . . , d/ax" at each point of U i form a basis of a vector space R". Let U j be another open coordinate patch on X such that the intersection U i n U j is nonempty, and let y', . . . , y"' be coordinate functions on U j . Then over each point of U in U j one has a/dyj = ( i J x ' / d y j )a/dx' . . . + (dx"/dyJ)a/dx" for j = I , , . . , m, so that the two copies of R" over each point of U i n U j are related by the usual linear action of the jacobian matrix (dxi/ayJ)E GL(m,R). This suggests that X has a well-defined real m-plane bundle associated to its differentiable structure: the tangent bundle
+
T ( X )Of
x.
Now let 2/2 be the field of integers modulo 2, and let 2/2 [[t]] be the ring of formal power series over 2/2. To each element f ( t ) E 2/2 [ [ t ] ] with leading term 1 E 2/2, and to each real vector bundle over a base space X , there is a naturally defined cohomology class us(<) in the direct product H * * ( X ; 2/2) of the singular cohomology modules H P ( X ; 2/2). In the special ) total Stiefel- Whitney cluss case j { t ) = 1 + t E Z/2[[t]] the class ~ ~ (is5the w ( < ) of <,which can be used to compute any of the other classes u,(<). For g ( r ) = ( 1 t ) - ' = 1 t + t Z + . . . E 2/2 [[t]] the class u,(<)E H * * ( X ; 2/2) is the (total)dual StieJeI-Whitney class W(<)of 5. Here is one of the many classical applications of such 2 / 2 characteristic classes. Let X be a smooth m-dimensional manifold, and let W ( t ( X ) )E H * ( X ; Z/2) [ = H * * ( X ; 2/2)] be the dual Stiefel-Whitney class of its tangent bundle T ( X ) .Then a necessary condition for the existence ofa smooth (proper) embedding of X into the euclidean space R 2 m - pis that W ( T ( X ) )vanish in each summand H 4 ( X ;2/2) for which q 2 m - p . Using this criterion one easily constructs a smooth closed m-dimensional manifold X with no smooth embedding in R2m-;r(m' , where a(m) is the number of 1's in the dyadic expansion of m.
<
+
+
One can define vector bundles 5 over arbitrary topological spaces. However, many useful properties of vector bundles depend upon further restrictions; in this book those restrictions are imposed directly on the base spaces themselves. The "category .94 of base spaces" is described in Chapter I, along with the category W of spaces of homotopy types of CW spaces and the category .U of smooth manifolds; since there are inclusions .Ic w' c 33 the restriction to vector bundles over base spaces X E .94 is not especially stringent. (For those specialists who prefer to consider numeruble bundles over arbitrary base spaces, any bundle over a space X E 93 is automatically numerable; numerability is discussed in Remark 11.8.4.) Let G x F + F be any effective action of a topological group G on a topological space F. If one substitutes F , G, and G x F F for R",GL(m,R ) , and GL(m,R) x R" R" in the earlier sketch of a definition of real m-plane -+
-+
3
Introduction
bundles, the result is afibrr bundle 5 withjber F and structure group G . For example, for each real m-plane bundle 5 there is a corresponding projectiue bundle P , , whose fiber is the real projective space RP"- and whose structure group is the projective group PGL(m, R). Fiber bundles (over base spaces X E 9 )are introduced in Chapter 11, which has several results needed in later chapters. One of the main results of Chapter I1 is that if the structure group of a fiber bundle is a Lie group G, then one can always replace G by any maximal compact subgroup H c G ; for example, the structure group GL(m,R) of any real m-plane bundle can always be replaced by the orthogonal group O(m) c GL(m,R). Another major result of Chapter I1 is the LerayHirsch theorem, which asserts for certain coordinate bundles E X that H * ( E ; 272) is a free H * ( X ; Z/2)-module; this is a crucial step in the later construction of E/2 characteristic classes. If X Y is any map in the category 9 of base spaces, and if q is any real m-plane bundle over Y, the pullbuck f ' q is a real m-plane bundle over X which is uniquely defined by q and the homotopy class [f] off. One of the main results of Chapter 111 is that there is a uniuersul real m-plane bundle 7" over the real Grassmanti manifold G"'(R") such that any real m-plane bundle 4 over any X E a is of the form f i ) " for a unique homotopy class [f] of maps X A G"(R"). For this reason G"(R") is also called the classgying space for real m-plane bundles. If E X represents a real m-plane bundle ( over X E a,then there is a well-defined zero-section X 5 E carrying each .Y E X into that point a(x) E E whose image under any of the local homeomorphisms n- I ( U i ) U i x R" with x E U i is (x, 0) E U i x R".If E* is the subspace E - a(X) of E , then there is a 2/2 Thom cluss U , E H"(E, E * ; 2/2) which is uniquely characterized in Chapter IV. The composition X 5 EL E, E* of the zero-section with the inclusion j then provides the h/2 Eulrr class e(4)= a*j*U; E H " ( X ; 2/2). Let 5 be a real m-plane bundle over X E d,and let P , be the corresponding projective bundle. The notation P , is used ambiguously for the total space ofany representative coordinate bundle P ; .+ X of P ; , and there is a canonical reul line bundle ( = real I-plane bundle) EL, with base space P;. One uses the Leray-Hirsch theorem to verify that H * ( P ; ; E/2) is a free H * ( X ; 2/2)-module with a basis fl,r(i.<),. . . , e(Ac)m-'j consisting of cup products of the 2/2 Euler class e(i,) E H 1 ( X ; 2/2); the cup product operation we(>.<) is itself an endomorphism H * ( P ; ; 2/2)-+ H*(P,; E/2) over H * ( X ; 2/2). The direct product H**(P,; 2/2) is similarly a free H * * ( X ; Z/2)-module, so that for any f(r) E 2/2[[t]] with leading term 1 E 2/2 one can define the E/2 characteristic class U/.(<)E H * * ( X ; E/2) to be the determinant of the induced free il(P(l
-
4
Introduction
<
For any real m-plane bundle over any X E the (total)Stiefel- Whitney class w(() is the class ur(<) for f ( t ) = 1 + t . One easily shows that w(() vanishes in all summands H P ( X ;B/2) c H * * ( X ; E/2) with p > m, so that w ( ( ) = 1 + wl(() . . . w,(<) E H * ( X ; B/2), where w,(() E H p ( X ;2/2) for p = 1, . . . , m ; furthermore w,(() is just the Z/2 Euler class e(<)E H " ( X ; 2/2). For any formal power series g(t) E E/2 [[t]] whatsoever, with leading term 1 E 2/2, there is an alternative computation of u,(() E H * * ( X ; Z/2) from the classes w,(<),. . . , w,(() alone, using the Hirzebruch multiplicative sequence associated to g ( t ) ; this construction is also given in Chapter V. According to one of the main results of Chapter 111, any real m-plane bundle ( over any X €99 is of the form f l y " for a unique homotopy class [J]ofmaps X 2 G"([W").One easily shows that w(<) = w(f!y") = f*w(y") E H * ( X ; 2/2) for the total Stiefel-Whitney class w(y") E H*(Gm(RaU); Zj2) of the universal real m-plane bundle 7"' over G"([W"), where H*(G"(R"); B/2) H * ( X ; 2/2) is induced by [f]. The cohomology ring H * ( G m ( R a ) B/2) ; is therefore of interest: it is the source of all B/2 characteristic classes. The main result of Chapter V is that H*(Gm(R");H/2) is the polynomial ring 2/2 [wl(y"), . . . , wm(ym)] with one generator w,(y") E HP(Gm(Ra);2/2) in each of the degrees p = 1, . . . , m. Chapter VI contains some of the many applications of 2/2 characteristic classes, including the nonembedding result sketched earlier in this Introduction. The vector bundles of primary interest in Chapter VI are tangent bundles r ( X )of smooth manifolds X .
+
+
<
The only prerequisite needed for the sequel is a modest background in singular homology and cohomology, as promised in the Preface. Such a background can easily be gleaned from any one or two of the following standard references: Artin and Braun [l, 21, Dold [8], Eilenberg and Steenrod [2], Greenberg [l], Hilton and Wylie [l], Hu [4,5], Massey [6], Spanier [4, Chapters 4 and 51, Vick [l], and Wallace [6, Chapters 1-41. Portions of differential topology, Lie groups, and homotopy theory will be introduced in detail as needed. References to other texts containing elementary introductions to topics in fiber bundles in general, vector bundles in particular, and characteristic classes, will also be given in appropriate later chapters.
CHAPTER I
Base Spaces
0. Introduction A vector bundle over a topological space X consists in part of a projection E + X from another topological space E onto X ;the space X is the base space of the given bundle. Vector bundles exist over arbitrary base spaces; however, interesting theorems are accessible only for bundles over appropriately restricted base spaces. Accordingly, in this chapter we introduce a large category 93 of topological spaces, and in the remainder of the book we consider only those bundles whose base spaces lie in 93. In $1 the category 93 is defined, and it is shown to be closed with respect to finite disjoint unions and finite products. In @2-7 one learns that 93 is indeed a large category, containing all spaces of the homotopy types of CW spaces, for example; reasonable topologists seldom ask for more. The most important feature of93 is that it is a category in which one can prove theorems by the Mayer-Vietoris technique, described and established in 69; several applications of the Mayer-Vietoris technique will appear in later chapters.
1. The Category of Base Spaces A mup from a topological space X to a topological space Y is any function X A Y that is continuous in the given topologies. If [0,1] is the closed unit interval, then any map X x [0,1] 5 Y induces restrictions to X x {O) 5
6
I. Base Spaces
I,
and X x { 1 which can be regarded as maps X 3 Y and X f Y, respectively; in this case fb is homotopic to f , . Two topological spaces X and Y are homotopy equivalent, or of the same homotopy type, whenever there are maps X 5 Y and Y 4 X such that the compositions g f and f g are homotopic to the identity maps X -+X'and Y -,Y, respectively. A topological space X is contractible whenever it is homotopy equivalent to the space {*} consisting of a single point *. The category 9l will be defined in two stages, the first being an induction on the natural numbers n 2 0. In describing the inductive step we shall suppose in part that a given topological space X can be covered by 4 families of open sets, for some natural number q > 0, where the sets in each one of the q families will be mutually disjoint; that is, for q unrelated index sets B , , . . . ,B, there will be q families { U,, Ifi, E B , ; , . . . , { UaqIfi,E B, 1 of open sets U P pc X, which collectively cover X, and for each fixed p 5 4 the sets in the family { U D PIfi, E BP}will be disjoint from one another. For notational y the q families of open sets, convenience we let { U l , a } a ., . . , { U q , y }represent without explicitly identifying the index sets; for each p 5 q the notation {Up,,},will represent the pth family. 0
0
1.1 Definition: A topological space is of 0th type if it is a disjoint union of contractible spaces. Suppose that a topological space X can be covered by the union offinitely many families { Ul,a}ar. . , , {U,,y}yofopen sets Up,,c X such that for each p q the sets Up,ain the family {Up,,},are disjoint from one another. Then X is of nth type whenever there is such a covering with the additional feature that all intersections of the sets in the covering are of ( n - 1)th type. A topological space is of j n i t e type if it is of nth type for some n 2 0.
The sets Up,, c X in the preceding definition are themselves of ( n - 1)th type since they are singleton intersections. The restriction concerning less trivial intersections is not unduly severe, however, since all intersections of more than 4 distinct sets in the covering {Ul,a}a,. . . , {U,,y}yare necessarily void. A metric on any set X is any function d from X x X to the nonnegative real numbers such that for any (x, y, z ) E X x X x X one has d ( x , y ) = d ( y, x), d ( x , y ) = 0 if and only if x = y , and d ( x , z ) S d ( x , y ) d ( y , z). For any x E X and any E > 0 there is a subset Ux,Ec X consisting of those y E X with d ( x , y ) < E, and the family { U x q E } (is x sthe E ) basis of the corresponding metric topology on X . A given topological space is metrizuble if its topology is the metric topology of some metric. We omit the elementary proofs that metrizable spaces have many familiar and desirable features. For example, a metrizable space is huusdo~f
+
7
1. The Category of Base Spaces
in the usual sense that any two distinct points lie in corresponding disjoint open neighborhoods, and normal in the usual sense that any two disjoint closed sets lie in corresponding disjoint open neighborhoods. Some equally elementary but less familiar properties of metrizable spaces will be presented as the need arises.
1.2 Definition: A topological space X is a base space if it is homotopy equivalent to a metrizable space of finite type. Morphisms in the category a of’ base spaces are arbitrary maps from one base space to another. A base space is not necessarily itself metrizable or of finite type; these properties are not preserved under homotopy equivalence. The principal reason for introducing the category B will appear in 99. In this section we show only that & isIclosed with respect to finite disjoint unions and finite products. 1.3 Lemma: I f X I and X , are metrizable spaces, then (i) the disjoint union X I X , is metrizable, and (ii) the product X I x X , is metrizable.
+
PROOF: (i) For any metric d on any set X there is another metric 6 with b(x, y) = d ( x , y ) / (1 + d ( x , y ) ) for all (x, y ) E X x X , as one easily verifies, and the topologies induced by d and S are clearly the same. Hence there are topologies on X , and X , induced by metrics d , and d , with values in the half-open interval [0, 1). There is then a metric d on X I X , given by
+
d,(.x, y ) d ( x , y) = d,(.x, y )
if (x, y) E X I x X I if (x, y) E X , x X , if ( x , ~ ) EX I x X, + X , x X , .
{2 The topology induced on X I + X , by d is the usual topology of X I X , , for the given topologies of X I and X , . (ii) For any metrics d , and d, on X I and X , there is a metric d on X I x X , given by
+
4 ( X l , X , ) , (Yl,Y,))
= d,(.x,,y,)
+ d,(x,,y,),
and an easy verification shows that the resulting metric topology on X I x X , is the usual product of the given topologies of X and X , .
,
1.4 Proposition: The category 99 of base spaces is closed with respect to jinitr di.$joint unions and jinite products.
PROOF: Since finite sums and finite products of homotopy equivalent spaces are themselves homotopy equivalent, Lemma 1.3 leaves only the task of verifying that finite sums and finite products of spaces of finite type are
8
I . Base Spaces
themselves of finite type. The assertion concerning sums is trivial since spaces of (n - 1)th type are automatically of nth type. It remains to show (by induction on n) that a product of two spaces of nth type is also of nth type. The case n = 0 is clear. Now let X be a space of nth type, covered by y families { U l , a ) a ., . . , { U q , y } of y open sets U p , pc X as in Definition 1.1, and similarly let Y be a space of nth type, covered by t families [ Vl.bjb , . . . , [ y.c]c of open sets c Y as in Definition 1.1. By the inductive hypothesis all intersections of the products U p , px K,c are of ( n - 1)th type, intersections of more than qt distinct such sets being void, and each of the families I F , , p x I/S,&}(B,&) in the open covering w l , a x ~ l , b l ( 3 , d ) 7 . . . Pq,y x K,[,()j.<) of X x Y is mutually disjoint. 3
2. Some Simplicia1 Spaces We now show that many familiar topological spaces are base spaces in the sense of Definition 1.2. Further examples will be given later. Let K O be any set. An abstract simplicia1 complex with vertex set K O is a family K of finite subsets of distinct elements i, . . . , i, E K O , subject to two conditions: (i) if {i,, . . . , i,] E K and {j,, . . . ,j,} c { i , , . . . , i,}, then { j,, . . . ,j,} E K ; and (ii) if i E K O ,then {i} E K . An element {io, . . . , i,) E K containing p 1 distinct elements of K O is a p-simplex. Condition (ii) permits one to identify vertices i E K O with 0-simplexes ( i ) E K . Let k , be the family of functions K O3 [0,1], with value x, E [0,1] on i E K O , such that x i = 0 except for finitely many vertices i E K O , and such that E i E K o x=i 1. The value xi E [0,1] is the ith barycentric coordinate of x E k,.For each vertex i E K O there is a unique element 6' E k, with value 8: = 1 (no summation), and any x E k, is uniquely of the form CiEKoxidi with C i e K o x= i 1. For each simplex {i,, . . . , i p } E K the corresponding geometric simplex li,, . . . , i,l c k , consists of those x E k , such that xi = 0 for i 4 { i , , . . . , i,). The simplicia1 space IKI c k, is the union over all simplexes { i,, . . . , ip) E K of the geometric simplexes li,, . . . , i,l c k,.The family goand the simplicia1space IK( c gocan both be regarded as subsets of the direct sum R, which consists of all functions K O5 R such that xi = 0 except for finitely many vertices i E K O ; that is, R is the real vector space with basis {6'(iE K O ] . There are at least two natural topologies on the vector space u K o R . One such topology arises from the norm R 2 R given by the finite sums llxll = for each x = xihi,the resulting metric and metric topology on R being given by setting d(x, y ) = IIx - yll. The metric
+
uKo
uK,,
uK,,
liEKol~ilCiaKO UKu
9
2. Some Simplicia1 Spaces
simpliciul spuce associated to an abstract simplicial complex K is the sim-
uKo uKo
R in the relative metric plicial complex K is the simplicial space IK1 c topology it inherits from the preceding metric topology of R. An abstract simplicial complex K is jinite-dimensional whenever there is a natural number 4 0 such that K contains no p-simplexes with p > q: K is y-climensionul if 4 is the least such number. The vertex set K O of a finitedimensional simplicial complex K need not be finite; however, if K O is finite, then K is a Jinite simplicial complex, trivially finite-dimensional. A simplicial space I KI is jnire, q-dimensionul, or jnite-dimensional whenever the underlying simplicial complex K has these properties. In this section we show that any finite-dimensional metric simplicial space is a base space in the sense of Definition 1.2. We shall later obtain the same result for any metric simplicial space whatsoever. Let K be an abstract simplicial complex with vertex set K O as before. The j r s r burycentric subdivision of K is an abstract simplicial complex K' with vertex set K , constructed as follows: for any distinct simplexes I , , . . . ,I , in K one has [ I , , . . . , I , } E K' if and only if one can relabel I , , . . . , I , in such a way that as subsets of K O they satisfy I , c I , c . . . c I , . One easily verifies that K' is indeed an abstract simplicial complex, and that K' is q-dimensional if and only if K is q-dimensional. Let k be the family of functions K 3 [0,1], with value X , E [0,1] on I E K , such that X , = 0 except for finitely many vertices I E K , and such that C l e KX , = 1. For each vertex I E K there is a unique element 6' E R with value b: = 1 (no summation), and any X E k is uniquely of the form C I EX,b' K with X I= 1. For each simplex { I , , . . . , I , } E K' the corresponding geometric simplex II,, . . . , I,/ c k consists of those X E I? such that Xi = 0 for I I$ [ I , , . . . , I , ] , and the simplicial space IK'I c k is the union over all simplexes { I , , . . . , I , } E K' of the geometric simplexes II,, . . . , I,[ c k. A metric topology is imposed on IK'I c k c R in R. the same way that a metric topology was imposed on IKI c k , c We now construct a homeomorphism IK'I 2 IKI. For any p-simplex I = [i,, . . . , i p ) E K the burycenter B' E (KI is the point ( p 1)-'6'" .. . ( p + l ) - ' S i p in the geometric simplex li,,. . . , i,l c IKI. For any q-simplex [ I , , . . . , 1,; E K', and for any point X,6' of the corresponding geometric simplex / I o , . . . , I,I c IK'I one easily verifies from the conditions x I t K X ,= 1 and X , = 0 for I $ [ I , , . . . , I , } that x I E K X , B 'E IKI; in fact, if l o c I , c . . . c I , c [i,, . . . , ipp! for a p-simplex {i,, . . . , i p } E K , then X,B' lies in the geometric simplex li,, . . . , i,l c IKI. Thus there is a X,d' E IK'I into well-defined set-theoretic map IK'I 2 IKI carrying any X,B' E IKI. A routine verification shows that is indeed a homeomorphism in the metric topologies of IKI and IK'I.
IIEK
uK uKo
+
xIEK
xIEK xIEK
IIEK
+
+
10
1. Base Spaces
Let i E K O be a vertex of an abstract simplicia1 complex K . The abstract stur of the 0-simplex { i } E K is the unique smallest subcomplex of K (in the obvious sense) that contains every simplex {i,, . . . , ip>E K with i E {i,, . . . , i,,}; the corresponding subset of IKI is the closed star of the point 6’ E IKI. In general the abstract star of { i } also contains simplexes { j,, . . . , j q } that do not contain i, and one obtains the open star of 6’ E IKI by removing all points of the corresponding geometric simplexes Ij,, . . . ,j,l in the closed star. Alternatively, the open star of 6’ E IKI consists of all points x j E K , , x j hIKI j ~ with x i > 0. Observe that the set of all points I j E K 0 x i h J ~ R with x i > 0 is an open convex set in uKO whose R intersection with ( K ( is the open star of 6’; hence the open star of 6’ is open in the metric topology of IKI, and it has a “convexity” property. If K ‘ is the first barycentric subdivision of K , then for any p-simplex I E K the open star of the point 6’ E IK’I consists of all points ~ I , E K ~ I . h ’ ’ such that x I > 0 which satisfy the following additional condition: if x,, > 0, then the subsets I c K Oand I’ c K O satisfy either I c I‘ or I’ c I (or both). The open star of 6’ E IK’I is denoted U p , ,in the proof of the following result, I being a p-simplex of K .
uK,
I
2.1 Proposition: Any finite-dimensional metric simplicia1 space KI is of’
.first type.
PROOF: Let K ’ be the first barycentric subdivision of K ; since there is a homeomorphism IK’I + IKI, it suffices to show that IK’I is of first type. For any p 2 0, let I and J be p-simplexes { i,, . . . , i,} and { j , , . . . ,j,} of K , and let Up,‘ c IK‘I and U p , Jc IK’I be the open stars of the points 6‘ E IK’I and dJ E [K’l,respectively. If the intersection Up,!n is nonvoid, any point x E Up,‘ n is of the form x I , E K ~ 1 . 8with 1 ’ both x I > 0 and x J > 0. I t follows that the subsets I c K O and J c K O satisfy either I c J or J c I , and since each of I and J has p 1 elements, either consequence is equivalent to I = J . Thus if fl denotes an arbitrary p-simplex of K , the family { U p : , } , of open stars U p , , c IK’I of points 6, E IK’I is mutually disjoint. If K IS 4 dimensional, one thereby obtains a covering { U O , a } a., . , { U , , y ) yof IK’I by 4 1 such families { U p , , ! , of open sets. All intersections of more than 4 1 distinct sets in the covering are void, and to complete the proof it remains only to show that all possible nonvoid intersections are contractible; but this is an immediate consequence of the “convexity” property of the open stars Up., c IK’I.
+
+
+
2.2 Corollary: Any finite-dimensional metric simplicia1 space IKI is a base
space.
PROOF: IKI is metric by definition and of first type by Proposition 2.1.
11
3. More Simplicial Spaces
A polyhedron is any metric simplicial space IKI whose vertex set K O is finite.
2.3 Corollary: Any polyhedron IKI i s a huse space.
PROOF: If K O is finite, then IKI is finite dimensional. Some elementary introductions to simplicial complexes and metric simplicial spaces are indicated in Remark 10.5.
3. More Simplicia1 Spaces Any metric simplicial space IKI whatsoever is a base space, even without the dimensionality condition of Corollary 2.2. However, rather than proving that IKI itself is of finite type, we construct a new metric space IKI* homotopy equivalent to IKI, and we show that IKI* is of second type; this implies that IKI is a base space as desired. We first replace the given metric simplicial space IKI by the simplicial space IK’I of the first barycentric subdivision K ’ of the underlying abstract simplicial complex K ; this is harmless since there is a natural homeomorCD phism IK‘I + IKI, as noted earlier. For any p-simplex 1 = { i o , . . . , i p } of K the dimension dim 1 is the number p . 3.1 Definition: For any metric simplicial space IKI the telescope function IK’I -5 [O, 1 ) assigns to each x = x I E k . x I bEf IK’I the value
El
For each x E IK’I both of the preceding sums are finite, and since x I = I , it follows that f ( x ) E [0, 1) as indicated.
eK
3.2 Lemma: The telescope ,function IK’I [0, I ) is continuous in the metric topology uf’ I K‘I and the usual real topology .f [0,1).
PROOF: The telescope function is the restriction to IK’I c H KR of a linear
+
I 1/(2 dim I ) ) . functional U K[w 5 R, also given by f ( x ) = x I E K ( xdim Since If’(x)l 5 llxll = for all x E R, it follows that the linear functional j ’ is bounded, hence continuous, in the metric topology induced by the norm 11 11; consequently the restriction off to 1K’I c R is continuous in the relative topology of IK’I.
~IEKl~,l
uK
uK
12
I . Base Spaces
In particular, since the image of the telescope function f lies in the halfopen interval [0, l), for any half-open subinterval [0, r ) c [0, 1) the inverse imagef-’[O,r) c is open in
IK’I
IK‘I.
3.3 Definition: The telescope IKI* of any metric simplicia1 space IKI is the subset
of the metric space IK’I x ( - 1, l), in the relative topology.
3.4 Lemma: Any metric simplicia1 space IKI is homotopy equioalent to its telescope IKI *. PROOF: Since there is a homeomorphism IK’I 4 IKI, it suffices to show that K’I is homotopy equivalent to IKI*. Let IKI* 5 IK’I be the restriction to c IK’( x (- 1 , l ) of the projection IK’I x (- 1 , l )-+IK’I, which is trivially continuous. If IK’I 3 IK’I is the identity map and IK’I [0, 1 ) the telescope function, then the image of IK’I 3 IK’I x [O,l) lies in IKI*, and by Lemma 3.2 the restriction IK’I 1:IKI* of (id,f) is continuous. Clearly the composition 1K’I 1:IK(* 5 IK’I is the identity on [K’l,and it remains to show that the composition IKI* 5 IK’I 5 IKI* is homotopic to the identity on 1KI*. For any point (x,s) E (K’I x ( - I, 1) and any point t E [0,1] let k,(x,s) = (x, (1 - t ) s tf’(x)) E IK’I x ( - 1, 1). If (x, s) E IK *, then k,(x, s) E IKI* for any t E [0, 11; furthermore, the restriction of k , to KI* is the identity IKI* -+ IKI* and the restriction of k , to IKI* is the composition h g, which completes the proof.
IK‘
+
13
+
3.5 Lemma: For any 4 > 0 the open set f’-’[O, q/(q 2 ) )c [K’I is of firsf type. PROOF: For any p >= 0 and for any p-simplex E K let U p , pc IK’I be the open star of lip E IK’[ as in the proof of Proposition 2.1, and let Vp,p= Up,pn f - ’ [ O , q / ( q 2)). Since f ( @ ) = p / ( p 2) it follows that the q 1 families { V,,,},, . . . , {V,,,}, of open sets cover f’-’[O, q/(q 2)). Since each family { U p , p ) is p mutually disjoint, exactly as in Proposition 2.1, it also follows that each family { Vp,p}pis mutually disjoint. It remains to show that nonvoid intersections of (at most q 1 ) of the sets Vp,pare contractible. Each open star Up.pc IK’I is the intersection with IK’I c [w of the convex set x,6’ E R such that x p > 0, and IK’I [w is the restriction to of points
+
+
+
XI uK IK’I c DKR of a linear functional uK [w
+
+
uK
R. Consequently any nonvoid
13
3. More Simplicia1 Spaces
intersection V,,,,.pc, n . . . n Vpp,flr (for which one necessarily has 0 5 r 5 y) is the intersection with IK‘I of an intersection of convex sets in R.The latter intersection contracts linearly to the barycenter of the geometric simplex Ib0, . . . , / j r l c IK’I, and the contraction restricts to a contraction of Vp,,,fl,, n . . . n V p r . f l ras , desired.
uK
3.6 Lemma: second type.
The telescope IKI* of uny metric simplicial space IKI is of’
PROOF: For any q > 0, let
u,,o
for the telescope function IK’I [0, I), so that IKI* = U , as in Definition 3.3. The two families .(Uq)qeven and {Uq},od?are each mutually disjoint, and together they cover IKI*. The only nonvoid intersections in the covering are the singleton intersections
u, = f - ’
[
0,q
:
2)
“($9
5)
and the intersections
+
since each f - ‘[O, q/(q 2)) is of first type by Lemma 3.5, all intersections in the entire covering { U,}, are of first type. Hence IKI* is of second type, as in Definition 1.1.
3.7 Theorem: Any metric simplicia1 space IKI is a base space. PROOF: According to Lemma 3.4, IKI is homotopy equivalent to its telescope 1K*l, which is of second type by Lemma 3.6. Since IKI* inherits its topology as a subspace of a product IK’1 x ( - 1 , l ) of metric spaces, IKl* is metrizable. Thus IKI is homotopy equivalent to a metrizable space IKI* of finite type, as required by Definition 1.2. 3.8 Definition: Let dl. denote the category of spaces of the homotopy types of metric simplicial spaces, morphisms being arbitrary maps. ’
Another characterization of w‘ will be given in Corollary 5.3. 3.9 Corollary: bases spaces.
The cutegory W . is a full subcategory of the category 49 of
14
I. Base Spaces
PROOF: This follows immediately from Theorem 3.7. The category W also satisfies an analog of Proposition 1.4.
3.10 Proposition: The category W is closed with respect to finite disjoint unions and finite products. PROOF: It suffices to show that finite disjoint unions and finite products of metric simplicial spaces are again metric simplicial spaces. The assertion about unions follows immediately from part (i) of Lemma 1.3. As for products, if K and L are abstract simplicial complexes with vertex sets K O and L o , respectively, then one constructs the obvious product complex K x L with vertex set K O x L o , observing that the resulting simplicial space IK x LI c Lo R is the image of IKI x ILI c R0 R under the canonical isomorphism R0 R+ Lo R of direct sums. The metric for IK x LI is precisely the product metric for IKI x ILI given in part (ii) of Lemma 1.3.
uKo
uKo uLo uKouKo uLo
A simplicial space 1KI is countable whenever the vertex set K O of the underlying abstract simplicial complex K is countable in the usual sense.
3.11 Definition: Let Wo denote the category of spaces of the homotopy types of countuble metric simplicial spaces, morphisms being arbitrary maps. 3.12 Corollary: The category Wois a full subcategory of the cutegory 3 of base spaces. PROOF: Clearly Woc W, and W c 23 by Corollary 3.9. The category Woalso satisfies an analog of Proposition 1.4.
3.13 Proposition: The category Wo is closed with respect to finite disjoint unions and jinite products. PROOF: If the abstract simplicial complexes K and L have countable vertex sets K O and L o , respectively, then K O + Lo and K O x Lo are both countable, and the remainder of the proof proceeds as in Proposition 3.10.
In summary, we have obtained inclusions Woc W c 23 of some useful categories of topological spaces, each category being closed with respect to finite disjoint unions and finite products. We shall identify W in another fashion in Corollary 5.3, and there is an analogous alternate description of W ; . The inclusions Woc W c 9 guarantee that spaces in the categories Woand W have all the properties of spaces in the category a,many of which are more easily established directly in 23, without the extra provisions of X i or W .
15
4. Weak Simplicia1 Spaces
4. Weak Simplicia1 Spaces In this section we give two new descriptions of the metric topology of a simplicia1 space IKI, and we construct a useful locally finite covering of IKI. Then we endow IK( with an entirely different topology, in general finer (more open sets) than the metric topology. Although the identity map carrying IKI in the new topology into IKI in the metric topology is not in general a homeomorphism, we shall show that it is always a homotopy equivalence, hence that IKI is a base space in either topology. Recall that if K O is the vertex set of an abstract simplicial complex K , with direct sum R over the real numbers R, then k oc R consists R with x i E [0,1] and xi = 0 except for finitely miny of those points x E vertices i E K O ,such that C i E K o x=i 1. The simplicia1 space IKI is a subset of k oas before. The metric topologies of R, k o ,and IKI arise from a norm R R, which will be denoted 11 in this section, given by llxlll = CicKolXil. There are other norms on R, one of which is the norm R It llm Iw given by llxllr = maxicKulxil.If K O is finite, then the metric topology on induced by (1 [Irn agrees with the metric topology induced by 11 However, in general the only related conclusion one can obtain for R itself comes from the obvious inequality IIx - yllx 5 IIx - yl(, for any x and y : the topology induced by 11 [IcI is finer than the topology induced by (1 [I1. The situation is better on the subset k oc R.
uKo uKo
uKo
uK,)
uKo
)II
-
uKo
uKo
uK<,(W
Ill.
uKo
uKo
For any vertex set K O the topologies induced by ugree on the subset koc R.
4.1 Lemma:
)I 1 ,
uKo
11
and
PROOF: The inequalities IIx - ylIx I (Ix - ylll imply that open sets in the 11 [I1 topology of K O are open in the 11 topology. Conversely, for any .YE k o and any E > 0 let U x , cc k o be the 11 Ill-open neighborhood of x consisting of those Y E k o such that IIx - ylll < E . If x lies in a geometric simplex 111 c ko for a p-simplex I = { i o , . . . , i,) c K O , then xi = 0 except for the p I vertices i E I . Let L'x,c c k obe the 11 [\,-open neighborhood of x consisting of those y E K O such that IIx - ylla, < ~ / 2 ( p+ 1). Then for any y~ Vx.,onehasxicJlxi- yil 5 ( p + 1)"x - yllm < ~ / 2 ; o n e a l s o h a s x ~>~ ~ y ~ E J x , - (t./2(p + l))] = 1 - 4 2 , which implies C i e , ( x i- yil = yi = 1 J', < ~ / 2 It . follows for any y E that
111
+
xi
xi,,
x
+ C [ X i - yil < E / 2 + E/2 = hence that V',t c U x , c .Consequently, open sets in the 11 [ I a i topology of k o are also open in the 11 (II topology. I(x - Ylll =
iel
[Xi
- yil
E,
i4I
16
I . Base Spaces
4.2 Proposition: Let K be an abstract simplicial complex with vertex set K O und simplicial space IKI c u K , , R ;then the norms 11 11, and 11 on u K , , R induce the same metric topology on IKI. PROOF: Since IKI c ko,this is an immediate consequence of Lemma 4.1. Thus the metric topology of IKl introduced in $2 via the norm 1) 11 ( = 11 [I1) can equally well be described in terms of the norm 11 l I x ; we simply speak of the metric topology of IKI. For each vertex i E K Othe ith barycentric coordinutefunction IKI -, [0, 13 carries each x = ~ i E K o ~E iIKI b iinto x i E [0, 13. 4.3 Corollary: The metric topology of a simplicial space IKI is the coarsest topology (fewest open sets) for which each barycentric coordinate function IKI -+ [O, 1 3 is continuous.
PROOF: Since IIx - yll, quence of the fact that I(
= maxiEK,,lxi - yil,
this is an immediate conseinduces the metric topology.
An open covering { U j i j of a topological space X is locally j n i t e whenever there is an open neighborhood V’ of each x E X with U , n V, void h except for finitely many indices j . If X -I,[w is a map with value 0 outside a R in terms of finite sums closed subset of U j ,then one can compute X on each V’. A family { h j } jof nonnegative such maps hj is a partition qf unity subordinate to { U j } jwhenever h j = 1. The following specialized partition of unity will be needed later.
cj
4.4 Proposition: For any metric simplicia1 space IKl there is u purtition of unity (hiI i E K O } subordinate to a locally j n i t e cover { Ui I i E K O ) indexed by the vertex set K O ,such that hi($) = 1 for each i E K O ;specijically, if x E IKI lies in the interior of a geometric simplex 111 c IKI fbr I = { i 0 ,. . . , i p ) E K , then x has an open neighborhood V, with Ui n V’ void for i 4 1. PROOF: For each i E K O let U i denote the set of those x E IKI such that 2xi > IIxIIII,.The family ( U i ( i EK O ) clearly covers IKI, and since 11 and the barycentric coordinate functions are continuous by Proposition 4.2 and Corollary 4.3, each Ui is open. To show that { Uil i E K O ) is locally finite suppose that x E IKI lies in the geometric simplex 111 = liO,. . . , ipl. Let V’ be the open neighborhood of x consisting of those y E IKI with IIx - y(( < then for any y E V, the triangle inequality gives
+I~X\\~,;
llyllm
2 llxllm - IIX
- Yllm
’llxlla
- 311xll,
= SIIXlltz.
If i 4 1, then x i = 0, so that 2
311xIIE,> 211x
- yll,
>= 21xi - yil = 2yi
for y E v,.
17
4. Weak Simplicia1 Spaces
Hence llyllx, > :[\xll,. > 2yi for any Y E V’, except for the finitely many vertices i E I ; that is, U i n V’ is void except for i E I , as required. Now for each vertex i E K O and each point x E IKI let g i ( x )E R be the maximum of the values 0 and 3xi - 211xllm.Then each gi is continuous by Proposition 4.2 and Corollary 4.3, vanishing outside an open subset of Ui. Since [ U iI i E K O }is locally finite, one can form the sum g = cisi,for which one has g(x) > 0 for all x E IKI; the desired partition of unity {hil i E K O }is then obtained by setting hi = gi/g for each i E K O .
If I is any simplex ( i o , . . . , i p } c K Oof an abstract simplicial complex K , then the metric topology induced on the corresponding geometric simplex 111 = li,, . . . , i,l c IKI is precisely the usual real topology of the subset of those points (xo,. . . , xp) E R p t l with x i E [0,1] and xo + . . * + x p = 1. Thus the topology of 111 can be independently defined. 4.5 Definition: For any simplicial space IKI the closed sets of the weak topology of IKI are those sets W c IKI for which W n 111 is closed in each geometric simplex 111 c IKI; in this topology IKI is a weak simplicia1 space.
We shall use the notations lKlmand lKlw to identify a simplicial space IKI in its metric and weak topologies, respectively. The subscripts m or w will be dropped whenever a topology is not specified or whenever the topology of IKI is clear in context. Observe that the barycentric coordinate functions IKI -, [0,1] are automatically continuous in the weak topology, so that by Corollary 4.3 every open set in lKlm is also an open set in IKlw. Hence the identity IKI+IKI induces a (continuous) map lKlw 5 IKI,,,. Although the inverse o f f is not necessarily itself continuous, as we shall soon show, there is nevertheless a (continuous) map lKlm3 )KIw such that f g and g f are homotopic to identity maps, so that lKlw and lKlmare homotopy equivalent; the proof of this result is the business of the remainder of the section. 0
Caution: In Definition 4.5, and in analogous situations which occur later, we use the terminology of J. H. C . Whitehead [3]: a “weak topology” on X frequently has more open sets than some other topology on X . The words “weak topology” are used elsewhere but not in t h i s book for the coarsest topology (fewest open sets) on X such that certain maps X + Yare continuous. Suppose that K O consists of all the natural numbers n 2 0, and that the simplicial complex K consists of the 0-simplexes {n} for all n 2 0 and 1simplexes (0,n } for all n > 0. Then each geometric 1-simplex 10,nl is canonically homeomorphic to the closed unit interval [0,1]. Let W c IKI be the set which intersects each IO,nl in the point corresponding to l / n E [0,1]. Then
18
1. Base Spaces
according to Definition 4.5 W is closed in IKl,; but since W does not contain the point 6' E IKI, the set W is clearly not closed in IKI,. Thus for this simplicial space IKI .the continuous bijection IKI, lKlm is not a homeomorphism; specifically, the set-theoretic inverse f - is not continuous. Despite the preceding example, certain maps ILI, 5 lKlm of metric simplicial spaces do remain continuous when one substitutes lKlw for IKI,. Suppose for each x E ILI that the image g(x) E IKI lies in only finitely many geometric simplexes of IKI. Then the metric and weak topologies of IKI induce the same relative topology on the image Img c IKI, and since the inclusion Img c IKI, is continuous the composition lLlm 3 Im g c lKlw is also continuous; that is, the original map g can also be regarded as a (continuous) map (LI, + IKl,. This observation is an essential ingredient in the proof of the following result.
s '
44.6 Proposition (Dowker[I]):
For any simplicial complex K the weak simplicial space lKlw is homotopy equivalent to the metric simplicial space IKlm.
PROOF: We have already observed that the set-theoretic identity map I IKI,,,, which is not necesIK( -+ IKI induces a continuous bijection IKI, + sarily a homeomorphism. Let { h iI i E K O }be the partition of unity constructed in Proposition 4.4, and suppose that x E IKI lies in a geometric simplex IllclKlforl={i,,.. .,i,)~K;thenh~(x)=Ofori$1,andsince~~,,,,h,(x)= 1, it follows that the point g(x) = CisKO hi(x)6'E R also lies in 111. Since the hi's are continuous in the metric topology, there is an induced (continuous) map lKlm 5 (KI, from (KI, to itself. By construction, each g(x) lies in only finitely many geometric simplexes in IKI, so that g is also continuous as a map lKlm+ lKlw, as in the preceding paragraph. To show that f g and g 0 .f are homotopic to identity maps let IKI x [O, I] 5 R be given by F , ( x ) = (1 - t)x tg(x). For any geometric simplex 111 c IKI we already know that if x E 111, then g(x) E 111, so that the image of F lies in IKI. Since g is continuous as a map from lKlm to itself F is a homotopy from the identity map lKlm 3 [Kim to the composition F , = f g, and since F restricts to maps 111 x [0,1] + 111, it also follows that F F is also a homotopy from the identity map lKlw -% IKI, to the composition F,=gcf.
UKr,
0
DK,
+
0
Some other expositions of Proposition 4.6 are indicated in Remark 10.7. 4.1 Corollary: The category w' of spaces of the homotopy types qf metric simplicial spaces IKI, is identical to the category of spaces of the homotopy
19
5. CW Spaces
rj,pes of weak simplicia1.space.s lKlw; u ,fortiori euerjl weak .simplicia1 spuce is a base spuce.
PROOF: We showed in Theorem 3.7 that every metric simplicial space is a base space, base spaces being defined in terms of homotopy type.
5 . CW Spaces We now describe a classical generalization of the category of weak simplicial spaces. It appears directly and indirectly throughout much of algebraic and differential topology, and it will appear in later chapters of the present work.
For any natural number n > 0 let R" be the standard real n-dimensional vector space, in the metric topology arising from the euclidean norm R" II II R, Il(rl.. . . , x,,)Il = . . . + x,'. The closed n-disk D" and (n - I)-sphere S"- c D" consist of those points x E R" satisfying llxll 1 and llxll = 1, respectively, in the relative topology of R"; in particular D' is a closed interval and So consists of two points. An (open) n-cell is any space homeomorphic to the interior D" - S"- I of D". Let iD:il be any family of homeomorphic copies of D" and [ S ; - l ; lthe corresponding family of homeomorphic copies of S"- c D", with disjoint D: and uolS:- I , respectively. For any map S!Y one unions denote the smallest equivalence relation in the disjoint union Y lets D: that identifies each x E with j ' ( x )E Y. The quotient topology D:/- is the finest topology (most open sets) such that the proof Y + jection Y + 0':+ Y + D:/ is continuous, and Y + D:/- is the udjuncrion spcice of j ' in this topology. The map f and its restrictions S:- ' Y are called atruching mups of the adjunction space, and the inclusion mapping Y into the adjunction space uttuches n-cells to Y; specifically. the attached n-cells are the homeomorphic images of members of the family [ 0':- S:- j z. A cell comp1e.u is a sequence X , -+ X I + X , + . . . of maps such that X , is a discrete space and each X,, X,attaches n-cells to XI,- '. The singleton subspaces of X , are the 0-cells of the complex, and for each n > 0 the n-cells attached to Xli- are the n-cells of the complex. Each point of X,_ is a member of one and only one q-cell for some 4 < n, and the complex X , -+ X I -+ X, . . . is closure-finite whenever for each n > 0 each attaching map y - I L.,Xll- I meets only finitely many cells. For any closure-finite cell complex let lim,,X, be the quotient of the disjoint union XI, by the smallest equivalence relation that identifies
-
6; +
u3 uz u1 ul
udl '
UlSr;-'
u1 -
-+
-+
un
uz
+
20
I . Base Spaces
,
,
each X,,- with its image under the map X,- -+ X , . Each point of limnX , is a member of one and only one n-cell for some n 2 0, homeomorphic to the interior D" - S"-' of D" in the case n > 0, and a set U c limn X , is open in the weuk topology of lim,, X,,if and only if the intersection of U with each ii-cell is open in the topology of D" - s"-'. A CW space is any topological space of the form limn X , , for a Closure-finite cell complex X , -+ X , -, X , + . . . in the Weak topology; the cell complex is the CW structure of the CW space. Closure-finite cell complexes are also called CW complexes. (In much of the literature the word "CW complex" is used both for a closure finite cell complex and for the resulting topological space. We use the terminology of Dold, who calls a space a space; see Dold [8, p. 891.) The simplest example of a CW space is the n-sphere s" itself, for any 11 > 0. One starts with a singleton space X , = (*) and attaches no cells of dimension less than n, giving X , = X I = . . . = X,- ; finally one attaches a single n-cell D" - S"- ' via the only possible attaching map S"- I + X,- = ( * ) .(Even the 0-sphere So can be constructed in this fashion as a CW space, with the obvious meaning of attaching a 0-cell.) For any CW complex X , -+ X , -+ X , -+ . . . and any n 2 0 there is an obvious inclusion of each X , into the CW space limnX,; X , is the n-skeleton of the CW complex, and its image is the n-skeleton of the CW space. A CW complex and the corresponding CW space arejnite-dimensional if the inclusion of the n-skeleton is a homeomorphism for n sufficiently large; in this case the dimension is the least m 1 0 such that the inclusion of X , is a homeomorphism whenever n 2 m. 5.1 Proposition: Any weuk simplicial spuce is a CW space.
PROOF: For any abstract simplicia1 complex K and any n 2 0 let K , be the subcomplex obtained by deleting all p-simplexes with p > n. The sequence K O -+ K -+ K 2 -+ . . . of inclusions then induces a corresponding sequence lKol lKll lK21 -+. . . of inclusions of weak simplicial spaces with IKI = lim,,[K,,as a set. As a topological space lKol is discrete, and since any geometric n-simplex 111 c R"' ' is trivially homeomorphic to the n-disk D" c R", it follows that lKol -+ lKll -+ lK21 -+. . . is a cell complex; closure-finiteness is obvious. The weak topology of IKI was described in terms of closed sets, a set W c (KI being closed if and only if W n 111 is closed in 111 for each 111 c 1KI. However, one can equally well replace each geometric n-simplex [iO, . . . , in[= 111 by its interior IZI', consisting of those points xodiO + . . . + x,Sin with all coefficients xo, . . . , xf, positive, which is canonically homeomorphic to the (open) n-cell D" - S"-'; then U c IKI is open in the weak topology of IKI if and only if U n 111 is open in 111' for each I E K . +
+
21
5. CW Spaces
There is a nontrivial partial converse to Proposition 5.1, which we shall not prove. We formulate the converse primarily to indicate the size of the category of base spaces. However, in order to keep the exposition as nearly self-contained as possible, we shall not apply the converse as such. 5.2 Theorem (J. H. C. Whitehead [4]): alent to u weak simplicial space.
Any C W space is homotopy equiu-
The proof of Whitehead’s theorem involves two steps. One first shows that a map Y -,X of CW spaces is a homotopy equivalence whenever it satisfies an apparently weaker condition; this step can be found in Gray [l, p. 1391, Lundell and Weingram [l, p. 1251, Maunder [l, pp. 298-3001, or Switzer [l, pp. 87-90], for example. The second step consists in assigning to any CW space X a weak simplicial space IKI and a map IKI -, X satisfying the preceding condition; this step can be found in Gray [l, pp. 145-1521 and Lundell and Weingram [l, pp. 102-1031, for example. The conclusion that the two preceding steps prove Theorem 5.2 is expressed in Gray [l, p. 1491 and Lundell and Weingram [l, pp. 126-1271.
5.3 Corollary: The category W of spaces of the homotopy types of metric simpliciul spuces lKlm is identical to the category of spaces of the homotopy types of C W spaces. PROOF: Corollary 4.7 asserts that W is identical to the category of spaces of the homotopy types of weak simplicial spaces, and Proposition 5.1 and Theorem 5.2 imply that the latter category is identical to the category of spaces of the homotopy types of CW spaces. (The category -I was + initially . introduced in Milnor [8] as the category of spaces of the homotopy types of CW spaces. However, the equivalent characterization of Definition 3.8 is implicit in Milnor’s paper.)
5.4 Corollary : Every C W space is a base space. PROOF: We showed in Theorem 3.7 that every metric simplicial space is a base space, base spaces being defined in terms of homotopy type. One can obtain a shorter proof of Corollary 5.4 by using just the material needed for the first step of the proof of Theorem 5.2; a sketch is given in Remark 10.6. In the remainder of this section we describe CW structures of some other useful CW spaces. We shall later give independent proofs that these spaces are base spaces, without invoking Theorem 5.2 or its corollaries.
22
1. Base Spaces
We shall describe the CW structures of real and complex projective spaces, which we now define. For any n 2 0 let R(n+' ) * be the space R"+ { 0 ) consisting of Iw"+I without its origin, in the relative topology. There is an in R'"+')* with x y whenever x = uy for some equivalence relation u E R*. The real projective space RP" is the quotient R("+'I*/ in the quotient topology. The space RP" can equally well be regarded as a quotient of S"c R"+ ' by the same equivalence relation.
'
-
-
-
5.5 Proposition: RP" is un n-dimensional CW space, with u cell structure consisting qj'one cell in each dimension q = 0, 1, . . . , n.
PROOF: Trivially RPo is a singleton space {*}. Let RPo
-+
RP'
-+
RP2
-+
. . . be the sequence ofinclusions induced by the natural inclusions R' + Rz R3--1... ; that is, for any q = 1 , . . . , n and any x = (xo,. . . , x Y - ' ) E S"' themapRP-' RP"carriesthepoint[x] =[(.yo,. . . , . x ~ - ~ E) ]R P Y - l into -+
-+
] Let S 4 - ' carry the point [ x @ 01 = [(xo,. . . , X ~ - ~ , OE) RP". S4-' ( cDq c Ry) into [x] E R P - ' , and let D43 R P carry .Y E Dq into [x 0d 1E RP"; t h e n 1 and g are continuous, and the inclusions S"' -+ D4 and RP4-' + RP4 provide a commutative diagram
.Y E
sq-1
I
.
RF-1
I
Dq A R F .
Jm
Since is positive whenever llxll < 1, the restriction g1D4 - S 4 - ' is a homeomorphism onto R P q - RP4-', and it follows that j' is an attaching map with adjunction space R F , as required. Projective spaces need not be finite-dimensional. Let iw' + R2 + R3 -+ . . . be the sequence of canonical inclusions and set R" = lim,R" in the weak fopology; that is, a set W c R' is closed whenever each W n R" is closed. Observe that R" is also a direct sum of real linear spaces R', hence itself a real linear space. Specifically, points of R " are sequences ( x o , x 1 , x 2 . .. .) of real numbers, only finitely many of which are nonzero, and addition and multiplication by scalars are defined as in the finite-dimensional spaces R". Let R x * = R' - (0) and observe that there is an equivalence relation in R'* with .Y y whenever x = aj' for some u E R*.The real projective space RP' is the quotient R2*/-, in the quotient topology.
-
5.6 Corollary: RP'
cell in euch dimension.
-
is a CW space, with u cell structure consisting of'onr
23
6 . Smooth Manifolds
PROOF: The inclusions RPo -+ RP' + RP2 -+. . . induced by the inclusions R ' -+ R2 + R3 + * * . provide a limit limnRP", in the weak topology, which is canonically homeomorphic to the definition R&*/- of RP". One can replace the real field R by the complex field C throughout the definition of RP" to obtain the complex projectioe space CP" = C=("'l)*/-, which can equally well be regarded as a quotient S2"+l/- of the (2n 1)sphere SZn+I c C"+ [S"" consists of those points z = ( z o , . . . , z,) E C"' such that 11z)12= 1z01*+ . . . + Iz,,IZ = 1.1
+
'.
'
5.7 Proposition: CP" is a 2n-dimensional CW space, with a cell structure consisting yfone cell in each even dimension 29 = 0, 2, , . . , 2n. PROOF: By analogy with Proposition 5.5 one uses the map 0 ' 4 3 CP4 carrying z E D Z 4(cCq)into [: 0 E C P , where 11 11 is the usual norm on C4. Since 41 - llzll' is real and positive for llzll < 1, the restriction yID2Y - S 2 4 - l IS a homeomorphism onto CP4 - C P - ' , so that the restriction g l S Z 4 - ' is an attaching map S Z 4 - ' CP4-' with adjunction space C P .
41-
'
The definitions of the projective spaces RP" and CP" suggest the obvious definition of the complex projective space CP", and the proofs of the preceding two results suggest the obvious proof of the following result: 5.8 Corollary: CP' is a CW space, with a cell structure consisting of one cell in each even dimension. Many other expositions of the study ofCW spaces in general are indicated in Remark 10.8.
6. Smooth Manifolds In this section we briefly describe the category L4tof (smooth) manifolds and we formulate some standard basic results of differential topology. These results quickly imply that any (smooth) manifold is a base space. A closed n-dimensional manijbld is a compact hausdorff space X with an I i E I } of homeomorphisms open covering f U iI i E I } and a family {Oi U i2Oi( Ui) onto open sets Oi( Ui) c R". The open covering is a coordinate covering of X by coordinate neighborhoods, and the family {Oi I i E I } of maps is an atlus for X . Suppose that the intersection U in U j c X of two coordinate neighborhoods is nonempty. Then Oiand O j restrict to homeomorphisms of U in U j onto nonempty open subsets Oi( U in U j ) c R" and Oj(U in U j )c [w". A
24
I. Base Spaces
closed n-dimensional manifold X is smooth whenever all the induced ho8 meomorphisms Qi(Uin U j ) Oj(Uin U j ) are diffeomorphisms of open subsets of R"; that is, the real-valued functions on Qi(Ui n U j )c R" and Oj(U i n U j )t R" which define O j Ql: and Qi 0 QJ:I , respectively, all have partial derivatives of all orders. In this case the coordinate covering and atlas form a smooth structure on X. Let X be a smooth closed m-dimensional manifold with coordinate covering { Uil i E I } and atlas {OiliE I},let Y be a smooth closed n-dimensional manifold with coordinate covering { blj E J } and atlas { Y j \j E J}, and let X Y be a map. Then f induces maps Y j f O; of open subsets Qi(U i nf - '( 5))c R" into R", and f is itself smooth whenever all the induced maps Y j o f 01: are smooth; that is, all partial derivatives exist as continuous functions on Qi(U i nf- '( 6)). The category of smooth closed manifolds consists of smooth closed manifolds and smooth maps. If a smooth map X Y has a smooth inverse Y % X, then f is a dgeomorphism; in this case we usually do not distinguish between X and Y . One example of a smooth closed n-dimensional manifold is the real projective space RP", whose cell structure was given in Proposition 5.5. To define the smooth structure of RP", for each i = 0, . . . ,n let U i c RP" be the open subset of all points [(x,, . . . , x,)] E RP" with xi # 0. There is a homeoR" with morphism Ui 0 Q - I
0
0
0
0
a
@i[(X,,
. . . , x,)]
=x
i YXO, . . . , xi- I ,xi+ 1, . . ,X"), '
is the diffeomorphism carrying points and each composition O j @:, ( y o , . . . , yi- l , y i + . . . , y,) E R" with yj # 0 into 0
Y ~ ' ( Y O. ., * Yi-l?l,Yi+l, *
1 . 9
Yj-liYj+l, * .
. ¶
Y n ) E [W".
Suppose that one replaces R" in the definition of smooth closed ndimensional manifolds by the closed subset (R")+ c R" of points (xI, . . . , x,) with x, >= 0; if x, > 0, then ( x l , . . . , x,) is an interior point of (R")', and if x, = 0, then ( x l , . . . , x,) is a boundary point of (IF!")+. The result of this substitution is the category of smooth compact manifolds. If { Ui I i E I} and {Oi1 i E I } provide the smooth structure of such a manifold, and if Oi(x)is a boundary point of (R")' for some i E I and some x E Ui, then necessarily Qj(x) is a boundary point of (R")' for all j E I such that x E U j . The set of such points is the boundary of X. One easily verifies that the smooth structure of X induces a smooth structure for which X is a smooth closed ( n - 1)dimensional manifold. The boundary X of a compact manifold X may be void, in which case X is a closed manifold. We occasionally use the language compact manifold with boundary to identify manifolds which are compact but not closed.
25
6 . Smooth Manifolds
An arbitrary hausdorff space X , not necessarily compact, is an n-dimensional manifold if it can be covered by denumerably many compact n-dimensional manifolds. Suppose that the manifolds in such a covering are smooth, and let lo,I i E I ) and {'Pj Ij E J ) be the atlases of two such smooth manifolds. Then X is a smooth n-dimensional manifold whenever every nontrivial If every point restriction Y j ' . 0; is a diffeomorphism of open sets in (IT)+. of a manifold X is an interior point of one of the manifolds in its denumerable covering by compact manifolds, then X is an open manifold; for example, [w" is itself a smooth open n-dimensional manifold. Smooth maps in the of' smooth manifolds are defined as before in terms of restrictions category to coordinate neighborhoods. Since a smooth manifold is covered by denumerably many compact manifolds of the same dimension, its underlying topological space X is necessarily second countable in the usual sense that there is a denumerable basis of the open sets of its topology. Conversely, any second countable hausdorff space X with a smooth structure, { U iI i E I } and {Qi 1 i E I } , is a smooth manifold. Equivalently, one can characterize smooth manifolds as hausdorff spaces with denumerable smooth structures, { U i1 i E N} and (Oi I i E N}, where N = {O, 1,2, . . . )-. This implies that any smooth n-dimensional manifold is of the form lim, X i for a sequence X , + X , + X , + . . . of inclusions of compact n-dimensional manifolds X , , X I , X , , . . . . Given an open covering ( U , [ iE I } of a topological space X , another open covering [ V,Ij E J } of X refines { Ui I i E I ) whenever there is a function J 1: I such that 5 c U p ( j for ) every j E J . A hausdorff space X is puracompact if any open covering of X has a locally finite refinement. The conclusion of the following lemma is somewhat stronger than paracompactness. 6.1 Lemma: Any open covering { Uil i E I } of a manifold X has a countable locally jinitr rejinement [ 51j E N} such that each closure is compact and satisfies c U p ( j , .
6
PROOF: Let X = lim, X , for a sequence X , + X , + K 2 + . . . of inclusions of compact manifolds of the same dimension, and let X, denote the interior X , - Xk0ofX k . For any open covering { Uili E I ) of X there is a refinement ( U in ( X , , , - Xk-,)l(!, k ) I ~x N ) , where X - , = X - , = 0. For each k E N the space X , - X k - is compact, and it can therefore be covered by finitely many of the open sets U , n (kk+ - x k - ,). The latter sets, for each k E N, form the desired refinement { 51j E N}.
,
,
Suppose that { h j l j E J } is a partition of unity subordinate to a locally finite cover [ j E J J of a smooth manifold X ; then {hjl j E J } is a smooth partition of' unity whenever each X 3 [0,1] is smooth. If { V j l j E J } happens
26
I . Base Spaces
to refine another open covering { Uili E 1). of X , then [hjlj E J ) is also regarded as subordinate to { U iI i E I ).
For unj1 open covering [ U ,I i E I ) of' u smooth mangold X tkrrr countuhle smooth purtition ojunity (hjlj E N} subordinate to fUili E I ) .
6.2 Lemma: is u
PROOF: Let R" s R be given by
where R" % R is the usual euclidean norm. Then j is a smooth function which is positive for llxll < 1 and zero for llxll 2 1. For any r > 0 and any point y E R" one easily adjusts f' to yield a smooth function R" k [0,1] which is nonzero precisely in the open disk DY,,of radius r about y. Let U be any open subset of R", and let I/ be any open subset with compact closure V c U . Then B can be covered by finitely many such open disks Dy., c U , and the sum of the corresponding finitely many functions&, is a nonnegative smooth function R" -+ R which is positive on V and vanishes outside of U . One can refine the given open covering { Ui I i E I } of X by taking intersections with coordinate neighborhoods, so that without loss of generality one can assume that f U iI i E I } is a coordinate covering; furthermore, since Lemma 6.1 provides a locally finite refinement of the latter covering, one may as well assume in addition that { Uil i E I } is itself a locally finite coordinate covering of X , with atlas ( Q i l i E I ) . A second application of Lemma 6.1 gives a countable locally finite refinement {vjlj E N) of {UiI i E I } such . there are open that each closure is compact and satisfies c U p ( j ,Thus sets @ p ( j ) ( Fand ) @ p ( j ) ( u p ( j ) ) in R" with @ p ( j ) ( v j ) = @ p ( j ) ( V j ) c @ p ( j ) ( U p ( j ) ) , SO that by the result of the preceding paragraph there are smooth nonnegative functions R" 2 R such that each gj is positive on (Dp(j,( and vanishes outside of U p ( j ) ) .Each composition U p ( j ) @'"' @ p ( j ) ( U p ( j ) ( U p ( j J R R that is positive on extends to a smooth nonnegative function X and vanishes outside of U p ( j ) Since . .(UiliE 1). and { vjlj E N} are locally finite one can compute the sum f' = CjE fj, and since .( 6 1 j E N ), covers X , it follows that f' is everywhere positive. The desired smooth partition of unity {hjlj E N) is then obtained by setting hj = j J j for eachj E N.
4
5
5)
'5
Suppose that a smooth map X & Y is a homeomorphism; then f' need not be a diffeomorphism. For example, if R R is given by f'(x) = x3, then f' is clearly a smooth homeomorphism: but since the derivative off vanishes at x = 0, the inverse is not smooth. The difficulty can be overcome by imposing the obvious condition on .f, which we do in a more general situation.
27
6. Smooth Manifolds
Let X and Y be smooth manifolds of dimensions m and n with smooth structures ( U i l i € I ) , ( @ i l i ~ I and ) ( v j l j ~ J ){Yjlj~J),respectively,and , let X 3 Y be a smooth map. By definition, the maps ' Y j a f o @ i ' carrying open subsets U in f - ' ( v j )c R" into R" are smooth in the sense that all partial derivatives exist as continuous functions on Ui n1'-'( Vj); in particular, one can compute the m x n jacobian matrix of Yj ;c ,f @il at each The map f is an immersion whenever each of these point of U i nf - '( jacobian matrices has rank m. Next suppose that X L Y is any injective map, not necessarily smooth. Then the image f ( X ) c Y , in the relative topology, need not be homeomorphic to X . For example, one easily constructs a (smooth) injective map R R2withj'(0) = (0,O)andf(x) = (l/x, sin(l/x))for x 2 1. In this example the inverse image of every neighborhood of (0, 0) contains arbitrarily large real numbers so that the relative topology of the image f(R) c R2 is not the usual topology of R; thus 1'is not a homeomorphism onto its image. An embedding of a smooth manifold X into a smooth manifold Y is a smooth injective immersion X -I, Y that induces a homeomorphism of X onto f ( X ) c Y , in the relative topology. (A map X Y of topological spaces is proper if and only if the inverse image of every compact set in Y is compact in X . One easily verifies that a smooth injective immersion X L Y is an embedding if and only if it is proper.) We shall extend the definition of embeddings somewhat. Recall from the preceding section that R" denotes the limit lim,R" of the sequence R' + R2 + R3 --+.. . of canonical inclusions, in the weak topology. For any smooth manifold X a map X R" is smooth whenever the composition X R" + R with each projection R" + R is smooth. If one regards points of R as row vectors with countably many entries, then for each coordinate neighborhood of X the jacobian matrix o f f consists of countably many columns, the number of rows being the dimension of X . A smooth map X R" is an immersion whenever the rank of each such jacobian matrix is the dimension of X . A smooth injective immersion X 3 RE' is an embedding whenever it induces a homeomorphism of X onto f ( X ) c R"', in the relative topology.
v).
6.3 Proposition: For. uny smooth munifold X there is at least one embedding
X+R'. PROOF: Lemma 6.1 provides a locally finite coordinate covering { Uil i E I ) of X such that each closure Biis compact. Let (Oi I i E I } be the corresponding atlas; if X is n-dimensional, then each homeomorphism Qifrom U i to the open set Q i ( U i )c R" is given by coordinate functions U j A R, . . . ,
28
I . Base Spaces
Ui R. According to Lemma 6.2 there is a countable smooth partition of unity (hjl j E N} subordinate to { U i l i E I } ; specifically, there is a funch tion N 5 J such that X 3 R vanishes outside of some open set vj c X , and such that c where ( V , l j E N} is also a locally finite covering of X. For each j E N and each k = 1 , . . . , n the product vj R therefore extends to a map X 3 R that vanishes outside of vj, and since { vjl j E N] is locally finite, it follows that the denumerably many maps yr induce a map X + R”. One easily verifies that the latter map is an embedding.
5
hJXk,cJ’~
6.4 Corollary:
-
Any smooth manifold X i s metrizable.
PROOF: As in (32there is a norm R“ II II R for which each \l(x,,, x x 2 , . . . ) 1) is the (finite) sum cjsNIxjl. For any embedding X R” there is then a metric d on X with d(x,y) = Ilf(x) - f(y)II. 6.5 Proposition: For any smooth compact manifold X there is at least one natural number N 2 0 for which there exists an embedding X + RN.
PROOF: If X is compact, then the modifiers “denumerable” or “countable” become “finite” throughout the proofs of Lemmas 6.1, 6.2, and Proposition 6.3. Propositions 6.3 and 6.5 are definitely not best-possible results. For certain values of n the following result is best possible. 6.6 Theorem (Whitney Embedding Theorem): Any smooth n-dimensionul manijdd X has at least one embedding X -+ R2”.
This theorem appears in Whitney [7, p. 2361; its proof is long. A detailed discussion of Theorem 6.6 and related embedding theorems is given in Remark 10.11. Any n-dimensional sphere S” has at least one smooth structure, constructed in the obvious way from two coordinate neighborhoods homeomorphic to R” itself. Since one can compose an embedding X -+ R2” with the inclusion Rz” Sz” of either coordinate neighborhood of S2”the following assertion is clearly equivalent to the Whitney embedding theorem: any smooth n-dimensional manifold X has at least one embedding X + S2”. -+
6.7 Theorem (Cairns-Whitehead Triangulation Theorem): Any smooth ndimensional manijdd X is homeomorphic to un n-dimensional metric simpliciul space K
I I.
In fact X has a “smooth triangulation” in the obvious sense. References for proofs of Theorem 6.7 are given in Remark 10.14.
29
6 . Smooth Manifolds
6.8 Corollary: Any smooth munijold is u base space; hence the category of'smooth munijblds is a f u l l subcategory ojthe category B of base spaces. c&'
PROOF: According to Corollary 2.2 any finite-dimensional metric simplicia1 space IKI is a base space. A strengthened version of Corollary 6.8 is described in Remarks 10.17 and 10.18; any (second countable) topological manifold whatsoever is a base space. The proof avoids the Cairns-Whitehead triangulation theorem, which is not available for topological manifolds. Recall that an embedding is a specialized immersion. Immersions are themselves of interest, however. For certain values of n the following result is best-possible, where SZn-' is the usual smooth sphere ofdimension 2n - 1. 6.9 Theorem (Whitney Immersion Theorem) : Any smooth n-dimensional munijbld X hus ut least one immersion X + Sz"-
'.
References to two proofs of Theorem 6.9 can be found in Remark 10.13, along with a detailed discussion of related results. As in the case of the Whitney embedding theorem there is a trivially equivalent alternative version of Theorem 6.9, but with a dimensional restriction: if n > 1, then any smooth n-dimensional manifold X has at least one immersion X -, R2"-' . The reason for the restriction is that S' itself clearly does not immerse in R'. Suppose that X is a smooth manifold of even dimension 2n, with smooth structure ( U i ( iE I ) and (OiliE I ) . The space Rz" is homeomorphic (in many ways) to C", so that the maps
a i ( U in U j )
8, 0 ; ' +
Q j ( U in U j )
can be regarded as maps from sets in C"to sets in @".The manifold X is a complex manifold of complex dimension n whenever there is a such a homeomorphism for each i E I for which all the maps aj 01: are holomorphic; that is, the n complex-valued functions on Qi(U i n U j )c @" which describe Oj QIT have local expansions about each point as Taylor series in n complex variables. We shall later show that are are many real even-dimensional smooth manifolds which have no complex structure in the preceding sense; for example, the 4-dimensional sphere S4 has no such structure. However, we have already encountered some topological spaces which do have such structures: the complex projective spaces CP",whose cell structures were described in Proposition 5.7. To construct a complex structure for CP" one merely substitutes C for R in the construction of a smooth structure for the real projective space RP", given at the beginning of this section. Incidentally, 0
'
'
30
I. Base Spaces
as topological spaces the projective spaces RP' and CP' are the circle S' and the 2-sphere S2, respectively; in particular, S2 does have the complex structure it receives from its identification with CP'. For later convenience we record an obvious consequence of Lemma 6.1. It is in part a specialized version of the shrinking lemma of Dieudonne [11, which also appears in Dugundji [2, pp. 152-1531. 6.10 Proposition: Any manifold X has a countable locally jinite covering [ U,In E N} by open sets U n c X with compact closures On, refining a given open covering of X . Furthermore, for such a covering {U,In E N], there is another countable locally finite covering { V, I n E N} by open sets V, with compact closures V, c U , for each n E N. PROOF: Lemma 6.1 itself guarantees the existence of (U,ln E N) as well as a countable open covering { W,In E N} and a map N 4 N such that W,c Up(,)for each n E N. Let p - ' ( n ) denote the set of those m E N with p(m) = n, and set V, = UrnEp-+,) W,, for each n E N. Then V, c U , and {V,ln E N] is an open covering of X . Since each V, is thus a closed subset of the compact set Dn,it is itself compact, and since { U,I n E N} is locally finite, so is { V,ln E
N 1. 6.11 Corollary: Let X be a smooth manifold with countable locally jinite open coverings { U , I n E N} and { V, I n E N} as in Proposition 6.10; then there is a smooth partition of unity { h, I n E N } such that each h, is positive on V, and vanishes on X - U,.
PROOF: One simply repeats the proof of Lemma 6.2, observing that the identity map N --* N now replaces the map N -f+ 1 used in Lemma 6.2 to describe { vjlj E N } as a refinement of { U i l i E I ) . Proposition 6.10 will be used to prove several approximation theorems, whose proofs also depend on the following best-known elementary approximation theorem. 6.12 Theorem (The Stone-Weierstrass Theorem): Let A ( 0 ) be any real algebra of continuous real-valued functions 0 3 R on a compact hausdorff space 0, and suppose that (i) A( 0) contains all real-valued constant junctions and that (ii) if x # y in 0 then there is at least one function g E A( 0) such that g(x) # g(y) in R. Then for any continuous function 0 R whatsoever, and for any constant E > 0, there is a g E A( 0 )such that Ig(x)- f ( x ) l < E for every XE
0.
PROOF: See Bartle [l, pp. 185-1861 or Royden [l, p. 1741, for example.
31
6 . Smooth Manifolds
6.13 Corollary: L e t X he any smooth manijbld, and let X (R")' he any (continuous)rnup.from X to the euclidean halflspace (R"')' ; then,for the usual R + and any c > 0, there is a smooth map X euclidean norm (R"') (R")' .suc/i [hut Ilg(s) - j'(x)ll < I: ,fiw t r n j ~.Y E X .
- -
J PROOF: The map ,f consists of m real-valued functions X A R, . . . , I Jrn R, X R'. In case X is a compact smooth manifold 0,the smooth real-valued functions D 4 R satisfy the conditions required of the algebra A ( 0 ) in the Stone-Weierstrass theorem, so that there are smooth functions 03 R, . . . , 1!7%R such that lgj(x) - fj(x)(< E/m for every x E 0,where j = I , . . . , m - 1; a minor modification of the Stone-Weierstrass theorem also provides a smooth function V L R ' such that one has lg,,(s) - j",(.~)l < E/m for every x E 0,and it follows that (g,, . . . , g), is a smooth map il -% (R"')' such that Ilg(x) - ,f'(x)Il< E for every x E 0. Now let X be any smooth manifold whatsoever. By Proposition 6.10 there is a locally finite covering [ Ur'InE N; of X by open sets U,, c X with compact closures Or,,so that the preceding argument provides a smooth map 0, (R"')+for each IZ E N such that Ilg"(x) - j'(x)ll == c for each x E Utn.However, according to Corollary 6.1 1 there is a smooth partition of unity {/i,,ln E N j subordinate to [U,llnE N;,and CneN /i,g" is then a well-defined map X 3 (R")' such that Ilg(x) - f(x)ll < E for every .Y E X , as desired. ./'8>1
Corollary 6.13 is a special case of a more general approximation theorem that implies that any (continuous) map X 5 Y of smooth manifolds is homotopic to a smooth map; we need only the latter implication. To start the proof one first uses Proposition 6.10 to find a countable locally finite covering [ U,"In E Nj of Y by open sets U,",each of which has a compact closure in some coordinate neighborhood of Y. Thus, if Y is rn-dimensional, there is an atlas {@,,InE N; of diffeomorphisms of open sets in Y onto open sets in (R"')' such that = @,,(U:) c (Rm)' for each n E N. One then applies Proposition 6.10 repeatedly to find a sequence { U: I n E Nj, UA In E Nj, [ U,Z I n E N), . . . of countable locally finite open coverings of Y such that each U;" has compact closure satisfying c U4,. The sets U ; and their closures appear throughout the following lemmas. Two maps X Y and X 5Y are homotopic relative to a subset X' c X if they are the restrictions to X x (0) and X x {l},respectively, of a map X x [0,1] 5 Y such that F(x,t ) is independent o f t E [0,1] whenever x E X'.
@,,(q)
6.14 Lemma: Let X 1; Y he a (continuous) map of smooth manifolds, and jbr each y E N let [ U ; I n E N) be the covering of' Y described above. Then there
32
I . Base Spaces
is a sequence (fo,fl,f2,. . .) of maps X -kY , beginning with fo = f,such that the following conditions are satisjied for each p E N: (i) fp is homotopic to fp+ relative to X - f; '( U ; + I), (ii) the restriction of fp+ to f; ' ( U i ) u . . * u f; 1(U;+2)is smooth, (iii) i f 0 sq Sn+2p,thenf,:l(UY,f2)cf;1(UY,+1) f o r a l l n E N,and (iv) if 0 5 q 5 n + 2p, then f;'(UY,+*) c .f;:l(UY,+l) for all n E N.
PROOF: We shall show that if a finite sequence (fo, . . . ,fp) satisfies all four conditions, then there is anfp+I such that (fo, . . . ,fp,fp+ ') satisfies the same conditions. Since is compact, it meets only finitely many members of the finite covering { U,"ln E N}. Hence there is an n p E N such that locally U," n UY,is void whenever n > n p ,for any q E N. Let {Onln E N} be the atlas described earlier, and for each U4, c Y set VY, = Op(U,"n UY,)c (!Rim)+. For each n E N and q E N the sets V:+ and V: - VY, are disjoint closed subsets of the compact closure V," = = Op(O,")c ([W"')', and they are therefore separated by a positive euclidean distance E: > 0, where EY, = 1 if one or both of the subsets is void. One then defines E > 0 by setting
' w)
E =
s
min { E Y , ~ O n Ow)
np, O 5 q 5 np
+ 2p + 2).
V," c (Rm)+ be the composition f; '(0;) Up0 Let J"i'(0;) --!%V:. One applies Corollary 6.13 to find a smooth map f; '(0,") (!Rim)+ such that IIgp(x)- gp(x)ll < E for any x E f; '(0:).Since and V," - V ; + are disjoint closed sets in V," the method of Lemma 6.2 yields a smoothmap Po --* [0,1] with restrictions h I = 1 and h 1 V : - V ; + ' = 0. Since V;+' and V : - Vg+' are separated by euclidean distance at least E > 0, the points ( 1 - th(g,(x)))g,(x)+ th(g,(x))g',(x) of (Rm)+ all lie in whenever ( x ,t ) E f; ' ( 8 : )x [0,1], and one thereby obtains a map J"; '(0;) x [0,1] 9 V,".By construction, G ( x ,t ) is independent of t E [0,1] whenever x E .f; (0:)- f ; '( UF+'), so that G is a homotopy from g p to a V:, relative to f;'(0:) - J;'(u;+'1. Consequently map .f; '( 0:) there is a unique homotopy X x [0,1] 5 Y from , f p to a map X 3Y, relative to X - .f; '(U;"), such that Flf;'(U:) x [0,1] = O;' G . The map f p + satisfies condition (i) by its very construction. Since g p + is clearly smooth on the union off; '(UP,") with any open set in V : on which g p is smooth, fp+ satisfies condition (ii). Finally, the definition of E and the property llgp - gpll < E imply both
'
'
,f;:l(O,"
n V Y , +cf;'(0: ~) n UY,+')
and
j';'(0: n U Y , +cf;:,(0,0 ~) n UY,+')
33
6 . Smooth Manifolds
whenever 0 5 VI 5 lip and 0 S q n p + 2p + 2 ; since 0;n Uj is void for n > n p , the same conclusions apply whenever n > n p , for any q E N. Since F is a homotopy relative to X - f ; ' ( U ; + ' ) , a fortiori relative to X 1'; '( the inclusions U j + 2 c UY," themselves imply both
o;),
0;) n ~j+l)
j';+!'((Y -
0;) n ~ j +c ~~' ; )' ( ( Y
f';'((Y
0;) n 1/j+2)c f ; + ! ' ( (-~0:)n ~ j + '
-
and -
for any )I and q whatsoever. Consequently both f;+'I ( U4,+2 , c f , '( U;+ I ) and f ; '( Uj'2) c f ' i 2 1 ( U j + ' ) whenever 0 5 q 5 n 2p + 2, as required by conditions (iii) and (IV).
+
6.15 Lemma: f;'(LT;+') c j ' ; ' ( U i ) f o rany p E N.
PROOF: This is a string
f;'(U;+')
cf;Qu;) c . .. c j ; ' ( u ; )c f ; ' ( U i )
of applications of condition (iii) of Lemma 6.14. 6.16 Lemma: f ; ' ( U ; p f 2 )c j ' p 1 ( U ; f 2 ) f o r a n y p EN.
PROOF: This is a string f;'(U;p+2)
c f ; ' ( U ; P + ' ) c . * . c f,"(u;+3) cf,'(u;+2)
of applications of condition (iv) of Lemma 6.14. 6.17 Lemma:
Thejbmily { U;P+21pE N) of open sets U ; p + 2couers Y.
PROOF: For any y E Y let ny E N be the largest number such that y E Ufs;ny exists because [ Uf I n E N j is locally finite. Since { U ; ' y + 1 p E N) also covers c Uz whenever q 5 2ny + 2, it follows that Y, and since )!E U i " Y + 2
6.18 Lemma: X.
u
u;n,,+2
u... u
u; u u; u . .. u u;s:+2.
u;syn,.+2c
Thefbmily [ f ; ' ( U ; + 2 ) l p E N} of o p e n s e t s f ; 1(U;+2)covers
PROOF: Sincef'; ' ( Y )= X , this is a consequence of Lemmas 6.16 and 6.17: f ; ' ( u ; u... u u ; P + 2 ) q ; ' ( U ; ) u . .. u f ; ' ( U ; P + 2 ) c .f';
6.19 Theorem: Any (continuous)map X homotopic to a smooth map X + Y .
'(u;)u .
* *
u f, '(Ui+2).
-5 Y of' smooth manifolds X
and Y is
34
I . Base Spaces
PROOF: Let ( f o , f , , f 2 , . . .) be the sequence of maps X 2 Y constructed in Lemma 6.14, and regard the homotopy of condition (i) as a map X x F [ p / ( p l), ( p l ) / ( p 2)] 2 Y; specifically, F , is a homotopy relative to X - f ; ' (U ; + ' ) . Sincef; '(U;") c J'; ' ( U ; ) for any p E N, by Lemma 6.15, and since i f ; '( U j ) I p E N} is a locally finite cover of X , it follows for any x E X that there is a px E N such that F,(x, t ) is independent of the choice of t E [ p / ( p l), ( p l)/(p 2)] for p > p x ; hence F , is the restriction to X x [ p / ( p l),(p l)/(p + 2)] of a well-defined homotopy X x [0,1] 5 Y, the restriction X x {O} 5 Y being the initial J' = fb. Condition (ii) of Lemma 6.14 implies that X x { 1 ) 5 Y is smooth on any open set f ; ' ( U ; + ' ) c X , and since { f ; ' ( U P , + ' ) l p ~N} covers X , by Lemma 6.18, X x { l ! . L Y is smooth on X , as required.
+
+
+
+
+
+
+
+
Many other expositions of the study of smooth manifolds in general are indicated in Remark 10.10.
7. Grassmann Manifolds The projective spaces RP", RP", CP", and CP" of $5 can be regarded as spaces of lines through the origins of the vector spaces R"+ ', R", C"+', and C",respectively; that is, their points are the 1-dimensional subspaces of the corresponding vector spaces. More generally, for any natural number m > 0 there are spaces whose points are the rn-dimensional subspaces of R""", R", Cm+",and C".These spaces play a crucial role in the theory of vector bundles, and the primary purpose of this section is to develop some of their properties, especially the fact that they are base spaces. 7.1 Definition: For any natural number m > 0 let V denote any of the real or complex vector spaces Rm+",R", C"'+", C";the usual topology is imposed on Rm+"and ern+", and the weak topology is imposed on R" and C".Let ( V x . . . x V ) * denote the set of m-tuples of linearly independent vectors in V , in the relative topology of the m-fold product V x * * x V , and let be ( y , , . . . ,y,) the equivalence relation in ( V x . . . x V ) * with (x,,. . . , x),
span the same m-dimensional subspace of 3
whenever the vectors x,, . . . , x, V as the vectors y,, . . . , y,. The Grassmann manifold G"(V) is the quotient ( V x . . . x V ) * / - , in the quotient topology.
For example, the Grassmann manifolds G'(R"+ '), G'(R"), G1(C"+') and G ' ( C " ) are precisely the projective spaces RP", RP", CP", and CP". Observe that the sequence R m f l-+ R m + 2 R m + 3--$.. . of canonical ') Gm(Rm+')-+ inclusions induces a corresponding sequence Gm(Rm+ -+
-+
35
7. Grassmann Manifolds
G'"(R"+3) + . . . of inclusions for which Gm(R")= limn Gm(Rm+") in the weak topology; similarly G"(C") = limn G"'(@"'+").We shall show that each of Gm(C"'+n) and Gm(@m+") is a smooth manifold, a fortiori a base space. More generally, if V is any of the vector spaces Rm+n,R", @"'+", @", then Gm(V) is a CW space, hence a base space; however, it will be more convenient to prove directly that G"(V) is a base space. We show that if V = Rm+n,then G"(V) has the structure of a smooth closed manifold of dimension rnn; the case V = @,+" is similar. Let @V be the tensor algebra generated over R by V , let I c @V be the two-sided ideal generated by squares x @ x E V @ V c @V, and let A V be the exterior algebra @V/l. The tensor algebra is graded by assigning degree rn to real linear combinations of elements x , 0 .. . @ x, E @V for any rn vectors x , , . . . ,x, E V , and since I is homogeneous, there is a corresponding grading of The real vector space of elements of degree rn in is the rn-fold exterior product f l V ; the image in of x, 0 . . . @ x, E @V is denoted
AV.
AmV
AV
X1 A " ' A X , .
One easily shows that m vectors x , , . . . , x , in V are linearly dependent if and only if x, A . . . A x, = 0 E A" V . A nonzero element of A" V is simple if and only if it is of the form x, A . . . A x, for m linearly independent vectors x,, . . . , x , in V . If (x,, . . . , x,+,) is a basis of V , then the (,mf") simple elements xi,A . . . A xi,, E A" V with i , < . . . < i, form a basis of A" V . If y , , . . . , y , and z , , . . . , z, span the same rn-dimensional subspace of V , then for some a # 0 one has y , A . . . A y, = az, A . . . A z, so that each point of G"( V ) is identified up to a nonzero factor with a simple element in A" V ; more specifically, there is an injective map G"( V )5 G 1 ( PV ) that is a homeomorphism onto the set Im F c G 1 ( A mV )of 1-dimensional subspaces spanned by simple elements of f l V , in the relative topology. Hence it is of interest to develop an algebraic criterion for simplicity of elements of A" V . For m = 1 there is a canonical isomorphism V + A' V , so that every nonzero element of A' V is simple. In case rn > 1 let f l - 'V 2 R be any real linear functional on the (m- 1)-fold exterior product V . One 0J easily verifies that there is then a unique linear map A"' V V with value 0 J (x, A . . . A x,) = ,( - lY-'@(x, A . . . A .fi A . . . A x,)xi E V on every simple element x, A . . . A x, E A" V , where Zi indicates that xi is deleted.
17'
7.2 Lemma : 11'm > 1 and X E A" V is nonzero, then X is simple if and only v 5 R. (f x A (0J X ) = o E 1 v jiw eoery linear jimctional +
PROOF:If X = x , A . . ' A X , , then any 0 -I X is a linear combination of . . . , x,, and since (x, A . . . AX,) A xi= 0 for i = 1, . . . , rn, it follows that
x,,
36
I. Base Spaces
X A (0_I X) = 0. Conversely, let V, c V be the subspace of all vectors of the form 0 J X.Since X is nonzero, one has dim, V, 2 m, and if X A x = 0 for every x E V,, then dim, Vx S m. Hence dim, V, = m, so that X = X , A . . ' A X , for some basis x,, . . . , x , of V,. For any simple element X E K V the identities X A (0J X) = 0 are the Plucker relations. If {xl, . . . , x,+,} is a basis of V , then any X E A" V is uniquely of the form a(i,, . . . , im)xi,A . * . A xi,
1
i l < . , .
for real numbers u ( i l , . . . , .),i For any permutation n of { l , . . . , m } let u(n(i,),. . . ,n(i,)) = e(n)a(i,,. . . , ,)i where ~ ( n=) f 1 according as TI is even or odd; if two or more of the indices i,, . . . , i, in { 1, . . . , m n } happen to agree let a ( i , , . . . , i,) = 0. Then for any ordered (m - 1)-tuple (i,, . . . , i , - ,) and any ordered (m 1)-tuple (j o , . . . ,j,) of indices in [ 1 , . . . , m n } the Plucker relations for simple X E V are of the form
+
+
m
1 (-l),u(i
+
A"
~ , . . . , i , - , , j k ) a ( j o ., . . ,.L.. .,j,)=O,
k=O
where j;, indicates that j , is missing. We use the Plucker relations in the latter form for the Plucker coordinates u(i,,. . , ,.),i
7.3 Proposition: For any natural numbers m > 0 and n > 0 the Grassmann manifold G"'(R"'+") has u smooth structure for which it is an mn-dimensional smooth closed manifold.
PROOF: Let {x,, . . , ,x,+,}
be a basis of
C
W""",and suppose that
a(i, , . . . , i m ) x i , A . . . A x i n ,
il<...
is a simple element of A" R""", representing a point in the image Im F of Gm(Rm+") 5 G , ( P R"""). For m fixed indices i , < . . . < i , among { l ? . . . , m + n }let ui,,,,,,i,,, c Im F be the open set of those points with a(i,, . . . , i,) # 0. For any other m indicesj, < . . . < j , among { 1 , . . . , m + nj suppose that there are m - p indices in the intersection {ill. . . , i,) n [j,,, . . . ,j,]. The Plucker relations imply that there is a real polynomial f{;; ; ; ;;;{ of degree p in mn variables whose value on the mn real numbers u(il, . . . , i,, . . . , i , , j ) a(il, . . . ,),i
is the quotient a ( j l t . . ,j,). u ( i , , . . . , i,)'
37
7. Grassmann Manifolds
here Tk indicates that i k is missing, and j is any of the n indices distinct from O i l , . . . , i, i , , . . . , i,. Thus the map U i l , , , i, + R"" carrying the point with projective coordinates a ( j l , . . . ,j,) E R into the point of R"" with mn coordinates ,,
is a homeomorphism. Since the ("':") open sets U i l , ,. . , i, c Im F cover Im F , they form a coordinate covering { U i l , , , i, I i , < . * . < i,,,}, with corresponding atlas {ail, . , i, I i l < . . < i,,}. The homeomorphism ,j m 0 a;,',. . . i, from ,,
,,
,,
1
ail,..
,
.i,,(uil,. ... i ,
U j l . . . . , j,)
c R""
to Qjl,.
. . , j m ( U i l . . . , i n , n ujl,.. . . j,, ) c R""
is given by the mn quotients
j;ji., .,.,.,3ii. . . . .j , d Ifi l , .. .... ,. Jn7 jl.
im
of polynomial functions, whose denominators are nonvanishing in ail,.. . , i,,,(uiI.. . . , i,,,
n uj1,.. . ,j,).
Since such quotients are trivially smooth this completes the proof. Alternative proofs of Proposition 7.3 are cited in Remark 10.21. Each complex Grassmann manifold G"'(Cmt") is similarly a closed complex manifold of complex dimension mn, hence real dimension 2mn. 7.4 Corollary: The Grassmann manijolds Gm(Rm+") and G"'(C"''") are base spaces.
PROOF: According to Corollary 6.8 any smooth manifold is a base space. There is an alternative way to show that Gm(Rmt")and G'"(C"'+") are and Gm(Ca):one shows base spaces, which applies equally well to Gm(RUL) that they are CW spaces, hence via Corollary 5.4 that they are base spaces. In fact, we have already carried out this program for the case rn = 1, the Grassmann manifolds G'(R"' l ) , G'(R"), G'(C"+'), G1(Cm)being the projective spaces RP", RP", CP", CP", respectively; CW structures of these spaces were given in Proposition 5.5, Corollary 5.6, Proposition 5.7, and Corollary 5.8. More generally, CW structures of arbitrary Grassmann manifolds are constructed i? Lundell and Weingram [l, pp. 13-15] and (in the real case) Milnor and Stasheff [l, pp. 73-81].
38
I . Base Spaces
There is also a direct proof that Grassmann manifolds are base spaces.
'"1, 7.5 Proposition: All the Grassmann manifolds Gm(Rm+"),Gm(R"), Gm(Cm Gm(C7) ure base spaces. PROOF: We consider Gm(R"), the other cases being similar. The space R'" is a direct sum of copies of R, which one endows with the usual inner product. If {xl,x2,x3,. . . } is an orthonormal basis of R" then the exterior product f l R x is also a direct sum of copies of R, with an orthonormal basis .(xi,A . . . A xi,,, I i , < . . . < i,}. As in Proposition 7.3 G m ( R a )can be regarded as the submanifold of those points of G1(/\, R") whose projective coefficients , i,) E R satisfy the Plucker relations. In the next two paragraphs we show that G"(R"') is of first type and metric, hence a base space. For each index ( i , , . . . , i,) let Uil. , _ _i,,,. denote the open set of those points in G"(R") with lu(il,. . . , i,)l > max(a(j,,. . . ,j,)l, where the maximum is computed over the finitely many indices ( j l , .. . ,J,) with u ( j , , . . . ,j,) # 0 and ( j , , . . . , J m ) # ( i , , . . . , i,). Similarly, for any set rI r i l . ,, . . i , , [ i l < * * . < i,} of mutually distinct positive numbers, let Kl, denote the open set of those points in G"(R") with T i , . , i m / a ( i .l ,. . , i max rjl, , j,,,la(Jl,. . . ,j,)I. Clearly { uiI., . . , i,, I i, < . . . < i,) is a mutually disjoint family, F l , , , i , < . . . < i,) is a mutually disjoint family, and the two families collectively cover G"(R"'); it remains to show that each Ui,. . . .i,,,, each y , ,. , ,,,, and each nonvoid intersection U i l . , . i,,, n . j,, is of 0th type. Each point of U i l , , . . i,,, can be regarded as a onedimensional subspace of E R ~ spanned ' by an element of the form (xi, y , ) A . . . A (xi,,, y,), where each of y , , . . . , y, is orthogonal to the subspace of R" spanned by xi,, . . . , xi,,. If t E [0,1], then (xi, + t y , ) A . * A (xi,,, ry,) also represents a point of U i l , , . so that Ui,. . i,,, trivially contracts to the point represented by .xi, A . . . A .xi,,,; a fortiori Uil. of 0th type. Similarly each bI,,,,j,,, is of 0th type. An inte i,, n yl,. , j , is nonvoid if and only if r i l , , , i, < r j l , .. . , j,,, and many other similar inequalities are satisfied. In this case points of j, are represented by points (xil t,xj, + y , ) A . . . A (xi,,, t,sj,,, J,,) E A" R', where each of y , , . . . ,y m is orthogonal to the subspace of R" spanned by x i , , . . . , xi,,,, xjl, . . . , xj,,!; the coefficients t , , . . . , t, satisfy r i I . , , i,n/rjl,, . , j,,, < I t , . . . t,l < I and finitely many other similar ty,) also inequalities. If t E [0, I], then (xil t,xjl + ty,). . . (xln,+ t,xj,,, represents a point of uil,.. . , i n , n vjl,. . .j , , , , SO that Ui1.. . , i,,, n Vl,. . . i,, contracts to a set represented by a set W c A" R" of points ( x i , + tlxjl)A . . . A (xin, + fmxjn,) in which t , , . . . , t, satisfy finitely many inequalities. Furthermore, W is a disjoint union of (finitely many) contractible relatively . , is a corresponding disjoint union open sets, so that Uil, . . n ,
, , ,
in,l
,,
,
, ,
, ,
, ,
, ,
,
+ +
+
1
,,
i,,l,
,
, ,
,
,,
+
,
+
+
,,
,
,
+
+
,
,,
,
,
,
39
8. Some More Coverings
of contractible open sets; that is, U , ] ,. . , In, n y,. , j, is of 0th type as required. To show that Gm(Ra)is metric, let A'" R" I' 'I2 R be the euclidean norm on the direct sum A" R", corresponding to the inner product used in the preceding paragraph. Since every element of the projective space G 1 ( PR") is an equivalence class [XI of nonzero elements x E A" R, one may as well choose representatives x that lie on the unit sphere: llxl12 = 1. There is then a metric G ' ( P Rm) x G 1 ( A mRm)5 R given by ,
-
,,,
d ( [ x I ,[ Y l ) = min IIx - Y112, X.Y
where the minimum is computed over representatives x , y of [ X I , [ y ] lying on the unit sphere, and the embedding Gm(R")5 G 1 ( PR") induces a metric on Gm(Ra),as desired.
8. Some More Coverings This section contains two unrelated results. The first result asserts that any base space whatsoever is homotopy equivalent to a paracompact space with an especially useful covering. The second result is a more specialized covering theorem. We start by quoting some classical general topology. Recall that a hausdorff space X is paracompact if and only if any open covering { U iI i E I } has a locally finite refinement { E J } , and that any partition of unity {hjlj E J ) subordinate to { j E J } is also regarded as being subordinate to { Uil i E I } .
8.1 Lemma : If X is paracompact, then there is a partition of unity subordinate to any open covering of X . The proof of Lemma 8.1 can be found in Dugundji [2, pp. 152-153,1701, for example. 8.2 Lemma (Stone [I]):
Any metric space is paracompact.
Proofs of Lemma 8.2 can be found in Dugundji [2, pp. 167-169, 1861, and Kelley [ l , pp. 129, 156-1601, for example.
8.3 Proposition : Any base space is homotopy equivalent to a paracompact hausdorjf space X E a for which there is a countable locally finite open covering jU,,ln E N} such that each connected component of' each U , is contained in a contractible open set in X .
40
I. Base Spaces
PROOF: Definition 1.2 states that any base space is homotopy equivalent to a metrizable space X of finite type. Since metric spaces are trivially hausdorff, Lemma 8.2 implies that X is paracompact. Definition 1.1 states that a space is of 0th type if it is a disjoint union of contractible spaces, of nth type if it has an appropriate covering by spaces of (n - 1)th type, and of finite type if it is of nth type for some n. It follows that X has a covering by contractible open sets, and since X is paracompact, Lemma 8.1 provides a partition of unity {hjlj E J } subordinate to that covering. For each j E J let Vj c X be the open set of points x E X with hj(x) > 0; then { V, Ij E J > is a locally finite covering of X such that each connected component of each vj is contained in a contractible open set in X. The covering { 61j E J } is not necessarily countable. For any finite subset I c J let W, c X be the open set of those x E X such that min h,(x) > max h,(x). ie:
i#I
Then W, is a subset of vj whenever j E I , so that each connected component of each W, is contained in a contractible subspace of X ; furthermore { W,}, trivially covers X . Let 111 be the number of indices in I , and suppose for I’ # I that (1’1= )ZI; then clearly W, n W,. = Hence for each natural number n > 0 each connected component of the open set U , = =, W, is also contained in a contractible open subspace of X. Trivially { U,ln E N} is a countable open covering of X , and it remains only to show that it is locally finite. Since { 51j E J } is locally finite, for each x E X there is an n > 0 and an open neighborhood of x which meets at most n of the sets 5;consequently that neighborhood of x does not intersect U , for any m > n, which completes the proof.
a.
u,,,
For later use we record the following minor improvement of a special case of Proposition 2.1. 8.4 Proposition : Let X be a smooth compactomanifoldwith interior 8. Then there is a finite covering { U , , . . . , U,} of X by open cells such that each nonvoid intersection U in U j is a cell U , in the covering, and such that the closures in X satisfy Ui n U j = Oi n O j .
PROOF : By the Cairns-Whitehead triangulation theorem (Theorem 6.7) X is homeomorphic to an n-dimensional metric simplicia1 space IKI, where n is the dimension of X . Since X is compact, one can assume that the vertex set K O of the abstract simplicia1 complex K is finite. For each vertex i E K O let Y,cU,,,Rconsistofall pointsCj,,,xj6’EUK,[Wsuch thatxi>(n +2)-’. Then the finite family { W,, . . . , W,} of all nonvoid intersections of the open convex sets Y, c R has the properties that each nonvoid intersection
uKo
41
9. The Mayer -Vietoris Technique
l4( n W, is of the form W, for some k = k(i,j ) , and that l4( n W, = n ITj. The same two properties are inherited by the family [ V , , . . . , V,) of intersections = W,n IKI : furthermore { V , , . . . , V,) is a covering of (KI” by open ti-cells. To complete the proof one chooses { U , , . . . , U,} to be the family of inverse images of the sets V , , . . . , V, under the homeomorphism X IKI. -+
9. The Mayer-Vietoris Technique We now develop a technique which will later be applied to prove certain theorems involving an arbitrary space X of finite type (Definition 1.1). Since the theorems will depend only on the homotopy type of X , one can regard the technique as a method for proving corresponding theorems for any space X ’ homotopy equivalent to a space of finite type; for example, X’ might be any base space (Definition 1.2). Specifically, the technique will be used to prove best-possible versions of the Leray-Hirsch theorem (Proposition 11.7.2) and the existence of Thom classes (Propositions IV.1.3 and IX.2.1). In a specialized form the technique will be used to prove the existence of fundamental classes (Propositions VI.1.3 and VII.2.3) and the PoincareLefschetz duality theorem (Propositions VI.2.3 and VII.2.7).
For any space X of finite type let O ( X )denote the category of open sets U c X , morphisms U -+ V in d‘(X) being inclusions, and let 6 ( X )1 :‘JJZ be any contravariant functor from O ( X ) to a category ‘JJZ of modules over a fixed commutative ring. Such a functor is udditiue whenever for any mutually disjoint family i U,) of open sets U , c X the module h ( u , U,) is the direct sum u , h ( U , ) . For any U E O ( X ) and V E O ( X )there are four morphisms
,
ui”.uuv V
~
U
U
V
and
U n v j ” - ~
Lin
vA
V
in 6(X ) and four corresponding morphisms h(U u V ) h(U) h(U u V ) 3h ( V )
and
A ( U )A ~ ( L n I V) h ( ~J’v. ) A(U n V )
in Y11. For any a E h( Ii u V ) let ic,va = i C a 0 ica E h( U )0 h( V ) ,and for any 0 y E h( U )0 h( V ) let j;.“(/? 0 y) = j $ / ?- j Z y E h( U n V ) . The induced sequence
h ( u~V )
h ( ~0)h ( ~jt.v. ) h ( n~ V )
of homomorphisms is automatically exact at the term h( U )0 h( V ) .
42
I . Base Spaces
-
Let (h" I m E Z} be any family of additive contravariant functors from CG(X) to $YJl, indexed by the integers Z,and suppose for each ordered pair ( U , V ) E O ( X ) x O ( X )and each m E Zthat there is a module homomorphism S;," for which
. . . & p - l ( U ) @ h m - I ( V ) % hm-l(U h"(U)@h"(V)
V)
p
ah" ( U n V ) a . . .
~
I
hm(U u V )
is a long exact sequence. In this case {h'"(mE Z)is a Mayer-Virtoris functor on e ( X ) . For example, if (h"lm E Z} is the restriction to O ( X ) of singular cohomology [H" Im E Z} with a given commutative coefficient ring Ho( [*)), then there are classical connecting homomorphisms S;," for which (h"Im E Z ) is a Mayer-Vietoris functor on O ( X ) . (The cohomology version of the MayerVietoris sequence can be found in Artin and Braun [2, pp. 137-1431, Hu [5, pp. 28-29], and Spanier [4, pp. 186-190, 218, 2391, for example. Since U and V are open sets in X , the pair ( U , V ) E O ( X ) x O ( X ) automatically satisfies the conditions required for the construction of connecting homomorphisms.) A natural transformation 8 from a Mayer-Vietoris functor (h" m E Z to a Mayer-Vietoris functor (k"lm E Z} consists of homomorphisms
I
I
h"( U ) 2k"( U )
in (YJl such that the diagram 1
v - ~ ( nu V ) sW ( U u V ) k h"(U) @ h"(V) k hm(U n V ~
I
fJuetfJv-
fir"
-I
/",,.r
P - ' ( un V ) a P ( U u V )
P(U)OP(V) P(U n V ) commutes for each U E O ( X ) , V E O ( X ) , and m E Z. I t will be assumed that H respects the additivity of the functors h" and k"; specifically, for any mutually disjoint family { U 6 } bof open sets Ud c X the diagrams
I
L16fJU6
Uak"(UJ commute. The main result of this section is that if X is of finite type and each h"( U ) --%k"( U ) is an isomorphism whenever U E O ( X ) is contractible, then each h"(X) % k"(X) is an isomorphism.
43
9. The Mayer-Vietoris Technique
--- ----
9.1 Lemma (The 5-lemma):
A,
B,
Let
A2
A,
A,
A,
B,
B,
B,
B5
hr u commutative rliagrum in a cutegory !lJl of modules over a j x e d ring, each row being exuc'r. I f H , , O , , 0,, 0, are isomorphisms, then 0, is also an isomorphism.
PROOF: This is an easy and classical diagram chase, which will be left to the reader. 9.2 Lemma: Let V he a space of' nth type, as in Dejnition 1.1, and let H he LI nuturtrl trunsformution of u Muyrr-Vietoris functor {h" Im E Z}on O( V ) to a Mayer-Vieroris functor (k"lm E Z}on U( V ) . Suppose .for each U E O( V ) 41 ( n - 1)th type thut each h m ( U )2k"(U) i s cin isomorphism; then each
k"( V ) is an isomorphism. h"( V ) PROOF: By hypothesis there is some finite 4 for which V is covered by 4 families [ U , 3L,. . . , ( U,,?; of open sets Up,,]E O( V ) , such that each family is mutually disjoint and such that each intersection ofany sets in the covering is of ( n - 1)th type. For p = 1 , . . . , 4 let U p denote the (disjoint) union U p . pE fi( V ) , and let {fir}rdenote the finite family consisting of all possible intersections of the sets U . . , , U , ; the sets U . . . , U , themselves are in the family [ firjr, which consists of at most 2 4 - 1 distinct open sets, possibly including the void set. Clearly each 0, is a disjoint union of spaces of (17 - 1)th type, although not necessarily itself of ( n - 1)th type. For each s > 0 let d5(V ) c O( V ) consist of those open sets W c V which are unions of at most s members of the finite family [ f i r ] , . . Then V = U , u . . . u U , E 9,(V ) . We shall show by induction on s that if W EA!s(V), then each (1 IV h"( W ) k"( W ) is an isomorphism.
u,,
,,
-
Since d , ( V ) is precisely the family (fir),.itself, it follows that any W E d , ( V ) is a disjoint union U 6 of open sets U 6 c V that are spaces of ( n - 1)th type. The additivity properties of the given functors and of 0 itself then provide commutative diagrams
u6
low
h"(W)
k"(W)
u6h"(Ud)
]WVd
2Usk"(Ua),
44
1. Base Spaces
and since each 8", is an isomorphism by the hypothesis of the lemma, each 8, is also an isomorphism, for any W E 5,(V ) . Now suppose that each Ow. is an isomorphism for any W' E 2s-l ( V ) , and suppose that W EAIs(V). Then W is a union W' LJ W" of some W' E L ~ - ~ (and V ) some W " E2 1 ( V ) ( c2 s - l ( V ) ) , and by the de Morgan laws W' n W" is also a member W"' E 2,- l(V). Consequently the four homomorphisms labeled either 8,. @ Ow., or Ow... in the commutative diagram 6"'- I W',W"+ hm- I ( w!) hm- I ( w!!)jtY'.W", hm- I(w!!f) h"( W 1 @
are isomorphisms by the inductive hypothesis. It follows from the 5-lemma that Ow is also an isomorphism, which completes the inductive step. Since V = U , L I ~ ~ * LUI q ~ 5 & V )each , h"(V)%k"(V) is therefore an isomorphism as claimed. 9.3 Theorem (The Mayer-Vietoris Technique): Let O ( X ) be the category of open sets on u space X of finite type, as in Definition 1.1, and let 0 be a natural transjormation of a Mayer-Vietoris ,functor {h"lm E Z } on O ( X ) to a k"( V ) is Mayer-Vietoris jhnctor {k" Im E Z } on O ( X ) . If each h"( U ) an isomorphism whenever U E O ( X )is contractible, then each h"(X) % k"(X) is also an isomorphism. PROOF: We shall show by induction on n that if V E O ( X ) is of nth type, then Ov is an isomorphism. If I/ E O ( X ) is of 0th type, then there is a mutually disjoint family { U6}d of contractible open sets UdE O ( X ) such that V = Ud, so that the additivity properties of the given Mayer-Vietoris functors and 8 itself provide commutative diagrams
ud
h"(V)
-
h"(U,)
45
9. The Mayer-Vietoris Technique
-
Since each O,, is an isomorphism by hypothesis, it follows that 8, is also an 0 L' k"( U ) is an isomorphism for isomorphism. Now suppose that h"( U ) each CJ E O ( X ) of ( n - 1)th type, and let V E O ( X ) be of nth type. Since V is open in X , it follows that each h"( U ) 3 k"( U ) is an isomorphism for each U E O( V )of ( n - 1)th type, so that each h"( V ) -% k"( V ) is an isomorphism by Lemma 9.2. This completes the inductive step, and since X is itself of nth type for some n 2 0, this also completes the proof of the theorem. As announced at the beginning of this section there will be several later applications of the Mayer-Vietoris technique. The applications will use slightly embellished versions of Theorem 9.3, and for convenience we present those versions as corollaries. For any space X of finite type one can regard a Mayer-Vietoris functor (h''(y E Zj as a single functor ( ( X ) ,el&@ carrying each U E O ( X ) into a direct sum jig( U ) of abelian groups, indexed by the integers 7 . Let R be a Z-graded ring that is commutative in the usual sense that if u E R and L' E R are of degrees p and q, then UL' = ( - 1 )p40uis degree p q, and suppose that 91lf is the category of 7-graded R-modules. Then 'Jn; c d d @ and we shall consider functors h4 from C ( X ) to 9 J f .
uqEI
+
u,
9.4 Corollary: Let lO(X) be the category of'open sets on a space X ojjinite type, let R he u graded commutative ring as above, and let h4 5 k4 be a degree-preserving natural transformation of' Mayer- Vietoris functors from P ( X ) to %IF. I f 9" is an R-module isomorphism whenever U E O ( X ) is u con9 tructible open set, then 0 is a naturul equivalence; in particular, h q ( X )A k 4 ( X )is an R-module isomorphism.
u,, u, u,
u4
PROOF: Immediate consequence of Theorem 9.3. The next variant of the Mayer-Vietoris technique is closer to Theorem 9.3 itself. 9.5 Corollary: Let H be a natural transformation of Muyer-Vietoris functors 1 h4 q E Z j und [ kq q E Z)as in Theorem 9.3, und let n E Z be a $xed integer. I / h4(U ) L I P (U ) is un isotnorphism ivhrtirler.q t I and the open set I / E 8(X ) 0 i s c~1ntr*trctihle,then hy(X)* li4(X ) is un isomorphism whenever q 5 ti.
I
I
PROOF: One proceeds exactly as in the proof of Theorem 9.3, observing that if p 5 q S n then the 5-lemma produces isomorphisms only when p < q 5 n.
In Chapters VI and VII we shall use a refinement of the Mayer-Vietoris technique, for which we single outoparticular base spaces. Let X be a smooth compact manifold with interior X . According to Proposition 8.4 there is a
46
I . Base Spaces
finite covering [ U l , .. . , U , } of 2 by open cells such that each nonvoid intersection U i n U j is a cell U k in the $overing, and such that the closures in X satisfy Ui n U j = O in U j . Let % ( X ) be the category whose objects are unions of the sets U , , . . . , U , . (One does not need “unions of intersections of the sets U , , . . . , U,” since each nonvoid intersection is automatically a union of the sets U , , . . , , U , . ) Morphisms in %(i) are inclusions. A MayerVietoris functor on %(k) is a family { h41 q E Z}of contravariant functors from 4 ( i ) to a category !JJl of modules for which there is a long exact MayerVietoris sequence for each pair ( U , V )E %(i) x .9(i),as before. The only difference is that we now restrict U and V to 2?(k); the category O ( 2 ) of all open sets in k is not needed as such. Here are two examples. Let H*(-) denote singular cohomology with coefficients H o ( {*}), and let H * ( X , -) denote relative singular homology with the same coefficients, where X is the given smooth compact manifold. 9.6 Proposition: Suppose that X is a smooth compact manifold; then there is u Mayer- Vietoris functor [ h4I q E Z)on %(i) with hq( U ) = H4( f o r every u E d(R).
e)
PROOF: It follows from Proposition 8.4that the closure in X of any element of %(i) is itself finitely triangulable. Hence one can construct the connecting homomorphisms d$,“ of the classical Mayer-Vietoris cohomology sequence
. .. +Hq-1(0 n V ) Hq(O H,(U” V)+ . ..
v)& H,( D)0
H4(
V)
for any ( U , V ) E %(g).Proposition 8.4also guarantees that K V = 0 n V , and since one always has U u V = 0 u V , this completes the proof. 9.1 Proposition: Suppose that X is an n-dimensional smooth compact mani-
I
j d d ; then thew is a Mayer-Vietoris functor { h4 q E Z}on % with hq( U )( = i) H,-,(x,x - U ) for every u E % ( X ) ,
PROOF: This time each X - U is itself finitely triangulable, so that one can construct the connecting homomorphisms of the classical relative MayerVietoris homology sequence to complete the proof. We shall construct a third Mayer-Vietoris functor on 2(k)in Chapter VI. Here is the specialized Mayer-Vietoris technique promised earlier. 9.8 Theorem (Mayer-Vietoris Comparison Theorem): Let X be a smooth compact manifold with interior 2, and let 0 be u natural transfurmation of Mayer- Vietoris functors { h4I q E H } and { k 4I q E Z}on 2(k).If h4(U i ) % k4(Ui) is an isomorphism for each of the open cells U . . . , U , in 2(2)when-
10. Remarks and Exercises
47
ever q S 0, it ,follows that h 4 ( i )3k q ( i ) is an isomorphism whenever 4 5 0. Simi/ur/y,i j ~P ( U J --!k kY(Ui) is an isomorphism for each of the open cells U , , . . . , U, it7 d ( i )jbr any q E Z,it follows that h 4 ( i )5k 4 ( X ) is an isotnorphism for un!’ q E Z.
PROOF: The 5-lemma applies exactly as in the proof of Lemma 9.2.
10. Remarks and Exercises 10.1 Remark: The category %? was identified in Osborn [6, pp. 745, 7541 as the natural category of topological spaces that serve as base spaces, in the sense described in the next chapter. Indeed, the defining properties of Definition 1.2, the closure properties of Proposition 1.4, the inclusion % c 28 of Corollaries 3.9 and 5.4, and the Mayer-Vietoris technique of Theorems 9.3 and 9.8 will be used not only in the next chapter, but throughout the entire book. The definition of %?, the inclusion W c B, and the Mayer-Vietoris technique were suggested by a construction in Connell [ l , pp. 499-5011; Connell attributes the construction to E. H. Brown. It is possible to replace the category by the category W (Definition 3.8 and Corollary 5.3) or by the category Wn (Definition 3.11 and the obvious analog of Corollary 5.3). However, many constructions appearing later in the book are most easily carried out in %? itself, rather than in -w‘ or in Wn, and there seems to be little point to frequent appeals to the inclusions %oc 7Y
cJ.
On the other hand, the relative size of %? is not itself a virtue. Although several major existence and uniqueness theorems (Theorems V.5.1, X.4.1, XI.6.1, and XI.7.3) are more easily established for the category 93,one is frequently interested in corresponding results (Theorems V.5.2, X.4.2, XI.6.2, and XI.7.7) for the category -4‘ of smooth manifolds. The inclusion . @ c J of Corollary 6.8 permits one to use the existence assertions of the former results to obtain corresponding existence assertions in the latter results; however, the uniqueness assertions in the latter results are somewhat more delicate. 10.2 Remark:
In Definition 1.1 the terminology “space of finite type” was chosen for convenience; the same phrase undoubtedly appears in many other contexts with different meanings, and no confusion is intended. However, Definition 1 . 1 itself may recall another definition, and there may indeed be some faint relation between the two concepts: a topological space X is of Ljusrernik-Sch,iirelmann category n 2 0 (according to one of the two
48
I . Base Spaces
most common conventions) if X can be covered by n + 1, but not by n, open contractible spaces. The Ljusternik-Schnirelmann category was introduced in Ljusternik and Schnirelmann [l] as a topological setting for variational problems; it is still used for that purpose, as in Ljusternik [l], Maurin [l], and Palais [l], for example. The homotopy invariance of the Ljusternik-Schnirelmann category suggests its importance in algebraic topology per se, and it has been investigated in that spirit by Fox [l, 21 and many later authors. Ganea [l] provides a 1962 survey of the subject, and there are more recent developments in Berstein [l], Borsuk [2], Coelho [l], Draper [l], Ganea [2, 31, Hardie [ 1,2,3], Hoo [11, Luft [l, 21, Moran [1,2,3,4], Ono [1,2], Osborne and Stern [l], Singhof [l, 21, and Takens [l, 21 for example; James [3] provides a 1978 survey of the subject.
10.3 Remark: The superficial similarity ofspaces of nth type (Definition 1.1) to spaces of Ljusternik-Schnirelmann category n has just been considered. However, the Ljusternik-Schnirelmann category ignores all properties of intersections of the sets in given coverings by open contractible sets, whereas such properties are of paramount interest in Definition 1.1. Intersections in open coverings are considered in Nagata [l, pp. 133-1371 in the form of “multiplicative refinements.” For example, a separable metric space is of dimension S n if and only if for every open covering there is a multiplicative refinement of length 5 n + 1 ; the appropriate definitions can be found in Nagata [ 11. (There is a corresponding result in Hurewicz and Wallman [l, pp. 54,66,67], which ignores intersections.) A somewhat more specialized result concerning minimal open coverings of manifolds, with well-behaved intersections, can be found in Osborne and Stern [l]. 10.4 Remark: There is an omission in the list B, W , Wo of categories considered in Remark 10.1. A hausdorff space X is compactly generated if a subset A c X is closed whenever all intersections A n B c X with compact subsets B c X are closed. The category X of compactly generated spaces ( = Steenrod’s convenient category) was introduced and investigated in Steenrod [6]; the definition and elementary properties can also be found in Cooke and Finney [l, pp. 86-1051, Gray [l, pp. 50-611, and G. W. Whitehead [l, pp. 18-20]. For example, any metric space is compactly generated. Some alternate “convenient categories” are introduced in Vogt [1,2] and compared to Steenrod’s convenient category. The categories of Steenrod and Vogt are indeed convenient for much of homotopy theory: Steenrod’s category suffices for more than 700 pages in G. W. Whitehead [l], for example. However, Steenrod’s definition is not itself homotopy invariant, and it would not serve especially well in the very
10. Remarks and Exercises
49
next chapter of this book, which uses frequent homotopy equivalences. One can perhaps replace 93 by the category % of spaces that are homotopy equivalent to compactly generated spaces: 93 c 92 since any metric space is compactly generated, and according to a result in May [l, p. 281, for any X E % there is a homotopy class of weak hornotopy equivalences X ’ -+ X relating some X’ E w‘ (cg) to X . However, 93 will suffice for the purposes of the present work. 10.5 Remark : Simplicia1 complexes and metric simplicia1 spaces, considered in $2, are also discussed in Cairns [6, pp. 65-89], Hilton and Wylie [l, pp. 14-52], Maunder [l, pp. 31-62], Spanier [4, pp. 107-1291, for example. 10.6 Remark: The telescope of Definition 3.3 was suggested to the author by a corresponding construction in Connell [l]. A similar construction is used for a related purpose in Bendersky [2, pp. 16-17]. Incidentally, the inclusion w‘ c 9 of Theorem 3.7 (and Corollaries 3.9 and 5.4) was proved more rapidly in Osborn [6], but by a more sophisticated method. Specifically, although Osborn [6] uses a simpler telescope IKI* than the one constructed in Definition 3.3, there is no direct construction of a homotopy inverse to the obvious projection IKI* .+ IKI. Instead, one observes that IKI* -+IKI is a weak homotopy equivalence, and according to a classical result of J. H. C. Whitehead [4] it follows that IKI* .+ IKI is a homotopy equivalence in the usual sense, as in this book. Whitehead’s theorem was cited following Theorem 5.2; it can also be found in Gray [l, p. 1391, Lundell and Weingram [I, p. 1251, Maunder [l, pp. 298-3001, and Switzer [l, pp. 87-90], for example.
10.7 Remark: The proof of Proposition 4.6 (Dowker [l]) can also be found in Lundell and Weingram [l, p. 1311, Milnor [8, p. 2761, and Dold [8, pp. 354-3551, for example. 10.8 Remark: CW spaces were first considered in J. H. C. Whitehead [3]. Other expositions can be found in Cooke and Finney [l], Gray [l, pp. 1131211, Hilton [l, pp. 95-1131, Hu [3, pp. 111-1491, throughout Lundell and Weingram [l], Massey [6, pp. 76-1041, Maunder [l, pp. 273-3101, throughout Piccinini [ 11, Rohlin and Fuks [l, Chapter 111, Spanier [4, pp. 400-4181, Switzer [ 1, Chapter V], and G. W. Whitehead [ 1, pp. 46-95]. There are CW spaces which are not simplicia1 in any sense (other than homotopy equivalence). Specific examples of such spaces can be found in Metzler [ 11 and in Bognar [ 13, for example.
50
I. Base Spaces
10.9 Remark: According to Corollary 5.4 every CW space is a base space, a fortiori homotopy equivalent to a metric space, so that Lemma 8.2 (Stone [I]) guarantees that every CW space is homotopy equivalent to a paracompact space. A stronger statement is available: every CW space is itself paracompact. This was established by Miyazaki [11, following partial results of Bourgin [l] and Dugundji [l]. Proofs of Miyazaki’s theorem can be found in Postnikov [l] and Lundell and Weingram [l,pp. 54-55]. 10.10 Remark: There are many elementary introductions to the category M of smooth manifolds, which was described very briefly in $6. See Auslander and MacKenzie [2], Boothby [l], Guillemin and Pollack [l], M. W. Hirsch [4], Hu [6], Lang [l], Milnor [15], Rohlin and Fuks [I, Chapter 1111, or Wallace [5], for example. 10.11 Remark: The Whitney embedding result of Theorem 6.6 is neither the easiest nor the best-possible embedding theorem for smooth manifolds. The “easy” Whitney embedding theorem asserts that any smooth n-dimensional manifold X admits at least one smooth embedding X -+ R2“+’. This result was announced in Whitney [l] and proved in Whitney [3]. Other proofs of the “easy” Whitney embedding theorem can be found in T. Y. Thomas [l, 21, in Whitney [9], Auslander and MacKenzie [2, pp. 106-1 161, de Rham [l, pp. 9-16], Greene and Wu [l], Guillemin and Pollack [l, pp. 39-56], and in the 1966 revised edition of Munkres [l], for example. The “hard” Whitney embedding result X -+ R2“of Theorem 6.6 appears in Whitney [7, p. 2361; its proof occupies all 27 pages of that paper. A long-standing “best-possible” embedding conjecture is that any smooth n-dimensional manifold X admits a smooth embedding X -+ [ W Z n - a ( f l ) + 1, where a(n) is the number of 1’s in the dyadic expansion of the dimension n. (See Gitler [l], for example.) One can definitely d o no better: in Chapter VI we shall show for each n > 0 that there is a smooth closed ; no stronger n-dimensional manifold which cannot be embedded in R2n-a(n’ counterexamples are known. Although the “best-possible’’ embedding conjecture remains unproved, there is at least one faint suggestion of its truth. According to a result of R. L. W. Brown [2, 31, every smooth closed n-dimensional manifold is equivalent to one which smoothly embeds in [ W 2 n - a ( n ) + .* th e equivalence is “cobordism,” which will be discussed in Chapter VI. In lieu of a proof of the “best-possible” embedding conjecture, there have been many improvements upon the “ h a r d Whitney embedding theorem. It is known for every n > 1, with the possible exception of the case n = 4, that every smooth orientable n-dimensional manifold embeds in R2”- 1 ., the case n = 4 is probably not an exception. It is also known for
51
10. Remarks and Exercises
every natural number n > 1 not of the form 2' that every smooth nonorientable n-dimensional manifold embeds in R2"- ; the cases n = 2' definitely me exceptions since the real projective spaces RP" do not embed in [W2"-' for these values of n. The fact that RP" is an exception for n = 2' will be established in Proposition VI.4.9, and the remainder of the preceding results will be considered in more detail in Remark VI.9.16, with appropriate references. 10.12 Remark: There are several analogs of the Whitney embedding theorems. For example, any n-dimensional separable metric space is homeomorphic to a subset of R2"+ ; proofs are given in Hurewicz and Wallman [I, pp. 60-631 and Nagata [l, pp. 101-1081. For any n-dimensional simplicial space IKI there is a simplicial embedding ( K (-+ R2"+ ; this analog of the "easy" Whitney embedding theorem is proved in Seifert and Threlfall [ l , German edition, pp. 44-46; English edition, pp. 45-47], and Cairns [6, pp. 78-80]. In case IKI is an n-dimensional triangulated manifold there is a simplicial embedding IKI + R2", due to van Kampen [I]. A necessary and sufficient condition for the existence of simplicial embeddings IKI -+ [W2" of n-dimensional simplicial spaces IKI in general is given in Wu [lo, p. 2371, and it is shown to be satisfied by n-dimensional triangulated manifolds in Wu [10, p. 2571; this provides another proof of van Kampen's analog of the " h a r d Whitney embedding theorem. Any two simplicial embeddings IKI 5 [ W Z n f 2 and IKI 3 R2n+2of an ndimensional simplicial space IK are always isotopic in the sense that there is a simplicial map IKI x [0,1] + R 2 n + 2such that each IKI % [ W 2 n + 2 is itself a simplicial embedding; if ( K (is a triangulated manifold of dimension n > 1, then any two simplicial embeddings IKI 2 R2"+ and IKI 3 R2"+' are isotopic. These results, also due to van Kampen [l], are proved in Wu [lo, pp. 207, 2581.
'
1,
10.13 Remark: The Whitney immersion result of Theorem 6.9 is neither the easiest nor the best-possible immersion theorem for smooth manifolds. The "easy" Whitney immersion theorem asserts that any smooth n-dimensional manifold X admits at least one smooth immersion X + S2". This result was announced (along with the "easy" embedding result) in Whitney [l] and proved in Whitney [3]. Another proof of the "easy" Whitney immersion theorem can be found in Auslander and MacKenzie [2, pp. 1061331. The "hard" Whitney immersion result X -+ SZn-lof Theorem 6.9 appears in Whitney [8, p. 2701; its proof occupies all 47 pages of that paper. The
52
I . Base Spaces
same result appears in M. W. Hirsch [l, p. 2701, with an alternative proof that has had far-reaching consequences. A long-standing “best-possible” immersion conjecture has recently been proved by Cohen [13, the proof depending upon results and techniques of M. W. Hirsch [I] and Brown and Peterson [4,5]: for any n > 1 any smooth compact n-dimensional manifold X admits an immersion X + p”- a(“), where a(n) is the number of 1’s in the dyadic expansion of the dimension n. This result will be described in more detail in Remark VI.9.14. One can definitely do no better: in Chapter VI, for each n > 0, we shall show that there is an easily constructed smooth closed n-dimensional manifold which cannot be immersed in R 2 n - a ( n ) As for embeddings, a weaker result of R. L. W. Brown [2,3] asserts that every smooth closed n-dimensional manifold is equivalent to one which smoothly immerses in R2”-‘(”): the equivalence is “cobordism,” which will be discussed in Chapter VI.
’.
10.14 Remark : The Cairns-Whitehead triangulation result of Theorem 6.7 was first proved for closed smooth manifolds in Cairns [l, 21. The result was extended to arbitrary smooth manifolds in J. H. C. Whitehead [ 11. Discussions and simplifications of the combined result can be found in Cairns [3,4,5], and other versions of the proof can be found in Whitney [lo, pp. 124-1351, in J. H. C. Whitehead [5], and in the 1966 edition of Munkres [ 11, for example.
10.15 Remark: One of the primary goals of differential topology is a reasonable classification (in some sense) of all smooth manifolds. One cannot expect a complete classification, however, even up to homotopy type, for the following reason. At the end of Chapter 7 of Seifert and Threlfall [ 11 one learns that for any prescribed finitely presented group G whatsoever there is a closed oriented 4-dimensional manifold X whose fundamental group is G ; the details of the construction can be found in of Massey [4, pp. 143-1441, for example. Quite independently of differential topology, Boone [1, 2, 31, Britton [I], and P. S. Novikov [l, 21 showed that the word problem for finitely presented groups is recursively unsolvable; this led Adyan [l] and Rabin [l] to conclude that the isomorphism problem for finitely presented groups is recursively unsolvable. The topological and group-theoretic pieces of the puzzle were juxtaposed by Markov [1,2,3] to conclude that no algorithm exists for classifying oriented 4-dimensional manifolds, a fortiori that no algorithm exists for classifying all smooth manifolds. There are some related results in Boone, Haken, and Poenaru [11. The lack of a universal recipe is no obstacle to useful partial results, however. The classification of compact 2-dimensional manifolds is a classical
10. Remarks and Exercises
53
result, for example, which can be found in Massey [4, pp. 10-181. Certain 6-dimensional manifolds are classified in Jupp [l]. There is a complete classification (up to homeomorphism) of closed ( n - 1)-connected (2n + 1)dimensional smooth n-dimensional manifolds for n > 7 due to Wall [2]; the same result is known for certain smaller values of n as a result of Wilkens [ 11 (and Wall [2]), and related classifications appear in Tamura [ 1, 2, 31. Finally, according to Cheeger and Kister [ 11 there are only countably many closed topologicul rnanifbfd.7; a fortiori, up to homeomorphism there are only countably many closed smooth manifolds. A 1975 survey of the classification problem appears in Sullivan [2], and a 1978 survey appears in T. M. Price [I]. 10.16 Remark : The topological manifolds mentioned in the previous Remark are merely locally euclidean hausdorff spaces whose topologies have a countable basis of open sets. For example, the underlying topological space of a smooth manifold is a topological manifold. Before one deals with the inverse question of finding smooth structures on a given topological manifold it is of interest to consider a broader question: can a topological manifold be triangulated? One-dimensional manifolds pose no problem, and the triangulation of surfaces was established by Rado [ 13, whose proof can be found in Ahlfors and Sario [ 1, pp. 44-46]. The triangulation of 3-dimensional manifolds was first established by Moise [ 11, and an alternative method was later supplied by Bing [ 11, with the same result. The 4-dimensional case remains a mystery. However, for n > 4 the following results were established independently by Kirby and Siebenmann [ l ] (reproduced in Kirby and Siebenmann [2, pp. 299-3061), and by Lashof and Rothenberg [2]. If X is any closed n-dimensional topological manifold ( n > 4), or any open n-dimensional topological manifold ( n > 5), and if H4(X;Z/2) = 0, then X can be triangulated; furthermore, ifH3(X; 2/2) = 0, then any two triangulations of X are equivalent in an obvious sense. The Z/2 cohomology conditions are known to be necessary. In fact, in every dimension n > 4 there is a closed topological manifold with no triangulation (as a manifold). An elementary discussion of the results described in this remark can be found in Schultz [3]. 10.17 Remark: The negative results of the preceding remark are offset by the following homotopy property of arbitrary topological manifolds, which remain locally euclidean hausdorff spaces whose topologies have a countable basis of open sets. Every topological manijold is homotopy equivulent to a s i m p k i d space IKI. More specifically, a simplicia1space IKI is locully$nite if and only if each point of IKI has a neighborhood which meets only finitely many geometric simplexes of IKI; equivalently, IKI is locally Jinite whenever the Dowker
54
I . Base Spaces
homotopy equivalence lKlw + lKlmof Proposition 4.6 is a homeomorphism. Y is proper if and only if the inverse image Furthermore, as in tj6 a map X of any compact set in Y is compact in X ; a proper homotopy equivalence is a homotopy equivalence in a given category of topological spaces and proper maps. According to a result in Kirby and Siebenmann [2, p. 1231 every topological manifold is proper homotopy equivalent to a locallyfinite simplicia1 space IKI.
10.18 Remark: By combining the weaker of the two preceding assertions with Theorem 3.7 one obtains the following generalization of Corollary 6.8: any topological manifold is a base space; hence the category of topological manifolds is a full subcategory of the category of base spaces. 10.19 Remark: The Cairns-Whitehead triangulation theorem (Theorem 6.7) precludes smooth structures for nontriangulable manifolds; in fact, there are even triangulable manifolds with no smooth structure, such as the 10dimensional manifold of Kervaire [3]. However, a topological manifold can also have more than one smooth structure, a result first established for the 7-sphere S7 in Milnor [l]. Since then, smooth structures on spheres have been thoroughly treated in Milnor [6, 71, Kervaire and Milnor [l], and Eells and Kuiper [3], for example; we shall consider some of this material in Volume 3 of the present work. Most products S P x S4 of spheres have more differentiable structures than the sphere S P + ¶ ,and more differentiable structures than the product of the corresponding numbers for the factors S p and S4. Some recent results concerning such products can be found in de Sapio [ 1,2], Kawakubo [1,2], and Schultz [ 1, 21, for example.
10.20 Remark: Here is a procedure that produces new smooth manifolds from old ones. Let X be a given smooth manifold, and let X -+ X be a smooth involution that is free in the sense that there are no fixed points; this provides an obvious equivalence relation in X for which the quotient X / - is a new smooth manifold with X as a double covering. For example, for any n > 0 the antipodal map S" S" of the standard n-sphere S" is a free involution for which the resulting quotient S"/- is the real projective space RP" of the same dimension. In the special case n = 3 Livesay [ 11 shows that the antipodal map is essentially the only free involution of S3: any free involution S3 -+ S 3 is smoothly equivalent to the antipodal map. However, according to Milnor [161, Hirsch and Milnor [I], and Fintushel and Stern [ 11, there are "exotic" free involutions of the standard spheres S7,S6, S5, S4 whose quotients S'/are not diffeomorphic to the corresponding projective spaces RP7, RP', R P 5 ,
-
-+
55
10. Remarks and Exercises
RP4. (The "exotic" involution S4 -,S4 is recent, although the quotient S4/had been constructed in Cappell and Shaneson [l] earlier by other means. Although S4/- is not diffeomorphic to RP4, it is homotopy equivalent to RP4.) More generally, Browder and Livesay [13 provide an invariant, further described in Livesay and Thomas [ 11, which leads to the following result of Orlick and Rourke [ I ] : for each k > 0 there are infinitely many distinct free (and of certain exotic spheres C4k+3).The quotients of involutions of S4k+3 S4k+3(and X 4 k + 3 ) provide a large supply of smooth closed manifolds of dimensions n = 4k + 3. There is a criterion for the existence of exotic free involutions of S" (and exotic spheres Y ) ,valid for any n >= 5, in Lopez de Medrano [I, p. 671. One can also use certain smooth involutions X + X that are not free to create smooth quotient manifolds X i - ; however, the results are not always new. For example, complex conjugation induces an involution C P 2 -+ C P z of the complex projective plane CP', for which one easily establishes that C P z / - is a smooth manifold; however, Kuiper [l] and Massey [5] independently established the disappointing result that C P z / - is merely the standard 4-sphere S4. Here are some easily constructed smooth manifolds which will play a role in later remarks; their construction is similar to the construction of RP" from the standard n-sphere S". Any odd-dimensional sphere S2"+ can be regarded as the subspace of those points (zo,zl,. . . , z,) E @ " + I such that Iq,(' + 1z1I2+ . . . + l : n l z = 1. Let q , , . . . ,4. be any integers which are relatively prime to a fixed integer p > 0, and let S Z n + l1:S2"+'be the diffeomorphism given by h ( z o , z , , . . . , 2") = (ezniipzo,e2niq1'p. . , e 2 n r q n / P A, n ) . Then h is periodic with period p , with no fixed points, and the quotient of S Z n + 1 by the resulting equivalence relation is a smooth closed (2n 1)dimensional manifold, the lens space L ( p ; q l , . . . , qn). The 3-dimensional lens spaces L ( p ;y) were first constructed in Tietze [ 11. Later work of Reidemeister [ I ] and J . H. C. Whitehead [2] established that L(7; 1 ) and L(7; 2), for example, are homotopy equivalent but not homeomorphic. R. Myers [ I ] generalizes the result of Livesay [ 11 concerning S3 by showing that all involutions of L ( p ;y) are equivalent to those induced by the action of the orthogonal group; similar results are valid for certain involutions of period greater than 2.
+
10.21 Remark: In 1844 Grassmann [l] described the Grassmann maniR""", the embeddings Gm(R"+")4; folds G"(R"'+''), the exterior products G 1 ( KR"'"), and the Plucker relations, all of which are used in Proposition
A"
7.3. Despite the 1862 amplifications in Grassmann [2], Grassmann's work was not fully appreciated until it was reexamined and further developed by
56
I . Base Spaces
Hermann Casar Hannibal Schubert, beginning in 1886; Schubert’s work is briefly cited in Remark V.7.3. By 1897 elementary differential geometry was easily presented in Grassmann’s setting, as in Burali-Forti [l]. In 1921 Corrado Segre, who had understood and applied Grassmann’s work as a student during the 1880s, wrote a much-respected encyclopedia article identifying Grassmann manifolds as the very source of higher-dimensional algebraic geometry; Segre’s enthusiasm appears in C. Segre [l, p. 7721. The Plucker relations and Proposition 7.3 can be found in Kleiman and Laksov [I], for example. Alternative proofs of Proposition 7.3 appear in Milnor and Stasheff [l, pp. 57-59] and in Lanteri [l]. 10.22 Exercise: Show that any connected base space X E B is pathwise connected. Hint: First prove the property for metric spaces of finite type, then observe that pathwise connectedness is a homotopy property. Some of the materials for this exercise can be found in Hu [3, pp. 84-90], for example. 10.23 Remark : The fundamental properties of Mayer-Vietoris sequences
were discovered long before the general notion of an exact sequence existed, in Mayer [l] and Vietoris [l]. Exact sequences as such were introduced by Eilenberg and Steenrod [l], and later by Kelley and Pitcher [l]. 10.24 Remark: Bott and Tu [l, pp. 42-53] prove a generalized version of
Proposition 8.4, using Riemannian metrics rather than triangulations, which they apply in their elegant presentation of the Mayer-Vietoris technique for smooth manifolds.
CHAPTER I1
Fiber Bundles
0. Introduction Let E X be a map onto any space X, and suppose for some space F that there is a homeomorphism E 5 X x F for which
commutes, where n 1 is the first projection. Then E 5 X represents a triuiul fiber hundlr. More generally, let E 5 X be locally trivial in the following sense, for some space F : there is an open covering [ U iI i E 1 ) of X and a corresponding family ('4'; I i E I } of homeomorphisms n- '( U i )-% U i x F for which each f l ( U i ) A U ix F
commutes. Then the projecrion E 5 X represents a j b e r bundle with total spuce E over the base spuce X,the j b e r being F.
57
58
11. Fiber Bundles
The preceding description is incomplete. If the intersection U in U i of two sets in the open covering { UiI i E I } of the base space X is nonvoid, then the restrictions of the corresponding homeomorphisms Y i and Y j to n - ' ( U i n U j )c E induce a homeomorphism Y j c 'I-',such :' that ( U i n U j )x F
Y , P; 0
' ( U i n U j )x F
U in U j commutes. Such a homeomorphism is necessarily of the form Y j Y; '(x, j ' ) = (x,${(x)(.f)),where ${ carries each x E U in U j into a homeomorphism *JW F of the fiber; in particular, $)(x) $i(x) is the identity for any x E U i n U j , so that $: and $/ carry any x E U i n U j into inverse homeomorphisms of the fiber. Since any transformations F -+ F whatsoever obey the associative law, it follows that each ${ can be regarded as a map of U in U j into a group G of homeomorphisms of the fiber, called the structure group of the given fiber bundle; the continuity of the compositions Y j \Y; ' imposes a specific topology on G for which the group operation G x G + G, the group inverse G 0 - 1 G, and the action G x F -+ F of G on F are are the transition functions of the continuous. The maps U in U j % G representation E X of the bundle, with respect to the covering { U i I i E I } and corresponding family { Y I i E I } . The structure group G is part of the definition of a fiber bundle, and its choice is critical. For example, if one is interested in preserving given properties of the fiber F , one does not choose G to consist of all homeomorphism F + F ; thus if F = R" and one wants to preserve vector addition in R" and multiplication by scalars, one chooses G to be the general linear group GL(rn,R), or one of its subgroups. On the other hand, if G consists only of the identity map F --* F, then any projection E X representing the given bundle is necessarily of the form X x F 3 X , so that the bundle is trivial. The classical Mobius band provides the simplest nontrivial fiber bundle. Let E S' be the projection of the Mobius band E onto the unit circle S',
-
0
-
E
0
S'
59
0. Introduction
as indicated in the accompanying figure, the fiber F being the closed unit interval [ - 1,1]. One can cover the base space S’ by two open sets U o , U 1 homeomorphic to open real intervals, for which there are obvious homeomorphisms f l ( U i ) 5 U ix [ - 1,1]. The intersection U o n U , consists of two disjoint open sets Vo and V , , which one can regard as subsets of U o , for example. The Mobius band E is completely described by requiring the homeomorphism ( U , n U l ) x [-1,1]
‘i’i
Yc i ’
( U , n U l ) x [-1,1]
to carry ( x , f ’ )E ( U o n U , ) x [ - 1,1] into ( x , j ’ )or (x, - f ) according as E V, or x E V , . In this case one takes the structure group G to be 2/2 in the discrete topology, which acts on [ - 1,1] via multiplication by + 1 or - 1. x
One traditional definition of fiber bundles is given in $1, followed in 92 by an equivalent description which is closer to the preceding sketch. Nothing in these two sections requires any properties of the base spaces X ; however, many later results do require some sort of restriction. Accordingly, fibre bundles have arbitrary base spaces X , and fiber bundles have base spaces X in the category $9 of Definition 1.1.2. Portions of the rationale for the eventual restriction to fiber bundles appear in %3,4, 5, and 7; beginning in $7 there are only fiber bundles. Contractible spaces were used in Definition 1.1.1 as building blocks for spaces of finite type, leading to the category B. In $3 it will be shown that fibre bundles over contractible spaces are trivial bundles, the building blocks for fibre bundles in general. In 94 it will be shown for any map X ’ X in the category $9 and for any fiber bundle over X that the corresponding “pullback” fiber bundle over X ’ depends only on the homotopy type off. The proof uses certain properties of the category d,which was defined in terms of homotopy types. We have already observed that the structure group G is an integral part of the description of a fiber bundle, and that it is desirable to “reduce” G to as small a subgroup K c G as possible. In $5 it is shown that if the structure group of a fiber bundle (base space in $9) is a Lie group with finitely many connected components, then one can “reduce” G to a compact subgroup K c G . In particular, in $6 it is shown that one can always “reduce” the linear groups GL(m,R), GL+(rn,R),and GL(n,C) to specific compact subgroups O ( m )c GL(rn,R), O+(rn)c GL+(m,R), and U ( n )c GL(n,C), as structure groups of fiber bundles. In 97 a basic step is provided for assigning cohomology classes in H * ( X ; A) to fiber bundles over base spaces X E 93,where A is an appropriate coefficient ring. The provisions describing the category B are essential for the result, which will be used several times in later chapters.
60
11. Fiber Bundles
1. Fibre Bundles and Fiber Bundles Some of the ingredients of a fiber bundle over a space X are a j b e r F and a structure group G of homeomorphisms F 3 F , as we have just learned. We shall assume that G acts on the lefl of F ; that is, ( g 1 g 2 ) f = g l ( g 2 f ) E F for any g 1 E G, g 2 E G , and f E F. Although the fiber F may be any topological space, the topology of G must be admissible: the group operation G x G + G, the group inverse G G, and the action G x F -,F of G on F must be continuous. The action G x F --t F must also be eflectiue: the identity is the only element g E G such that gf' = f for every f E F. The space X is the base space of the bundle; we shall later require X to belong to the category B of base spaces, as in Definition 1.1.2. For any map E $ X of a space E onto a space X , and for any x E X , let E x denote the inverse image x - l ( { x } ) of {x} c X under 7c. A nonvoid set h S, of homeomorphisms E x + F is G-related whenever for any h E S,, h' E S,, h-l and g E G the compositions Ex F 3 F and F Ex 1;F belong to S, and G, respectively. Equivalently, for any fixed h E S,, the set S , consists of all compositions E x 5 F 5 F , where g E G. Given two maps E 5 X and E' 5 X ' onto spaces X and X ' , and given points x E X and x' E X ' and G-related sets S, and S,. of homeomorphisms h' E x -$ F and E',. -, F , respectively, a G-related isomorphism is any homeohW' h' morphism E x + such that every composition F E x + E',. + F belongs to G. If
-
-
El,,
E
X
f
+ E'
+ X'
commutes and x E X , then for Ex = x - ' ( { x } ) c E and E;,,, = x'-'( {f'(x))) c E' it is clear that f induces a map E x + E;,,,. 1.1 Definition: Given a fiber Fand a structure group G of homeomorphisms
F + F , a fumily of' jibers over a space X is a surjective map E 5 X and an assignment to each x E X of a G-related set S, of homeomorphisms E x + F. If E 3 X and E' 5 X ' are families of fibers with the same fiber F and same structure group G , then a morphism from E 5 X to E' -5 X ' is a pair of maps
61
I . Fibre Bundles and Fiber Bundles
X
X ' and E 1,E such that E
f
X
r
t
E'
t
X'
commutes, and such that for each x E X the induced map E , + E;,,, is a G-related isomorphism. For any family E 5 X of fibers one calls E the total space, 71 the projection, and X the base space. For the moment the base space X may be any topological space; the restriction X E L@ will be imposed later. For any given fiber F and structure group G it is clear that families of fibers are the objects ofa category V(F,G) whose morphisms are commutative diagrams. Here is a way to construct new such objects and morphisms. 1.2 Definition: Let E' 5 X ' be a family of fibers in the category V(F,G), let X 1; X ' be any map, and let E c X x E' consist of those (x, e') E X x E' with j ( x ) = ~ ' ( e E' )X ' , in the relative topology. The pullback of' E' 5 X ' ulong ,f is the restriction E X to E c X x E' of the first projection X x E' 3 X .
1.3 Lemma: Let E' 5 X ' be a family of'j b e r s in the category W ( F ,G), and X 1; X ' he a map. Then the pullbuck E 5 X also belongs to W ( F ,G), and there is u canonical map f such that let
E
f
X
s
+ E
t
X'
is u morphism in (6( F ,G).
PROOF: This is a direct verification, in which E E' is defined as the restriction to E c X x E' of the second projection X x E' 3 E'.
Here are the most obvious families of fibers. 1.4 Definition: Given a fiber F and a structure group G , as before, the product fumily q/',fiber.sover a space X is the first projection X x F A X ;
62
11. Fiber Bundles
for each x E X the G-related set S , of homeomorphisms E x -, F consists of themapsE,={x} x F = F z F , f o r a l l g E G . 1.5 Lemma: Pullbacks of product families of fibers are product families oJ’
Jibers. PROOF: Let X’ x F -% X‘ be a product family, and let X 3 X ’ be any map. The total space E of the pullback along g consists of those (x, (x’, f ) ) E X x ( X ’ x F ) with g(x) = n,(x’,f ) = x’, which is canonically homeomorphic to X x F, and the map E -1: X is the first projection X x F -%X . The isomorphisms in the category %?(F,C) are clearly those morphisms E
f
X
s
E’
+
X‘
for which bothf and f are homeomorphisms. However, we shall be interested primarily in those isomorphisms in which X = X’ and f is the identity; such isomorphisms are of the form
for a homeomorphism g. 1.6 Lemma: Let X X‘ be any map. Then pullbacks along f of isomorphic jamilies of ,fibers over X ‘ are isomorphic families of fibers over X .
PROOF: Given a morphism
in which g’ is a homeomorphism, the corresponding diagram
63
I . Fibre Bundles and Fiber Bundles
for the pullbacks is just the restriction of
to E c X x E’, which consists of those (x,e’)E X x E’ with f ( x ) = 7c’(e’). The map E 5 is then the restriction to E of a homeomorphism id x g’, and hence itself a homeomorphism. We now pass from the category V(F,G) of families of fibers, for a given fiber F and structure group G, to the category of isomorphism classes of families of fibers; as before we consider only isomorphisms of the form E
g
+ E
Lemma 1.6 guarantees that if 4 denotes an isomorphism class of families of fibers over a space X ’ , then the pullbacks along any map X 1s X ’ also form ’ such an isomorphism class, denoted f’t.One simply calls f’!( the pullback X’ .% X” is a sequence of maps then for of’ 5 along X X’. Clearly if X any isomorphism class 5” over X ” , then one has f ! g ! ( ” = (g f)’5” as isomorphism classes over X. 0
1.7 Definition: Let X be an arbitrary topological space. The trivial bundle with fiber F and structure group G is the isomorphism class of the product family X x F 3 X of fibers.
5 over X
X’ be 1.8 Lemma: Let 5‘ be u trivial bundle over u space X ‘ , and let X un urbitury nzap; then the pullback f’!<‘ is the corresponding trivial bundle over X . PROOF: Lemmas 1.5 and 1.6. Now let 4 be an arbitrary isomorphism class of families of fibers over a space X , and let U X be an inclusion U c X . The pullback i!( is the restriction 5 ) U of 5 to U c X. 1.9 Definition: For a given fiber F and structure group G , an isomorphism class of families of fibers over a topological space X is a j b r e bundle over
<
64
11. Fiber Bundles
X whenever there is an open covering { U , I i E I } of X such that each restriction 5 I U , is a trivial bundle. A jiber bundle is any fibre bundle over a space X in the category B of base spaces, as in Definition 1.1.2.
For the moment we work with fibre bundles in general. However, the major theorems of 44 and 45 will be proved only for fiber bundles, and the remainder of the book concerns only certain fiber bundles. 1.10 Proposition: Let X -5 X‘ be any map and let 5‘ be u jibre bundle over X ‘ f o r u given jiber und structure group. Then the pullback f ’ 5 ’ is (I jibre bundle over X for the same fiber and structure group.
PROOF: By hypothesis X ’ has an open covering .( U ; I k E I ) such that each restriction 4’ I U ; is trivial. If Uk = f - I ( U;) c X , then [ Uk 1 k E I } is an open covering of X , and for each k E I the map f restricts to a map Uk 3 U ; . Let U k X and U ; 1, X’ be the inclusion maps, so that 9 +
u;
commutes. Then and since each 5’1 U ; is trivial by hypothesis, Lemma 1.8 guarantees that each f ’ t ’ I U k is trivial, as required. One frequently identifies a fibre bundle over X by choosing a single representative E 1:X of the isomorphism class 5 of families of fibers, arbitrarily calling E the total space and II the projection of the bundle. This is a harmless abuse of language since E 1:X is related to any other representative E‘ 5 X by a commutative diagram
in which f is a homeomorphism inducing a G-related isomorphism E, + EL for each x E X. It will be useful to simplify the preceding criteria for E 5 X and E‘ 5 X to represent the same fibre bundle over X .
65
I . Fibre Bundles and Fiber Bundles
Recall that the topology of the structure group G is admissible, meaning that the group operation G x G + G , the group inverse G- 0 - 1 G, and the action G x F + F of G on F are continuous. Recall also that G acts effectively on F , meaning that the identity is the only element g E G such that gJ = f' for every J' E F ; that is, a given homeomorphism F + F is induced by at most one element of G.
I . 11 Lemma : Let f he uny mup such that X X F - X fX F
\ n'
/
\X J
commutes, und which induces u homeomorphism F then f i s u homeomorphism.
+F
i n G Jbr euch x.
E
X;
PROOF: Since G is effective, there is a unique map X 3 G such that f ( s , / ' )= ( . ~ , $ ( . x ) ( f ' ) ) for every ( x , f ' ) E X x F. Since the topology of G is admissible, the composition X % G 2G is continuous. Hence one can f-' define a continuous inverse X x F X x F by setting f '(x,,/') = (s,$(.x)- ' ( f ' ) ) for every ( x , f ' )E X x F. 1.12 Proposition: Let E X unrl E' 5 X represent jibre bundles ( und (' with j h r r F und structure group G oiwr the sume space X . I f there i s 11 mup f such thNt f t E' E
cornmutes and induces G-reluted isomorphisms E x + E; jbr euch x E X , then f i s necessarilj~u homeomorphism, ~ i n dhence 4 = 4'.
PROOF: There are open coverings [ Ui I i E I ] and [ U j l j E J such that the restrictions (1 U ;and 4'1 U j are trivial; hence for K = I x J and U , = U,i,j, = I / , n U j there is a single open covering ( U k I kE K ) of X such that the restrictions (1 U , and (' 1 U k are trivial. The latter restrictions are therefore represented by product families, for which f induces maps fk such that each U, x F \
U, x F /
66
11. Fiber Bundles
commutes, inducing a homeomorphism F + F in G for each x E U k . By Lemma 1.11 each fk is a homeomorphism, so that f is necessarily bijective. To show that f is a homeomorphism it remains only to show that it is an open map; but if V c E is open, then f ( V ) c E’ is a union of open sets f ( V n n - ‘ ( U k ) )c E’, also provided by Lemma 1.11. 1.13 Corollary: Given a jiber F and structure group G, let the firmilies E 5 X and E’ 5 X ‘ of ,fibers represent j b r e bundles 5 over X and [‘ over X ‘ , respectively. Then for any morphism
E
/1 X
of’ firmilies of
f
I
t
E‘
>
X‘
,fibers 5 is the pullback f ‘5’ of’ 5‘ along f’.
PROOF: The total space E” of the pullback E” X of E’ 5 X‘ consists of those (x,e’) E X x E‘ with f ( x ) = lc‘(e’)E X . Since f n = n’ c f, there is consequently a map E 5 E” with g(e) = (lc(e),f(e))for each e E E . Trivially 0
commutes, where lc” is induced by the first projection X x E‘ 5X . Furthermore, for each x E X the induced map E , -+ E:’ carries e E E x into (x,f(e))E E:’; therefore, since f induces G-related isomorphisms E, + E:, the maps E, + E:’ are also G-related isomorphisms. Hence g is a homeomorphism by Proposition 1.12.
2. Coordinate Bundles The informal description of fiber bundles given in the Introduction to this chapter began with something more concrete than Definition 1.9. Although the informal description was used primarily as motivation for the structure group G, it too can be molded into a formal definition of fiber bundles, equivalent to Definition 1.9.
67
2. Coordinate Bundles
Assume that a fiber F and structure group G are given, where G has an admissible topology and acts effectively on F . Let E 5 X be a map onto a space X . Suppose that there is an open covering { Uil i E I } of X , let El U i= n - ' ( U i ) for each U , , and suppose that there is a corresponding family { Y 1 i E I } of homeomorphisms Y i such that each ElU.
2Ui x
\I
/
F
commutes. If the intersection U i n U j of two sets in the covering is nonvoid, then Y i and Y j induce a homeomorphism Y j 'J 'PI: such that
'
( U i n U j )x F
P
y-1
(Uin Uj)x F
commutes, and one necessarily has Y j Y1:'(x,f) = ( x , $ { ( x ) ( f ) )for each ( x , f )E ( U i n Uj) x F , where t,bi carries each x E Ui n U j into a homeomorIL'W F. phism F 4
2.1 Definition: If each of the preceding $ps is a (continuous) map from Ui n U j to the structure group G of homeomorphisms F -,F , then E 5 X is a coordinate bundle with respect to the covering { Ui I i E I } of X . The maps E I Ui 3Ui x F required for Definition 2.1 are the local :X , and the induced maps triuia1i~ation.s of the coordinate bundle E 1 Uin Uj G upon which the definition is based are the transitionfunctions. As before, TL is itself the projection of the total space E onto the space X , and for each .Y E X one lets E , denote the ,fiber n- '( {x)) ouer x . In $1 we considered arbitrary families E 5 X of fibers, fibre bundles being locally trivial equivalence classes of such families. A coordinate bundle is clearly just a locally trivial family of fibers.
2.2 Proposition: Let ( be a jibre bundle, consisting of equivalence classes of families E 5 X oj'jibers as in Definition 1.9; then every family E 1:X representing ( is a Coordinate bundle.
68
11. Fiber Bundles
PROOF: Let E 5 X represent 5. By definition there is an open covering { U iI i E I } of X such that each g I Ui is a trivial bundle, so that for E I Ui = 7c- '(Ui) one has a homeomorphism Yi such that
EIUi
2U ix F /
\
Ui Y commutes and induces G-related homeomorphisms E x -b ( x ) x F . If Ui n U j is nonvoid, it follows that the homeomorphisms Yi and Y j induce Y,"Yy;' a composition ( U in U j ) x F + (Ui n U j ) x F carrying ( x , f ) into (x, $ { ( x ) ( f ' ) ) ,where t+b{(x)(f)E F depends continuously on ( x , f ) ,and where each ${(x) is a homeomorphism. Since each E x 5 {x} x F is G-related, Y;' E x 5 {x) x F belongs to G , by definition each composition {x} x F of G-relatedness. Hence ${ is a (continuous) map Ui n U j -+ G, as required.
-
Thus if one identifies a fibre bundle over X by choosing a single representative E 5 X of an isomorphism class 5 of families of fibers, that representative is always a coordinate bundle. Proposition 2.2 also provides the desired relation between Definition 1.9 and the informal description of fibre bundles given earlier.
2.3 Corollary : Any j b r e bundle is an equivalence class of' coordinate bundles. PROOF: The equivalence relation is the isomorphism E
r
of families of fibers over the same space X , now restricted only to those families which happen to be coordinate bundles. We already know from Proposition 1.10 that pullbacks of fibre bundles are fibre bundles, so that Proposition 2.2 implies that pullbacks of coordinate bundles are coordinate bundles. Here is a more explicit version of the same result. 2.4 Proposition: Let E' 5 X ' be a coordinate bundle with respect to an open covering {Uili E I ) of' a space X ' , let E 5 X be its pullback along a map X 3 X ' , and let Ui = g - ' ( U ; )for each i E I ; then E 5 X is a coordinate
69
2. Coordinate Bundles
-
bundle M ' i t h respect t o the open covering [ Ui I i E I } of' X . SpeciJcally, if' Y;' Uj n U > G are the transition functions of E' 5 X', then the compositions $jj g are the transition functions Uj n U j 2G of' the pullback E 5 X . PROOF: Let
E
B
E'
X
Y
X'
be the pullback diagram. Since pullbacks of products are products, the local trivializations E'I Uj 5 Uj x F of E' 5 X' pull back to corresponding local trivializations E I U i-% Ui x F of E X . For any nonvoid intersection Uj n U j c X this provides a commutative diagram ( U jn U j )x F
YJ,
YJ
E I U i n U j A ( U in U j ) x F
gxid
gxid
J.
J.
( U j n U;) x F
J
L E ' I U ;n u; A (uj n u;)x F.
-
Each of the compositions Y j Y ; and Y ; Y j - ' is described via a transition function, t,b! and $fj, as in Definition 2.1, and the outer rectangle of the diagram maps any (x,f)E ( Ui n U j ) x F as indicated: 0
(x,f') T
(x,$!(x)f') T
g x id
g x id
J
J
(dx),.f)t---* (g(x),t,bIj(s(x))f).
The result $i(.x)f = $fj(g(x))f' for any (x,f) $:j g, as claimed.
E
(Ui n U j ) x F implies $;
-
=
Suppose that E 5 X is a coordinate bundle with an open covering Y U iI i E I j of X and a family {Y I i E I ) of local trivializations E I Ui Ui x F . Then if U in U j n uk is nonvoid, the restrictions of Y i , Y j , and Y k triviallysatisfy(Yk Y,: I ) ( Y j Y ; ') = Y k Y ; asself-homeomorphisms of ( U j n U,i n U , ) x F to itself; consequently the corresponding transition functions satisfy $!(x)$;(x) = $!(.x) in G for every x E uin uj n uk.Conversely, this condition suffices for the construction of a coordinate bundle. I
70
I I . Fiber Bundles
In the following construction we assume as before that a fiber F and structure group G are given, that G acts effectively on F , and that G has an admissible topology.
2.5 Proposition: Let { U i I i E I } be an open covering of a space X , and suppose f o r every nonvoid intersection U i n U j that there i s a map U i n U j +’ -L G . The maps ${ are the transition ,functions of a unique coordinute bundle with jiber F and the given action G x F + F i f and only i f for every nonvoid intersection U in U j n Uk and every x E U i n U j n Uk the identity $j”CX,$i(X)
=
4 m
is satisfied in G .
PROOF: We have just learned that the condition is necessary. To prove the converse, observe that if i = j = k, then the condition becomes $i(x)$:(x) = $i(x), so that $i maps any x E U iinto the identity 1 E G ; similarly if one merely assumes i = k, one has $i(x)$:(x) = $$x) = 1, so that $f(x) = (${(x))-’ for every x E U i n U j . Let be the relation in the disjoint union u i ( U i x F ) for which (xi,h) ( x j , f j ) whenever x i = x j E U in U j and fj = ${(xi)J E F . The hypothesis of the proposition guarantees that is transitive, and the preceding consequences of the hypothesis guarantee that is reflexive and symmetric. Hence is an equivalence relation, and one sets E = u i ( U i x F ) / - in the quotient topology. The first projections Ui x F A U i induce a map E 5 X , and one easily verifies that E X is the desired coordinate bundle, uniqueness being trivial.
-
- -
-
In the next chapter we shall use Proposition 2.5 to construct new fibre bundles out of old ones, sometimes changing both the fiber and the structure group.
2.6 Definition: Let G x F -+F and G’ x F‘ + F‘ be effective actions of transformation groups G and G’ on topological spaces F and F’, respectively, the topologies of G and G’ being admissible. A morphism of’ transformution groups is a pair (r,@) of maps G G’ and F % F’ such that is a group homomorphism and the diagram G x F - F
G’x F‘ commutes.
-
F
71
2. Coordinate Bundles
In the following result a morphism of transformation groups is applied to a fibre bundle 5 over a space X to construct a new fibre bundle t' over the same space X . One chooses any coordinate bundle E 5 X representing t, for a family [ Y I i E I j of local trivializations with respect to an open covering [ U ,I i E I ) of X , and one constructs a new coordinate bundle E' 5 X , for a family [ Y i l i I~) of local trivializations with respect to the same open covering. One then verifies that the fibre bundle 5' represented by E' 5 X depends only on 5 itself. r
cg
G', F + F ' ) he u morphism of transjormation groups, und let X he a topological space. Then t o any jibre bundle t over X with structure group G and,fiber F the morphism (r,@) assigns a uniquefibre bundle 5' over X with structure group G' and fiber F , satisfying the following condition: jbr unj' coordinate descriptions qj' 5 and 5' (as above), there is a f projection-preserving map E -+E' whose restriction fl U i to each E I U ic E provides u coniniutative diugram
2.7 Proposition: Let (G
-+
Elui
E'IUi
:
+
Ui x F
-I Y:
id x 8
U i x F'.
PROOF: Let U i n U.i G be the transition functions corresponding to j Y i ( iI),andlet ~ Ui n Uj--+G'bethecompositionsUi *;' n Uj%G5G', for nonvoid intersections U i n U j . Since r is a group homomorphism, the conditions $;(x)$!(x) = $f(x) imply that $F(x)$ij(x)= $ik(x) whenever U i n U j n U k is nonvoid, for any x E U i n U j n uk. By Proposition 2.5, the maps $; are the transition functions of a unique coordinate bundle E' 5 X with fiber F' and group action G' x F' -+ F'. The total spaces E and E' are quotients of disjoint unions u i ( U i x F ) and u i ( U i x F),and the Y U i x F and E'I U i y; U i x F' arise from the local trivializations E I U i 4 projections of u i ( U i x F ) and u i ( U i x F ' ) onto each U i x F and U i x F', respectively. The relation ( x i , j J ( x j , j j ) in u i ( U i x F ) means that xi = s j E U i n U j and j , = ${(.q),A. E F , and since
-
-
GxF-F
G' x F'
-
F'
72
11. Fiber Bundles
commutes, it follows that both x i = xj E U i n U j and qfi = @(t,b{(xi)j;) = (rt,b!(x))(qjJ= t,blj(x)(@h);that is, the relation ( x i ,@jJ (xj,@jJ is satisfied in u i ( U i x F'). Consequently F f F' induces a map u i ( U i x F ) + Ui x F'),which in turn induces a projection-preserving map
-
ui(
The remainder of the proof consists of direct verifications.
2.8 Definition: Let (G f, G', F f F') be a morphism of transformation groups. Then for any fibre bundle 4 over a space X , with structure group G and fiber F , the induced bundle with respect to (r,@) and 5 is the bundle 5' of Proposition 2.7. It is clear that the construction of induced bundles is functorial in the following sense, for the morphism (r,0) and bundle 4 of Definition 2.8: for any map 3 X one has ( ~ ' 4 )= ' g'5' over r?. The proof consists of direct verifications, similar to the verifications omitted from the proof of Proposition 2.7. This property will henceforth be used with no further comment. The following result will be used in the next chapter to verify that different morphisms of transformation groups sometimes lead to the same induced bundle.
x
2.9 Proposition: Let (I-, 0)be a morphism oj' transjimnution groups consisting of' (I group uutomorphism G 5 G and an uction F % F o j G, und let 5 be any jibre bundle with j b e r F and structure group G over u spuce X ; then the induced bundle 5' over X satisfies 5' = 5.
PROOF: Proposition 2.7 provides a commutative diagram I
E
\
* E'
/
\
/
X of coordinate bundles representing 5 and <', respectively, so that according to Proposition 1.12 one need only verify that f induces a G-related isomorphism E x + El for each x E X. I t suffices to show that if x E U , n U , n U k , v and if Y,, YL, and f, denote restrictions of the maps E l U , A U , x F, E'I U I , 3U kx F , and El U , 2E'I U , of Proposition 2.7, then the f E x iE: 3(x) x F is induced by an element composition (x) x F of G. Since f I U , = Y:-' (id x 0) Y, by Proposition 2.7, the preceding (id x @) Y, Y; I , composition is the restriction to ( x ) x F of Y; c so that one must show for the transition functions U , n U , G and 6
73
3. Bundles over Contractible Spaces
IL?W ID IL;w G that the composition F + F is a transformation U, n U , in G ; but since +:(x) E G and IC/:’(x) E G, this is an immediate consequence of the hypothesis 0 E G .
3. Bundles over Contractible Spaces In this section we prove a result which directly implies that any fibre bundle over any contractible space is trivial.
3.1 Lemma: Let v] be a j b r e bundle over the product X x [0,1] of an arbitrary topological space X and the closed unit interval [0,1],and let t be any point i n [O, 13. Zj each of rhe restrictions v]lX x [O,t] and v] 1X x [ t , 1 1 is trivial, then q is itself‘ trivial.
PROOF: Let F and G be the fiber and structure group as usual, and let E 5 X x [0,1] represent q. By hypothesis there are trivializations E IX x . re[ O , t ] ~ X x [ O , t ] x Fand E I X x [ f , l ] ~ X x [ t , l ] x FThe strictions of Y oand ‘PI to E J X x i t ) induce a homeomorphism X x { t ) x F Y I Y i i I X x ( t } x F carrying any ( x , t , f ) E X x { t } E F into ( x , t , $ ( x ) f ) E X x x F for a unique “transition function” X x { t } 3 G, just as in the preceding section. (The quotation marks indicate only that the domain X x i t ) of is not open in X x [0, I ] ; however, there is no change in the existence and uniqueness of +.) There is then a map X x [ t , 13 x F 5 X x [ t , 13 x F given by Y ( x , s , j )= ( x , s , + ( . ~ ) - ~for j ) which the composition Y Y is a new trivialization El X x [ t , 1 1 3X x [ t , 13 x F . The induced Yi Y6l homeomorphism X x [ t } x F X x ( t } x F carries (x, t , f ) E X x i t ) x F into (x, t , +’(x)f) E X x ( t } x F for a new “transition function” X x ( r } 3 G with constant value 1 E G , so that Yo and ‘Y; are restrictions of a common trivialization El X x [0, I ] + X x [0,1] x F , as required.
-
+
-
3.2 Lemma: Let [ W , , . . . , W,!, be u ,finite open covering of [0,1] such that euch i s an intervd, and let v] be a j b r e bundle over X x [0,1],for any space X . If’ ouch of’ rhe restrictions v] I X x W , , . . . ,q I X x W, i s trivial, then v] i s itself’ triz>ial.
w
PROOF: Renumbering W , , . . . , W, ifnecessary, thereareq + 1 real numbers t o , r , , . . . , t, with 0 = t o < r , < . . . < t, = 1 such that [ t i _ , , t i ] c for i = 1,. . . , y. The inclusions X x [ti- t i ] c X x i4( imply that each restriction v] I X x [ti- ,,ti] is trivial, and the result is then an obvious iteration of Lemma 3.1.
,,
w
74
11. Fiber Bundles
3.3 Lemma: Let X be any space, and let r] be any jibre bundle over the product X x [0,1]. Then there is at least one open covering { U , )i E l } of X such that each restriction r] U i x [0,1] is trivial.
I
PROOF: By Definition 1.9 there is an open covering { $1 j E J } of X x [0,1] such that each restriction r] $ is trivial. Any point (x, t ) E X x [0,1] has a neighborhood basis of sets of the form V x W for an open neighborhood I/ of x and an interval W c [0,1] that is open in [0, I] and contains t in its interior. Hence each (x, t ) E X x [0,1] lies in at least one neighborhood of the form V x W , where V x W c rj for at least one j E J . Since restrictions of trivial bundles are trivial (as in Lemma 1.8) it follows that there is a covering of X x [0,1] by sets of the form V x W such that each restriction r] I V x W is trivial, where W is an interval that is open in [0,1]. For each x E X let ?iY, be the family of all such products V x W for which x E V and I/ x W c rj for some j E J . Since [0,1] is compact, there is a finite subfamily { V, x W,, . . . , V, x W,> of ?iY, such that { W,, . . . , W,} covers [0,1], and for U , = V, n * * * n V, each of the restrictions r] I U , x W,, . . . , r] I U , x W, is trivial. Lemma 3.2 then implies that q ( U , x [0,1] is itself trivial, so that { U , I x E X } is a covering of the desired form.
I
3.4 Proposition: Let X be any space, and let r] be a jibre bundle over the product X x [0,1] such that the restriction r] X x (0) is triuial. Then 11 is itself' trivial.
I
PROOF: Let E + X x [0,1] represent q, and let El X x (0) 5 X x (0) x F be a fixed trivialization of q I X x (0).By Lemma 3.3 there is an open covering { Uil i E I} of X such that each r] U ix [0,1] is trivial, and one can use the procedure of Lemma 3.1 to guarantee that there are trivializations El U ix Y [0,1] U i x [0,1] x F such that each restriction El U i x { O } Ui x {O} x F coincides with the corresponding restriction of Y. Hence the transition functions U in U j x [0,1] A G have the constant value $!(x,O) = 1 E G on each nonvoid intersection U in U j x {O}. For each i E 1 let El U ix [0,1] 3 U i x {O} x F be the composition of Y j with the map U ix [0, I] x F + U i x { 0} x F that carries (x, t, f)into (x, 0, f).Since &(x, 0) = 1 for each nonvoid intersection U i n U j , it follows that the restrictions El U in U j x f [0, I] iU in U j x (0) x F and El U in U j x [0,1] 2 U i n U j x (0) x F agree, so that there is a well-defined map f for which the diagram
I
E
-
-x
x x [O, 11
r
x {O} x F
J
xx
(0)
4. Pullbacks along Homotopic Maps
75
commutes, wheref‘(x, t ) = (x, 0 )for each (x, t ) E X x [0,1]. By Corollary 1.13, q is then the pullback along X x [0, I] X x {O) of the trivial bundle r] I X x { 0}, so that by Lemma 1.8 r] is itself trivial, as asserted. Here is the main result of the section.
3.5 Proposition: Let 5 be any ,fibre bundle over any contractible space X ; then 5 is trivial. PROOF: By hypothesis X is homotopy equivalent to the singleton space (*}, so that there is a composition X 3 { * ) 5 X that is homotopy equivalent to the identity map X + X ; that is, there is a map X x [0,1] 5 X whose restrictions X x {O} X and X x (1) 2 X are the composition h g and the identity map, respectively. Let r] be the pullback f’
a
0
4. Pullbacks along Homotopic Maps Let X ” 3 X and X” 4 X be homotopic maps into an arbitrary space X , and let be any fibre bundle over X . We shall show that if X ” is homotopy equivalent to a paracompact space then = .fit over X”.In particular, iffb and .fi are homotopic maps in the category % of base spaces, then fbr = f ! , t over X ” E 4? for any fiber bundle 5 over X E g.
<
4.1 Lemma: Let E
fa<
Y represent any j b e r bundle r] over an arbitrary space Y , and suppose thut Y 5 Y is a map that restricts to the identity outside of some open set I/ c Y . l j the restriction r] 1 V is trivial, then there is a morphism +
E
K
Y
4
+ E
F
Y
in the sense o j Definition 1.1 such thut g restricts to the identity outside of E I V .
76
11. Fiber Bundles
PROOF: Let the restriction of g to El Y - V be the identity, and for any trivialization El V 5 V x F of ql V let the restriction of g to El V be the composition Y - (g I V x id) Y. 0
‘-1
4.2 Lemma: Let X ‘ be a paracompact space, let E -,X ’ x [0,1] represent any jibre bundle q ouer the product X ’ x [O, 13, and let X’ x [0,1] 3 X’ x { 1) carry any (x’,t ) into (x’, 1). Then there is a morphism
E
EIX‘ x {I}
!
1
X’x [0,1] A X’ x { l ) in the sense of’Dejinition 1.1.
PROOF: According to Lemma 3.3 there is at least one open covering ( U i ( iE I } of X’ such that each restriction q 1 Ui x [0,1] is trivial. Since X’ is paracompact, one may as well assume that { U iI i E I ) . is locally finite and that there is a partition of unity { h i [i E I } subordinate to { U i l i E I } . For any well-ordering of the index set I , and for each i E I , let k i - = hj and let y- c X’ x [0,1] consist of those (x’,t)E X’ x [0,1] such that t 2 ki- ,(XI); in particular, for the initial element 0 E I one has Yo = X’ x [0,1]. Define yi- I 5 yi c yi- by setting gi(x’, t ) = (x’, max(ki(x’),t ) ) , so that gi is the identity outside of the open set ( U i x [0,1]) n I.;:-1. Since the restriction q I U i x [0,1] is trivial, it follows from Lemma 4.1 that there is a morphism
cj.i
,
yi-l
-
y
81
in the sense of Definition 1.1. Since { U i l i E I ) is locally finite, each x’ E X‘ has a neighborhood U,. such that the restrictions of gi and g i to U,. and E I U,. ,respectively, are identities except for finitely many indices i l , . . . ,i, E I . Furthermore (U,. x [0,1]) n = U,. x { l } whenever i, < i, . . . , i, < i, so that the composition E = E I Y o 5 -.-+Ep-,aElY,-
I
X’x [O, 13 = Y, 5 . . *
- x-I 1 1
A
y-
..‘
...
77
4. Pullbacks along Homotopic Maps
is a well-defined morphism of the desired form
E
EIX’ x {l}
X ’ x [0,1] A X’ x ( 1 ) . Here is a special case of the main theorem of the section.
a
4.3 Proposition: Let X’ 2 X and X ’ X be homotopic maps of a purucompuct space X’ into an arbitrary space X . Then ,fb( = f !,5 over X‘ for any jibre bundle t over X . PROOF: By hypothesis there is a map X’ x [0,1] X with restrictions f b and f , to X ’ x (0) and X’ x (I}, respectively. Hence for r] = f ! t one has ‘1I X’ x (0) = ,fit and q I X’ x (1) = f i t . Let E + X’ x [0,1] represent r]. The left-hand morphism in the composition E I X ’ x (0)
X‘ x (0)
h
E BE I X ’ x { l }
h
X’ x [0,1] 9X ’ x (1)
is the pullback diagram of an inclusion, and the right-hand morphism is provided by Lemma 4.2. Since X ’ x {O} X’ x { 1) is the identification homeomorphism and El X ’ x (0) 2El X’ x i 1) induces G-related isomorphisms in each fiber, it follows from Proposition 1.12 that r ] l X’ x ( 0 ) = r] I X ’ x [ 1) ; that is, f b t = f i t , as asserted. In order to extend Proposition 4.3 to the case in which X‘ is replaced by any space X ” homotopy equivalent to X‘ we first refine Lemma 3.3, which was used in the proof of Proposition 4.3. 4.4 Lemma:
Let X “ be any space, let ‘1 be any jibre bundle over the product X ” x [0, I], und let f Uil i E I } be any open covering of X” such that each restriction ‘11 U ix ( 0 ) is trioiul. Then each restriction r] I Ui x [0,1] is also trivial. PROOF: In Proposition 3.4 replace X and q by Ui and 91 U ix [0,1], respectively, for each i E I .
78
11. Fiber Bundles
4.5 Lemma: Let X ’ 3 X‘ be a homotopy equivalence of a space X” with a h paracompact space X’, with homotopy inverse X ’ -+ X“. Then g!h![= [ .for any jibre bundle [ over X”.
PROOF: Since X’ is paracompact, there is a locally finite open cover { Kl i E I } of X ’ such that each restriction h!cI of the pullback h!( over X’ is trivial, and there is a partition of unity {kil i E I } subordinate to { Kl i E I ) . Let U i = g-’(K), and let X” 3 R be the composition ki 0 g, for each i E I . Then, even though X” is not itself necessarily paracompact, { Uil i E I } is a locally finite cover of X ” such that each restriction g!h![l U i is trivial, and {hi)i E I } is a partition of unity subordinate to { Uil i E I } . Since g and h are f homotopy inverses, there is a map X ” x [0,1] + X” whose restrictions X” x {0}5X ” and X” x { 1) 5X ” are the composition h g and the identity map, respectively. Let q be the pullback f ! c over X” x [0, I], and observe that q I X ” x ( 0 ) = f = g!h!< and q I X ” x (1) = fie = Since each restriction g’h’clU i is trivial, that is, since each restriction q I U i x ( 0 ) is trivial, Lemma 4.4 implies that each restriction ql Ui x [0,1] is trivial. Hence if E” X” x [0,1] represents v], then one can use the partition of unity { h i /i E I } to construct the right-hand morphism in the composition 0
c.
-+
X” x ( 0 )
-
X” x [O, I]
-
X” x { 1)
exactly as in the proof of Lemma 4.2; the left-hand morphism of the same diagram is the pullback diagram of an inclusion. It follows as in Proposition 4.3 that v] 1 X” x { 0 } = v] I X ” x { 1) ; that is, g!h!c = as asserted.
c
Here is the main theorem of the section. 4.6 Theorem: Let X“ be homotopy equivalent to a paracompact space, and
X be homotopic maps of X “ into an arbitrary let X” 5X and X“ space X . Then fat = j’\( for any fibre bundle over X .
<
PROOF: Let X” 3 X’ be a homotopy equivalence from X “ to a parah compact space X’, with homotopy inverse X‘ -+ X”, so that y‘hjf’bt = fat and g!h!f\t = f i t over X” by Lemma 4.5. However, the compositions X’ 5 X” 3X and X ’ 5 X ” A X are homotopic maps from the paracompact space X’ to X , so that Proposition 4.3 yields h!fbt = ( f o h)!<= ( f , h)!t = h!f,(.Consequently,fbt = g ! ( h ! f b t )= g!(h!f\t)= f \ t asclaimed. ?
rj
79
4. Pullbacks along Homotopic Maps
We shall use only the following special case of Theorem 4.6. Recall from Definition 1.9 that a fiber bundle is any fibre bundle whose base space lies in the category 9 of Definition 1.1.2. 4.7 Proposition: Let X“ --%X and X“ X be homotopic maps in the category d of base spaces (Definition 1.1.2),and let 5 be a fiber bundle over X ;then f bt = f ! 5, over X”.
PROOF: One of the provisions of Definition 1.1.2 is that any base space X” E 9is homotopy equivalent to a metrizable space X’. Since any metrizable space X‘ is paracompact, by Lemma 1.8.2, Theorem 4.6 applies. The product of any base space X E d by the closed unit interval [0,1] is also a base space X x [0,1] E B, by Proposition 1.1.4.
E
&?
4.8 Corollary : Giveti a base spuce X E d,any fiber bundle q over the product X x [0,I] is the pullbuck n!,( along the .first projection X x [0,1] 5X, fi)r some fiber bundle 5 over X .
PROOF: Let X 3 X x [0,1] be the inclusion x ~ ( x , Oand ) set 5 = iaq. Since the composition X x [0,1] 5X 4X x [0,1] is homotopic to the identity, Proposition 4.7 gives n\< = n!,iaq = (io ,’ n,)$ = q as claimed.
4.9 Corollary : Given a base spuce X E &?, any fiber bundle over the product X x [0, I ] cun be represented by u coordinate bundle of the form E x [0,1] X x [0, I] f o r a coordinate bundle E % X over X . PROOF: Immediate consequence of Corollary 4.8.
x Corollary 4.9 gives a clearer view of Proposition 4.7. Let X ” x LO, 11 i be a homotopy of maps f0 and f; from X” E &? to X E 93,and let 5 be any fiber bundle over X . Then the pullback f ! < can be represented by a coorn” x id dinate bundle of the form E ‘ x [0,1] X” x [0,1] for some coordinate bundle E” 5X”, so that j’a< and j\< are represented by E” x {O} n“ x Id n” x id X ” x [O) and E“ x { I ) X” x { l}, respectively; a fortiori
-
.f!0.E
-f15 ’! .
4.10 Proposition : Let
-
80
11. Fiber Bundles
be a morphism of coordinate bundles such that g is a homotopy equivalence in the category LBof base spaces; then g is also a homotopy equivalence.
PROOF: Let ( be the fiber bundle represented by E 1X , with pullback g!< represented by E' 5 X ' , and let X 5 X ' be a homotopy inverse of g. Then X 3 X is homotopic to the identity map, so that h!g!<= ( by Proposition 4.7; hence there is a pullback diagram h
E
E'
h
X
+
X'
for some h, which provides a composed morphism h
E
g'
X
o'h
+ E
In
+x
of coordinate bundles. Let X x [0,1] X be the homotopy from g h to the identity map X -+ X . By Corollary 4.9 the pullback f!( is represented nxid by a coordinate bundle of the form E x [0,1] X x [0,1], whose restrictions E x {O} X x ( 0 ) and E x ( 1 ) -+ X x { 1 ) represent h!y!
-
-+
E x [0,1] R x id
I
x x [0,1]
AE
In x, -
where f provides a homotopy from E Similarly there is a homotopy from E which completes the proof.
s
g h
h g
E to the identity map E E . E' to the identity map E' -+ E', -+
5. Reduction of Structure Groups Let ( be a fibre bundle with structure group G and fiber F over a space X , and let K be a subgroup of G . The action G x F -+ F restricts to an action
81
5 . Reduction or Structure Groups
K x F -, F . Hence, for the inclusion homomorphism K f, G and the identity is a morphism of transformation groups in the map F F , the pair (r,@) sense of Definition 2.6. 5.1 Definition: The structure group G of a fibre bundle
<
over X can be reduced to a subgroup K c G if and only if 5 can be induced, as in Definition 2.8, by applying (r,@) to some fibre bundle over X with structure group K . For example, the structure group G of any trivial bundle can obviously be reduced to the trivial subgroup { 1) c G. The reduction theorem of this section concerns any fiber bundle with structure group G over any X E B.It asserts that if H c G is a subspace homeomorphic to a euclidean space RP, and if K c G is a closed subgroup such that every element of G is uniquely of the form hk for h E H and k E K , then the structure group G of 4 can be reduced to K . The homeomorphism H + Rp need not be a group isomorphism. A Lie group is any.topologica1 group G that is also a smooth manifold, ( ) - I G for which the group operation G x G + G and the operation G carrying elements into their inverses are both smooth maps. For example, the general linear groups GL(rn,R) and GL(n,C)are trivially Lie groups. At the end of this section we quote the very general Iwasawa-Mal'cev theorem, which asserts that if G is a Lie group with only finitely many connected components, then there is a decomposition G = H K as in the preceding paragraph, in which K is any maximal compact subgroup of G; hence the reduction theorem permits one to replace any such structure group G by a cornpuct Lie group. Although we do not prove the full Iwasawa-Mal'cev theorem, which can be found in several references given later, we do devote the next section to the special cases required in this book, in which G is either a complex general linear group GL(n,C),a real general linear group GL(rn,R), or the subgroup GL+(m,R) c GL(rn,R) of those elements in GL(m,R) with positive determinants; the corresponding maximal compact subgroups are the unitary groups U ( n )c GL(n,C),the orthogonal groups O ( m ) c GL(rn,R), and the rotation groups O'(m) c GL+(m,R), respectively. Before embarking on the proof of the reduction theorem, we rephrase Definition 5.1 directly in terms of coordinate bundles. Let E 5 X be a coordinate bundle representing <,and let {Uil i E I } be an open covering of X for which there are trivializations El U i 3 U ix F ; the transition functions U in U j *; G are defined as usual by the requirement that Y j i. Y; '(x, f )= (x,$!(x),f)E ( U i n U j ) x F for any ( x , f ) E ( U i n U j ) x F. For any family (iili E I i.of maps U i3 G there are new trivializations El U i U ix F
<
-
-
82
- Y
A,-
11. Fiber Bundles
I
consisting of compositions El U i Ui x F U i x F in which u),! Ai(x,,f) = (x, &(x)f).There are then new transition functions U j n U j 4 G for the same coordinate bundle E 1X and same covering { U j1 i E I ) of X , defined by the requirement that Vi '4';- ' ( . x , f ) = A,:' T i ' ' V; ' Ai(x,,/') = ( x , ( i j ( x ) - ' . $j(x). Ai(.x))j)= (x,$;j(x)f) for any ( x , f ) e i U i n U j ) x F ; that is, $ij(x) = Aj(x)-' $j(x) . i i ( x )E G for any x E U in U j . Clearly the structure group G can be reduced to a subgroup K c G if and only if one can find a family {AiI i E I } of maps U i5 G such that Aj(x)-' . $i(x) Aj(x)E K for each x E U j n U j . To prove the reduction theorem it therefore suffices to find such a family {Aj I i E I } of maps U i 2 G. I
5.2 Definition: Let E 1X be any coordinate bundle over a space X ; a section of E : X is any (continuous) map X 5 E such that X 5 E 5 X is the identity on X . In the following lemma the fiber is any euclidean space RP; however, the unspecified structure group does not necessarily act linearly on Rp.
5.3 Lemma: Let X be a paracompact hausdorf space for which there i s (I countable locally finite open covering { U,,I n E N} such that each connected component of each U , is contained in a contractible open set in X . Then lor any coordinate bundle E 5 X with,fiberRp there is a section XG E . PROOF: Any paracompact space is normal (as in Dugundji [2, p. 1631, or Kelley [I, pp. 158-1691, for example), so that there is a countable locally finite refinement (Kin E N} of { U,ln E N } by open sets V, whose closures satisfy V,, c U , . Since each connected component ofeach U,,is contained in a contractible open set in X it follows from Proposition 3.5 and Lemma 1.8 that y,> U , x Rp, hence local trivializations there are local trivializations El U,El V f l 5 x Rp. Let W, = u . . . u for each I I E N, and observe no El W, is given by co(x)= VyO'(.x,O) that since W, = V,, a section W, I for any x E W,. For any section W,,-' El W,-' the restriction 0,- I V,, n W,- can be composed with V,, to yield a map
v0
- cl
On-
wl-
and since V, n is a closed subset of a normal space, the Tietze extension theorem (in Dugundji [2, pp. 149-1501, for example) permits one to -J,,-I ( V , n W , - , ) x Rp to a map V,,: V, x Rp. It extend V, n W,,-' Y-1 follows that the composition V,,2 V, x R P ) 1 El V, is the restriction to V, c y , of a section W, E I W, whose restriction to W,-, c W, is 0,Since { I n E N;covers X each x E X lies in some and since [ U , 1 n E N Ytt
cl
wl,
83
5. Reduction of Structure Groups
is locally finite, one has CT,,+l(x) = CT,,(X)for n sufficiently large. Hence there is a well-defined section CT = limn-,, G,,,as claimed.
-
Now let K be any closed subgroup of a topological group G, and let be the equivalence relation in G with 9' g if and only if g' = g k for some k E K . The quotient G/-, in the quotient topology, is the homogeneous space GIK. Since K is not necessarily a normal subgroup of G, the homogeneous space G / K is not necessarily a group. However, there is a natural action G x G / K -+ G/K of G on G / K given by left-multiplication. The kernel of the natural action is the largest subgroup K O c K that is normal in G , so that for 6 = G / K , there is an effective transformation group G x G / K + G / K . Let E 5 X be any coordinate bundle with fiber F and structure group G, with respect to some open covering ( U i l i E I > of X . Any local trivializations El U i U ix F provide transition functions U in U j 3G as always: Y j - Y,:'(x,,f) = (.x,Ic/j(x)f) for (x, f ) E ( U in U j ) x F. For any closed subgroup K c G let G 5 G / K , = 6 and G 3 G / K denote the canonical surjections, where is in fact a group epimorphism. If one sets &(x) = T(Ic/i(x))E 6 for any x E U i n U j , then the conditions Ic/:(x). Ic/j(x) = Ic/:(x) imply $!(x). I&) = @(x) for any x E U i n U j n U k ;hence according to Proposition 2.5 there is a unique coordinate bundle l? 5 X with fiber G / K and structure group defined with respect to the covering ( Uili E I } by $' 6. We temporarily call means of the transition functions U i n U j E'5X the ussociuted bundle with respect to the given coordinate bundle E : X and given closed subgroup K of the structure group G.
-
-
5 X be any coordinate bundle with structure group G, and suppose that H c G is a subspace homeomorphic to a euclideun spuce Rp and that K c G is a closed sub groupsuch that every element of G is uniquely of the form hk for h E H und k E K . Then there is u section X 5 I? of the associated bundle E' 5 X . 5.4 Lemma: Let X satisjy the conditions of Lemma 5.3, let E
PROOF: The hypotheses guarantee that the homogeneous space G / K is homeomorphic to H , hence also to Rp. Hence by Lemma 5.3 there is a section X 5 i,as claimed. 5.5 Lemma: Let G he u topological group with u subspaceH c G and closed subgroup K c G us in Lemma 5.4, und let G 3 G / K be the canonical surjection. Then there is a mcip G / K 5 G such that the composition G/K 4 G G / K is
the identity.
PROOF: It has already been noted that G / K is homeomorphic to H , and one can identify any h E H as an element of G ; let z denote the composition GIK + H c G.
84
I I . Fiber Bundles
If the coordinate bundle E 5 X is described in terms of a given covering [ Uil i E I } of the space X , then the associated bundle E 5 X can clearly be described in terms of the same covering; that is, if there are local trivializations E I U i 3U i x F of E 5 X, then there are also local trivializations El U i
5U i x G / K of ,!? 5 X. Although Y iand qi are not directly related, the
compositions Y j Y;-1 and q j q; ' provide transition functions U i n U j G and U i n U j -% ?; that are related by the very definition of E 5 X ; specifically, $/ is the composition U i n G 5 G. For each i E I let ,!?I U i 2 G / K be the composition U i 3U i x G / K -% G / K , where n2 is the second projection. 0
vj
5.6 Lemma: For any x E U i n U j and any P E in GIK.
Exone has $ / ( x ) $ ~ ( P ) = $,i(S)
Gj
PROOF: By definition of J iand one has qi(9= (.x,$,(Z)) and qj(a= (x, & j ( Z ) ) , and by definition of the associated bundle ! , 5 X the trivializations qi and q . are related only by the requirement that ( U i n U j ) x +, i , J I ( U i n U j ) x G / K carry any (.x,,f) into (x, &(.x)f), for,FE G I K . G/K Hence
-
(x,&(,Y)$i(i!))
= +j+;'(x,$i(z)) =
+j(S)
= (XJj(Z)),
to which one applies n2 to complete the proof,
5.7 Definition: Let the coordinate bundle E 5 X satisfy the conditions of Lemma 5.4, and let E 5 X be the associated bundle with fiber G / K and structure group If E G X and E 5 X are defined with respect to an open covering { U iI i E I ) of X , and if the local trivializations E I U i 5U i x G / K are given by \ k i ( Z ) = ( Z ( P ) , $ i ( P ) ) , then the reducing maps U i 5 G are the compositions
c.
Ui
*
El U i 3 G / K A G,
where 31 U i is the restriction to U i of the section X 5 where G / K G is defined in Lemma 5.5.
of Lemma 5.4, and
5.8 Lemma: Let U i n U j %G be the transition junctions and U i 5 G the reducing maps qfthe coordinate bundle E 5 X of Dejinition 5.1; then 4(1/4(x). i i ( . x ) ) = 4(Aj(x))E G / K 0
for any
.Y
E
Uin Uj,
where G -P G / K is the canonical surjection. Hence ij(x)- . ${(.x) . i i ( . x ) lies in the kernel of 0.
85
5 . Reduction of Structure Groups
PROOF: Since G / K 5 G 2 G/K is the identity by Lemma 5.5, it follows from Lemma 5.6 that @(+j(x). A i ( X ) ) = & ( X ) @ ( A i ( X ) ) = $j(x)$,(a(x)) = qj(a(x)) = @(/Ij(X)).
5.9 Lemma: Let the coordinate bundle E X satisfy the conditions of Lemma 5.4; then its structure group G can be reduced to the given closed subgroup K c G. PROOF: If U i n U j G are transition functions for E X with respect to the covering { U,ln E N ) of X , then for any maps U i5 G whatsoever one obtains new transition functions U i n U j d4J G for the same coordinate bundle by setting II/IJ(x)= R j ( s ) - . cl/{(x) . &(x) E G for any x E U i n U j . In particular, if one chooses U i 2 G to be the reducing maps of Definitions 5.7, then it follows from Lemma 5.8 that the canonical surjection G 5 G/K carries each +fJ(x)into the point @ ( K )E G / K , hence that +ij(x) E K , as required.
-
5.10 Theorem (Reduction Theorem): Let 5 be anyjiber bundle with structure group G over uny base space X E d,and suppose that there is a subspace H c G homeomorphic to a euclidean space Rp, and a closed subgroup K c G, such that every element of G is uniquely o f t h e form hk for h E H and k E K . Then the structure group G of 5 can be reduced to the subgroup K c G.
PROOF: By Proposition 1.8.3 there is a homotopy equivalence X 3 X’ of any X E d with a space X’ E for which there is a countable locally finite covering [ U , I n E N} such that each connected component of each U,, is contained in a contractible open set in X ’ ;that is, X ’ satisfies the conditions of Lemma 5.3. If X’ 1:X is a homotopy inverse of g, then the pullback h!< of ( over X ’ is represented by a coordinate bundle X’ that satisfies the conditions of Lemma 5.4. Hence by Lemma 5.9 the structure group G of the fiber bundle h!( can be reduced to the subgroup K c G. Since 5 = g!h!t, by Lemma 4.5, it follows that the structure group G of ( can also be reduced to the subgroup K c G, as claimed.
E‘z
Suppose that 5 is a fiber bundle whose structure group is an arbitrary Lie group G with only finitely many connected components. We shall quote one of the major triumphs of the theory of Lie groups, which permits one to apply the preceding reduction theorem to obtain a very satisfying result. As promised earlier, however, explicit proofs will be given in the next section for the groups GL(n,C), GL(m,R), and GL+(m,R); in these cases Theorem 5.10 will permit one to regard any fiber bundle with one of these groups for its
86
2. Fiber Bundles
structure group as a fiber bundle whose structure group is one of the compact Lie groups V ( n )c GL(n,C),O(m)c GL(m,R),or O + ( m )c GL+(m,R),respectively. Any Lie group G with only finitely many connected components possesses a maximal compact subgroup K c G , and one can show that any other maximal compact subgroup of G is of the form g - Kg for some g E G . This property of Lie groups is closely associated with the following theorem.
’
5.1 1 Theorem (Iwasawa-Mal’cev Decomposition Theorem) : Let G be any Lie group with only jinitely many connected components, and let K E G be any maximal compact subgroup of G . Then there are one-parameter subgroups H , , . . . , H , of G , each isomorphic to the additive group R’, such that every element of G is uniquely of the form ( h , . . . h,)k for h , E H , , . . . , h, E H , , and k E K .
Proofs of Theorem 5.1 1 can be found in Iwasawa [13, Cartier [1, pp. 22-15-22-161, Mostow [l, pp. 47-48], and in Hochschild [l, pp. 180-1861, for example. Theorem 5.11 is usually called the “Iwasawa decomposition theorem”; however, a rationale for including Mal’cev’s name is given in Remark 8.22. Let 5 he ajiber bundle over any base space X E B, whose structure group G is any Lie group with only jinitely many connected components; then the structure group G can be reduced to u compact subgroup K c G. 5.12 Proposition:
PROOF: This is an immediate consequence of Theorems 5.11 and 5.10. One can obtain a slightly weaker version of Proposition 5.12 without appealing to the Iwasawa-Mal’cev decomposition theorem : see Corollary 6.14.
6 . Polar Decompositions In this section we show that if G is any of the linear groups GL(n,C), GL(m,R), or GL+(m,R), then there is a subspace H c G diffeomorphic to a euclidean space Rp, and a (maximal) compact subgroup K c G, such that every element of G is uniquely of the form hk, for h E H and k E K . The compact subgroups K are the unitary groups U(n)c GL(n,C),the orthogonal groups O(m)c GL(m,R), and the rotation groups O+(m)c GL+(m,R), respectively, which are easily shown to be compact. Hence the reduction theorem (Theorem 5.10), and the results of this section, imply for any fiber
87
6 . Polar Decompositions
bundle whose structure group G is one of the groups GL(n,C), GL(m,R), or GL+(rn,R) that G can be reduced to one of the compact subgroups V(n), O(m),or 0 (m),respectively. Let G L ( n , @ )be the general linear group of invertible n x n matrices of complex numbers, acting as usual on the left of the complex vector space C"of column vectors. One assigns GL(n,C)the relative topology as an open subset of C"',so that it is a complex manifold of complex dimension n', hence real dimension 2n2, covered by a single coordinate neighborhood. The group product GL(n,C)x GL(n,C)-+ GL(n,C ) and group inverse GL(n,C) 0 - 1 GL(n,C)are trivially smooth, so that GL(n,C)is a complex Lie group. The usual hermitian inner product C"x C" ( . ) C is given by setting (x, y) = x l r l . . . xnLl . . * x,?, for column vectors x and y with entries x I ,. . . , x, and L1,. . . , y,, where L1, . . . , y, are complex conjugates of y l , . . . , jn.The adjoint A* E EndC" of any A E EndC" is defined by requiring that ( A x , y) = (x, A * y ) for all (x, y ) E C"x C",and A is self adjoint with respect to ( , ) if A* = A . A self-adjoint element A E End C"is positive if ( A x , x) 2 0 for all x E C",with equality holding only for x = 0. It is clear that positive elements A , B E End C"belong in the general linear group G L (ti, C)c End C" and that the sum of two positive elements A,B E GL(n,C) is a positive element A + B E GL(n,C). For any endomorphism A of C",a classical existence and uniqueness theorem for ordinary differential equations provides a unique map R :End @" such that (d/dt)Y = A Y , with prescribed initial value Y ( 0 )E End C";the uniqueness theorem also implies that Y ( s ) Y ( t )= Y ( s + t ) for any s, t E R. The solution Y of the initial value problem (d/dt)Y = A Y for Y ( 0 )= I E GL(n,C)is the exponental of A , written Y ( t ) = eAf.Thus eAseAf= e A ( ' + ' )for all s, t E R, and since eAu= I it follows that e-AteAf= I, hence that eA1E GL(n,C)for all t E R. If B E End @" is self-adjoint, then the differential equations (d/dt)Y = BY and (d/dt)Y = B*Y are one and the same, so that eBr = eB" = (eBt)* for all t E R. since eBI = eBt/2eB1i2 = (eBt/2)*eBri2 , it follows for any x E C" that (eB1x,x ) = (eBri2x, e B f i 2 x2 ) 0, with equality if and only if x = 0. Hence eBrE GL(n,C) is positive for any self-adjoint B E End C"and any t E C. For any positive A E GL(n,C) and any s E [0,1] the sum 1(1 - s) As is positive, hence an element of GL(n,C),and the logarithm of A is defined by setting +
-
+
+
+
+
+
In A
=( A-
I)
6.1 Lemma: For any positiwe A
J'
s=o
E
(1(1 - s)
+ AS)-'
ds.
GL(n,C)one has e" A = A .
88
11. Fiber Bundles
+
PROOF: Set B = A
- I , so that I Bst is positive for any s E [0,1] and [0,1]. Then ln(I + Bt) = Bt Jf=o(I+ Bst)-' ds, the value at t = 1 being 1nA. Since
t
E
t
d Bt(I dt
+ Bst)-' + dsd ( I + Bst)-'
-
= 0,
-
it follows that
- S . =i o dds ( I + B s t ) - ' d s = B t ( I + B t ) - l
t -dl n ( l + B t ) = dt hence that
d -
dt
ln(I
+ Bt) = B(I + B t ) - ' ;
consequently d eh(I+Bt)= B(I +
Bt)-leh('fB')
dt
+
for all t E [0,1], the value of at t = 0 being I . Since I Bt is another solution Y of the initial value problem (d/dt)Y = B(I Bt)-' Y for Y ( 0 )= I , the uniqueness theorem for ordinary differential equations gives el"("" = I Bt for all t E [0,1] ; in particular, for t = 1 one has e'" A = A, as claimed.
+
+
6.2 Lemma: For any selfadjoint B E End C"one has In es = B.
PROOF: Compute d
d
i
- lneB' = - 6 = o ( e " - I)(I(l - s) dt dt
= BeB'
i d s(I(1 l=o
= BeB'e-B1= B
- s)
and
+ eB1s)-'ds
+ eB's)-' ds In eB" = 0 ;
another solution Y of the initial value problem (d/dt)Y = B for Y(0) = 0 is given by setting Y ( t )= Bt, so that the uniqueness theorem for ordinary differential equations gives In eBt= Bt for all t E R, including the case t = 1. Lemmas 6.1 and 6.2 together imply that the exponential map is a diffeomorphism from the self-adjoint elements B E End C" onto the positive elements 2 E GL(n,C); the logarithm is the inverse map.
89
6 . Polar Decompositions
The square root of any positive element of GL(n,C) is given by setting E End@" is self-adjoint; it follows from Lemmas 6.1 = eB. For any A E G L ( n , C ) one has ( A A * x , x ) = (A*.u, A * x ) 1 0 for any x E C",with equality only for x = 0, so that AA* is positive. The square root IAl = E G L ( n , C ) is the modulus of A. Clearly [ A [ is a positive element es E GL(n,C), where B = In AA*/2, by Lemmas 6.1 and 6.2. A n element B E GL(n,C ) is unitary whenever BB* = I ; that is, B-' = B*. If B and C are both unitary, then (BC)(BC)*= BCC*B* = BB* = I , and also B - ' ( B - ' ) * = B - ' B = I , so that the unitary elements of G L ( n , C )form a subgroup U ( n )c GL(n,C ) , the unitary subgroup.
Jt.8 = eBiz,where B and 6.2 that Jr"@
6.3 Proposition (Polar Decomposition of GL(n, C ) ) : Any element A GL(n,C ) is uniquely of the form eBC,where es is positiue and C E U(n).
E
PROOF: The modulus \ A ] is uniquely of the form es for the self-adjoint element B = In AA*/2, so that A = esC for C = IAI-'A. Since IA[* = [ A [ , one has CC* = IAI-'AA*IAI-' = IAI-'IA1'IAJ-' = I , so that C is unitary. For any other such decomposition eFG of' A the equality esC = A = eFG gives e - F e " = G C - ' , which is unitary, so that e - F e 2 B e - F = ( e - F e B ) ( e - F e B ) * = I . Hence ezB = e Z F ,so that B = 41n ezB = &IneZF= F , which in turn implies C = G. 6.4 Corollary: For any n > 0 there is a subspuce H c GL(n,C )difeomorphic to R"' .such thut every element of' G L ( n , C ) is uniquely of' the form hk ,jbr h E H and un element k in the unitary subgroup U ( n )c GL(n,C).
PROOF: We have already observed that Lemmas 6.1 and 6.2 imply that the exponential map is a diffeomorphism from the self-adjoint elements B E End C" to the positive elements eB E GL(n,C), and the self-adjoint elements B E End C"form a real vector space R"'. Since GL(n,C ) is of real dimension 2n2, it follows from Corollary 6.4 that the unitary group U ( n ) is of real dimension n 2 ; it is not itself a complex manifold. 6.5 Corollary: For any n > 0 the inclusion U ( n )+ GL(n,C ) is a homotopy rquiuulenc~r. PROOF: Let GL(n,C)+ U(n) project eBCE GL(n,C ) onto C E U(n).Then U(n)+GL(n,C)+ U ( n )is the identity, and the map GL(n,C)x [O,l]+GL(n,C) taking (eBC,t ) into eB'Cis a homotopy from the identity GL(n,C) GL(n,C ) to the composition GL(n,C) + U ( n )+ GL(n,C). --f
90
11. Fiber Bundles
We now consider real analogs of the preceding results, replacing the hermitian inner product @" x @" C by the usual euclidean inner product R" x R" R. The adjoint A* E End R" of any A E End R" satisfies ( A x , y) = ( x , A * y ) for all ( x , y) E R" x R", and self-adjoint elements B E End R" and positive elements A E GL(m,R) are defined exactly as in the complex case, as are the exponential eB E GL(m,R) of any B E End R" and the logarithm In A E End R" of any positive A E GL(m,R). Real analogs of Lemmas 6.1 and 6.2 then imply that the exponential map is a diffeomorphism from the self-adjoint elements of End R" to the positive elements of GL(m,R). The orthogonal subgroup O(m)c GL(m,R) consists of those A E GL(m,R) such that AA* = I ; that is, A - ' = A*.
<
?
)
6.6 Proposition (Polar Decomposition of CL(m, R)): Any element A GL(m,R) is uniquely of the form eBC,where eB is positive and C E O(m).
E
PROOF: This is a real analog of Proposition 6.3. 6.7 Corollary: For any m > 0 there is a subspace H c GL(m,R) difleomorphic to Rm("+l ) i 2 such that every element of GL(m, R) is uniquely of the jbrm hk jor h E H and an element k in the orthogonal subgroup O(m)c GL(m,R). PROOF: The real analogs of Lemmas 6.1 and 6.2 imply that the exponential map is a diffeomorphism from the self-adjoint elements B E End R" to the positive elements eBE GL(m,R), and the self-adjoint elements B E End R" form a real vector space Rm("+I)". 6.8 Corollary: For any m > 0 the inclusion O(m)+ GL(m,R) is N homotopy equivalence.
PROOF: This is an obvious real analog of Corollary 6.5. Recall that GL+(m,R)is the subgroup of those elements in G L ( m , R ) with positive determinants; the rotation group is the subgroup O+(m)= O(m)n GL+(m,R)c GL+(m,R). 6.9 Proposition (Polar Decomposition of CL+(m,R ) ) : Any element A E GLt(m, R) is uniquely of the jbrm eBC,where eB is positive and C E O+(m).
PROOF: This is an immediate corollary of Proposition 6.6. 6.10 Corollary: For any m > 0 there is a subspace H c GL+(m,R) difleomorphic to Rrn("+' ) I 2 such that every element of GL+(m,R) is uniquely of the form hk for h E H and an element k in the rotation subgroup O t ( m ) c GL+(ni,R).
PROOF: See Corollary 6.7.
91
6. Polar Decompositions
6.11 Corollary: For unjq m > 0 the inclusion O + ( m )-+ GL+(m,R) is u homoropy equivulrncr.
PROOF: See Corollaries 6.5 and 6.8. In order to present the main result of this section, it is necessary to know that the subgroups U(n)c GL(n,C), O ( m )c GL(m,R), and O f ( m )c GL+(m,R) are closed; in fact, they are even compact. 6.12 Proposition: For any n > 0 and m > 0 the unitary group U(n), the orthogonal group O(m),and the rotation group O + ( m )are compact.
PROOF: For any matrix (a;) E U ( n )one has a$$ + . . . + a;Zii = 6 for the 5 1, Kronecker delta 6,,, by definition of U(n).For p = q it follows that so that (u;) E C"'lines in the compact subset (0')"' c Cn2,where D2 c C is the closed unit disk. Since limits of points (a;) E (D2)"' satisfying the algebraic relations ajc(; . . . a:c(i = 6,, themselves satisfy the same relations, U ( n ) is a closed subset of the compact set (D2)"',hence compact. Analogous proofs apply to O(m) and O+(rn).
2I ; : [
+
+
6.13 Theorem (Linear Reduction Theorem): Let 4 be any jiber bundle over uny buse spuce X E d whose structure group G is one of' the linear groups GL(n,C ) , GL(m,R), or GL+(m,R); then G can he reduced to one of' the compact subgroups U ( n )c GL(n,e),O(m) c GL(m,R), or O'(m) c GL+(m,R), respectivelj*.
PROOF: By Proposition 6.12 the given subgroups are compact, a fortiori closed, and it remains to apply the general reduction theorem (Theorem 5.10) to Corollaries 6.4, 6.7, or 6.10, respectively. 6.14 Corollary: Let 5 be any fiber bundle over any base space X E 98 whose structure group G is a linear Lie group with only jinitely many connected components; specifically, G is a subgroup OJ' GL(m,R) jbr some m > 0. Then ( C U H be regarded as a ,fiber bundle with a compact structure group.
PROOF: One simply regards 5 as a fiber bundle with structure group GL(m,R), to which Theorem 6.13 applies. Corollary 6.14 is essentially a weak version of Proposition 5.12. However, its proof does not require the Iwasawa-Mal'cev decomposition theorem, and in any event most interesting Lie groups with at most finitely many connected components are linear. (The first example of a connected nonlinear Lie group appeared in Birkhoff [l]; the example appears as an exercise in Hochschild [ I , p. 2251. A method for constructing other nonlinear Lie groups is given in J. F. Price [I, pp. 119-121, 156-1571.)
92
11. Fiber Bundles
Fiber bundles with structure groups GL(n,C),GL(m,R), GL+(m,R), or equivalently U(n),O(m),O+(m),will occupy the rest ofthe book. It is therefore reasonable to begin looking at the topologies of these groups. For the moment we merely count their connected components.
6.15 Lemma: Let H be a connected closed subgroup of a topological group G such that the homogeneous space G / H is connected; then G is connected. PROOF: If G is covered by two nonvoid open sets U and V , then their images U' and V' in G / H are two nonvoid open sets covering G / H . Since G/H is connected, there is an x E G whose image x' E G / H lies in the intersection U' n V ' . The inverse image W of the set {x'} c G / H is trivially homeomorphic to H , hence connected, so that the two nonvoid sets U n W and V n W , which cover W , have a nonvoid intersection; a fortiori U n V is nonvoid.
6.16 Lemma: The roration group O+(rn)c O(m) is connected for every m > 0. PROOF: We proceed by induction of m, observing that O + ( l )consists of a single point. Regard O+(m)as a transformation group, acting via rotations of the ( m - 1)-sphere S"-' c R", and embed O + ( m- 1) in O + ( m )as the subgroup leaving some fixed unit vector e E S"- invariant. For any other f E S"- there is at least one g E O+(m)with ge = f E Sm-',and an easy verification shows for any h E O+(m)that he = f E S"- if and only if h and g have the same image in O+(m)/O+(m - 1). Since the image S"- ' of the induced one-to-one map O + ( m ) / O + ( m- 1) -+ S"-' is compact, the map is a homeomorphism. Since S"-' is connected, so is O+(rn)/O+(m- I), and since O+(m- 1) is connected by the inductive hypothesis, O+(m)is connected by Lemma 6.15, as required.
'
'
6.17 Lemma: every m > 0.
The orthogonal group O(m)has two connected components .for
PROOF: Multiplication of the connected component O + ( m )c O(m) by an element in O(m) with determinant - 1 provides a diffeomorphism from O+(m)to the complement O(m)- O + ( m ) .
6.18 Lemma:
The unitury group U ( n )is connected for every n > 0.
PROOF: The construction of Lemma 6.16 is easily modified, beginning with the circle U(1) = S', to yield a homeomorphism of the homogeneous space U(n)/U(n- 1) with the (2n - 1)-sphere St"-' c C";since V ( n- 1) is connected by the inductive hypothesis, V ( n )is connected by Lemma 6.15.
93
7. The Leray-Hirsch Theorem
6.19 Proposition: For uny n > 0 und unjl m > 0 the group GL(n, C) has one connected component, and GL+(m,R) is one ~f the two connected components of GL(m,R).
PROOF: By Lemmas, 6.16-6.18, the statements are valid for the subgroups U(n) c GL(n, C), O(m)c GL(rn,R), and O + ( m )c GL+(m,R), and by Corollaries 6.5, 6.8, and 6.1 1 the inclusions are homotopy equivalences.
7 . The Leray-Hirsch Theorem We henceforth consider only fiber bundles rather than fibre bundles, both of which appeared in Definition 1.9; that is, all base spaces, now and forever more, belong to the category 98 of Chapter I. Suppose that E 5 X is a coordinate bundle for some X E 98, representing a fiber bundle 5 over X. For any commutative ring A with unit let H * ( X ) and H * ( E ) be the singular cohomology rings H*(X; A) and H * ( E ; A), respectively, with coefficients in A. Since H*(-) is a contravariant functor, there is an induced ring homomorphism H*(X) 5 H*(E), and one can use n* to regard H * ( E ) as a left H*(X)-module, the product p . u E H * ( E ) of a E H * ( E ) by a scalar E H*(X) being the cup product n*P u c( E H*(E). Up to canonical isomorphisms the H*(X)-module H * ( E ) depends only on the fiber bundle (, and not upon the particular coordinate bundle E 3 X chosen to represent 5. For if E' 5 X is any other coordinate bundle representing 5 there is a homeomorphism j ' such that
E'
I
+ E
commutes, so that H * ( E ) 1:H*( E') is an isomorphism of H*(X)-modules, as claimed. There is even more latitude in the construction of the H*(X)-module H * ( E ) , up to canonical isomorphisms. For any map X' -5 X in the category .# of base spaces, and for any coordinate bundle E 3 X, there is a pullback diagram E'
n'
B
I
1
X'
rE
1 4
' x,
94
11. Fiber Bundles
hence a commutative diagram
H*(X) 9 . H*(X’)
rl
H * ( E ) A H*(E’) of ring homomorphisms. One can regard g* as a module homomorphism over the ring homomorphism g*; that is, g * ( p . a) = g*(x*p u a) = g*n*P u g*a = x’*g*p u g*a = ( g * p ) . (g*a) for any a E H * ( E ) and p E H * ( X ) . I f y is a homotopy equivalence, then H * ( X )% H * ( X ’ ) is a ring isomorphism: this is an immediate consequence of the homotopy axiom for singular cohomology.
7.1 Lemma: If X ‘ 5 X is a homotopy equivalence in g, then jor uny courdinate bundle E 5 X the homomorphism H * ( E ) H * ( E ) is an isomorphism over the ring isomorphism H * ( X ) % H * ( X ‘ ) . PROOF: In the pullback diagram E’
B
+ E
X‘
9
+x
the map g is also a homotopy equivalence, by Proposition 4.10. In the following result H*( ) continues to denote singular cohomology with coefficients in a fixed commutative ring A with unit. For any coordinate bundle E 5 X over any X E a, and for any x E X , let E x 2 E denote the inclusion of the fiber E x over x ; then H * ( E ) 4 H*(E,) is a homomorphism of modules over the ground ring A. An element a E H*(E) is homogeneous whenever a E Hq(E) for some fixed index q E Z,in which case j z a E H Y ( E x ) for each x E X .
7.2 Theorem (Absolute Leray-Hirsch Theorem): Let E 5 X be a coordinate bundle over anj’ buse space X E A!,?, let H*(-) be singular cohomology H*(-; A) with coeflcients in u commutative ground ring A with unit, and suppose that there are finitely many homogeneous elements al, . . . ,a, E H*( E ) such that for each x E X the A-module H*(E,) is free on the basis ( jza,, . . . ,j;a,} ; then the H*(X)-moduleH * ( E ) is free on the basis ( a , , . . . , a,,I .
95
7. The Leray Hirsch Theorem
PROOF: First suppose that X ' % X is any homotopy equivalence, with pullback diagram E'
R
X'
Y
,x
as usual. For any s'E X ' the map g induces a G-related isomorphism E:. JS Ey,s.j of fibers, hence a A-module isomorphism H*(E,,,.,) H*(EI,). Since j.T,(g*a)= gfj&.,a for any c1 E H * ( E ) , it follows that each Amodule H*(EI.) is free on the basis (j,*.(g*Cc,),. . . ,j,~(g*Cc,)J.Hence by Lemma 7.1 one can substitute E' 5 X ' for the given coordinate bundle E X whenever X ' .% X is a homotopy equivalence. In particular, since X E %, there is a homotopy equivalence X ' 5 X such that X ' is metrizable and of finite type, by Definition 1.1.2, so that one may as well assume throughout the remainder of.the proof that X is itself of finite type, as in Definition 1.1.1. Suppose that ct1 E H " ( E ) , . . . , c1, E H + ( E ) for integers q l , . . . , q,, let q be any integer, and for each open set U c X let h 4 ( U )be the direct sum H 4 - q 1U ( )0. . . @ Hq-4v(U ) of A-modules, where H 4 - Q i ( U )= 0 for q yi < 0. If R is the cohomology ring H * ( X ) of the base space X itself, then one can use the ring homomorphism R = H * ( X ) -+ H * ( U ) induced by the e ( )as a graded R-module, inclusion U c X to regard the direct sum U q E Z h U via cup product in H*( U ) . It follows from the classical Mayer-Vietoris cohomology exact sequence that one thereby obtains a Mayer-Vietoris on the category O ( X ) of open sets U c X to the category functor UqEZhY 9liF of L-graded R-modules, as in Corollary 1.9.4. We now construct another Mayer-Vietoris functor UqEz kY from O ( X ) to 91iz,and we later construct a natural transformation uqezkq to which to apply Corollary 1.9.4. For any open set U c X let El U = n- '( U ) as usual. and for any integer y let P ( U ) be the A-module H4(EI U ) . If El U 3 U is the restriction to El U of the original coordinate bundle E 1:X (over the space X of finite type), one can then combine the homomorphism R = H * ( X ) + H * ( U ) % H*(EI U )with cup product in H*(EI U ) to regard the direct sum UqEZ ky(Li) (=H*(EI U ) )as a graded R-module. It follows from the classical Mayer-Vietoris cohomology sequence that one thereby obtains another Mayer-Vietoris functor, k4, from 8 ( X ) to !Ill:, as promised.
UyeZh4
uqEz
96
11. Fiber Bundles
Now recall that hq( U ) = H 4 - 4 1 ( U0 ) . . . @ H 4 - 4 r ( U )for each q E Z and U E O(X), and that the hypothesis of the theorem provides elements a , E H 4 ' ( E ) ,. . . , a, E Hqr(E). Let El U 5 E be the inclusion map, and recall that El U 3 U denotes the restriction of E X. Then for each pi E H4-"(U) one has n$Pi E H4-4i(EIV ) and j:ai E H4'(EI U ) , for i = 1, . . . ,r, so that there is a well-defined map h4(U ) + k4( U ) carrying each p10 . . . O p, E h4(U ) into the sum .;Pi u j$ct, E kq(U). One easily verifies that the direct sum of such maps, over all q E Z,is an R-module homomorphism U y E z h 4 ( U2 ) U q e m k 4 ( Uand ) that the family {&)" of such homomorh4 5 UqEz k4 of Mayer-Vietoris phisms is a natural transformation functors on O(X) to ! I l l : . The final step in the proof of Theorem 7.2 is to show that 9 is a natural equivalence, a fortiori that Ox is an H*(X)-module isomorphism. If U E U ( X) is contractible, then the restriction E I U U represents a trivial bundle over U , by Proposition 3.5, so that there is a homeomorphism f such that
Uqek
commutes, where F is the fiber and n 1 is the first projection. If U contracts to x E U , then UqEH h4(V )2 k4(U) is canonically equivalent to the ) . . O H * ( { x } )+ H*(E,), which carries p1 homomorphism H * ( { x } 0 . . .O p, into RT pi u j,*aiE H*(E,), where H * ( { x ) ) = H o ( { x } )= A. Since H*(E,) is free on the basis { j:alr . . . ,j:a,), by hypothesis, it follows that 6" is an isomorphism for contractible U E O(X). Since X is of finite type, the Mayer-Vietoris technique applies in the form of Corollary 1.9.4, with the consequence that the H*(X)-module homomorphism H*(X)0 . . . H*(x) H * ( E ) carrying any p1O . . . 8, E ~ 4 - 4 " ~ 0) .. . O l F q r ( X ) into n*pl u a1 . . . IT*& u a, E H 4 ( E ) is an isomorphism; that is, H * ( E ) is a free H*(X)-module with basis [ a l , .. . , a,), as asserted.
UyEZ
c:=,
o
+
+
o
In a certain sense Theorem 7.2 is a generalization of a Kunneth theorem, as follows.
7.3 Corollary: Let X E 93 be m y base space, and let F be any space whose cohomology H * ( F ) ( = H * ( F ; A)) is a free A-module on Jinitely many homogeneous generators; then for the projections R , and n2 of X x F onto X and F , respectively, the map H*(X) @, H * ( F ) + H * ( X x F ) currying 0 a into RTP v nrci is rn H*(X)-module isomorphism.
8. Remarks and Exercises
97
PROOF: Apply Theorem 7.2 to the trivial coordinate bundle X x F 3 X . The label attached to Theorem 7.2 suggests that there is a corresponding relative Leray-Hirsch theorem. There is such a theorem, its proof being virtually identical to that of Theorem 7.2 itself. We require only a very specialized relative Leray-Hirsch theorem, however, which will be formulated in a later chapter.
8. Remarks and Exercises 8.1 Remark: The first organized account of the material of this chapter appeared in Steenrod [4, Part I], in 1951. Steenrod profoundly influenced the development of fiber bundles, and Part I of his book, at least, remains surprisingly modern. More recent introductory accounts of fiber bundles (in general) can be found in Bore1 and Hirzebruch [l, Chapter 111, Auslander and MacKenzie [2, Chapter 91, Holmann [l], Husemoller [l, Part I], Liulevicius [2, Chapter I] and Liulevicius [3, Chapter I], in the beginning pages of Lees [l], Eells [ 13, Porter [2, Chapter 21, and Rohlin and Fuks [1,Chapter IV], for example. Expository accounts of more general fiber spaces (or fibrutions), considered later in these Remarks, can be found in Cartan [3, Expos& 6, 7, 81, Schwartz [l, Part I], Hu [2, Chapter 1111, May [l, pp. 1-30], Switzer [l, Chapter 41, and G. W. Whitehead [l, pp. 29-75], for example. 8.2 Remark: The first explicit definition of a coordinate bundle is due to Whitney [2], and it was further amplified in Whitney [4, 5,6]. The importance of Whitney’s brief original paper and its successors was quickly recognized. Related notions of fiber spaces (without structure groups) were soon independently introduced by Hurewicz and Steenrod [11 and by Eckmann [l], followed by Fox [3]; fiber spaces will be discussed further in Remarks 8.5-8.8. Simultaneously Ehresmann and Feldbau [13, and later Ehresmann [3,4] considered alternatives to Whitney’s construction, with structure groups. The general definition of coordinate bundles with arbitrary fibers F and appropriate structure groups G had become mathematical folklore before its first appearance as an incidental feature of Steenrod [2], and the equivalence relation providing fibre bundles (and fiber) bundles in the sense of the present chapter was equally well understood before its first publication in Steenrod [4].
8.3 Remark: The distinction between fibre bundles and fiber bundles is a convenient technical device whose introduction is justified by the appearance
98
I I . Fiber Bundles
of the category 9 in several major results of this chapter: Proposition 4.7, the general reduction theorem (Theorem 5.10) and its application (Proposition 5.12), the linear reduction theorem (Theorem 6.13), and the LerayHirsch theorem (Theorem 7.2).These are some of the reasons for introducing the category 9? in the first place. The author apologizes to those readers who might prefer an interchange of the spellings “fibre” and “fiber.”
8.4 Remark: One can achieve some of the results of the preceding remark by restricting the bundles themselves, rather than their base spaces. According to Dold [ 5 ] a fibre bundle over a base space X (not necessarily in a) is numerable if there is a covering { U ii(E I } of X and a corresponding locally finite partition of unity Chili E I } such that (i) the covering i ‘h; ‘(0,1] I i E I } refines { U iI i E I } and (ii) each restriction I U iis trivial. Clearly if X is metric (a fortiori paracompact) and of finite type (as in Definition I.1.1), then Proposition 3.5 implies that any fibre bundle over X is numerable. By Definition 1.1.2 any space in 9? is homotopy equivalent to such a space X , so that one can use Lemma 4.5 to conclude that any fiber bundle whatsoever is automatically numerable. An equivalent definition of numerability occurs in Derwent [3].
<
<
8.5 Remark: A map E G X has the covering homotopy property with respect to a space Y if any commutative diagram
Y x (0)
-
E
Y x [O, 11
A
x
(with left-hand inclusion map) admits a homotopy lifting f for which
Y x [O,
11
-
x
is also commutative. Borsuk [ 13 introduced homotopy liftings and the covering homotopy property. One of the principal features of the fiber spaces E 5 X of Hurewicz and Steenrod [ l ] is the covering homotopy theorem, which asserts that such fiber spaces have the covering homotopy property with respect to any space Y whatsoever; the proof requires restrictions on the base space X’. Fox [4] showed conversely that any map E 5 X
99
8. Remarks and Exercises
satisfying the conclusion of the covering homotopy theorem is a fiber space in the sense of Hurewicz and Steenrod; Fox’s result requires further restrictions on X . The covering homotopy theorem is also valid for the more general fiber spaces of Hu [l]. Accordingly, Hurewicz [l] (and later, Curtis [l]), defined even more general fiber spaces as “locally trivial” maps E X satisfying the conclusion of the covering homotopy theorem; the base space X is required to be paracompact. 8.6 Remark: A Hurewicz Jibration is any projection E 1 :X whatsoever that satisfies the covering homotopy property with respect to any space Y . This definition ignores the “local triviality” condition of Hurewicz [13; one can easily construct Hurewicz fibrations which are not “locally trivial,” as in Husch [l], for example. However, there are easily defined conditions which do guarantee “local triviality” of Hurewicz fibrations, some of which can be found in Raymond [l]. Other recent information about Hurewicz fibrations is in Arnold [I] and in Jaber and Alkutibi [ 11.
8.7 Remark: Serre [ 11considers projections E 5 X that satisfy the covering homotopy property just with respect to polyhedra Y; in fact, it suffices to restrict the polyhedra Y to be cubes [0,1]“ of arbitrary dimension n, as in Spanier [4, pp. 374-3761, for example. R. Brown [ 11 gives an example of such a Serre jibration that is not a Hurewicz fibration.
8.8 Remark: One of the implications of Hu [l] is that every fibre bundle over a suitable base space satisfies the covering homotopy theorem; an independent proof of the same result appears in Steenrod [4,pp. 50-541. Since then Huebsch [l, 21 and Derwent [l] provided simplified proofs of the covering homotopy theorem for fibre bundles over arbitrary paracompact base spaces; a special case (of interest for Chapter 111) is in Szigeti
PI. 8.9 Remark: We have deliberately avoided some of the most natural and beautiful fiber bundles since they will not be used explicitly in the sequel; however, they will appear in some later exercises. Any topological group G has an obvious effective action G x G + G upon itself, for which the topology of G is trivially admissible: left-multiplication. A principal G-bundle tpis any fiber bundle in the sense of Definition 1.9, in which a topological group G serves both as the structure group and as the fiber, the action G x G + G being left-multiplication. If E X is any coordinate bundle with structure group G acting on a G defined with respect fiber F , and with transition functions U in U j
*
100
11. Fiber Bundles
to an open covering { Uil i E I ) of the base space X , then there is a principal coordinate bundle EP X with fiber G, obtained by letting the transition functions ${ act on G rather than F . Principal G-bundles were first suggested by Ehresmann [3, 41, in terms of the fibrations of Ehresmann and Feldbau [l].
8.10 Exercise: Show that the preceding construction can be used to replace any fiber bundle 5 with structure group G by an associated principal G-bundle lp, which is independent of the specific coordinate representations E 5 X and EP % X.
z
8.11 Exercise: Let EP X represent a principal G-bundle <,, and observe that there is a well-defined action EPx G + EP in which G acts on the right of Ep.Suppose that there is also an effective action G x F -+F of G on the left of a topological space F , for which the topology of G is admissible. Then the product EP x F has an equivalence relation with (e',f') ( e , f ) whenever ( e ' , f ' )= ( e g - ' , g f ) for some g E G, and there is a quotient E = E, x F / - in the quotient topology; furthermore, the projection EP-%X induces an obvious projection E $ X . Show that E $ X is a coordinate bundle with structure group G and fiber F .
-
-
8.12 Exercise: Let EP5X represent a principal G-bundle t,, and let 5 be the fiber bundle represented by the coordinate bundle E 5 X at the end of the preceding exercise. Show that 5 depends only on the given principal X representing G-bundle tp,independently of the coordinate bundle E , SP.
8.13 Exercise: Show that the constructions of Exercises 8.10 and 8.12 are inverse, in the obvious sense. That is, show that the composition of the construction of Exercises 8.10 and 8.12 yields the given bundle of Exercise 8.10, and that the composition of the constructions of Exercises 8.12 and 8.10 yields the given principal G-bundle tPof Exercise 8.12. (Caution:It is always assumed that the action of G on the fiber F is effective.)
<
8.14 Remark: To some extent Exercise 8.13 permits one to concentrate exclusively on principal G-bundles, and many books on differential geometry (and other subjects) use this consequence to advantage. Principal G-bundles are defined and studied in Bishop and Crittenden [l, pp. 41-45], Kobayashi and Nomizu [l, pp. 50-541, and Koszul [l, Chapter 111, for example, in each case bejore the general definition of a fiber bundle is introduced.
101
8. Remarks and Exercises
However, the choice of G itself as a fiber is not always a virtue, for the same reason that the study of an effective transformation group G x F -+ F is frequently easier than the study of the left-multiplication G x G + G. For example, the usual action GL(n,R) x R" + R" of the general linear group is much clearer, in terms of the geometry of R", than is the left-multiplication GL(n,R) x GL(n,R) + GL(n,R). 8.15 Exercise: Let G be a topological group and let E $ X represent a principal G-bundle with a section X 4* E . Show that is the trivial principal G-bundle, represented by the first projection X x G -% X . (See Steenrod [4,p. 361.)
<
<
8.16 Exercise: Let G be a topological group and let E 5 X represent any principal G-bundle <. Show that the pullback d < is the trivial principal G-bundle over E. 8.17 Exercise: According to the preceding exercise, 7r!( can be represented by a coordinate bundle E x G 5 E, and the composition E x G % E X represents a principal (G x G)-bundle q over X , not necessarily trivial. Observe that the pullback diagram
E x G " x ' d X x G
E
A
X
for the trivial principal G-bundle over X also provides a principal (G x G)bundle r j over X with the same representation E x G 5 E 5 X . Show that in general r j and 5 are nevertheless dijlerent principal (G x G)-bundles. (See Akin [2].) 8.18 Remark: One of the most useful features of principal fiber bundles is that they can be clussijied in the following sense. For any topological group G there exists a universal principal G-bundle yG over a classifying space BG E 28 such that any principal G-bundle over any base space X E 9?l is a pullback , f ! y c , with respect to a map X A BG that is uniquely defined up to homotopy. We shall establish an analogous homotopy classijication theorem in the next chapter, which will suffice for all the remaining chapters.
102
11. Fiber Bundles
Chern and Sun [I] provided the first homotopy classification theorem, for any classical linear Lie group G, applying to principal G-bundles over certain base spaces. Heller [ 11 described an algebraic setting which established the existence (but not the construction) of universal principal Gbundles for an arbitrary topological group G, base spaces being suitably restricted. The first general construction of a universal principal G-bundle for an arbitrary topological group G was given by Milnor [2], thereby providing a homotopy classification theorem for principal G-bundles over moderately restricted base spaces; the restrictions were somewhat relaxed by Lusztig [11, and further relaxed by Gel’fand and Fuks [11 and Segal [11. An entirely different construction of universal principal G-bundles appears in Cartan [4], in a simplicia1 setting. It was observed in tom Dieck [I] that Milnor’s construction applies to numerable principal bundles over arbitrary base spaces. Since any fibre bundle over a base space X E is automatically numerable, by Remark 8.4, it follows that M ilnor’s construction classifies any principal bundle over any
x E B.
8.19 Remark: If EG + BG represents a universal principal G-bundle tc, then the total space EG is contractible. Conversely, if the total space EG of a principal G-bundle 5, is contractible then 4 is universal. An early version of this characterization of universality appears in Steenrod [4, pp. 102-1031, and the general case is in Dold [5]. Also see tom Dieck [l] and Liulevicius [2, p. 531 for portions of Dold’s result. 8.20 Remark: One can pass from the category of topological groups to the larger category of associative H-spaces with unit in carrying out Milnor’s construction. This was done by Dold and Lashof [l] (who had further applications in mind), and improvements to their work were added by Fuchs [l] and Gottlieb [l]. The Dold-Lashof construction led to an entirely different construction of universal bundles, by Milgram [ 11, and it was observed in Steenrod [8] (and by an earlier reviewer) that Milgram’s construction could be reformulated to provide a satisfying identity for classifying spaces: BG x BH = B(G x H). Steenrod also observed that in case G is an honest topological group, then the total space EG of Milgram’s universal principal G-bundle is itself a group containing G as a subgroup, for which the homogeneous space EG/G is precisely the classifying space BG. Milgram and Steenrod define the universality of EG + BG via contractibility of EG, as in Remark 8.19, for any associative H-space G. An excellent alternative version of the Milgram--Steenrod construction is given in May [l, pp. 31-54], and a sketch of the original version is given in Porter [2, pp. 132-1651.
8. Remarks and Exercises
103
One can even pass beyond the category of associative H-spaces with unit to the category of u11 topological spaces, for which there exist the “classifying spaces” of Segal [ 13. A very minor alteration of Segal’s construction, applied to an honest topological group G , simply reproduces the original construction of Milnor [2]; this was observed by Accascina [l] and others. Incidentally, if G is a topological group, and if BG and B’G are classifying spaces in the category d of base spaces, each satisfying the homotopy classification theorem with respect to principal G-bundles over base spaces in 98,then BG and B’G are trivially homotopy equivalent. However, if BG and B’G apply to different categories of base spaces, then their homotopy equivalence is less clear. A very general homotopy equivalence theorem is developed in tom Dieck [2, 31, to assist anyone with a desire to stray outside the category d.
8.21 Remark: In Thom [4] two coordinate bundles E X and E’ 5 X over the same base space X are j b e r hornotopji equivalent (or of the same ,fiber hornotopy type) whenever there is a homotopy equivalence of E and E‘ via maps that preserve projections onto X . The definition applies equally well to more general fibrations, and fiber homotopy equivalence has been studied extensively by Dold [1, 51, Fadell [ 1, 2, 31, and Stasheff [l, 2, 31, for example. Specifically, Fadell [3] shows that any Hurewicz fibration E 5 X over any X E % 0 (Definition 1.3.11) is fiber homotopy equivalent to a coordinate bundle, for some fiber F and structure group G. According to Allaud [I] the same result is true for Hurewicz fibrations over any X E ”W (Definition I.3.8),and it is probable that the same result is also true even more generally for Hurewicz fibrations over any base space X E B whatsoever (Definition I. I .2). The classification of fiber homotopy types of fiber bundles began in Dold [I] and continued in Curtis and Lashof [ I], Dold and Lashof [11, Fuchs [ 11, Stasheff [I, 2, 31, and Siege1 [l]. Although we shall not attempt to describe any classifications here, the reader is hereby warned that many results of later chapters concern properties of fiber bundles that are in fact merely properties of their fiber homotopy types. This phenomenon will be identified as it arises. 8.22 Remark: The lwasawa-Mal’cev decomposition theorem (Theorem 5.1 1) is the culmination of many efforts by Chevalley and others to generalize the familiar polar decompositions of Propositions 6.3, 6.6, and 6.9. Theorem 5.1 1 was first published in Mal’cev [l], in 1945; however, Chevalley, who reviewed Mal’cev’s paper, identified several gaps and obscurities. Iwasawa, who knew Chevalley’s review, but who could not obtain a copy of Mal’cev’s paper even as late as 1949, published a new proof of Theorem 5.1 I , which is
104
11. Fiber Bundles
formulated in Iwasawa [1, p. 5301. Iwasawa’s proof can be found in Cartier [l, pp. 22-15, 22-16], in Mostow [l, pp. 47-48], and in Hochschild [I, pp. 180-1861. Since Mal’cev directly inspired Iwasawa’s work, and since Mal’cev’s efforts probably contain the bulk of a correct proof, Mal’cev certainly deserves at least hyphenated credit for the result. Hence Theorem 5.1 1 is the Iwasawa-Mal’cev theorem. 8.23 Remark : The Leray-Hirsch theorem (Theorem 7.2) was established by Leray [l, 2, 3,4, 51 and G. Hirsch [l, 2, 3,4, 5,6], with mild restrictions on the base space. Leray’s work coincides with the development of spectral sequences; Hirsch’s constructions are somewhat more geometric. The clarification of spectral sequences by Serre [l, 21 paved the way for simplifications of Leray’s technique, with results which can be found in Kudo [1,2], Blanchard [l], E. H. Brown [l], Dold [4], Vastersavendts [l], and Dold [6], for example. A particularly important special case of the Leray-Hirsch theorem (for sphere bundles) is implicit in much earlier work of Gysin [l]; explicit geometric proofs of this special case were given by Thom [11 and by Chern and Spanier [l]; the latter proof was extended by Spanier [l]. Another historically important special case (for bundles over spheres) appears in Wang [ 13. Since the Leray-Hirsch theorem is independent of the structure group, it is clear that it applies equally well to more general fibrations; for example, a corresponding result for Serre fibrations is given in G. Hirsch [8]. The first application of a Mayer-Vietoris method to prove even a special case of the Leray-Hirsch theorem appears in Milnor [3, pp. 136-1421, material which is repeated in Milnor and Stasheff [l, pp. 105-1141; the base space is severely restricted. The same technique is applied in Spanier [4, pp. 258-2591 and in Husemoller [l, 1st ed., pp. 229-2301 to prove the Leray-Hirsch theorem in general, but with the same restrictions on the base space. The Mayer-Vietoris proof of the Leray-Hirsch theorem over any base space X E 9 lfirst appeared in Osborn [6]; it is based upon the method of Connell [l, pp. 499-5011, which Connell attributes to E. H. Brown. 8.24 Remark: According to Proposition 3.5, any fibre bundle over a contractible base is trivial. Hence, if E $ X represents a fibre bundle, then the Ljusternik-Schnirelmann category of the total space E is clearly related to the Ljusternik-Schnirelmann category of the base space X.This observation leads to several results, some of which are in %arc [l, 2, 31, Ginsburg [l], Varadarajan [l], Hardie [l], Palais [l], Ono [2], and Moran [l, 21.
CHAPTER 111
Vector Bundles
0. Introduction A vector bundle is any fiber bundle whose fiber and structure group are a vector space and its general linear group, respectively, the general linear group acting in the usual way on the vector space. Vector bundles arise naturally in differential topology. For example, if X is a smooth m-dimensional manifold, then there is a well-defined fangent bundle over X with fiber R" and structure group GL(m,R). For notational convenience we first consider real vector bundles, described and illustrated in the first ten sections of the chapter. The main result is that there is a "universal" real vector bundle y" over the base space G"(R") E !a,such that any vector bundle with fiber R" and structure group GL(rn,R) over any base space X E 39 is a pullback f'y" of y" along some map X 5 G"(R"); the map ,f is uniquely defined up to homotopy. Analogous descriptions and illustrations of complex vector bundles are provided in $1 1 merely by substituting the complex field C for the real field R throughout, with occasional complex conjugations; in particular, there is a "universal" complex vector bundle over the base space G"(C") E 93. Some relations between real and complex vector bundles are explored in 912.
1 . Real Vector Bundles This section consists of definitions and the rationale for the name "vector bundle." 105
106
111. Vector Bundles
1.1 Definition: A real vector bundle of rank m, or simply a real m-plane bundle, is any fiber bundle whose fiber is the real rn-dimensional vector space R" and whose structure group is the general linear group GL(m,R) of invertible m x m matrices, acting in the usual way on R". A real line bundle is a real vector bundle of rank 1.
Throughout the bulk of this chapter we frequently omit the adjective "real"; complex vector bundles are first introduced in $11. Since the fiber R" of an m-plane bundle 4 has the structure of a vector space, which is preserved under the action of the structure group GL(m,R), one expects for any coordinate bundle E 5 X representing 4 that the fibers E x are also mdimensional vector spaces. 1.2 Proposition: If a coordinate bundle E X represents a real m-plane bundle 4 over a base space X E-B,then thejbers E x are real m-dimensional vector spaces in a natural way; furthermore, if E' 5 X ulso represents 4, then there is a natural vector space isomorphism E x -,E: for each x E X .
PROOF: As in $11.1, E X is a family of fibers F = R" with structure group G = GL(m,R), and for each x E X there is a GL(m, R)-related family S , of h homeomorphisms E x -, R"; that is, if h E S,, h' E S,, and g E GL(m,R), then h-I the compositions E x R" 5 R"' and R" --+ E x 5 R" belong to S , and GL(m,R), respectively. For any r , , r z E R and e l , e2 E E x one can use any h E S, and the linear structure of R" to obtain K 1 ( r l h ( e l + ) r,h(e,)) E E x . For any other h' E S , the composition h'h- ' E GL(m,R) preserves the linear structure of Rmso that h'h-1(r,h(e,)+r2h(e2))=rlh'(e,)+r,h'(ez) E R"; that is, h - ( r h(e ) + r2h(e 2 ) )= h' - ( r h'(e ) + r,h'( e 2 ) )E E x . Thus h ( r h(e ) + r2h(e2))is an element r l e , + r2e2 E E x that is independent of the choice of h E S,, and one thereby obtains a linear structure in E x ; trivially E x 5 R" is itself an isomorphism with respect to the linear structures of Ex and R", which completes the proof of the first assertion. If E' 5 X also represents <, then there is a commutative diagram E
f
+
E'
for a homeomorphism f that induces a GL(m,R)-related isomorphism Ex Ex for each x E X , as in $11.1; that is, if h E S, and h' E S:, then the
107
I , Real Vector Bundles h-'
composition R" d E x E: 5 R" belongs to GL(m,R). It follows from the preceding definition of the linear structures of E x and E: that E x 5E: is a vector space isomorphism, as required. Let E 5 X und E 4 Y be coordinate bundles representing 5 over X E J und u reul n-plane bundle v] over Y E 9?, and suppose for some p 2 0 that there is a comniututive diagram 1.3 Proposition:
a reul m-plune bundle
+ E
I
E
X
' Y
such thut euch induced map E x -% Eg(,,i s lineur of rank p ; then if E' E' 5 Y ulso represent 5 und u], there is u commututive diagram
5X
und
I'
E' 11'
Y
X
' Y
such thut euch induced map Ex 5E$,,, is ulso linear of rank p. PROOF: As in Proposition 1.2 there are commutative diagrams E
f
+
E'
E
T
'2
for homeomorphisms f and T such that each E x k E: and Ey ,!? is;a vector space isomorphism. It suffices to set g' = fc g f - ' and to observe that each fg(,, g, f; is of rank p . 0
Proposition 1.3 leads unambiguously to morphisms in a category whose objects are real vector bundles. Let X 3 Y be a map of base spaces and let 5 and v] be vector bundles over X and Y, represented by coordinate bundles E X and 1.4 Definition:
108
111. Vector Bundles
2 Y, respectively. A bundle homomorphism 5 + q of rank p resented by any commutative diagram E
g
+l?
9
.Y
2 0 is rep-
II
X
such that each induced map E , 5 ie(x, is linear of rank p.
2. Whitney Sums and Products For any real vector spaces R" and R" the direct sum R" 0 R" and tensor product R " 0 R" are also vector spaces, R"'" and R"", and one expects corresponding constructions for vector bundles; these constructions and their properties are the goal of this section. Specifically, let 5 be an m-plane bundle over a base space X E B , let 5' be an n-plane bundle over a base space X ' E 93,and recall from Proposition 1.1.4 that 93 is closed with respect to products; we shall construct an ( n + m)-plane bundle 5 + 5' and an nmplane bundle 5 x 5' over the base space X x X' E & In !I. case X = X ' these constructions will lead to an (rn n)-plane bundle 5 0 5' and an mn-plane bundle 5 0 5' over X itself, and for any vector bundles 5, <',5'' over the same X E 9the identities (5 0 5') 0 5" = 5 0 (5' 0 t"),50 4' = 5' 0 5, (5 0 5') 0 5" = 5 0 (5' 0 5'7, 5 0 5' = 5' 0 5, and (5 0 5') 0 5" = (5 0 5") 0 (5' 0 5") are easily verified.
+
We start with addition of vector bundles. Given coordinate bundles E X and E' 5 X ' that represent an m-plane bundle 5 over X E 93 and an n-plane bundle 5' over X ' E @, respectively, the Cartesian product E x E' 2X x X' is a coordinate bundle with fiber R" x R" and structure group GL(m,R) x GL(n, R). Since X x X' E 9 by Proposition 1.1.4, E x E' 5 X x X' represents a fiber bundle q over X x X ' , in the sense of Definition 11.1.9, which is trivially independent of the choices of coordinate bundles representing 5 and 5'. The fiber R" x R" of q is the underlying topological space of the direct sum R" 0 R", so that there is a homeomorphism R" x R"%R"'@[w" of topological spaces. There is also a group homomorphism GL(m,R) x GL(n,R) f, GL(m + n, R) with
109
2. Whitney Sums and Products
for any invertible m x m matrix A E GL(m,R) and invertible n x n matrix B E GL(n,R). Clearly r and 0 form a morphism (r,0)of transformation groups in the sense of Definition 11.2.6, which one can apply to q to induce a new bundle over X x X ' , as in Proposition 11.2.7. 2.1 Definition: Let 5 be an rn-plane bundle over X E L4?, let 5' be an n-plane bundle over X ' E B, and let q be the fiber bundle with fiber R" x R" and structure group GL(m,R) x GL(n,R) over X x X ' E 2,as in the preceding discussion. The sum 5 + 5' is.the (rn + n)-plane bundle over X x X ' induced of transformation groups, as in from q by the preceding morphism (r,@) Definition 11.2.8, in which GL(m,R) x GL(n, R) 5 GL(m m,R) and R" x R"5 R"' 0 R".
+
One can obviously reduce the verbiage of Definition 2.1. As it stands, however, there is an entirely analogous construction of a product of the m-plane bundle 5 and the n-plane bundle 5' beginning with the same fiber bundle q over X x X ' , but using a different morphism of transformation groups. Let R" x R" 2 R" 0 R" ( = Rmn)be the map @((x', . . .
y " ) ) = (xly', . . . , x'y"; . . . ; xmyl, . . .
and let GL(m,R) x GL(n,R) f, GL(mn,R) be the group homomorphism with
uYB . . . for any invertible m x m matrix
and any invertible n x n matrix B E GL(n, R). Clearly the new pair, r and 0,also forms a morphism (T,0) of transformation groups in the sense of Definition 11.2.6. 2.2 Definition: Let 5 be an rn-plane bundle over X E B, let 4' be an n-plane bundle over X ' E g, and let 9 be the fiber bundle with fiber R" x [w" and structure group GL(rn,R) x GL(n,R) over X x X ' E L4? as in the earlier discussion. The product 5 x 5' is the mn-plane bundle over X x X ' induced from q by the preceding morphism (I-,@) of transformation groups, as in Definition 11.2.8, where GL(m,R) x GL(n,R) 5 GL(rnn,R)
and
R" x R" f R " 0 R".
110
111. Vector Bundles
There is a noticeable asymmetry in Definition 2.2. What happens if one replaces the given morphism (r,a) of transformation groups by another obvious choice (l-', a')?Specifically, let R" x R" 2 R" 0 R" ( = R"") be the map W( ( X I , . . . , X"), ( y ' , . . . ,y " ) ) = (x'y', . . . , x"y1 ; . . . ; x l y " , . . . , x"y"),
and let GL(m,R) x GL(n,R) f; GL(mn,R) be the group homomorphism with T ' ( A , B )=
(
' . * Ab,'
Ab;
. . . Ab:
(; ")
for any invertible m x m matrix A
E
GL(m,R) and any invertible n x n matrix
... * "
)
Abf
= B E GL(n,R).
b:
Clearly the pair (I-',@') forms a morphism of transformation groups in the sense of Definition 11.2.6. 2.3 Proposition If one substitutes the preceding morphism (I?, a') oftransformation groups for the morphism (r,O) of transformation groups used in Definition 2.2, one obtains the same mn-plane bundle 5 x 5' over X x X ' . PROOF: Observe that there are factorizations r' = F'' and 0'= 0'' Q, for a morphism (l-",0'') induced by a change of basis in R"",where l-" is an automorphism of GL(mn,[w) and W' itself belongs to GL(mn,R). It follows from Proposition 11.2.9 that (l-",W') leaves 5 x 5' unchanged, as required. For any topological space X the diagonal map X 5 X x X carries each X into (x,x) E X x X . In the following definition we implicitly use the following special case of Proposition 1.1.4: if X E B, then X x X E 9?.
x
E
2.4 Definition: Let 5 and 5' be vector bundles over the same base space X E 9,and let 5 5' and 5 x 5' be their sum and product over X x X , as in Definitions 2.1 and 2.2, respectively. The Whitney sum 5 0 5' over X and the Whitney product 5 0 5' over X are the pullbacks A!(< + 5') and A!(< x 5') along the diagonal map X 5 X x X ,
+
There are many equivalent ways of defining the Whitney sum 5 0 5' and Whitney product 5 0 5' of an m-plane bundle 5 and an n-plane bundle 5' over the same X E 93.For example, if E X and E' 5 X represent 5 and t', respectively, so that E x E' n x n' X x X represents a fiber bundle q over
-
111
2. Whitney Sums and Products
X x X E J with fiber R" x R" and structure group GL(m,R) x GL(n,R), then there is a pullback [ = A$ along the diagonal map X 5 X x X , which can itself be used to define 5 05' and 5 05'. Let (re,@@) denote the morphism of transformation groups used in Definition 2.1, with G L ( m , R )x 1@a GL(n, R) G L ( ( m+ n, R) and R" x R" R " 0 R", and let (r@,@@) denote the morphism of transformation groups used in Definition 2.2, with @@ GL(mn,R) and R"' x R" R" 0R". (GL(m,R) x GL(n,R)
-
-
@
-
2.5 Proposition: Given an m-plane bundle 5 and an n-plane bundle 5' over X E A, let [ be the preceding bundle A$ over X , with jiber R" x R" and structure group GL(m,R) x GL(r2,R). Then 5 04;' is the bundle induced from [ by the morphism (r@, @@) of trunsjormation groups, and 5 04;' is the bundle induced from i by the morphism (I@,@@) oj' tranbformation groups. PROOF: According to the discussion following Definition 11.2.8, the construction of fiber bundles induced by a given morphism of transformation groups commutes with the construction of pullbacks. 2.6 Proposition: For any real vector bundles 5, t', 5" over the same base spuce X E d one has ( 5 05') 05" = 5 0(t'0t"), 5 0 5' = 5' 05, (5 0 4') 0 5" = 5 0(5' 0 t"), 5 0 4' = <' 0 5, and ( 5 0 5') 0 5" = ( 5 0i") 0(5' 05") over X .
PROOF: These identities are all immediate consequences of Proposition 11.2.9. We prove the commutative law 4; 05' = 5' 05, for example. If 5 is an m-plane bundle and 5' is an n-plane bundle, then ( 05' is induced from the bundle [ of Proposition 2.5, with fiber R" x R" and structure group GL(m,R) x GL(n,R), by applying the morphism (r@,@@) of transformation CD groups. Let R"'" + R"'" be the change of basis O(xl,. . . , .xm, y , , . . . , y,) = . . . ,y,,, x l , . . . ,x"), inducing an inner automorphism GL(m + n, R) 5 G L ( m + n, R), and observe that 0 itself belongs to GL(m n, R). Then 5' 05 is induced from the same bundle [ of Proposition 2.5 by applying the @ 0 @@) = (r,@) (I-@,@@), and since ( r , O ) is a morcomposition (r I'@, phism of transformation groups satisfying the hypotheses of Proposition 11.2.9, one has 4; 05' = 5' 05 as claimed. (Incidentally, the commutative law 05' = 5'0t for products was essentially proved by a similar method in Proposition 2.3, except for pulling back the resulting identity along the diagonal map X 5 X x X . ) Proposition 2.6 asserts that the set of real vector bundles over a fixed X E S satisfies most of the axioms required of a commutative ring, with respect to the operations 0 and 0. In fact, there are even neutral elements with respect to sums and products. ()I,,
+
0
112
111. Vector Bundles
E 2 and any natural number m 2 0, the trivial real m-plane bundle E" over X is the vector bundle represented by the first projection X x R" 5X .
2.7 Definition: For any base space X
Clearly the trivial 0-plane bundle 8' and the trivial line bundle E' satisfy = ( and 5 0 E' = 5 for any real vector bundle 5 whatsoever over the same base space. Consequently, except for additiue inverses the real vector bundles over a given base space X E 98 form a commutative ring with unit, with respect to 0 and 0 .There is an analogous situation for complex vector bundles, which will be further exploited in Volumes 2 and 3.
5 0 8'
3. Riemannian Metrics An inner product on a real vector space R" is any symmetric bilinear map R" x R" R such that ( e , e ) > 0 for all nonzero vectors e E R". We shall establish the existence and demonstrate the usefulness of corresponding "inner products" for any vector bundle over any base space
x E a.
For any real m-plane bundle 5 over X E 2 let 5 be the bundle A'q over X with fiber R" x R" and structure group GL(m,R) x GL(m,R), as in Proposition 2.5, for 5' = 5 . If { U iI i E I is an open covering of X,with local trivializations E I U i .-% U i x R" of a coordinate bundle E 1:X representing 5, then { U ix U j ) ( ij, ) E I x I } is a corresponding covering of X x X for the bundle q. Since the subfamily { U ix U i ( iE I } covers the image of the diagonal X 5 X x X , it follows that there are corresponding local P;' U ix (R" x R")of a corresponding representation trivializations E" 1 U i JI' E" X of the bundle 5. Specifically, transition functions U i n U j A GL(m,R) describing the coordinate bundle E 5 X provide transition functions U in U j (JI1*JI') * GL(rn,R) x GL(rn,R) describing the coordinate bundle E" 5 X . If Ex is the fiber over x E X of the coordinate bundle E A X , then Ex x E x is the corresponding fiber E:' of the coordinate bundle E" 5 X . Suppose that E" 2R is any map which restricts to an inner product Ex x E x = E:' R over each x E X . If E'"'. X is any other representation of 5, and if E"' 3 X is the corresponding representation of (, then the fiber E:" = EL x Ex over each x E X is related to the fiber E:' = E x x E x by an element in the image of the diagonal GL(m,R) + GL(m,R) x GL(rn,R). Hence if E"' 1,E" is an isomorphism of coordinate bundles representing [,
-
-
1 I3
3. Riemannian Metrics
-
and if E"' R is the composition E"'5E" ' . 1R, it follows that ( , )' also restricts to an inner product EL x EL = E:' ( )k R over each x E X . Consequently the following definition is independent of the coordinate bundle E 5 X chosen to represent 5. 1
3.1 Definition: Let 5 be a real m-plane bundle over X E 33,represented by X represent the preceding a coordinate bundle E A X , and let E" bundle [, with fiber R" x R" and structure group GL(m,R) x GL(m, R). A riemannian metric on 5 is any map E" < . ) R that restricts to an inner product E x x E x + R for each x E X .
-
3.2 Lemma: If X E 9? is paracompact, then there is a riemannian metric on any real vector bundle 5 over X .
z
-
PROOF: Let E A X and E" X represent 5 and [ as before, and suppose that { Ui I i E I ) is an open covering of X with local trivializations E I U i y , U ix R". Since X is paracompact, one may as well suppose that { Ui I i E I } is locally finite, and that there is a partition of unity { h i ( iE 1 ) subordinate Y;' to { Uil i E I ) . For the corresponding local trivializations E'I U i Ui x (R" x R"), and for any fixed inner product R" x R" R, there is a map h,( U ix (R" x R") R carrying each ( x , ( e l , e 2 )E) U i x (R" x R") into the real number h i ( x ) ( e , e, z ) i .Since hivanishes outside U i ,each composition hi( , )i 3 4' ;' extends to a map E' ha( s) 'Pi' + R that vanishes outside U i , and one trivially verifies that the well-defined sum hi( , )i0 Yi' is a riemannian metric.
-
-
<.>#
)I
I
1
xiel
3.3 Lemma: Let ( , ) be a riemannian metric on an m-plane bundle 5 over any X E 93,and let X ' 3 X be any map in 99;then there is a riemannian metric ( , )' on the pullback f '5 over X ' E W .
-z
PROOF: If E 5 X and E" X represent the bundles 5 and [, as before, n"' X ' represent the pullbacks f '5 and f '[, then and if E' 5 X ' and E"' there is a commutative diagram E"
f
X'
I
E"
+
x,
as in Lemma 11.1.3, for which the composition desired riemannian metric on f '5.
E"'5E"
<.)
R is the
114
111. Vector Bundles
3.4 Proposition: Given any real vector bundle 5 over any base space X there is a riemannian metric on 5.
E
.g,
PROOF: By Definition 1.1.2 the space X is homotopy equivalent to a metrizable space X ' of finite type; by Lemma 1.8.2 (Stone [l]) the metrizable space X ' is paracompact. Thus there is a homotopy equivalence X 3 X ' of X with a paracompact space X ' . Let X ' X be a homotopy inverse of g. Since X' is paracompact, the pullback over X ' has a riemannian metric by Lemma 3.2, so that the pullback g!h!
E
B
+.i
X
B
> Y
such that each restriction E x -% Eg(x)is linear of rank p . In particular, if E' 2X and E $ X represent vector bundles 5' and 5 over the same X E 93, then a vector bundle morphism 5' + 5 of rank p is represented by any commutative diagram E' B
such that each restriction EL 5 E x is linear of rank p .
3.5 Definition: Let 5'
-.+
5 be a vector bundle morphism E'
such that each restriction subbundle of 5.
i?
E i 3 Ex
t E
is a monomorphism; then 5' is a
115
3. Riemannian Metrics
('
c2
For example, let and be real vector bundles of ranks p and q, respectively, over X E a, with Whitney sum 5' 0t2 of rank p + q. If n' x nz EL X and E 2 3X represent 5' and t2,then E' x E 2 X x X represents a fiber bundle v] with pullback i= A$ along the diagonal map X 5 X x X as before; the fiber and structure group of iare RP x Rq and GL(p, R) x GL(q, R). One obtains a coordinate bundle E 5 X representing 5' 0t2 by applying (rQ,@@)to i,as in Proposition 2.5. The fibers Ex are direct sums E i 0E: for each x E X , and there is a well-defined map E' 5 E that identifies E i as a direct summand of E x for each x E X ; continuity of g is an easy exercise. Thus 5' is a subbundle of t10 t2,as expected.
-
3.6 Proposition: Let 5' be a subbundle of a vector bundle 5 over a base space X E a;then there is another subbundle t2of 5 such that ( = t' 0t2.
PROOF: Suppose that 5' is of rank p , represented by a coordinate bundle E' A X , and that is of rank rn, represented by a coordinate bundle E 5 X . By definition there is a commutative diagram
<
-
such that each restriction E i 3 E x is a monomorphism. Let E" 0 R represent a riemannian metric on 5, inducing an inner product E x x E x = EY *, R over each x E X . One identifies Ek as a linear subspace of E x via the monomorphism g,, and there is an orthogonal complement E: of E: with respect to ( , ),, of rank q = rn - p . The projection E 5 X restricts to a projection E 2 3X of the union E 2 = UXEX E: c E . We shall show that E 2 2 X is a coordinate bundle representing a vector bundle t2; trivially one then has 5' 0t2 = 5. Let [ Uil i E I } be any open covering of X for which there are local trivializations El U i .% U i x R", so that there are corresponding local y' (,) U i x (R" x R"). The riemannian metric E" trivializations E"I U i < R induces a riemannian metric U i x (R" x R") R for each i E I , which can be regarded as a nonsingular symmetric matrix whose entries are real-valued continuous functions on U i. By the classical Gram-Schmidt process one can choose an orthonormal basis of R", for each i~ I , such that the first p basis elements span the image under Y i of E' I U i c El U i , scalars being real-valued continuous functions on U i ;the remaining q basis elements, for each i E I , span the image under 'Pi of E 2 I U i c El U i .
-
9
)t
-
116
111. Vector Bundles
Suppose that U in U j is nonempty and that the preceding bases are chosen for each of the indices i E 1 and j~ 1. The transition functions U in U j % GL(m,R) for the coordinate bundle E 1X can then be regarded as a family of linear transformations Rp 0 W4 JI:(X) RP@ R4 that depend continuously on x E U in U j and preserve the summands Rp and R4. The restrictions R4 R4then also depend continuously on x E U in U j , and they provide the transition functions U in U j G L ( ~R), of a coordinate bundle representing a q-plane bundle 5'. Since the transition functions $is' arise from local trivializations E2 1 U i U ix R4 of the projection E 2 2X, it follows that E 2 3X is itself a coordinate bundle representing a q-plane bundle 4' as required.
-
a
2
Proposition 3.6 is one of several classical applications of riemannian metrics. One can also use riemannian metrics to show, for example, that the structure group GL(m, R) of any real m-plane bundle over any X E 93 can be reduced to the orthogonal subgroup O(m)t GL(rn,W); however, since this is a special case of the linear reduction theorem (Theorem II.6.13), the details will be left as an exercise. (See Remark 13.17.)
4. Sections of Vector Bundles For any base space X E 93 let Co(X)be the ring of real-valued continuous functions X R. In this section we develop a one-to-one correspondence between real vector bundles over X and (isomorphism classes of) certain Co(X)-modules. --f
We begin with a special case of Definition 11.5.2. 4.1 Definition: Let E
X represent a vector bundle over X E & aIsection ; of E $ X is any (continuous)map X % E such that X 5 E $ X is the identity on X. According to Proposition 1.2, if E $ X represents a vector bundle, then the fiber Ex over each x E X is a vector space in a natural way. We use this structure to turn the set of sections X 5 E of such a bundle into a Co(X)module. 4.2 Lemma: I f E 4 X represents a vector bundle, then for any two sections a+n' X 5 E and X 2 E, and for uny .f E C o ( X ) ,there are unique sections X L E and X 3E such that (a a')(x)= a(x) a'(x) and (fb)(x)= .f(x)a(x) in E x .for each x E X.
+
+
117
4. Sections of Vector Bundles
PROOF: Although uniqueness is automatic, one must verify that a + a' and fa are continuous. If E 5 X represents a real m-plane bundle, for example, there is an open covering { Ui I i E I } of X with local trivializations E I U i 2Ui x R", and it suffices to verify for the restrictions 01 U i ,0' 1 U i ,and f I Ui that
ui
Yt (alU,+o'lU,)
+uix
R"
and
ui
Y, (
f IU,)(al U , )
+
uix
R"
are continuous for each i E I . However, Y i a[U i carries x E U i into (x,(al(x) ,..., o,(x)))E Ui x Rmforol,..., a , ~ C ~ ( U , ) , a n d ~ ~ ~ a ' ( U ~ c a r r i e s x E Ui into (x,(ai(x),. . . , oa(x))E Ui x R" for a;, . . . ,a; E Co(UJ, and sums at + a',,. . . , a,,, a; and products f i T t , . . . ,fa, of continuous functions Ui -+ R are continuous. J '
+
4.3 Proposition: I f E 5 X represents a vector bundle, then the set 9 of sections X 5 E ,forms a Co(X)-modulewith respect to the addition and scalar multiplication of Lemma 4.2.
PROOF: Let the zero-section X 5 E carry each x E X into 0 E E x , and for -a any section X 5 E let X .-+ E carry each x E X into - a(x) E E x . One verifies as in Lemma 4.2 that 0 and - a are continuous, and that a + ( - a) = 0. The remaining module axioms are trivially satisfied. 4.4 Definition: If E 5 X represents a vector bundle over X , then the Co(X)module 9'of Proposition 4.3 is the corresponding module of sections. Clearly if E 5 X and E' 5 X represent the same vector bundle X E g,then the isomorphism
5
over
of coordinate bundles induces a Co(X)-module isomorphism B -+ B' of the corresponding modules of sections. Hence a vector bundle 5 over X EA!9. induces an isomorphism class of Co(X)-modules. By abuse of language one frequently picks a particular coordinate bundle E 5 X representing 5 and calls the corresponding Co(X)-module B the module of sections of 5.
118
111. Vector Bundles
One feature of Lemma 4.2and Proposition 4.3deserves further comment. If Co(X; U,)denotes the ring ofrestrictions to U ic X ofcontinuous functions X -+ R , then the local trivialization E I U i5 U ix R" turns the Co(X)module 9 into the free Co(X; U,)-module ( C o ( X ;Ui))"of rank m. Furthermore, there is another way of describing both C o ( X ;U i ) and (Co(X; U,))". Let I ( U i )c C o ( X ) be the ideal of all continuous functions X 2 R with f ( x ) = 0 for x E U , ; then C o ( X ;U,) is isomorphic to the quotient CO(X)/I(Ui), and ( C o ( X ;U,))" is isomorphic to the quotient of the Co(X)-module(CO(X))" by the submodule I( U,)(C0(X))".More generally, a Co(X)-module 9 is locally ,free of rank m if there is an open covering { U iI i E I } of X such that each C O ( X ) / IU,)-module ( 9 / 1 (U i ) 9 is free of rank m. Thus if E X represents an m-plane bundle, then the corresponding Co(X)-module of sections is locally free of rank m. 4.5 Proposition: For any X E let 9 be a locally free Co(X)-moduleof rank m. Then 9 is the module of sections of a coordinate bundle E S X representing a real m-plane bundle 5 over X ; jurthermore, 5 is unique.
PROOF: Let { U i I i E I ) be an open covering of X such that each $ / I ( U i ) 9 is free of rank m, and fix a basis for each 9 / I ( U i ) 9 ;the choices of bases induce a C o ( X ;U,)-module isomorphism 9 / I ( U i ) 9-,(Co(X; U,))" for each i E I . If U in Uj is nonempty, then 9 / I ( U i n U j ) 9 is also free of rank m, and it inherits a basis from 9/1(U i ) 9and a basis from 9 / I ( Uj)S;hence there is an invertible m x m matrix of elements in C o ( X ;U in U j )relating the two JI' GL(m,R), bases. Such a matrix is precisely a continuous map U iA U j A and since the basis of each 9 / f ( U i ) 9 is fixed, the conditions $?(x)$j(x) = $ f ( x ) of Proposition 11.2.5 are trivially satisfied; that is, the maps $! are the transition functions of a coordinate bundle E X representing a real mplane bundle 5 over X. Uniqueness of 5 is clear. 4.6 Theorem: For any X E B and any m 2 0 there is a one-to-one correspondence between the real m-plane bundles 5 over X and the isomorphism classes of locally free Co(X)-modules 9 of rank m.
PROOF: Since any vector bundle is itself defined as an isomorphism class of coordinate bundles, this is an immediate consequence of Propositions 4.3 and 4.5. If 9is a module of sections of a real m-plane bundle over X E 99,then the = Homco(x)(9, Co(X))is itself locally free of rank m. dual Co(X)-module9* In light of Proposition 4.5 it is reasonable to identify the m-plane bundle as its module of sections. that has 9*
119
5. Smooth Vector Bundles
4.7 Proposition: For any X E 23 let 9 he a locally free Co(X)-module of runk m, und let 3*be i t s dual HomCocx,(S,Co(X)). Then 9 and 9*are modules of sections of coordinute bundles representing the same real m-plane bundle ooer X. PROOF: By Proposition 4.5, 9 is the Co(X)-module of a coordinate bundle E 5 X representing a real m-plane bundle 5 over X, and by Proposition 3.4 there is a riemannian metric on 5. As in Definition 3.1 the riemannian metric is defined in terms of inner products E x x E x -R that depend continuously on .Y E X, so that any two sections a E 9 and T E B determine an element (a, r) E Co(X)with value (CT(.Y),T(X))~ on x E X. Consequently there is a nondegenerate bilinear map 9 x 9 Co(X) over Co(X), which induces an isomorphism 9-9*carrying any T E 9 into the element F Co(X) of Hom,o~x,(9,Co(X)). Thus 9 and 9*are isomorphic Co(X)-modules, and the result follows from Theorem 4.6.
-
A nowhere-uunishing section of a vector bundle 5 over X E 98 is any section X E of a coordinate bundle E : X representing such that a(x) # 0 E Ex for each .Y E X.
<
>
4.8 Proposition: 11'un in-plune bundlr 5 h u s u nowhere-vunishing section, then = I:' @ rl j01' thc triviul line bundle e ' untl sonie (m - I)-plane bundle '1.
<
PROOF: Let E : X represent (, let X 5 E be the nowhere-vanishing section, let E' c E be the set of all points of the form ra(x)for any real number r E R, and let E' 2 X be the restriction to E' of E 5 X. Since every a(x) is nonzero, it follows that every element of E' is uniqut4y of the form r a ( x )for some real number I' E [w, so that there is a coordinate bundle isomorphism
lX.7
E'
R'
Thus the inclusion E' -,E identifies I:' as a subbundle of (, and Proposition 3.6 implies the desired result.
5. Smooth Vector Bundles For any smooth manifold X the algebra C'( X) ofsnioorh functions X -, R can be substituted in the preceding discussion for the algebra C o ( X ) of continuous functions X + R, locally free C ' (X)-modules being substituted for locally free Co(X)-modules.A smooth analog of Theorem 4.6 then relates
120
111. Vector Bundles
isomorphism classes of locally free Cm(X)-modules to corresponding smooth vector bundles. Some smooth vector bundles of primary interest in differential topology are defined via this relation in the next section. We begin with a discussion which leads directly to the definition of smooth vector bundles. The underlying topological space of the general linear group GL(rn,R) of invertible rn x rn matrices is an open subset of R"', in the relative topology, so that GL(m,R) is itself a smooth manifold of dimension mz : it can be covered by an atlas consisting of a single coordinate neighborhood. Furthermore, the group product GL(rn,R) x GL(m,R)+ GL(m,R) and group inverse or' GL(rn,R) GL(m,R) are trivially smooth maps, so that GL(m,R) is a Lie group. Clearly the usual action GL(m, R) x R" + R" of GL(m,R) on the vector space R" is also smooth. Recall that coordinate bundles E -1: X were described in general via local trivializations in Definition 11.2.1. The local trivializations were homeomorphisms, transition functions were required to be continuous maps of open subsets of X into the given structure group G , and G was required to act continuously on the given fiber. According to the preceding paragraph, if X is a smooth manifold one can reasonably specialize Definition 11.2.1 to coordinate bundles E X with fiber R" and structure group GL(m,R), but with the additional requirements that local trivializations be diffeomorphisms and that transition functions be smooth maps of open subsets of X into GL(m,R). Specifically, suppose that E 5 X is a smooth map of smooth manifolds, and that there is an open covering [ U i I i E I ) of X with local trivializations Y i that are diffeomorphisms such that each i U . . UI E Y x R"
-
<
/
commutes, where El U i = f ' ( U i ) , as always. If the intersection U in U , of two sets in the covering is nonvoid, then there is a composed diffeomorphism Y j Y ; such that 0
Y .Y-'
( U i n U j ) x R" A ( U i n U j ) x R"
\
/
121
5. Smooth Vector Bundles
-
= ( . q $ { ( x ) j ' )for each ( x , f ' )E commutes. One necessarily has Y, Y; '(x,,f) *:CX, ( Ui n U,) x R", where R"' R" a diffeomorphism for each .Y E Ui n U j .
5.1 Definition: If each of the preceding ${s is a smooth map from Ui n U j to the group G L ( m ,R),acting in the usual way on R", then E X is a smooth coortlinute bundle over the smooth manifold X, with fiber R" and structure group GL(m,R). The terminology of coordinate bundles in the sense of Definition 11.2.1 applies equally well to the specialized smooth coordinate bundles of Definition 5.1. Specifically, the diffeomorphisms El Ui 5 Ui x R" are local trivialixztions and the smooth maps Ui n U j -% GL(m, R) are transition firncrions, which automatically satisfy the conditions $?(x)${(x) = $t(x) of Proposition 11.2.5. Two smooth coordinate bundles E 5 X and E' 5 X with fiber R" and structure group GL(m,R) over the same smooth manifold X are isomorphic if there is a diffeomorphism f such that the diagram E
f
r E
commutes and induces a vector space isomorphism E x -% EL for each x E X . Isomorphism of such smooth coordinate bundles is trivially an equivalence relation.
5.2 Definition: A smooth real vector bundle of' rank m over a smooth manifold X is any isomorphism class of smooth coordinate bundles E : X with fiber R"'and structure group GL(m,R), for the usual action G L ( m , R ) x R"
+ R"'.
Since smooth maps are also continuous, it follows that smooth vector bundles can also be considered as vector bundles in the sense of Definition 1.1. In fact, we shall learn in $9 that any vector bundle whatsoever over a smooth manifold can be represented by a smooth coordinate bundle, a result which suggests that Definition 5.2 is superfluous. However, if one is given a smooth vector bundle, one can study its smooth properties as such.
5.3 Proposition: Let E 5 X ' be a smooth coordinute bundle representing a smooth m-plane bundle 5' over X ' . Then the pullback E X of' E 5 X' along any smooth map X 3 X ' is also a smooth coordinate bundle; a fortiori, the pullback g't' is a smooth m-plane bundle over X .
122
111. Vector Bundles
PROOF: Let E' 5 X' be defined as a coordinate bundle with respect to an open covering { U t ) iE I } of X'; by Definition 5.1 it is smooth if and only if the transition functions U ; n U J % GL(m,R) are smooth. If one sets Ui = g- ' ( U i ) for each i E I, then according to Proposition 11.2.4 E 5 X is a coordinate bundle with respect to the open covering { Ui I i E I} of X , its transition functions U i n U j 2 GL(m,R) being the compositions t,bij g1 Ui n U j . Since y is smooth, the latter compositions are smooth, so that E 5 X is smooth by Definition 5.1. ~J
5.4 Definition: Let E 5 X be a smooth coordinate bundle, representing a smooth vector bundle over a smooth manifold X. A smooth section of E 1X is any smooth map X E such that X 5 E 5 X is the identity on X.
s
s
The set 8 of smooth sections X E of any smooth coordinate bundle E 1X representing a vector bundle forms a C"(X)-module, as in Proposio'E 8,andf E P ( X ) one defines o o' tion 4.3. Specifically,for any o E 9, and fa by setting (o o')(x) = o(x) o'(x) E E , and ( f b ) ( x )= f ( x ) o ( x )E E , for each x E X; the verifications that o o' E 9 and j b E 8 are analogous to corresponding verifications in Lemma 4.2, with CX(X)instead of C o ( X ) .
+
+
+
+
5.5 Definition: If E 1X represents a smooth vector bundle over a smooth manifold X, then the preceding C"(X)-module 8 is the corresponding module of' smooth sections of E 1X.
For any smooth vector bundle 5 over a smooth manifold X one frequently chooses a particular smooth coordinate bundle E 3 X to represent 5, calling the corresponding C" (X)-module 9 the module of smooth sections of 5, by abuse of language. A C"(X)-module 9 is locully free of rank m 2 0 if there is an open covering [ LI, I i E 1 ) of X such that each CK(X)/I(UJ-module 9 / I (U , ) 9 is free of rank m, where I(Ui)c Cn(X)is the ideal of those functions that vanish on U , . 5.6 Proposition: For arty smooth manifold X und any m 2 0 there is u one-to-one correspondence between the smooth real vector bundles 5 of rank m over X und the isomorphism classes of locally jree C"(X)-modules 8 oj runk m.
PROOF: This is the smooth version of Theorem 4.6, which one proves merely by substituting Cm(X)for C o ( X )throughout the proofs of Propositions 4.3 and 4.5.
123
5. Smooth Vector Bundles
We shall later illustrate Proposition 5.4 with specific examples. An alternative characterization of riemannian metrics was used, without identifying it as such, in the proof of Proposition 4.7. We shall soon prove a smooth analog of Proposition 4.7, and for variety we begin with the smooth analog of the alternative characterization of riemannian metrics.
5.7 Definition: For any smooth manifold X, and for any locally free Ca(X)module .Fof fixed rank m >= 0, a smooth riemannian metric is a symmetric bilinear map 9 x 9 ( * ) C a ( X )such that for each a E 9and each x E X one has ( a , o)(x) > 0 whenever ~ ( x#) 0.
-
-
The proof of the following existence theorem is essentially a smooth version of the proof of Lemma 3.2.
5.8 Proposition: For any smooth manijbld X, and for any locally free C' ( X ) tnodule S of jixed rank m 2 0, there is at least one smooth riemannian metric Sx 9 CZ(X). PROOF: Let { U iI i E I ) be an open covering of X such that each Cm(X)/l(Ui)module @ / I ( U i ) 8 is free, where I ( U i ) c C w ' ( X )is the ideal of smooth functions vanishing on U i . Since X is paracompact, by Lemma 1.6.1, one may as well suppose that { U iI i E I } is locally finite; by Lemma 1.6.2 there is a smooth partition of unity ( h i l i e I ) subordinate to ( U i l i ~ I )For . each i E I the C" ( X ) / I (U,)-module 9 / I ( U i ) 9 is isomorphic to (Ca(X)/I(Ui))", and one constructs a symmetric bilinear map
9 / I ( U i ) 9 x 9/1(U i ) 8 3Crn(X)/I(U i ) via the usual m x m matrix with 1's down the diagonal and 0 s elsewhere. Since hi vanishes outside U i , each ( , ) i induces a symmetric bilinear map 9 x .F ht< C T ( X ) for , which h , ( x ) ( a , a ) ( x )> 0 whenever both hi(x)> 0 and o(x) # 0. Since .(hiI i E I : is a partition of unity, the sum hi( , ) i is then a well-defined map 9 x 9 Cz'(X)of the desired type.
+
)t
Cia,
5.9 Proposition: For m y smooth manifbld X let 9be a locally free C'(X)module of fixed rank m >= 0, and let 5*be the dual, Hom,,,,,(S,C"(X)). Then 9 and S* are modules of' smooth sections of smooth coordinate bundles representing the same smooth m-plane bundle over X.
PROOF: This is the smooth version of Proposition 4.7, and its proof is identical to that of Proposition 4.7, except that one uses the smooth riemannian metric 9 x 9 % Cx(X)of Proposition 5.8 in place of a merely continuous riemannian metric.
124
111. Vector Bundles
Let E 1X be any smooth coordinate bundle that represents a vector bundle over a smooth manifold X, and let X 5 E be a continuous section of E 1X. Can one approximate o in some sense by a smooth section X 5 E ofE<X? In order to pose a more precise question let 9 x 9 2Ca(X) (cC o ( X ) )be a riemannian metric for the C"(X)-module 9 of sections of E 1X, as in Proposition 5.8, and observe that 9 can also be regarded in the obvious way as a Co(X)-module. The riemannian metric induces a euclidean norm 9 Co(X), which carries each section o E 9 into the positive square root q m ' E Co(X), assigning the classical euclidean norm Ila(x)llx= (o(x),~(x)), to the vector a(x) E E, for each x E X. Clearly lloll = 0 E Co(X) if and only if o E 9 is the zero-section; furthermore the Schwarz inequality I(a, z)l S llall llzll and resulting triangle inequality \lo zll 5 11o11 1 1 ~ 1 1are satisfied in C o ( X )for any sections o,z E 9 because they are satisfied in each E x . We shall show for any continuous section X 5 E and any strictly positive function E E C o ( X )that there is a smooth section X 5 E such that - 011 < E ; that is, we shall show that Ilz(x) - o(x)II, < E ( X ) for each x E X.
2
+
+
\IT
x [w" 0 be the product m-plane bundle over a smooth compact manifold 0, let 9be the Co(0)-module of continuous sections 0-, 0 x Iw", and let 9 Co(0)be any euclidean norm. For any strictly positive function E E C 0 ( 0 )and any continuous section 0 0 x R" there is then a smooth section 0 5 0 x R"such that 117 - oII < E .
5.10 Lemma: Let
2
PROOF: The given section o can be regarded as an m-tuple (o',. . . , a") of elements oi E Co(0), and the euclidean norm arises from a riemannian metric which can be regarded as a corresponding m x m symmetric matrix (gij) of functions gij E C 0 ( U )such that each ( g i j ( x ) )is positive-definite. In particular gii > 0 in Co(U),and one can set M i = max,." g i i ( x ) ;the maximum value exists because 0 is compact. Similarly one sets c0 = min,, r, E ( X ) > 0. The Stone-Weierstrass theorem (Theorem 1.6.12) applies to the algebra A ( 0 ) = C n ( 0 ) ,so that for each ai E Co(0) there is a zi E C a ( U ) such that Izi(x)- oi(x)l< Eo/mMi for all x E 0.Let 02 0 x R" be the section corre,o'., . . . , om),for i = 1 , . . . ,m ; in parsponding to the m-tuple (z', . . . , ticular o = oo and a, is a smooth section t,so that it remains only to show that 117 - 011 < E . However,
$'
(oi - oi- 1 ,
oi
- oi- 1) = gii(z' - o')2,
125
5. Smooth Vector Bundles
1X be any smooth coordinate bundle representing an m-plane bundle over a smooth manifold X , let 2F be the Co(X)-module of (continuous) sections X 5 E, and let 2F (III\ C o ( X )be any euclidean norm. For any strictly positive function E E C o ( X )and any continuous section X 5 E there is then u smooth section X E such that - (ill < E. 5.1 1 Proposition: Let E
>
]IT
PROOF: For any covering of X by open sets U such that each restriction E I U is trivial, Lemma 1.6.1 provides a countable locally finite refinement [ U;,I n E N) by open sets U ; with compact closures 0;, and it follows that there is a smooth local trivialization E I U;,3U ; x R" for each n E N. By Proposition 1.6.10 one can shrink {U:,InE N} to a new open covering (U,ln E N} with compact closures 0,c U ; , and by Corollary 1.6.1 1 there is a smooth partition of unity {h,ln E N; subordinate to {U,ln E N}. For each n E N the local trivialization Y , and the given euclidean norm 11 11 induce on the Co(0,)-module 5 1 0,of cona euclidean norm 9I 0,5 Co(0,) y alO,, tinuous sections 0,, On x R", with IIY, 0 ~,,I[,(x) = 1101 Dnll(x) for each restriction 0, 3E I D,and each x E D,.Since 0,is compact, Lemma 5.10 provides a smooth section 0,2 E I D,, such that - (i I ~T,,ll(x)< E ( x ) , for each n E N and each x E On. The partition of unity {h,ln E N} consists of smooth functions X 3 [0,1] such that h, vanishes outside of U n for each n E N, so that each smooth section z, can be extended to a smooth section X 3E that vanishes outside U , . Since { U,ln E N} is locally finite, the sum h,z, is a well-defined smooth section X 5 E of E X . Finally, for any x E X let N , be the finite subset of those indices n E N with x E U , , ,and observe that since h,(x) = 1, one has
-
GI
\IT,
126
111. Vector Bundles
As in Proposition 4.8, a section X 3 E of a coordinate bundle E X representing a vector bundle over a base space X E is nowhere-vanishing whenever a(x) # 0 E E x for each x E X.
5.12 Proposition: Let E X be any smooth coordinate bundle representing u vector bundle over a smooth manifold X. Then if there is a continuous nowhere-vanishing section X 5 E there is also a smooth nowhere-vunishing section X E . PROOF: Let 9 Co(X)be a euclidean norm as before. Since X E is nowhere-vanishing, one has Ilall(x) > 0 for every x E X, so that there is a strictly positive function E = 411~~11 E Co(X). By Proposition 5.1 1 there is then a smooth section X A E with - 011 < E = fllall, for which the triangle inequality gives llzll 2 llall - IIz - all > tllall > 0, as required.
[IT
6. Vector Fields and Tangent Bundles There is an obvious contravariant functor from the category of smooth manifolds to the category of real-valued function algebras, which we have already used: to any smooth manifold X one assigns the algebra C"(X) of smooth real-valued functions X 4 R, and to any smooth map Y -+ X one assigns the algebra homomorphism Ca(X)-,Cm(Y)carrying X iR into the composition Y -,X R. There is also a contravariant functor that assigns a specific C"(X)-module b(X) to each smooth manifold X, and that assigns a module homomorphism b ( X ) -,b(Y ) over the ring homomorphism Ca(X) -, Cm(Y ) to each smooth map Y -,X. We shall show that if X is m-dimensional, then b ( X ) is a locally free C"(X)-module of rank m, so that Proposition 5.6 provides a canonical smooth m-plane bundle z(X) over X , assigned to X itself. The properties of z(X) characterize much of the differential topology of X itself. The next few paragraphs are devoted to the construction of the dual 8 * ( X ) of &'(X),which will lead to &(X)itself, and to some elementary properties of the resulting bundle z(X) over X. 6.1 Definition: For any smooth manifold X , a smooth vector ,field on X is any real linear map CyJ(X)5 P ( X ) such that L(fg) = ( L f ) g + f ( L g )for any f E C n ( X )and g E Ca(X).
For example, if X is the smooth manifold R", and if xl, . . . , X" E Cm(Rm) are the projections Rm + R that provide the usual coordinate functions on
127
6 . Vector Fields and Tangent Bundles
R", then the partial derivations ?/ax', . . . , d/dx" are vector fields on R". Similarly, if X is the smooth submanifold (Rm)+c R" with nonnegative mth coordinate, then (7/?x', . . . , (7/i)xmare vector fields on (R"')'. If L and M are vector fields on a given smooth manifold X, and if g E Cx(X),then there are vector fields L + M and gL on X that carry any f ' C~x ( X )into ( L f ) ( M f )E C " ( X ) and g ( L f )E C5(X), respectively. It is clear that the set of all vector fields on X forms a C" (X)-module with respect respect to these definitions. For example, if y', . . . , g" E Ca(R"'), then y1 ?/?x' + . . . + y" (7/dx"' is a vector field on R".
+
6.2 Definition: For any smooth manifold X, the C"(X)-module 8*(X) of' smooth ~iectorjields on X consists of smooth vector fields L , with respect to the preceding addition and scalar multiplication. The goal of the next few lemmas is to show that if X is an m-dimensional smooth manifold, then &*(X)is a locally free C"(X)-module of rank m.
6.3 Lemma : Vector jields annihilute constant junctions. PROOF: Let 1 E C'(X) be the constant function on X with value 1 E R. ForanyvectorfieldC'(X)fi C"(X)onXonethenhasL(l)= L(12) = 2L(1); hence L( I ) = 0, so that L(c1) = cL(1) = 0 for any c E R. 6.4 Lemma: For uny smooth munijbld X und uny open subset U c X, let E C" (X) have common restrictions f 1 U = y I U E C" ( U ) . Then jix uny vector ,field C" (X) fi C" (X) on X the images L j E C r ( X ) and ~g E C ' ( X ) huve common restrictions L f I u = Lgl U E C = ( U ) .
,f' E C ' ( X ) und g
PROOF: For any x E U Lemma 1.6.2 provides an h E C x ( X ) that vanishes outside U and satisfies h ( x )= 1. Then (.f - g)h = 0, so that (Lf - Lg)h (1'-g)Lh = 0. Evaluation at x yields ( L f ) ( x ) (Ly)(x)= 0, and since x is an arbitrary point of CJ, one has LfI U = Lgl U .
+
Lemma 6.4 asserts that any vector field L is local in the sense that & f I U depends only on f I U ; we shall rephrase the result with this in mind. If U is any open set of a smooth manifold X, and if I ( U ) c Ca(X) is the ideal of smooth functions X + R that vanish on U , then the inclusion U c X induces a homomorphism CJ'(X) Cz(CJ) whose image is the quotient algebra C'(X)/I(U ) . One defines smooth vector jie1d.s C" (X)/I(U ) 3 Cx(X)/I(U) by the obvious requirement that M ( ( f 1U ) ( g lU ) )= W f I U ) ( g (U ) + ( j ' l U)M(yI U ) for anyf U and 81 U in C"'(XVI(U). .+
6.5 Lemma : Let U br uny open set of (1 smooth munifold X, and let C" (X) C"(X)/I(U) he the restriction mup currying any f EC"(X) into f I U
.+
E
128
Ill. Vector Bundles
C m ( X ) / I ( U ) Then . any vector jield C z ( X )1; C a ( X )on X induces a unique vector jield Ll U such that L
Ca(X)
'CX(X)
commutes.
PROOF: This is just a rephrasing of Lemma 6.4, as promised. 6.6 Lemma: Let xl, . . . , xmE C" ((R"')') he the usual coordinate functions on the submanifold (Rm)+c R" with nonnegative mth coordinate. Then for any vector j e l d C30((Rm)')1; CZ((Rm)+) one has L = (Lx')d/dx' + . . . (Lx")d/dx"for the functions Lx', . . . , LxmE C"((Rm)+).
+
PROOF:The line segment joining any two points xo E (Rm)+and x1 E (Rm)+ lies entirely in (Rm)+,so that for any f E Cm((Rm)')and each index i = 1, . . . , m one can define a function f; E C"((Rm)+ x (Rm)+)by setting
for xo = (x;, . . . , xt ) and x1 = (x!, . . . , xy). For fixed xo E (Rm)+the latter identity becomes I
c (xi m
.f
-
f(x0) =
- x~,f)(xo;)
i= 1
in Ca(Rm)+), to which one applies L and Lemma 6.3 to conclude that m
m
129
6. Vector Fields and Tangent Bundles
for any x 1 E ( R m ) + In . particular, ifx, is any point of(Rm)+and x 1 = x o , then
so that Lf = xy=l(Lx')(dJ/dx')for any f I ( L . ~(J/dx') ') as claimed.
E
C o c ( ( R m ) + that ); is, L =
I f U and U' are open sets of a smooth manifold X such that the closure
0 of U satisfies D c U', then every restriction f I U E C a ( U )of an element
C r ( X ) is trivially the restriction gI U E C a ( U ) of the element g = f I U' E C m( U ' ) . The following lemma provides a useful converse assertion. J
E
6.7 Lemma: Let U and U' be open sets of a smooth manijold X such that D c U ' ; then the inclusion U' t X induces an isomorphism C m ( X ) / l ( U+ ) C' (U')/I(U ) of the restrictions to U of the ulgebras of smooth functions on X and U ' .
PROOF: The induced homomorphism C"(X)/I(U) + C%(U')/I( U ) is trivially injective, and it remains to show that the restriction g I U E Coo( U ) of any y E C" ( U ' )is also the restriction ,fl U E Ca(U ) of some f E Ca(X).Since X is paracompact, by Lemma 1.6.1, it is also normal. (See page 163 of Dugundji [2], for example.) Hence there is an open set U" c X such that 0 c U" and 0"c U'. By Lemma 1.6.2 there is a smooth partition of unity subordinate to the covering of X by the two open sets U" and X \ U ; in particular, there is an h E P ( X ) such that hl U = 1 and h(X\D") = 0. For any g E C r (U ' )the product ( h I U ' ) g E C"(U ' )vanishes on the open set U'\D", and since U'' c U', the function ( hI U')g is the restriction f(U' of a function j" E C" (X) that vanishes on X\D". Since hl U = 1, it follows that U = g1 U as required.
fl
6.8 Proposition: For any smooth m-dimensional manifold X the Cm(X)module 8*(X) of' smooth vector Jields on X is locally ,free of rank m.
PROOF: By Proposition 1.6.10there is a countable open covering { Ubl n E N} of X and an atlas {@,,InE N}, which consists of homeomorphisms Ui, % VL onto open sets @,(UiI)= V:, c ( R m ) + ;the homeomorphisms @, are in fact diffeomorphisms, by definition of the smooth structure of X. By Proposition 1.6.10 one can also shrink [UillnE N} to a new open covering (U,llnE N) with 0,c U ; for each n E N, and the diffeomorphisms 0, provide new open subsets On(U,J= V, c ( R m ) +with c Vk for each n E N. We shall show for each n E N that the C"(X)/I(U,)-module d*(X)/ I(U,)€*(X) is free of rank m. Since C a ( X ) / I ( U , ) + Cw(Ub)/I(U,) is an isomorphism, by Lemma 6.7, and since vector fields are local, by Lemma 6.5,
r,
130
Ill. Vector Bundles
it suffices to show for each n E N that the C7(Ub)/I(U,,)-module (s.*(U',)/ I ( U,)c"*( Uh) is free of rank m. Since the diffeomorphisms U ; 3 V:,c (R"')' restrict to diffeomorphisms U , -% V, c (R")', it therefore suffices to show for any open sets V c ( R m ) + and V ' c ( R m ) + with V c V' that the C x (V ' ) / I (V)-module &*( V')/I(V ) 6 * (V ' ) is free of rank m. Since Cm((Rm)+)/Z(V ) + C"( V ' ) / I (V ) is an isomorphism, by Lemma 6.7, and since vector fields are local, by Lemma 6.5, we are left with the task of showing for any open set V c (Rm)+ that the Cm((R")+)/I( V)-module 6*(( R m ) + ) / IV)&( ( (R")') is free of rank m. However, this is an immediate consequence of Lemma 6.6, which asserts that the Ca((R")+)-module &*((R")+) is free of rank m, one basis being J/ax',. . . , c?/dx". According to Proposition 5.6, if X is a smooth manifold, then any locally free C"(X)-module 9 of rank rn determines a unique real m-plane bundle 5 over X ; specifically, the elements of 9 are the smooth sections of some coordinate bundle E: X representing 5. According to Proposition 6.8, if X is of dimension m,then a particular locally free C"(X)-module 6'*(X) of rank m is available for such an application. For any smooth manifold X the tangent bundle 7(X) is the smooth real vector bundle over X represented by the smooth coordinate bundle whose sections are vector fields L E a*(X).
6.9 Definition:
We shall give a more direct construction of a particular smooth coordinate bundle E X whose sections are vector fields, which will give a better understanding of the tangent bundle 7 ( X )it represents. Let { U i I i E I } be an open covering of the smooth m-dimensional manifold X by coordinate neighborhoods U i , chosen as in Proposition 6.8, and let { Oi I i E I } be a corresponding atlas of diffeomorphisms U i5 ai(Ui),where each Oi(Ui)is open in (Rm)+. For any nonvoid intersection U i n U j the @, @; Ui n Uj) * Oj(U i n U j ) is a diffeomorphism of open composition Oi( sets in (R")' that can be described directly in terms of coordinate functions: for any (x', . . . ,x") E Oi(Ilin U j )one has ,
. . . , x"), . . . , ~ " ( x ' ,. . . , x")) E Oj(ui n U j ) for uniquely defined smooth real-valued functions y', . . . ,7" on Qi( U i n U j )c Oj c O,: '(x',
. . . , x")
= (~'(x',
@. .@:
I
(R"')'. The inverse diffeomorphism Oj(U i n U j ) ' ' + Oi(Uin U j )has a similar description: for any ( y ' , . . . ,y") E O j ( U in U j )one has Oi ') @ :, ' ( y ' , . . . ,y") = ( ~ ' ( y ' ., . . , y"),
. . . , ~ " ( y '.,. . ,y"))E
for uniquely defined smooth real-valued functions X',. ..,Y' onQj(U i n U j ) c (Rm)+.The partial derivatives iIjF/dxP and dXp/dyq provide m x m jacobian
131
6. Vector Fields and Tangent Bundles
matrices
and
of smooth functions on U in U , c X , each being the inverse of the other. In particular, since (c?Tq/dxPr~ Oi) is invertible, it can be regarded as a smooth map U in U j %GL(m, R) carrying any M E U in U j into ((aL4/dxP)(Qi(u)))E GL(rn,R). If U i n U j n U , is nonvoid, the chain rule implies for any u E U i n U i n U , that @ ( u ) . Il/!(u) = t,bf(u) E GL(m,R), so that by Proposition 11.2.5 the smooth maps U in U j -% GL(m,R) are the transition functions of a unique coordinate bundle with structure group GL(m,R) and fiber R". Since the structure group GL(m,R) acts on the left, as always, we regard elements of R" as column vectors. 6.10 Proposition: Let X be a smooth m-dimensional manifold with an open I i E I } . For each nonuoid coordinute covering i U iI i E I ; and utlus {Qi U i n U j c X let ( X I , . . . , x") H (?'(XI, . . . , X"), . . . , ?"(XI, . . . , x"))
be the coordinute description of the dflkomorphism Oi(Uin U , )
a,, a,- '
n Ujh
+ @j(ui
let U in U j -% GL(m,R) be the trunsition junctions given by the jacobian matrices, $!(u) = (((?7Q/dxP)(cDi(u))) for u E U in U , , as above. The smooth srctioiis of' the resulting smooth coordinate bundle E 5 X are the elements of the C'(X)-module B * ( X ) ?/'vector fields on X , us in Dqfinition 6.2.
cind
PROOF: I t remains only to identify the sections X -+ E of the resulting bundle with the vector fields on X . The usual coordinate functions on (Rm)+ provide a distinguished basis of vector fields on (UP)+, the partial derivations with respect to the coordinates, which restrict to corresponding bases
132
111. Vector Bundles
U in U j ) c of vector fields on Qi( (d/axl, . . . , a/axm)and (i7/dy1, . . . , (R"')' and Oj(Ui n U j ) c (R'")', respectively. For any f E C"(Ui n U j )let
and let
so that ( L l , . . . ,L,) and ( M I , . . . ,M,) each form a basis of the vector fields on Ui n U j , which are restrictions to U in U j of corresponding bases of the vector fields on Ui and U j , respectively. The restrictions satisfy relations
for any f E Cm(U in U j ) ;that is,
where the matrix entries merely serve as coefficients for the vector fields L , , . . . , L,. Suppose that a section X + E is given, inducing restrictions U i5 E I Ui and U j 1,E I U j .For fixed local trivializations El Ui -% Ui x R"' and E I U j 2U j x R" each of the restrictions 0 and z can be regarded as a column vector and
('!I)
0,
(1') Srn
of smooth real-valued functions on Ui and U j , which can be identified with
0'L1
+ . . + O m L , = (L1,.. . , L,) *
F) Is"
and with
133
6. Vector Fields and Tangent Bundles
as vector fields on U iand U j , respectively. One must verify that these vector fields have the same restrictions to U in U j . Since
by definition of the transition functions t,b{
= (8Ji"/SxP
Qi), one has
over U , n U , , as desired, the two jacobian matrices being inverses. Thus any section of the coordinate bundle E X constructed from the given transition functions determines a unique vector field on X , as required; the converse assertion is trivial. One can regard Proposition 6.10 as an alternative description of the tangent bundle t(X) of the smooth manifold X . However, both 6.9 and Proposition 6.10 describe a specific coordinate bundle whose isomorphism class is the tangent bundle t(X); one can easily represent z ( X ) by other coordinate bundles. For example, Proposition 5.9 asserts that one can replace the C'(X)-module & * ( X ) of vector fields C m ( X ) : Cm(X) by its dual & * * ( X ) = Hom,,,,,(&*(X), C x ( X ) ) we ; shall do so in the next result. For any module 9 whatsover over a commutative ring, C"(X), for example, there is always a canonical homomorphism from F into the second conjugate .F** = Hom~,,,,(Hom,.,,,(3, C x ( X ) ) Cw(X)), , carrying any 0 E J into that element 6** E F** that maps any L E &* into the value of L on H. If B is a free C"(X)-module of finite rank, then the canonical homomorphism 9+ F**is trivially an isomorphism; similarly, if F is merely locally free of finite rank, then one can use smooth partitions of unity in an obvious way to conclude that F F**is an isomorphism. For this reason we shall use the notation € ( X ) rather than € * * ( X ) to describe the dual of & * ( X ) ; in Remark 13.20 and Exercise 13.21 the notation € ( X ) is justified by a direct construction. --$
134
111. Vector Bundles
6.11 Definition: For any smooth manifold X the C"(X)-module &(X) of dlfferentials on X is the dual Hom,,,,,(b*(X),C"(X)) of the C"(X)-module &*(X)of smooth vector fields on X. There are some classical differentials. For any g E Ca(X)there is a C"(X)-linear map I*(X) 2 Cm(X)carrying any vector field L E &*(X) into the function Lg E Cm(X);consequently dg E b(X). Similarly, for any natural number p 2 0 and any 2p elements f l , . . . ,f p , gl, . . . ,g p E P ( X ) there is a differential f l d g l +...+fp dg,~B(X)carryinganyL E b*(X)intofl(Lgl) . . . fp(Lgp)E C"(X). In case X = R" the m coordinate functions x', . . . , X" lead to m differentials dx', . . . , dx". Since axi/axj = 1 or axi/dxj = 0 according as i = j or i # j , it follows that (dx', . . . , dx") is the basis of I ( R m )dual to the basis (a/ax', . . . , d/axm)of S*(rWm).A similar remark applies to (R")' and to any open subset of (R")'.
+
+
6.12 Proposition: For any smooth manifold X the tangent bundle s(X) is the smooth real vector bundle over X represented by the smooth coordinate bundle whose sections form the C"(X)-module &(X)of diferentials on X.
PROOF: Sinceb(X)is defined as the dual I**(X) = Hom,,,,,(&*(X), C x ( X ) ) of the C"(X)-module b*(X) of vector fields on X, this is an immediate consequence of Definition 6.9 and Proposition 5.9. In view of Proposition 6.12 it is of interest to have an analog of Proposition 6.10, using &(X)in place of b*(X). We use the notation of the discussion preceding Proposition 6.10. In place of the jacobian matrices (ajjq/axP0 mi) and ( Z p / d y 4 @,) displayed earlier, we shall use the transposed jacobian matrices
and
135
6. Vector Fields and Tangent Bundles
each of which is the inverse of the other. In particular, since '(i?XP/2vq O j ) is invertible, it can be regarded as a smooth map Ui n U j 3GL(m,R) carrying any u E Ui n U , into '( (ZIXP/2yq)(Oj(u))) E GL(m,R). If the intersection Ui n U j n U , is nonvoid, the chain rule implies for any u E Ui n U j n U , that t,bi(u).i,h;(u)= $:(u) E GL(m,R), so that by Proposition 11.2.5 the smooth maps Ui n U j -% GL(m,R) are the transition functions of a unique coordinate bundle with structure group GL(m,R) and fiber R". As before, since the structure group GL(m,R) acts on the left, the elements of R" will be regarded as column vectors. c'
6.13Proposition: Let X be a smooth manijold with an open coordinate covering { Uil i E I ) and atlas (mili E 1 ) . For each nonvoid Ui n U j let ( y ' , . . . ,y") H (XI(y ' , . . . , y"), . . . , X"( y ' , . . . , y")) be the coordinate de0 0-1 scription of' the difleomorphism Oj(Ui n U j )a O i ( U in U j ) , and let U in U j % GL(m,R ) be the transition functions given by the transposed ,jacobiait mutrices, t,hj(u)= '((dXP/dyq)(Oj(u)))jiw u E U i n U , , as uboue. The smooth sections qf the resulting smooth coordinate bundle E 3 X are the elements of'the C"(X)-moduleb ( X ) of' differenrials on X , as in Dejnition 6.1 1.
PROOF: The coordinate functions on (R")' restrict to bases ( d x ' , . . . , dx") and ( d y ' , . . . , djl") of the differentials on Oi(Ui)c (R")' and Oj( V,) c (R")', and we use their further restrictions to Oi(Ui n U,) and Oj(Ui n U j ) . In particular, ( d ( x ' Oi),. . . , d(x" OJ) and ( d ( y ' O,),. . . ,d( y" 0,)) are bases of the differentials on Ui n U,, and by the chain rule they satisfy the relations L
3
(J
that is,
Suppose that a section X + E is given, inducing restrictions Ui 5 El U i and U , 5 El U j . For the fixed local trivializations El Ui 5 U i x R" and El U j 3 U j x R" each of the restrictions (T and 7 can be regarded as a
136
111. Vector Bundles
column vector,
f)
and
('!I),
om
T"
of smooth real-valued functions on Ui and U j , which can be identified with
C'
d(x'
0
ai)+ . . . + o " ' ~ ( x * Oi) = (d(x' 0
0
Qi), . . . ,d(xm
0
i"' j
ai))
om
and
as differentials on Ui and U j , respectively. One must verify that these differentials have the same restrictions to U in U j . Since
by definition of the transition functions t,bi = '(dXP/dyq
0
(d($
1
Q j ) ,.
= (d(.x'
.
*
, d(4'"
i
Q j ) ,one
has
Oj))
a+),. . . , d ( x m a+)) J
(q om
over Ui n U j , as desired, the two transposed jacobian matrices being inverses. Thus any section of the coordinate bundle E 5 X constructed from the given transition functions determines a unique differential on X , as required; the converse assertion is trivial.
137
6 . Vector Fields and Tangent Bundles
YC'
I t is worth noting that the differing transition functions U in U j --+ GL(m,08) appearing in Propositions 6.10 and 6.13 are transposed inverses of each other, where the resulting smooth coordinate bundles both represent the same tangent bundle T(X):in Proposition 6.10 one has t+b{ = (dL4/ilxP Qi) and in Proposition 6.13 one has ${ = '(M'/8P Q j ) . Although the map GL(m,R)5 GL(m,R) carrying each invertible m x m matrix into its transposed inverse is indeed a group automorphism, r is not part of a morphism (r,0)of transformation groups to which one can apply Proposition 11.2.9to conclude directly that the resulting coordinate bundles represent the same vector bundle. On the other hand, the preceding situation is part of a more general phenomenon : if coordinate bundles E X and E' 5 X are described $I' by transition functions U in U j A GL(rn,R) and U in U j GL(rn,R ) that are transposed inverses of each other, then E 2 X and E' 5 X represent the same real m-plane bundle over X, for any X E .%?.(See Exercise 13.24.) Throughout this section we have considered the C"(X)-modules &*( X) and€(X) associated with smooth representations ofthe tangent bundle z ( X )of a smooth manifold X. One can replaceb*(X)and €(X) by C o ( X )@cm(x)b*(X) and Co(X)@c..cx, &'(X), however, which represent the same tangent bundle T ( X ) ;'that is, one can replace smooth sections of coordinate bundles representing t ( X ) by continuous sections of the same coordinate bundles. Thanks to Propositions 5.11 and 5.12, the use of continuous sections presents no serious handicap, however, as the following result suggests. X is the smooth coordinate bundle whose sections Suppose that E X 1,E are vector fields Ca'(X) 3 Cm(X)on a given smooth manifold X. The sections X 1,E are smooth maps, by construction; however, one can also consider sections X 5 E that are merely continuous, as just noted. Accordingly, we distinguish between smooth vector fields and continuous vector fields on the same smooth manifold X. 0
-
6.14 Proposition: Suppose that X is u smooth manifold that admits a nowhere-
vunishing continuous vector j e l d L ; then X also admits a nowhere-vanishing smooth vector j e l d M .
PROOF: Continuous and smooth vector fields are continuous and smooth E and X A E, respectively, of that smooth coordinate bundle sections X E X whose smooth sections are smooth vector fields, elements of b*(X). Hence the result is a special case of Proposition 5.12. In the opening paragraph of this section we indicated that there is a contravariant functor on the category of smooth manifolds that assigns to each smooth manifold X and to each smooth map Y -,X a C"(X)-module &(X)
138
111. Vector Bundles
and a module homomorphism &'(X)--f &( Y ) over the algebra homomorphism C" ( X ) + C x ' Y), ( respectively. We have constructed the C"(X)-module t ' ( X ) , somewhat indirectly; however, we have not yet constructed the homomorphism & ( X ) + &( Y). Since both constructions will be carried out more directly in Remark 13.20 and Exercises 13.21 and 13.22, we only sketch the latter homomorphism here; it will not be required in the sequel. Suppose that Y 2 X is a smooth map of smooth manifolds. The induced algebra homomorphism C" ( X ) 3 C'&(Y ) carries any smooth function x R into the composition Y R; that is, @*f = ,f , 0.Following Definition 6.11 it was noted that for any natural number p 2 0 and any 2p elements , f l , . . . ,f,, gl, . . . , g p E C",(X)there is a differential f l dy ... @g, E €(X)carrying any L E & * ( X ) into fl(Lgl) . . . fp(Lg,) E C x , ( X ) . There is a corresponding differential ( f ,"0) d(g @) . . (jb .@) d(g, 0) E €(Y), and one easily verifies via local coordinate systems and smooth partitions of unity that there is then a unique module homomorphism & ( X )2a(Y ) over C a ( X ) CaJ(Y ) such that @*(fldgl . . . f,dg,,) = ( f l ' @ ) 4 g y @ ) + . . . + ( f p @)d(g,,,@) for any j ; , . . . , f p , g l, . . . , g , ~ & ( Y ) will appear in C' (XI.A more satisfying construction of & ( X I Remark 13.20 and Exercises 13.21 and 13.22, as promised earlier.
a
,+ +
+ + +. +
+ +
We now compute a concrete example of a tangent bundle. Let R* c R denote the nonzero real numbers and let [W("+l)* denote the nonzero vectors in R"", in the relative topologies. The real projective space RP" is the quotient [W("+l)*/- of R("+')* by the equivalence relation with x' x whenever x' = a x E R("+l)* for some u E R*, as in 51.5: alternatively, RP" is the Grassmann manifold GI(R""), which is smooth by Proposition 1.7.3. Let E c R("+' ) * x R " + l consist of those pairs (x, y) E R("+' ) * x R n + I such that y is orthogonal to x in the usual inner product of R"". There is then a quotient E / z of E by the equivalence relation z with (x', y') z ( x , y) whenever x' = ux E R("+l)* and y' = uy E R"" for the same u E R*. If ( . ~ y' ' ) = (x,y), then x' x , so that the first projection R("' I ) * x R"' I -% R("+l)* induces a projection E / z RP" of quotient spaces, which is continuous in the quotient topologies.
-
-
-
6.15 Proposition: The preceding projection E / z 5 RP" is a smooth coordinute bundle thut represents the tungent bundle r(RP") of the real projective spuce RP".
FROOF:According to Definition 6.9, r ( R P " ) is represented by the smooth coordinate bundle whose sections are vector fields C r ( R P " )1; C ' ( R P " ) ;we shall identify El= 1RP" with that coordinate bundle. First observe that
139
6. Vector Fields and Tangent Bundles
C'(RP")can be regardedas the subalgebra ofthose functionsf'EC?(R("+')*) that are homogeneous of degree 0. Thus if R'"+')* %R("+')* is multiplication by u E R*, one has f E C 7 ( R P " )(cCr(R("+')*))if and only if,f m, = f for each II E R*; equivalently ,f E C y ( R P " ) ( cCx(R("+')*)) if and only i f f is constant on each line through 0 E R"". For any 0
one then has ( I ( y o
+ . . . + yR?/?.x")j
M, = (yo ?/?YO = (y"?/ix"
+ . . . + y" ?/?x")(f
ni,J
+ . . . + 4'"(7/&")/
for any f E C ' ( R P " ) (c C : * ( R ( " + ~ ) * ) )by , the chain rule. Hence, if x ' = ox E R'""'*, one obtains a vector field Q:' ( R P " ) C " ( R P " ) only by imposing the second condition y' = U ) ? E Rf'+' in the equivalence relation (s',1,') 2 ( Y , described earlier, for the same u E R*. Since any f E C"(RP") is constant along each line through 0 E R"+ I , it follows that L,j(.x) = 0 for each Y E R("+l)*,so that L, vanishes at the equivalence class in RP" of Y E R("+l'*. However, the values of the vectors L, for any y E R"+' orthogonal to x span an n-dimensional vector space at the preceding point of RP", and since RP" is an n-dimensional smooth manifold this completes the proof: if E c R("+')* x R " + I consists of those pairs (x, y) E R("+')* x R"" such that y is orthogonal to x, then E / % RP" represents the tangent bundle T ( RP"),as claimed. \I)
There is a useful corollary of Proposition 6.15. This time we replace E c @"+'I*x Rf'+' by the enrire space R("+')* x R"" and let E" be the quotient R('l+l'* x R"' '/= by the equivalence relation 4 with (x, J,') t (x, y) whenever .Y' = U.Y E R("+I)* and J,' = E R"+ I for the same a E R*, as before. The first projection R("+' ) * x R"+ 5R'"' ' ) * still induces a projection E" 5 RP" of quotient spaces, which is continuous in the quotient topologies. UJI
'
6.16 Corollary: The preceding projection E" 5 RP" is u smooth coordinute bundle thut represents the Whitney sum t ( R P " )@ c' of the tangent bundle
s(RP")and the tritliul line bundle c1 over the real projective space RP". PROOF: The product representation RP" x R ' " I ,RP"ofthe trivial bundle can be regarded as a projection of quotients in an obvious fashion: there is
140
111. Vector Bundles
an equivalence relation z in R('l+l)* x R' with (x',.~')z (.Y,s) whenever x' = ux E R("+ ')*, for some u E R*, and s' = s, independently of u E R*, and the first projection R("+' ) * x R ' -% R("+' ) * induces RP" x R' a RP". However, in place of R(''+l)* x R ' one can substitute the space E' c @,I+ 1 I* [w" t 1 of points of the form (x, ): for any .Y E R("+l)* and any scalar multiple z = sx of x. The corresponding equivalence relation x in E' then sets (x',z') x (x, z) whenever x' = ax and z' = az for the same a E R*. The first projection R("+')* x IT"' 5 R("+')* induces a projection 5 RP" that can be identified with RP" x R' RP" by observing that each (x, 2) E E' is of the form (x, sx) for a unique s E R. The coordinate bundle E / z 5 RP" representing t ( R P " ) in Proposition 6.15 and the coorn' -, RP" just constructed to represent c 1 are of the same dinate bundle nature; the only difference is that E c R("+l)* x R"+'consists of those pairs (s,y) such that y is orthogonal to .Y, while E' c R("+ ' ) * x R"+ I consists of those pairs (x,:) such that z is a scalar multiple of x. The equivalence relation = is defined in the same way in each case, and since any element of R"+ is uniquely of the form y z for y orthogonal to x E R("+')* and z a scalar multiple of the same x E R("+"*, it follows that the Whitney sum n" . r ( R P " ) @ c l is represented by the quotient map [ W ( " + l ) * x R"+'1% R'"t 1 ) * as R(,I+ l ) * / - induced by the first projection R("+' ) * x R"+ claimed.
*
El/=
El/=
+
-.
7. Canonical Vector Bundles For any natural numbers rn > 0 and n > 0 the real Grassmann manifold G"'(Rm+")is a smooth closed rnn-dimensional manifold, as in Proposition 1.7.3;in particular, the real projective space RP" = G'(R"+') is a smooth
closed n-dimensional manifold. In this section we construct a smooth coordinate bundle representing a particularly useful nontrivial rn-plane bundle 7: over each Gm(R"+"); applications will occur later. We also show that (11 + 1 ) ~ : = T ( R P " ) @ E 'where , ( n + 1 ) ~ :is the Whitney sum of 11 1 copies of the canonical line bundle y: over RP", where r(RP")is the tangent bundle of RP", and where e l is the trivial line bundle over RP".
+
We briefly recall the construction of the Grassmann manifolds Gm(Rm+"), which first appear (for V = Rm+")in Definition 1.7.1. Let (Rm+")m*denote the set of ordered rn-tuples ( x l ,. . . , x), of linearly independent vectors Sl,...,, [W"l + , in the relative topology ofthe m-fold product R"+" x . . . x R"+", and let be the equivalence relation in (Rm+n)m* with ( X I , . . . , .Y,) ( y l , . . . , y,) whenever (xl, . . . , x), and ( y l , . . . , y,) span the same rn-plane
-
11
-
141
7. Canonical Vector Bundles
in R"'+f t .Then G"'(R'"+'') is the quotient ( R m + f t ) m * /in + the quotient topology; that is, Gm(Rmt")is the set of m-planes in R"'" in an appropriate topology. If ( . Y ] , . . . , xnt)and ( y l , . . . , y,) span the same ni-plane in [w"'", then there is an m x m invertible matrix (a;) E GL(m,R) such that x p = a;y, for each p = 1, . . . m ;that is,
. . , x,)
(XI,.
= (y1..
. . , y),
("i
4
cy,
... .*.
4
where each of x l , . . . , x,, . . . , y m is a column vector with m + n entries. We shall write this relation in either of the equivalent forms (xl,. . . , x,) = ( y l , . . . , yf,,)(u;)or . . . , y,) = ( x l ,. . . , x,)(a;)-' for (a;) E GL(rn,R). Elements of the product space ( R m + n ) m x* [w" are of the form j l l ,
(
~
9
'
.
for
and we write
whenever both ( y l , . . . , y),
= (s,, . . . , . ~ J ( U ;and )-~
for the same ((14,) E G L ( m ,R). Trivially 2 is an equivalence relation in ( R t n + f ' ) mx* R", and we temporarily let E denote the quotient space (1Wnl+"),* x R m / = ,in the quotient topology. Clearly if
*
in (R"l+")nl* x R", then ( s l , . . . , x m ) ( y l , . . . , y,) in (Rmtn)"*,so that the first projection (R"tt")m*x R" (Rm'n)m* induces a projection E 1
I42
111. Vector Bundles
Gm(Rm '") of quotient spaces. Furthermore, if
then for some (cry,) E GL(m, R) one has
so that the equivalence class e E E of
defines a specific point slxI+ . . . + sn'.q,, in the m-plane n(e)c R"+" spanned by ( x I ,. . . , x,,,)E (Rm+'l)m*. Thus E consists of pairs (n(e),e) in which n(e)E Gm(Rm+") is an ni-plane in Rmt"and e is a point in the m-plane n(e). 7.1 Lemma: The projection E 5 Gm(Rnl+")is u sniooth coordinutr bundle with structure group GL(m,R) undjber R", reprrsenting u smooth real m-plrtne bundle over the Grussmunn munifold Gm(Rrn+").
PROOF: Recall from Proposition 1.7.3 that the smooth structure of Gm(R m + " ) was obtained via the canonical embedding Gm(Rm+') 5 G I ( P Rrn+")and a fixed basis {xl,. . . ,x,+,,J of R"+". Each open set U i I , , ,i,,, c Im F consisted of projective equivalence classes of points ,
C j , < . . . < j , , , u ( j l , .. .,
,
~ , ) X ~ , A ~ ~ ~ A X ~ , , E / \ ~ [ W ~ + ~ ~
such that u ( i l , . . . , i,) # 0, and the Plucker relations implied that all ratios u ( j , , . . . ,jm)/u(il,. . . , i,) are polynomial functions of the mn ratios
143
7 . Canonical Vector Bundles
-
.?
u(il, . . . , i,,
. . . , i,, j)/u(il,. . . , i,). Here is another description of the same set . .. in terms of the inner product R"+" x R"'" < . ) R with (.xi..x,i) = (sij: U i l , i,,, consists of the projective equivalence classes of all points of the form (.xi, + . Y ; , ) A . . . A (.xi,,, + J'~,,,), with automatic Plucker relations, where the nonzero elements among y i l , . . . , yin,are orthogonal to the subspace spanned by .xi,, . . . , -xi,,,. In this description one has a ( i , , . . . , i,) = 1, each u ( j l , . . . ,j,) is the coefficient of xj, A . . . A xj, in the expansion of ( x i , y i l )A . . . A (xi,,, yimJand each u(j , , . . . ,J,) is a . . , ,i j ) . polynomial function of the mn coordinates u(il, . . . , As above, each point of El U i l , , . i,,, is a pair (n(e),e) in which n(e)is the m-plane spanned by xil yil, . . . , .xi,,3 + J ' ~ and ~ , ~ e is a point uij..
in,,
,
+
+
rk,.
, ,
+
there is a local triviahition El U l l , , , ;,,, (n(e),e) into , ,
Y,
Uil,,
, ,
, ;,,
x R" carrying
If U j l . , , jv,2 c Im F is any other set of the preceding form, then U i l , , . n U j l . , . is automatically nonvoid, and one must show that the transition '.;I. . . . . 3 GL(rn,R) induced by Y j Y; is function Ui,, . . . . i,,, smooth. Suppose that the intersection ( i l , . . . , i,) n [ j l , . . . ,j,} contains m - p . . . , i, = j,, for indices, which we temporarily assume to be i p + = j,, notational convenience; the adjustment required for the general case will be indicated later. In the notation relative to U i , , . . ;,,, one has , ,
,
,
,,
j,,l
Jnt
:I
,
(.xi,
+
.Yil)A.
. ' ~ ( s i ,+, ,J'i,,,) E
,
U i l . .. . , j,,, n u,jl... . , j,,,
if and only if the coefficient a(.jl,. . . ,j,) of s j , A . . . A xj, in the expansion of ( x i ]+ yjI) A . . . A (.xi,,, + yi8,J is nonzero. This happens if and only if there is a nonsingular matrix
B=
144
111. Vector Bundles
+
+
for the ( m - p) x ( m- p) identity matrix I , such that ( x i , y i l ,. . . ,xi,, yi,,,)= (xjl,. . . , .u,,,)Bmodulo the subspace R"-, c R"'" orthogonal to the subspace spanned by x i , , . . . ,xi",, xj,, . . . , xj,. Up to a sign each bi is the coefficient u(i,, . . . , i a , . . . , i,, j ) of xi, A . . . A ?,A * * . A x i , A xj in the expansion of ( x i , yil) A . . . A (xint yinJ Since a ( j , , . . . ,j,) is a homogeneous polynomial of degree p in the inn coordinates u ( i , , . . . , i k , . . . , i,,j), it follows that
+
+
a ( j , , . . . ,j,)B-' has homogeneous rational entries of degree zero in the same coordinates. Furthermore, if A = ( u ( j , ,. . . ,J,)B-')-', and if ( x i , y i , , . . . , xi,, yiJ E (Rm+"Y*is a Ui,, , ,_-description representinga point in the intersection U i l . , ,i,,, n Ujl.. , ,,,, then(xi,+yi,, . . . , X ~ , ~ ~ + J ~ ~ ~ , ) A is the corresponding Ujl,. , jn;description representing the same point. Now recall that the total space E of 7:: is a quotient (Rmf")'"* x R m / z , where
+
+
, , ,
,,
,,
,,
whenever ( u , , . . . , u,) = ( u , , . . . , u,)A-
I
and
-
for the same nonsingular matrix A . Consequently the transition function U i l , , . ,i,,,n U j , , . j,,, *; GL(m,R) carries any point with U i l , , , i,a-coordinates u ( i , , . . . , i,, . . . , i,, j ) into the preceding nonsingular matrix A with homogeneous rational entries of degree zero in the same coordinates; in particular ${ is smooth. To complete the proof of Lemma 7.1 it remains only to discard the specialized hypothesis i,+ = j,+ . . . , i, = j,. Whenever ti,, . . . , i,} and f . , I , , . . . ,,j,) have m - p elements in common, one can reorder the entire set (1, . . . , m + n } to produce the specialized hypothesis, and the only effect of undoing the reordering is a permutation of the rows of A and a possibly distinct permutation of the columns of A. This does not affect the assertion of the lemma. ,
, ~,
,
,,
,,
7.2 Definition: The smooth coordinate bundle E Gm(Rm+") of Lemma 7.1 represents the canonical real m-plane bundle 7:: over the Grassmann manifold Gm(Rmt").In the case m = 1, for which G'(R"+') = RP", the smooth coordinate bundle E 5 G'(R"+') of Lemma 7.1 represents the canonical r e d lint.
bundle
i)!
over the real projective space RP".
145
7. Canonical Vector Bundles
We next establish the identity (11 + I)?: = t ( R P " )@ e l mentioned at the beginning of the section. The method will be to find a coordinate bundle E' 2; RP" representing ( n + l)y: that can be compared to the coordinate RP" of Corollary 6.16, representing z(RP")0 E ' . Although bundle E" the construction of E' 5 RP" will appear to differ from that of E" 2RP", we shall show that they are nevertheless identical coordinate bundles. Let R("+I)* be the space of nonzero vectors in R"+l, in the relative topology, let R* be the nonzero real numbers, let % ' be the equivalence relation in R("+')*x R"+' with(x',y')z'(x,y) whenever bothx'=axE R("+l)* and y' = u - ' y E R n + l for the same u E R*, and let E' be the quotient space R("+l)* x R " + ' / z ' , in the quotient topology. If (x', y') z ' (x, y), then x' and x satisfy the equivalence relation x' x used to define the real projective space Rn+1 R(n+l)* R("+l)*/- = RP", so that the first projection R'''+l)* induces a projection E' 5 RP".
-
7.3 Lemma: The preceding projection E' bundle thut represents the Whitney sum (11 cuiioiiicul lirie bundle A!; over RP".
a
RP" is a smooth coordinute of' n I copies of the
+ 1);i;
+
PROOF: We briefly recall the general construction of the total space of the One introduces an equivcoordinate bundle E G"( R""") representing alence relation z in R(m+'l)m*x R" with .:I;
whenever both ( yl, . . . ym) = ( x I, . . . , .u,,)(uY,)
~
and
for the same (cry,) E GL(m,R), and one sets E = R("'+")"* x Rm/%. In case 1 one can regard multiplication by u - E R* as the action of (a;) E GL(m,R), so that the total space of the coordinate bundle E 1 :RP" representing 7: can be described as follows: one introduces an equivalence relation t in R("+I ) * x R 1 with (.Y, s) z (y, I ) whenever (JJ,t ) = (ux, a - 's) for some LI E R*. and one sets E = R("+I ) * x R 1 / z . A point of E is then a pair (n(e), e), consisting of a 1-dimensional subspace n(e) c R " + l spanned by some .y RIP1 + 1 ) * and a point s x = e E n(e);if (y,t ) = (ux,u-'.s), then one has the same subspace n(r)E R " + l and the same point t j = ( u - ' s ) ( u x ) = sx = e E n(e).In order to conform to the notation of Corollary 6.16 we write x' in in =
146
111. Vector Bundles
place of y. so that yf has a total space E = R("+ ')* x R'/= with (x', 1) % (x, s) whenever (x', t ) = ( a x , a ~ ' sfor ) some a E R*. The corresponding total space E' of the Whitney sum (n + l)yL of n + 1 copies of 7; is then the quotient of R('I+I ) * x R"+ by the equivalence relation = ' with
whenever both .x'
= ux E
R("+')* and
for the same u E R*. The quotient E' = R("+')* x R"+'/E' and the obvious projection E' 5 RP" are precisely the definitions used in the statement of the lemma, in the notation
7.4 Proposition: Let t(RP")0E' be the Whitney sum of' the tangent bundle
T ( R P " )and the trivial line bundle E' over the real projective space RP", and let (n 1)y: be the Whitney sum of n + 1 copies of' the canonical line bundle over RP"; then r(RP")@ E' = (n + l)y!.
+
PROOF: According to Corollary 6.16 r(RP")0E' is represented by a coordinate bundle E" % RP" with E" = [w("+')* x [w'"' / E , where (x', '2 (x, !!) E RW+1 I* x R"+ 1 whenever (x', y') = ( u x , cry) for some u E R*. According to Lemma 7.3, ( n + l)y! is similarly represented by a coordinate bundle RP" with E' = R("+')* x R " + ' / z ' , where (.y',y') z ' ( x , y ) E R("+')* x R " + l whenever (x', y') = (us, a- 'y) for some u E R*. Each of the projections E" 5RP" and E' 5 RP" is induced by the first projection R("+')* x R"' 5R("+')*; however, one must reconcile the evident difference in the equivalence relations 2 and E ' . Observe that if S"c R("+')* is the usual unit sphere, then one can alternatively describe RP" as the quotient Srz/-, where .Y' x E S" whenever x' = ux for some u E R* as before, and since .Y and x' are both unit vectors, one either has u = 1 or u = - 1 ; that is, in this description of RP" one can replace the group GL(1, R)= R* by the subgroup ~ 3 ' )
E's
-
147
8. The Homotopy Classification Theorem
O ( l ) = [ + l . - l ] c R*. Thus E " = S " x R " " / a , where ( x ' , y ' ) z ( x , y ) ~ S" x R"' I whenever (x', y')= (ux,u y ) for some u E O( I), and similarly E' = S" x R"+ I / z '. where (x', J,') z ' (s, J,) E S" x R"+ I whenever (.Y', = (us, ( I - 'ji) for some u E O(1), the projections E" RP" and E' 5 RP" both being induced by the first projection S" x R"+ -% S". Since u - l = u for both elements LI E O(l), it follows that E" 5 RP" and E' 5 RP" are the same coordinate bundle, hence that T ( R P " )0 I:' = ( n l);)!, as claimed. ~
3
'
)
+
8. The Homotopy Classification Theorem One can replace the canonical real rn-plane bundle y; over the Grassmann manifold G"(Rmf") E 9 by a corresponding real rn-plane bundle y" over the Grassmann manifold G"(R') E 39.We shall show that if 5 is any real m-plane bundle whatsoever over an arbitrary base space X E B, then 5 is the pullback f!?" of 7" along a map X G"(R') that is uniquely determined up to homotopy. For any real rn-plane bundle 5 over a compact space X E d (or over a finite-dimensional simplicia1 space X = IK( E J , or over a smooth manifold X E , d c .9),one can substitute an appropriate finite-dimensional Grassmann manifold G'"(R"'+'') for Gm(R"); the homotopy uniqueness of X 5 G"'(R') is then replaced by an "ersatz homotopy uniqueness" of the corresponding map X + G"'(R"''). In the latter cases it follows for some I I > 0 that there is an rz-plane bundle q over X for which 5 0 q = E""'. The real linear space R" was defined in $1.5 as the direct limit limn+ R", in the weak topology; elements of R' are sequences of real numbers that contain only finitely many nonzero entries. It will be useful, although not essential, to regard elements x E R' as infinite column vectors
for evident typographical reasons we shall occasionally abandon this convention. The Grassmann manifolds G"( R"'"') and G"( R a ) were both defined in Definition 1.7.1; by Proposition 1.7.5 they both belong to the category B of base spaces. Briefly, if (R" )"'* is the set of ordered rn-tuples ( x l ,. . . , x,) of linearly independent vectors xl, . . . , x, in R", then there is an equivalence ( y l , . . . , y,) whenever ( x l , . . . , x,) relation in (Rx)"'* with ( X I , . . . , x,) and ( y 1 , .. . , y,) span the same rn-plane in R', and one sets Gm(R") = (Rr)m*/- in the quotient topology. As in the previous section, the relation
-
-
148
111. Vector Bundles
-
. . , x), ( y , , . . . ,y,) is the same as requiring that (xl,. . . , x), = ( y l , . , . , y,)(a4,) for a unique invertible rn x rn matrix (a;) E GL(rn,R); equivalently, (xl, . . . ,x), (yl, . . . ,)y, E (Rm)m*whenever ( y , , . . . ,),11 = (xl, . . . , x,)(aY,)for a unique (a;) E GL(rn,R). We mimic the construction of the canonical rn-plane bundle y: over G'"(R'''+"), which will lead to the corresponding m-plane bundle y" over Cm(Rm).Elements of the product space (Rm)"'* x R" are of the form (xl,.
for (xl,. . . , x),
-
E
(Rm)"'* and
i';) E
R".
Srn
We write
whenever both ( y , , . . . , y,)
= (x,,
. . . , x,)(u4,)-
and
for the same (a;) E GL(rn,R), and we let E" denote the quotient space (Ra)"'* x Rm/= in the quotient topology. As in the finite-dimensional case, if
-
in (R")"'* x R", then (x,, . . . , x,) ( y l , . . . , y,) in (RQ)"'*, so that the first projection x R" 5(R5)"'* induces a projection E" 5G"(R"); as before, the preceding hypotheses imply flyl + . * + Py, = SIX, +* ..+ smx, E R", so that elements of E" may be regarded as pairs (7rS(e),e) in which e is a point SIX, + . . . + xmx, in the rn-plane n*(e) c R"' spanned by (XI,. . . * Xm)*
8. The Homotopy Classification Theorem
149
nL
8.1 Lemma: The projection E" G"(R") is a coordinate bundle with structure group GL(m,R) and jiber R", representing a real m-plane bundle over the Grassmann manifold G"(R').
Y
PROOF: One constructs local trivializations El U , , , U , , . . ., x R" with resulting transition functions U , , ,", n U,, Jm % GL(m, R) that consist of invertible m x m matrices exactly as in the proof of Lemma 7.1 ; although smoothness of the latter entries is not defined, one easily verifies that they are continuous, as required. 8.2 Definition: The coordinate bundle E" 3 Gm(Ra) of Lemma 8.1 represents the unioersal real m-plane bundle y" over the Grassmann manifold Cm(R").In the case m = 1, for which G'(R') = RP", the coordinate bundle E" 5G1(Rx) of Lemma 8.1 represents the universal real line bundle over the real projective space RP". We now embark upon a sequence of lemmas that will lead to the main theorem of the section. They will be formulated in the following terminology. 8.3 Definition: Let E -5X be a coordinate bundle with structure group GL(m,R) and fiber R" over a base space X E a,representing a real m-plane bundle over X . A Gauss map is any map E 3 R" that restricts to a linear monomorphism E x --%R" for the fiber E x c E over each x E X .
Morphisms of arbitrary families of fibers were characterized in Definition 11.1.1, and coordinate bundles are merely locally trivial families of fibers. In particular, for the structure group GL(m,R) and fiber R", a morphism from
the coordinate bundle E 5 X of Definition 8.3 to the coordinate bundle E" 5 G"(R') of Definition 8.2 is a commutative diagram
E
X
f
'
+
E"
G"(R")
such that for each x E X the restriction E x of real n-dimensional vector spaces.
E&, is a linear isomorphism
8.4 Lemma: Given u coordinute bundle E 5 X as in Dejnition 8.3, there is a one-to-one correspondence between Gau.s.s maps E 3 R" and morphisms ,from E x to E" 3 G"(R").
150
1.11. Vector Bundles
PROOF: Let E 5 R" be a Gauss map for the given coordinate bundle. Then for any e E E the fiber En(e)over n(e)E X is mapped isomorphically onto an m-dimensional subspace g(En(J c Ra', and e E E is itself mapped into a point g ( e ) E g(En(p)).Since elements of Gm(Rm)are m-planes in R", one can define X Gm(Rm)by settingf(x) = g(E,) for each x E X , and since elements of the fibers E,-(,, over f ( x ) E Gm(R") are points of the m-plane g(E,) c R", one can define E 5 E" by setting f(e) = ( g ( E n ( J , g ( e ) ) for each e E E. The resulting commutative diagram E
X
f
'
* G"(RX)
is the corresponding morphism of coordinate bundles with structure group GL(m,R) and fiber R". Conversely, suppose that one is given such a morphism of coordinate bundles. Then the composition E 5 X 3 Gm(Ra)carries any e E E into an m-dimensional subspace f ( n ( e ) )c R", and by commutativity of the diagram defining the morphism one has f(e) E f ( n ( e ) )the ; composition E A f f ( n ( e )c) R" is itself the corresponding Gauss map E 3 R". In the next two lemmas we shall show for any Gauss maps E 5 R" and E --%R" that there is a Gauss map E x [0,1] 4 R" for the coordinate nxid bundle E x [0,1] X x [0,1], such that g l E x {0}= g o and such that g I E x { 1) = g , . We temporarily write elements of R" as row vectors rather than column vectors, for typographical reasons.
-
8.5 Lemma: Let E 5 X represent u real m-plane bundle over a base space X E 28, let E 5 R' be a Gauss map carrying any e E E into (yo(e),g'(e), g2(e),. . .) E R", and let E --%R" be the Gauss map carrying any e E E into (0,yo(e),0,y'(e),0,g2(e),. . .); then there is a Gauss mup E x [0, I] 5 [w' for n x id the coordinate bundle E x [0,1] X x [0,1], carrying any point (e,t ) E E x [0,1] into (1 - t ) go(e) + t g , ( e ) E R".
-
PROOF: The map g is trivially linear on each fiber ( E x [0, l])(x.t)rand the restrictions gl E x {0}and gl E x (1) are monomorphisms since go and g , are Gauss maps; it remains to show for any t E ( 0 , l ) that the restriction gl E x { t } is a monomorphism. Suppose that g(e, t ) = 0 E R" for some e E E and some t E [0,1] such that t # 0 and 1 - t # 0. The vanishing of the 0th entry of g(e, t ) would imply that go(e) = 0; hence the vanishing of the first entry of g ( e , t ) would imply that g'(e) = 0; hence the vanishing of the
151
8. The Homotopy Classification Theorem
second entry of g ( e , t ) would imply that g2(e)= 0; and so on, which would imply the contradiction g(e) = 0 E R". 8.6 Lemma: Let E -5 X represent a real m-plane bundle over X E 99,and let E 3 R'" and E -% R" be any Gauss maps. Then there is a Gauss map n x id E x [0, I] 5 R" .for the coordinate bundle E x [0, I] X x [0,1], such t h u t g l E x YO) = g o a n d g l E x { l ) = g , .
-
PROOF: For any P E E let g,(e) = (gE(e),gh(e),gi(e),. . .) and let g l ( e ) = (gy(e),g:(e),g:(e), . . .). By Lemma 8.5 one can replace g o by a Gauss map E R" with &(e) = (O,g$e),O,gA(e),O,. . .), and one can similarly replace g , by a Gauss map E -% R" with g l ( e ) = (gy(e),O,gi(e),O,g:(e), . . .). There is then a Gauss map E x [0,1] 4 R'"' given by setting g(e,t ) = (1 - t)go(e) t g , ( e ) for any (e, t) E X x [0,1], for which 21 E x {0}= 8, and E x {l}= E l , as required.
+
x
8.7 Lemma : Let X 2G"( R" ) und 2G"( R l ' ) be maps of an arbitrary huse space X E %9 into the Grassmunn munifold G"(R"), and let f by"' and f \j+" be the corresponding pullbacks over X of' the universal m-plane bundle y" over Gm(R*');then f by" = f !,f' if und only if,fois homotopic to f,. PROOF: If f 0 is homotopic to j',, then j'i);)"'= f \y" by Proposition 11.4.7. Conversely, if = ,f!,y"'. then there is a coordinate bundle E 5 X that represents both j ' ~ ~ j ) "and ' f ; ; ~ " ,and by Lemma 11.1.3 there are morphisms ,/'b;1"
fo
E
(I X
SO
In,
E'
+
E
fl
.]
and
I*-
+ E"
X J ' 1 Gm(Ra)
G"(R7)
from E : X to E x 5G"(R"). Let E 2 R" and E 2 R" be the corresponding Gauss maps, as in Lemma 8.4. By Lemma 8.6 there is then a Gauss map E x [0, I ] 5 R ' for the coordinate bundle E x [0, I] 3 X x [0,1] such that g l E x [ O ) = g o and g l E x ( 1 ) = g , , and Lemma 8.4 provides a corresponding morphism E x [0,1] n
x
id
1
X x [O,1]
f
E"
+
G"(R").
152
111. Vector Bundles
Since gl E x { O )
= go and
E x (0;
g I E x [ 1) = g , , the restrictions
Ex
1
na
X x (01
I"
and
* Gm(Rx)
X x (1)
are the given morphisms, so that X x [0,1] to .1;.
3 G"(R")
fl
k
Gm(R7)
is a homotopy from
i b
8.8 Lemma: Let E X represent any real m-plane bundle t over a base space X E YB; then there is ut least one Gauss mup E 5 R".
PROOF: According to Proposition 1.8.3 any base space X E a is homotopy equivalent to a paracompact hausdorff space X ' E a with a countable open covering [Ukln E N} such that each connected component of each U ; is X ' be the homotopy contained in a contractible open set in X ' . Let X equivalence, with homotopy inverse X' .% X , and let 5' be the pullback y!<. Since X ' is paracompact, one has f ! < ' = f ' ! g ' t = over X by Lemma 11.4.5, so that if E' 1;X ' represents t', then there is a morphism
<
E
f
X
S
n
+
X'
ofcoordinate bundles, as in Lemma 11.1.3. Since each restriction Ex E;,,, is an isomorphism of real m-dimensional vector spaces, the composition E A E' 5 R" off with any Gauss map E' 5 R" for E' 1;X ' will be a Gauss map E 5 R" for E X , as desired. Thus it remains to construct a Gauss map E' 5 R" ; that is, one may as well suppose at the outset that E X itself represents a real m-plane bundle bundle t over a paracompact hausdorff space X E @ with a countable locally finite open covering { U , I n E N} such that each connected component of each U,, is contained in a contractible open set in X . Since any fibre bundle over any contractible space is trivial, by Proposition 11.3.5, each restriction [I U , is trivial, and one chooses a trivialization El U,, yn U , x R" for each n E N. Since X is paracompact. there is a partition of unity (h,,(nE N)
-
153
8. The Homotopy Classification Theorem
-
subordinate to the countable locally finite open covering [ U,In E Nj, and .LEFJ~~~'~' (Rm)x= R" carrying any e E E there is then a well-defined map E into the sequence
( / l " ( n ( 4 ) Y d rh) l, ( n ( 4 ) Y l ( o )h,A n ( e ) ) Y *.,. . I
E (R"')x;
trivially the map CIlcN hl,Y,,is itself a Gauss map g. 8.9 Theorem (Homotopy Classification Theorem): Any real m-plane bundle oiwr LI husr spuce X E .?B is ( I pullhuch f '7"' o j tlir universal r e d nz-plane hundlr ;$'"over G"'(R ) E J , ulony CI mup X Gm(Rz) thut is unique up to homotop!,.
<
PROOF: Let E 5 X be any coordinate bundle representing 4 . By Lemma 8.8 there is a Gauss map E
3 R', so that Lemma 8.4 provides a morphism F
E R
1
X
I
-
* G"'(R
)
R'
to the coordinate bundle E" G"(R") representing 7": hence 4 = j ' y " for the map X G"'(R"), by Corollary 11.1.13. The uniqueness of .f up to homotopy was proved in Lemma 8.7. The homotopy classification theorem justifies the extravagant language of Definition 8.2: the rn-plane bundle ym is indeed universal. The base space Gm(R' ) of is itself dignified by a suitable name: it is the clussijyiny space for real m-plane bundles. Specifically, according to the homotopy classification theorem there is a one-to-one correspondence between the real mplane bundles over any base space X E 9 and the set [X, G"( R" )] of homotopy classes of maps X + G"( R ' ). I n the proof of the next result we use an alternative description of the coordinate bundle E' 5 G"(R') of Lemma 8.1, which represents the universal real m-plane bundle y". The total space E" is the quotient (R' )"* x R m / z ,as before; however, there is another way of expressing the equivalence relation 2 .One has ;I"'
154
111. Vector Bundles
A"
in (R')m* x Rm ifand only ifx, A . . . A X , = a y , A . . . A ynlE R", for some a E R*, and s l s l . . . S ~ X , ,= f l y l . . . t"y, E R'. The subspace E' * of nonzero fibers in E' is then the quotient (R7),* x Rm*/=, for the same equivalence relation =. It follows that one can use a change of basis to represent any equivalence class
+
+
+
+
in the fashion
for the (nonzero) vector yn, = sixl + . . . + smx,,E Rr*.Thus E x * consists of pairs (J',, P,- ,) for y , E IF**, where P,- is the ( m - 1)-plane spanned by yl, . . . , y,-,; that is, P,-, is any ( m - 1)-plane not containing y,.
L"
For any m > 1 let E" G m ( R mrepresent ) the uniwrsal real m-plane bundle ym, and let E * be the space qj'nonzerojbers in E' . Then there i s a homotopj) equivalence G"- I(RRI) E X* such that the composition Gm-'(R") 1:E x * G"(R") classijes the Whirney sum ;lrn-' 0 over G"- '(R* ), 8.10 Proposition:
:
PROOF: Let R," be the subspace of all vectors in R" with 0th entry 0 E R; then the inclusion Rg c R" is a homeomorphism whose inverse is the obvious translation of coefficients. Since the inclusion is linear, there is an induced inclusion G"-'(R,") G"-' (Rm) that is also a homeomorphism. If y , E R'* is the fixed vector with 0th entry 1 E R and all other entries 0 E R, there is a map G"-'(R,") 2 E m * that carries any ( m - 1)-plane P,- I E G"-' (R,") into the pair (y,,P,-l). Let Grn-'(RZ) 5 E" * be the composition ho i - The second projection, E"'* to P , , , - ~E G " - ~ ( [ W Z )is, a map E,* 5 G ~ - I ( R ~ ) mapping (Y~,P,-~)E for which the composition k h is trivially the identity map on G'"-'(R7). To show that h k is homotopic to the identity map on Em* one lets F E'"* x [O, I ] --* En'* carry (( y,, P , _ 1), t ) into ( ( 1 - t ) y , + rye, P , _ l). Then F , is the identity map on Em* and F , is the composition h i k ; thus h is a homotopy equivalence. I
155
8. The Homotopy Classification Theorem
Since IT' ( y l ,. . . , y m ( t0, I 1 4'1
h is induced by the inclusion R(m-l)*-+ Rm* carrying into ( y o , y , , . . . , J > , ~ - which induces a (linear) isomorphism
+ . . . + t " - ~ ~ ' m - l ) ~ ( t o y o + t+l y. l. . + t m - l y m - l )
over each [ ( ~ l , . . .,
Ym-l)]~[(yo,ylr...rym-l)],
it follows that there is a pullback diagram
G"-'(R')
n'
G"(R")
h
in which the left-hand vertical arrow represents e 1 @ y m - ' ( = y m - @ e l ) . Hence (7c7 'J h)'ym = 1'"- I @ e l ; that is, 7c" h classifies y"-' @ e l , which completes the proof. 1)
Proposition 8.10 omits the case m
8.11 Proposition: I] E'
5 RP'
=
1; here it is.
represents the universal real line bundle
y ' , then the space E m * of nonzero fibers in E" is homeomorphic to R"*; furthermore, R" * is contractible.
This time Ex'* is a quotient R"'* x R * / x with ( x ,s) x ( y , t ) if and only if sx = t y E R" *; that is, E"* = R"*. Now let R"* 1, R"* be the obvious shift such that for every x E R"'* the initial coefficient of j(x) E R" * is 0 E R. One easily verifies that (1 - 2t)x + 2tj(x)E R"* for every ( x ,t ) E R" * x [O,$], and, if xo E R" * has initial coefficient 1 E R, then (2t - l ) x o + ( 2 - 2t)j(x)E R" * for every (x,t ) E R" * x [$, 13. Hence there is a contraction R'* x [O, 11 + R" * to xo E R" *. PROOF:
There are other homotopy classification theorems that apply to restricted categories of base spaces, or to vector bundles other than the real m-plane bundles of Definition 1.1. Here is one example of such a theorem; there will be other examples later. 8.12 Proposition: For any compact base space X E 28 and any m > 0 there is an n > 0 with the following property: any real m-plane bundle 5 over X is a
156
111. Vector Bundles
pullback j ! y : of' the canonical m-plane bundle y: over the Grassmanti manyold ~ m ( ~ m +E n a, ) along some map x 5 ~ m ( ~ m + n ) .
PROOF: Let E 5 X represent 5. Since X is compact, one can replace the countable open covering (V,ln E N) used in Lemma 8.8 by a finite open covering { U 1 , . , . , U p } of X , with the same properties. In this way one obtains a Gauss map E 3 RmPc R", and for n = m ( p - 1) the evident analog of Lemma 8.4 then provides a corresponding morphism f
E
where E'
+
E'
1;G"'(Rrn+") represents y:.
5 be any real m-plane bundle over a compact base space X E 9; then for some n > 0 there is an n-plane bundle q over X such that the Whitney sum t 0q is the trivial bundle c"+" over X .
8.13 Corollary: Let
PROOF: Recall that G"'(R"'+") consists of m-planes V c R"'"',
so that if V1 c R"+" is the orthogonal complement of V with respect to a fixed inner product on R"'+", then there is a diffeomorphism G"(R"'+') 3 G"(R"'+") carrying V1 into V , as one easily verifies. Hence the total space of the bundle g'yk over G"'(R"'+") consists of pairs (n(g(e')),e')in which n ( g ( e ' ) )is an mplane I/ c R"+" and e' is a point of the orthogonal n-plane V' c R""". Since the total space of y r consists of pairs (n(e),e)in which n(e) is an mplane V c R""" and e is a point of V , it follows that the total space of y: 0g'y; consists of pairs (n(e),e e') in which n(e)is an m-plane V c R"+" and e e' is any point of Rm+";this is precisely the description of the trivial bundle E ~ + " over Gm(Rrn+"), so that 7: 0 g'yk = c""". Consequently if 5 = ,f!y;, as in Proposition 8.12, then one can set r] = f ! g ' y ; to conclude that
+
+
5 09 = f ! ( y ; 0g!yL) = f p + "= E m + " , as desired. Proposition 8.12 lacks one feature of the homotopy classification theorem (Theorem 8.9). The classijying map X L Gm(Ra)of Theorem 8.9 is unique up to homotopy; but the finite classifying map X L Gm(Rn'+")of Proposition 8.12 is in general not unique up to homotopy. Fortunately there is a simple remedy in the latter case.
157
8. The Homotopy Classification Theorem
Let E -I, G m ( R m + "represent ) the canonical m-plane bundle :y over Gm(Rm+'),and for any n" 2 n let E" 5 Gm(Rm+n") represent the canonical The usual linear inclusion Rm+"+ Rm+"" m-plane bundle yy,,over Gm(Rm+""). then provides a commutative diagram B,Z.n''
E
Gm(Rm+n)
[Ju.ll''
+ E
,G m ( R m t r i ' ' )
that is a (linear) isomorphism in each fiber, so that gn,n..is a finite classifying map for :y itself; that is f ' = g~.,..y:,.. Clearly if n 5 n' 5 n", then one has gn..n.8 gn,n.= gn,n..and :y = g"qn.g;.,n..yF.. The maps gn,n.,s.~,~..,and gn.,nf.are finite classifying extensions. I
1
r
8.14 Proposition (Ersatz Homotopy Uniqueness Theorem): Let 5 he a real m-plune bundle over X E W such that 5 = f a y : and ( = f i r ? for maps x 2~ m ( ~ m + and n ) x 3~ m ( ~ m +).n 'Then there is an n" 2 max(n,n'), with .finite classifying extensions gn,n..and gn..n.r,respectively, such that the compositions A Gm(Rm+n) Gm(Rm+n")
x
and
x -& G m ( R m + n ' ) 5Gm(Rrn+n" ) ure homotopic maps from X to Gm(Rm+""), .for which (gn.n,, c
f o ) ! ~ F s= 5 = ( g n s . n , ,
0
fl)!~Fs.
PROOF: One may as well assume that n' 2 n, in which case one sets n" = ni + 2n'; then m + n" = 2(m + n') 2 2(m + 11). If E 4 X represents (, then the maps ,fo and 1; are equivalent to finite-dimensional Gauss maps E -, p + n C [Wm+n'andE+Rm+n',respectively, so that the compositions gn,n..c' fo and LJ~~.,~...1; both correspond to finite-dimensional Gauss maps E -+R Z ( m + n ' ) in which the last rn + n' entries are identically zero. The latter condition provides just the right amount of elbow room for obvious finite-dimensional analogs of Lemmas 8.5 and 8.6; the homotopy from the composition gfi,fI.. fo to the composition y,,.,,.. c. f l is then constructed as in Lemma 8.7. ~
Proposition 8.14 was discovered in 1949, among the unpublished papers of the late Friedrich Adolph Karl Ersatz.
I58
111. Vector Bundles
9. More Smooth Vector Bundles We prove that any vector bundle whatsoever over a smooth manifold can be represented by a smooth coordinate bundle, as announced in $5. The first step is a simplicial analog of Proposition 8.12, which does not require compactness. 9.1 Proposition: Given natural numbers m > 0 and q > 0, set n = mq. Then any real m-plane bundle 5 over any q-dimensional metric simplicial space IKI is a pullback f l y : of' the canonical m-plane bundle :y over the Grassmann along some map IKI Grn(Rrn+'). manijold Grn(Rrn+'),
PROOF: Since there is a canonical homeomorphism IK'( 2 IKI, for the first barycentric subdivision K' of the simplicial complex K , one may as well regard 5 as a bundle over IK'I. According to Proposition 1.2.1, IK'I is of first type. Specifically, since K is q-dimensional, there is a covering (UO,ar}a,. . . , (U,,y}y of IK'I by q 1 families of contractible open sets U p , Dc IK'I, the sets in each family being mutually disjoint. Let U p= c IK'I for each p = 0, . . , ,q, so that { U , , . . . , U , ) is a finite open covering of (K'I. Since any fibre bundle over a contractible space is trivial, by Proposition 11.3.5, it follows that each restriction 5 1 U p , Dis trivial, and since each family { U p , , } s is mutually disjoint, it follows that each restriction 5 1 U p is trivial. Thus if E J IK'I represents (, then one can choose q + 1 specific trivializations E I u03 u0 x I W.~. .,, E 1 U , LU , x and a partition of unity ( h o , .. . , h4) subordinate to { U o ,. . . , U,j to obtain ho'€'o +. . . + hqYq a Gauss map E + R r n ( , + l )as in Lemma 8.8. The analog of Lemma 8.4 then provides a corresponding morphism
+
up
E
f
+ E'
(K'(J ' Grn(R"'+''),
where E' 5 Grn(Rrn+") represents 7:. Since Proposition 1.4.6 (Dowker [I]) provides a homotopy equivalence lKlm -, lKlw from any metric simplicial space IK(, to its weak counterpart IKlw, Proposition 9.1 applies equally well to q-dimensional weak simplicial spaces. Hence, one may as well omit the metric condition from the statement of Proposition 9.1.
9. More Smooth Vector Bundles
159
9.2 Corollary: Let 5 be anj1 real m-plane bundle over any finite-dimensional simplicia1 space IKI; then for some n > 0 there is an n-plane bundle q over IKI such that the Whitney sum [ @ q is the trivial bundle E ~ + "over IKI. PROOF: Substitute Proposition 9.1 for Proposition 8.12 in the proof of Corollary 8.13. If the simplicial space IK( in Corollary 9.2 is q-dimensional, then Proposition 9.1 permits one to set n = mq. However, there is also a direct proof, using a general position argument, in which one can set n = q. The second step toward the main theorem of this section combines Proposition 9.1 with Theorem 1.6.7 (the Cairns-Whitehead triangulation theorem) and Theorem 1.6.19 (that any map of smooth manifolds is homotopic to a smooth map); one also uses Proposition 1.7.3 (that the Grassmann manifold Gm(Rm+") is a smooth closed mn-dimensional manifold).
9.3 Proposition: Given natural numbers m > 0 and q > 0, set n = mq. Then any real m-plane bundle 5 over any smooth q-dimensional manifold X is a pullback f;;~:of the canonical m-plune bundle :y over Gm(Rm+")along a smooth map x ~ m ( ~ m ). + n
a
PROOF: By the Cairns-Whitehead theorem X is homeomorphic to a qdimensional metric simplicia1 space IKI, so that by Proposition 9.1 4 is a By Proposition 11.4.7 one pullback f a y : along some map X 5Gm(Rm+"). can replacef, by any map X 2Gm(Rm+" ) homotopic to f,, and by Theorem 1.6.19 one can choose a smooth such map j ; . 9.4 Corollary: Let 5 be any real m-plune bundle over any smooth manifold X ; then j b r some n > 0 there is an n-plane bundle q over X such that the Whitney sum 5 @ i s the trivial bundle em over X .
PROOF: Substitute Proposition 9.3 for Proposition 8.12 in the proof of Corollary 8.13. If the smooth manifold X in Corollary 9.4 is q-dimensional, then Proposition 9.3 permits one to set n = my. However, there is also a direct proof, using the transversality theorem, in which one can set n = q. (See pages 99-101 of M. W. Hirsch [4], for example.) The main result of this section is another corollary of Proposition 9.3.
9.5 Theorem: Given a smooth manifold X , any real vector bundle 5 over X can be represented by a smooth coordinate bundle E X ; that is, any real vector bundle over X is itselfsmooth, as in DeJinition 5.2.
160
111. Vector Bundles
PROOF: If 5 is an rn-plane bundle, then 5 = f ! ? ; for some n > 0 and some smooth map X i G"(R"+'), by Proposition 9.3. Since 7:: is smooth by Lemma 7.1, it follows from Proposition 5.3 that its pullback f ' y ; is smooth.
10. Orientable Vector Bundles The familiar parlor-trick "one-sidedness" of the Mobius band E reflects the failure of the tangent bundle r ( E )to be orientable, in the sense described in this section. To define orientability of any real rn-plane bundle 5 over any X E &9 one first constructs a real line bundle A" 5 over X , from which one obtains a fiber bundle o(5) over X with structure group 0(1)( = Z j 2 ) and fiber So ( = Z/2); the bundle 5 is orientable if and only if o(5) is trivial. Following the construction of o(5) we shall show that 5 is orientable if and only if it is of the form f ' y " for a map X 3 Gm(Ral)that factors through the total space of the bundle o(y") over Gm(Ra').Finally we show that the canonical line bundle y : over RP' (= S ' ) is not orientable, and that the tangent bundle T ( E )of the Mobius band E is not orientable. Let E 5 X represent a real m-plane bundle 5 over X E 98, and for any natural number p > 0 let E x . . . x E X x . . . x X be the product of p copies of E .!!,X . The pullback of the product along the diagonal map X 5 X x . . . x X is a coordinate bundle E' 1;X with fiber R" x . x R"' and structure group GL(m,R) x . . . x GL(m,R), and we recall from Proposition 2.5 that one can construct the Whitney sum 0 .* * 0 5 and product 5 0 .. . 0 5 by applying the morphisms (re,@@)and (T@,@@) to E' 5 X ; these bundles have structure groups GL(mp,R), GL(mP,R) and fibers R" 0 . . .0 R", R" 0 * . 0 R", respectively. We now modify E' 5 X for another purpose. Let GL(m,R) itself act on the left of R" x * . . x R", with g(.x,. . . . , x,) = (yx,, . . . , gx,) for every g E GL(rn,R) and (x,, . . . , x,) E R" x . . . x R". If the original coordinate bundle E X is defined with respect to a covering *J (U,li E 1 ; and transition functions U in U j -r GL(rn,R), one can then 1"' construct a new coordinate bundle E" X with respect to [ Uil i E 11, with fiber R" x . . . x R" and structure group GL(rn,R): one uses the same transition functions $!and the preceding action of GL(m,R) on R" x . . . x R". Now let R " x ~ ~ ~ x [ W "be~ the ~ map R m carrying ( x l , . . . , x p ) ~ R" x . . . x R" into the exterior product x 1 A . . . A x p E A"", and let GL(m,R) 5 GL(();, R) be the group homomorphism carrying g E GL(rn,R) I I X " ' X R
1
-
161
10. Orientable Vector Bundles
into that element of GI,((;), R) with value gx, A * * * A g x p on any x, A . . . A xp E R". The pair (I-,@) is a morphism of transformation groups in the sense of Definition 11.2.6, so that by Proposition 11.2.7 one can apply (I-, @) to the coordinate bundle E" X to obtain an (;)-plane bundle over X , which is independent of the coordinate bundle E 3 X chosen to represent the m-plane bundle (.
Ap
10.1 Definition: Given an m-plane bundle 5 over X is the preceding (;)-plane bundle over X . power
E 3, the
pth exterior
If nr = p, then A" 5 is a line bundle over X , and since gx, A . * * A gx, = (detg)x, A ' . ' A X , for any (xl, . . . , x,) E R" x * * * x R" the group homomorphism GL(m,R) f, GL(1, R) merely carries g E GL(m,R) into its determinant det g, acting via scalar multiplication on A" R". Let (A"Rm)* c A" R" consist of the nonzero elements y E A" R", observe that (A"Rm)* is preserved under the action of GL(1, R), and let be the equivalence relation in (A" Rm)*with y' y if and only if y' = uy for some a > 0. The canonical surjection (A"Rm)*.%(A" R")*/- maps (A" Rm)*onto a space (A" R")*/- with just two elements, and if GL '( 1, R) c GL(1, R) is the subgroup consisting of multiplications by positive real numbers the canonical epimorphism GL(1, R) 5 GL(1, R)/GL+(l,R) has image 2/2, which acts on (A"R")*/- by interchanging the two elements. The pair (I?,@') is another morphism of transformation groups in the sense of Definition 11.2.6, carrying the transformation group GL(1, R) x (A"Rm)*+ (A" Rm)*into the transformation group 2/2 x (A"Rm)*/- + (A" Rim)*/-.
-
-
10.2 Definition: Let 4 be any real m-plane bundle over any X
E
3, and let
(A"'t)*be the fiber bundle with fiber (A" Rm)*and structure group GL(1, R), obtained from the mth exterior power A"( by removing the zero-section. The orientation bundle o(4) over X is induced by applying the preceding morphism (l-', 0') of transformation groups to (A" 4)*, as in Definition 11.2.8. There is another way of describing o(t), which we sketch. According to the linear reduction theorem (Theorem 11.6.13) one can reduce the structure group GL(m,R) of ( to the orthogonal subgroup O(m) c GL(m,R), and Proposition 3.4 provides a riemannian metric ( , ) for any coordinate bundle representing 5. One easily alters ( , ), if necessary, in such a way that the action of O(m) on each fiber E x preserves the inner product Ex x E, Iw; in fact, this is automatically the case if one uses ( , ) itself to reduce GL(m,R) to O ( m )c GL(m,R), as suggested at the end of $3. Hence one can replace GL(m,R) by O(m),and E by the subspace of fibers of
162
111. Vector Bundles
unit length, to obtain the sphere bundle associated to <,with structure group O(m)and fiber S"As part of Definition 10.1, we introduced a transformation group GL(m,R) x (R" x . . . x R") + R" x * x R", and we applied a morphism (r,0)carrying any g E GL(m,R) into det g E GL(1,R) and any (xl, . . . , x,) E [w" x . . . x R" into the exterior power x 1 A * * AX, E A" R" = R'; in effect, however, we replaced R" x * * x R" by the subspace (Rm)"* c R" x . . * x R" of ordered bases of R". One can apply the same morphism (r,@) to the transformation group O(m) x (S"-')"* + (S"-')"*, where (,"I- '),* c (Rm)"* consists of orthonormal bases of R"; in this case (I-,@) carries any y E O(m) into detg E O(1) and any ( x l , .. . , x,) E (S"-l)"* into a point x 1 A * . * AX, in the 0-sphere So E R'. The orientation bundle o(<)of Definition 10.2 is clearly the result of applying the preceding morphism (l-,@) of transformation groups to the sphere bundle associated to the given m-plane bundle. The preceding description of the orientation bundle o(<)suggests that one should regard its structure group as the orthogonal group O( 1) ( = 2/2) and its fiber as the 0-sphere So ( = 2/2). In any event, the coordinate bundle repn(5) X ; it is the double covering of X resenting o(<) will be denoted O ( 5 ) associated to 4.
'.
-
-
-
<
10.3 Proposition: Let X ' X be any map of base spaces, and let be any real m-plane bundle over X . Then o(f '5) = f !o(<)over X ' ; that is, there is a map O ( f ) preserving the action of O( 1) such that the diagram
-
O(f!<)
X'
OCS)
s
O(0
b X
commutes.
PROOF: Any pullback diagram E'
for
f
f
< and f '5 induces a corresponding pullback diagram for (A"<)* and
!(A"<)*, and one obtains the result by applying (l-',@').
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10. Orientable Vector Bundles
Observe that since the fiber of o(5)consists of two points, o(5) is trivial ifand only if there is a section X 5 O ( < )of the coordinate bundle representing
45). 10.4 Definition: A real m-plane 5 is orientable if and only if the orientation bundle o(t) is trivial; that is, 5 is orientable if and only if the mth exterior power A" 5 is the trivial line bundle e l . An orientation of 5 is a specific section X 5 O ( < ) ,which orients 5. We shall later describe orientability of 5 more concretely in terms of the transition functions U i n U , 3GL(m, R) for any coordinate bundle representing 5.
10.5 Proposition: Any orientation a of a real m-plane bundle 5 over X E 93 induces an orientation a'of the pullback f!5 of 5 along a map X ' X in 93; thus, if 5 is oriented, then so is any pullback f!5. PROOF: We use the diagram of Proposition 10.3. For any x' E X ' the point a(f'(x'))E O ( 5 )is the image under O(f)of precisely one of those two points of O ( f ' 4 )that project onto x', and one lets a'(x') E O ( j ! t )be that point. The
composition X ' s O(f! 5 ) continuity of 0' is trivial.
n(S!5)
X ' is the identity, by construction, and
Since the total space O(5)of the orientation bundle o(5)of a real m-plane bundle 5 over X E 2 is a double covering of X , it is trivial to verify that O(<) itself belongs to B. Hence it makes sense to pull back bundles along the projection map O ( 5 ) n(C) X ; one can even pull 5 itself back along ~ ( 5 to) obtain an m-plane bundle n(<)!rover O(5) E 93.
-
10.6 Proposition: l f O(5) 3X represents the orientarion bundle o(5) of a real in-plane bundle 5 over X , then the pullback .n(()!< is an orientable m-plane bundle over O ( 0 , oriented in a canonical way. PROOF: The commutative diagram of Proposition 10.3 becomes
O(5)n o
x
in this case. Let 41 be any point in the lower left-hand copy of O(5).The same point y in the upper right-hand copy of O(5)is then the image of precisely
164
111. Vector Bundles
one of the two points of O ( n ( t ) ! [whose ) projection under n ( x ( l ) ! cis) y , and one lets a ( y )E O(n(t)!<) be that point. The composition O(5) 5 O(n(()!<) n(n(;)'t) ---+ O(<) is the identity, by construction, and continuity of 0 is trivial. We have observed that if 5 is a real m-plane bundle over X E g,then O(5) E &?.In particular, if y" is the universal m-plane bundle over the Grassmann manifold G"(R"), as in Definition 8.2, then O(y") E &?. 10.7 Definition: For any m > 0 let @(R") be the total space O(y") of the orientation bundle o(y") of the universal real m-plane bundle ym over Gm(Ra); that is, @'(R50) Cm(Ra)represents o(y"). The pullback n(y")!y" over @(R' ) is the universal oriented m-plane bundle 7"'.
According to Proposition 10.6, 7" has a canonical orientation. 10.8 Proposition (Homotopy Classification Theorem): Let { be a real mplane bundle over X E 9. Then 5 is orientable if and only if 5 = j or some map x G"(R~).
77"
3
PROOF: If 5 = f!jj'", then 5 is orientable by Proposition 10.5. Conversely, suppose that 5 is any orientable bundle over X EB,and let X G"(R") classify as a real m-plane bundle, so that 5 = f ! y " ; such an ,f exists by Theorem 8.9. The pullback diagram of Proposition 10.3 is then of the form
X
J"
+ G"(R').
Since any orientation of 5 is a section X 5 O(5) of the bundle o ( l ) ,by definition, the composition X 5 O(5) 3 X is the identity, so that f = n(y") r O j j ) 0 0. If X -5 Gm(Ra) is the composition O ( j )n 0 , then 5 = (n(y") .f)!y" = ,f!n(y")!y" = as desired. 0
77"
-
Briefly, Proposition 10.8 asserts that 5 is orientable if and only if it can be classified by a map of the form X &""Rou) N Y m ) G"(R"), where @'"'Rm) is the double covering of G"(Ra) with respect to y". The map.7 is the oriented classijying map, determining a specific orientation of 4. One can give direct constructions of @"(R") and the universal oriented m-plane bundle 7'" over @'( Rm) that superficially resemble the corresponding unoriented constructions more closely than does Definition 10.7. According to Definition 1.7.1 the unoriented Grassmann manifold Gm(Rm) can be regarded as the quotient ([Was)"*/of (Rm)"* by the equivalence
165
10. Orientable Vector Bundles
-
-
relation with (x,, . . . , x), ( y , , . . . , y,) whenever x, A . . . AX, = a y , A . . . A y, E A"' R" for some a E R*. If one replaces the latter condition by the requirement that a > 0, then the result is a new equivalence relation - ' for which one easily establishes (Rm')m*/-' = @""R"). Similarly, one can describe the total space E" of the universal oriented m-plane bundle 7"' as a quotient Rm* x R m / z 'in which
if and only if x , A ~ ~ ~ A x ~ = u ~ , A for ~ ~some ~ Aa >~ 0,, Eand~ R ~ , =fly, . . . t"y, E R". The subspace Em*of nonzero fibers in ,? is then the quotient (RZ)"'* x R m * / z ' ,for the same equivalence relation z'.
six, + . . . + smx,
+
+
2Gm(R") represent the universal 10.9 Proposition : For any m > 1 let oriented m-plane bundle y"', and let EU* be the space of nonzero jibers in E x . Then there is a homotopy equivalence @'-'(R") Em*such that the composition & - ' ( R x ) ; Em*5 @""R") classijies the oriented Whitney sum j j r n - l 0 e 1 ouer @ - I ( R 1. ~ PROOF: The proof is virtually identical to that of Proposition 8.10, using the preceding description of E x -% @'"'R") in place of the corresponding nr description of E" Gm(RZ). (One also uses the property that if m is even, then c1 0 ym-' and y"-' 0 1.:' have opposite orientations.)
-
-
Let 5 be any real m-plane bundle over any X E g,as before. The canonical n(5) involution of the double covering O(5) X is the well-defined homeomorphism O(5) AO(<) that interchanges the two points of the fiber of o(<) over each x E X. If 5 is orientable, then for any orientation X 5 O(5) the r a opposite orientation is the composition X O(5); the resulting opposite[y oriented m-plane bundle is usually denoted - (.
-
Since @""R')
= O(y"),
there is a canonical involution of @(Ra).
10.10 Proposition: If m i s odd the canonical involution @(Ra) 5 c m ( R x ) is homotopic to the identity.
,;
PROOF: Let ( . . . ; x r r x r t . . .) denote any element (xo,x,; x,,x,; . . . ; x,, x r t I ; . . .) E R' , where r is even, and let R" x [0,1] R a be the map that carries any ( ( . . . ; x,, x,+ ; . . .), t ) into
, ( . . . ; x, cos nt - x, , sin nt, x, sin nt + x, , cos nt ; . . .). ~
166
111. Vector Bundles
The restriction f I R" x ( t } is a linear isomorphism for each t E [0,1], the restriction f l R " x {O} is the identity, and the restriction ,flRL x (1) carries (. . . ;xr,xrt . . .) into (. . .; - x r , -xrt . .). Hence if E G"(R") represents the universal bundle y", then there is an induced homotopy -1 E x [0,1] 5 E from the identity E + E to the map E E that carries into -e E En,-,,, where n( - e ) = n(e). If m is odd and E' % each e E G"(R") represents the mth exterior power A" y" over G"(Ra), F induces a -1 E'. The restriction homotopy F' from the identity E' + E' to the map E' of F' to any fiber is an isomorphism for each t E [0,1], so that there is a corresponding homotopy of maps of the total space of the bundle (A"y")* (Definition 10.2), which induces the desired homotopy of maps of the total space 6 m ( ~ofao(ym). i)
,;
-
-
We now replace R"' by R"'" for m
+ n < 00.
10.11 Definition: For any m > 0 and n > 0 let @'"'R"+") be the total space O(y;) of the orientation bundle o(yr) of the canonical real m-plane bundle y r over the Grassmann manifold Gm(Rm+");that is, @'(Rmtn) ,(YIP) Gm(Rm+")represents o(y;). The pullback n(y;)!y; over @'(R'"+") is the
-
canonical oriented m-plane bundle 7;. Since Gm(Rm'") is a smooth closed mn-dimensional manifold, by Proposition 1.7.3, the same is true of the double covering @""R"""). Furthermore, the bundle over @""Rm'") is canonically oriented by Proposition 10.6.
jjr
10.12 Proposition: Let X E &? be homotopy equivalent either to a compact space or to a finite-dimensional simplicia1 space, and let 5 be any real m-plane bundle over X . Then 5 iqorientable if and only if 5 = for some finite n > 0 and some map X @""R"""). I f X is a smooth mangold, then 5 is orientable i j and only if a smooth such exists.
f'jjr
7
PROOF: By Propositions 8.12,9.1, or 9.3 one has 5 = f ' y ; for an appropriate map X G"( R"' "), and one substitutes R" " for R" throughout the proof of Proposition 10.8. 10.13 Proposition: 1.f m and n are both odd natural numbers, the canonical involution @'(R'"+tl) A @"''R'''~") is homotopic to the identity.
+
PROOF: Since m n is even, one can substitute Rm+"for R" in the proof of Proposition 10.10.
167
10. Orientable Vector Bundles
10.14 Corollary: For any odd m and any n > 0 the canonical real m-plane bundle $' over G'"(R'"+'') is nonorientuble. PROOF: The inclusion Rm+'-, Rm+ninduces a map Gm(Rm+')-5 G'"(R'"+") for which 7'; = f ' y ; , so that nonorientability of 7'; implies nonorientability of yr, by Proposition 10.5. Hence it suffices to consider only the case that n is the odd number 1, for which Proposition 10.13 guarantees that 6'"(Rm+')1,6'"(R'"+') is homotopic to the identity. If 7'; were orientable, then since the double covering Gm(R'"+')of Gm(Rm+l)is the total space O(y;") of the orientation bundle o(y';),it would consist of two disjoint copies of G'"(R'"+ I ) , and z would interchange the two copies of @(R'"+'); however, such a map T would not be homotopic to the identity. '
Since trivial vector bundles are clearly orientable, Corollary 10.14 incidentally guarantees that the bundles 7; are not trivial when m is odd. The simplest case is the canonical real line bundle 7 : over RP', the base space RP' being diffeomorphic to the circle S'. We now develop a more concrete characterization of orientability. Let So be the 0-sphere { + 1, - I ) , and let the orthogonal group O(1) act on So as usual, via multiplication by + 1 or - 1. Any coordinate bundle E 5 X with fiber So and structure group 0(1)over a space X E 9?is a double covering of X in the sense defined earlier; in fact, since 0(1) c GL(1, R), one easily constructs a real line bundle over X such that E .5 X is precisely the double covering ~ (3 i ) X.
10.15 Lemma: Let E 5 X be u double covering of a space X E 9, a coordinate bundle with respect to some open covering { Uil i E I ) of X . Then E X JI' is trivial $and only fi there is a jumily of transition functions U in U j 4O(1) each of which has the constant value 1 E O(1). PROOF: Clearly E 5 X is trivial if and only if there is a section X 5 E . Let {'PiI i E I ) be a family of local trivializations E I U i5 U ix So. If a section 0 exists, then each composition U i El Lri 3 U i x So maps each x E U iinto (x, 1) E U ix So, and one can alter the local trivializations in such a way that ('Pi 01 U i ) ( x )= (x, + 1) for each i E I and each x E Ui. The transition functions 'Pi are defined by the requirement that the compositions Y , Yx-' + U i n U j x So maps (x, I) into (x, $i(x)( f 1)); U i n U j x So hence, using the altered local trivialization 'Pi, one has $i(x) = 1 E O(1) for x E U in U j , as required. Conversely, if the latter conditions are satisfied, then there is a unique section X 5 E such that for each i E I the composition Y i 01 U i carries each x E U iinto (x, + 1)E U ix So.
a
0
168
111. Vector Bundles
The following property of orientable vector bundles is frequently used as the definition of orientability, where GL+(rn,R) c GL(rn,R) is the subgroup of elements with positive determinants, as usual. 10.16 Proposition: A real rn-plane bundle 5 over any X E a is orientable
if and only if the structure group GL(m,R) can be reduced to the subgroup GL+(rn,R) c GL(rn,R).
PROOF: Let E 4 X be a coordinate bundle representing 5, with respect to some open covering { Ui I i E I } of X . Then according to the discussion following Definition 11.5.1 we must show that 5 is orientable if and only if there is a family of transition functions U i n U j 3GL(rn,R) whose values all lie in the subgroup GL+(rn,R) c GL(rn,R). The coordinate bundle E 1X induces a double covering E' 5 X that represents the orientation bundle o(5) with respect to the same covering { Uil i E I } . Specifically, for each $! the corresponding transition function U i n U j *," O( 1) for E' 5 X is given by setting $ i j = det ${/ldet $!I. By Lemma 10.15 the orientation bundle o(5) is trivial if and only if there are transition functions $I' for E' 5 X such that $Ij(x) = 1 E O(1) for each x E U i n U,. Hence the bundle 5 is orientable if and only if there are transition functions $i for E 5 X such that det $i(x) > 0 for each x E U i n U,, as claimed.
-
10.17 Corollary: A real rn-plane bundle 5 over any X E G3 is orientable i f and only if the structure group can be reduced to the rotation subgroup O'(rn) c GL(rn, R).
PROOF:By the linear reduction theorem (Theorem 11.6.13) the structure group GLf(rn,R) can always be reduced to the subgroup O'(rn) c GL+(rn,R). 10.18 Corollary: Let 5 be any real m-plane bundle over any base space X E 39;then the Whitney sum 5 0 5 is orientable.
PROOF: If 5 is represented by a coordinate bundle using some open covering {U,l i E I } of X and transition functions Ui n U j --%GL(rn,R), then 4 0 5 is also represented by a coordinate bundle using the same covering ( U i l i E I } and transition functions U i n U j *:W1 GL(2m,R); however, det(${ 0 $!) = (det $!)' > 0 over Ui n U j .
-
One can easily strengthen Corollary 10.18:5 0 5 has a natural orientation, which will appear just before Proposition 12.8.
169
11. Complex Vector Bundles
10.19 Corollary: Let 5 and 5‘ he real vector bundles over anj’ base space X E d,and suppose that 5 is orientable; then 5‘ is orientable i f and only i f 5 05’ is orientable.
PROOF: If 5 and 5’ are represented by coordinate bundles using open coverings jUili E I } and ( U , l j ~5 ) of X , respectively, then they can both be represented by coordinate bundles using the common open covering ( U i n U j 1 ( i , j ) E I x 5)ofX;thelattercoveringwillbedenoted ( U i l i € I J f o r convenience. By hypothesis, the transition functions Ui n U j A GL(m,R) for the representation of 5 can be chosen in such a way that det $; > 0 over Ui n U j . If U in U j ‘ ; I GL(n,R) are transition functions for the representation of C’, then U in U j GL(m n, R) are transition functions for a representation of 4 05’; however, det(${ 0 1+9ij)= (det $i)(det $iJ), where det Ic/{ > 0, over Ui n U j . ‘J
-
+
The Mobius band was introduced for motivation at the beginning of this section. We now sketch the proof that its tangent bundle is indeed nonorientable. First, recall that the Mobius band was described in 811.0 as the total space E of a coordinate bundle E 5 S ’ ( = R P ’ ) whose fiber is the closed interval [ - 1, + 11 c R and whose structure group is 2/2, acting on [ - 1, + 11 via multiplication by 1 or - 1 ; that is, 2/2 = O(1) c GL(1,R). It is clear that E 5 RP’ is merely a restriction ofa coordinate bundle E‘ 5 RP’ representing the canonical line bundle 1,: over RP’. In any event, one can pull the tangent bundle t ( E ) or r(E’) back along the zero-section RP’ + E c E‘ to obtain a 2-plane bundle & ( E ) over RP’, and a direct computation shows that & ( E ) = 7 ; 0 E ’ for the trivial line bundle E ’ . However, y i is nonorientable by Corollary 10.14, so that a!t(E) is nonorientable by Corollary 10.19, so that T ( E )is nonorientable by Proposition 10.5, as claimed.
+
1 1. Complex Vector Bundles One can replace the real field R by the complex field C throughout the development of vector bundles, with virtually no other changes. The replacement is more than an idle exercise, however, even if one is only interested in real vector bundles. For example, “complexifications” of real vector bundles will be used in Volume 2 to compute the real cohomology rings H*(Gm(R’X ); R) of real Grassmann manifolds; Theorem 8.9 suggests
I70
111. Vector Bundles
the real importance of such cohomology rings. Complex vector bundles are also used in the very definition of K-theory, in Volume 3, which is used to solve the classical problem of real vector fields on spheres. However, complex vector bundles are of geometric interest in their own right, especially if one studies complex manifolds by looking at their tangent bundles. As always, the base space of a fiber bundle belongs to the category 9 of base spaces.
11.1 Definition: A complex vector bundle of rank n, or simply a complex n-plane bundle, is any fiber bundle whose fiber is the complex vector space C"and whose structure group is the general linear group GL(n, C)of invertible n x n matrices, acting in the usual way on C".A complex line bundle is a complex vector bundle of rank 1. The obvious complex analog of Definition 2.4 or Proposition 2.5 provides the Whitney sum i0 i'and product i0 i'of complex vector bundles iand [' over the same base space, with the associative, commutative, and distributive properties described for real vector bundles in Proposition 2.6. If one substitutes hermitian inner products C" x @" ( , ) C for real inner products R" x R" R in Definition 3.1, then the result is a hermitian metric for a given complex n-plane bundle i.The obvious complex analog of Proposition 3.4 then guarantees that any complex vector bundle (over a base space X E .9d, as always) has a hermitian metric. Consequently there is a complex analog of Proposition 3.6: for any complex subbundle i' of a given complex vector bundle ( there is another subbundle 1' of isuch that [ = i'0 1'. One can also use hermitian metrics, as in the real case, to show that the structure group GL(n, C)of any complex n-plane bundle over any X E $28 can be reduced to the unitary subgroup U(n)c GL(n, C);however, since this is a special case of the linear reduction theorem (Theorem II.6.13), the details will be left as an exercise (Exercise 13.19). Definitions 7.2 and 8.2 have obvious complex analogs, which provide the canonical complex m-plane bundle y r over the Grassmann manifold Gm(Cm+") and the universal complex m-plane bundle y" over the Grassmann manifold Gm(@ a'), respectively; in particular there is a canonical complex line bundle 7 ; over the projective space CP", and a universal complex line bundle y' over the projective space CP'". We already know from Proposition 1.7.5 that Gm(Cmf") E .9d and Cm(Cm)E g.
-
-
11.2 Theorem (Homotopy Classification Theorem): Any complex m-plane bundle iover a base space X E $28 is a pullback f'y" of the universal complex
171
1 1 . Complex Vector Bundles
m-plune bundle 7"' ouer Gm(@" ) up to homotopj,.
E
4, dong a mup X
A G m ( @ "thut ) is unique
PROOF: Substitute C for R throughout the proof of Theorem 8.9.
n"
11.3 Proposition: For any n > 1 let E' G"(C' ) represent the universal complex n-plane bundle y", and let E" * he the space of' nonzero Jihers in E x . Then there i s u homotopji equivulence G n - ' ( C z )+!- E m* such that the comn= G"(C7) classijes the Whitney sum y"-' 0 E' position G"-'(@') A E'* over G"- I ( C ' ).
-
PROOF: Substitute C for R (and n for m ) throughout the proof of Proposition 8.10. 11.4 Proposition: I f E" 2CP" represents the universal complex line bundle ill, then the space EL* of nonzero ,fibers in E" is homeomorphic to the contructihle space C"*. If' E : CP" represents the canonical complex line bundle if:,f;w u given n > 0, then the space E* of' nonzero Jibers in E is homeomorphic to @'"+ I ) * , trivially homotopy equivalent to the (2n 1)-sphere S2"+
+
'.
PROOF: As in Proposition 8.11, E x * is a quotient C X * x C*/%,with (x, s) % ( J:t ) if and only if sx = t y E @" *; similarly E* is a quotient C("+ I)* x @*/z,with (x,s) = ( y , t ) if and only if sx = t y E V+')*. The proof that @' * is contractible follows the pattern of the corresponding proof for Rz*, given in Proposition 8.1 1. The homotopy classification theorem for complex vector bundles has finite analogs, just as in the real case. Here are some corresponding complex uniqueness and existence results, in that order. 11.5 Proposition (Ersatz Homotopy Uniqueness Theorem): Let i be a complex m-plune bundle over X E d such that [ = j'ay'," and i= f'iv',"' jbr mups X 2 G m ( @ m +und " ) X 2Gm(Cm+"'), where r'," and :y are canonicul complex m-plane bundles. Then there is an n" >= max(n, n') with Jinite classi~ m ( ~ m + n "Llnd ) ~ m ( c m + n ' ) ~ m ( ~ m + ~ ~ ,jj;llgexterlsiorls ~ m ( , n l + n ) such thut gn,n.. f b and g,I.,n.. j1 ure homotopic maps from X to Gm(Cm+""), fiir which (y,.,.. " j0)'y;. = i= (g,,,,,. ' f J y : . . #:
PROOF: Substitute C for R throughout the proof of Theorem 8.14. 11.6 Proposition: Let X E d be u compact space, a .finite-dimensionul simpliciul spucv, or u smooth manijold, and let i be u complex m-plane bundle
" )
172
111. Vector Bundles
=-
over X . Then there is a natural number n 0 such that [ i s a pullback .f '7: of' the canonical complex m-plane bundle j$' over Gm(Cmfn) along a map X Gm(C"'+n).
PROOF: Substitute C for R throughout the proofs of Propositions 8.12,9.1, or 9.3, respectively. 11.7 Corollary: Let X E 98 be a compact space, a ,finite-dimensional simplicial sptrce, or u smooth manifold, and let [ be a complex m-plane bundle over X . Thenjor some n > 0 there is a complex n-plane bundle [' over X such thut the Whitney sum ( 0 [' is the trivial complex bundle d'"'"over X .
PROOF: This is the complex analog of Corollaries 8.13, 9.2, or 9.4, respectively. 11.8 Theorem: Any complex m-plane bundle [ over a smooth manfold X can be represented by a smooth coordinate bundle E X ; that is, any complex vector bundle over X is itself smooth.
PROOF: By Proposition 11.6, [ is a pullback f ' ! f of the canonical complex m-plane bundle y: along some map X Gm(Cm+"), and by Proposition 1.6.19 one can choose f to be a smooth map. One then substitutes C for R throughout the proofs of Lemma 7.1 and Proposition 5.3 to conclude that 7: and f ' y : are smooth.
12. Realifications and Complexifications For any complex n-plane bundle [ over a base space X E 9fthere is a corresponding oriented 2n-plane bundle laover X ; one can also ignore the orientation of la and simply regard it as a real 2n-plane bundle over X . Similarly, for any real m-plane bundle 5 over a base space X E there is a corresponding complex m-plane bundle ta:over X . The constructions ( )Iw and ( )a: are studied in this section. Recall from Definition 11.2.6 that a morphism from a transformation of group G x F + F to a transformation group G' x F' -+ F' is a pair (r,@) 0 maps G I ,G' and F -,F' such that is a group homomorphism and the obvious diagram commutes. For any fiber bundle [ with structure group G and fiber F over a space X , the morphism (r,@) induces a new fiber bundle [' with structure group G' and fiber F' over the same space X , as in Proposition 11.2.7 and Definition 11.2.8.
I73
12. Realifications and Complexifications
Let G = GL(n,C).and let r assign to any (h4, + ic4,) E GL(n, C)the real 2n x 2n matrix consisting of 2 x 2 blocks
(:;-3.
n.
where bz and cz are real and i = Similarly let F = C", and let C"5 R2" carry column vectors with pth entry x p + iyp into column vectors whose (2p - l)th and (2p)th entries are x P and y p , respectively, where p = 1, . . . , n. One easily verifies that (F,@)is a morphism from the transformation group GL(n,C)x C"+ @" to the transformation group GL(2n, R) x R2" ---t R2". 12.1 Definition: For any complex n-plane bundle [ over a base space X E J the retilijicution iR is the real 2n-plane bundle over X induced by the preceding morphism (r,0). Let I E GL(n, C) be the identity element, so that il E GL(n, C)is scalar multiplication by i = E C. Since (il)' = - I E GL(n, C), the element J = F(i1)E GL(2n, R) satisfies J 2 = - I E GL(2n, R); furthermore J commutes with every element in the image of GL(n, C) 5 GL(2n,R).
J-1
12.2 Definition: Let E 5 X be a coordinate bundle that represents a real m-plane bundle 5 over X E B. A vector bundle morphism E
J
is a comples structure in E 5 X whenever the restriction E x A E x over each .Y E X satisfies J t = - I for the identity element I E GL(m, R). Clearly a complex structure in one coordinate bundle representing 5 induces a complex structure in any other coordinate bundle representing 5, so that one can regard a complex structure as a structure in 5 itself. Equally clearly, the endomorphism J = T ( i I )E GL(2n, R) induces a complex structure in the realification is of the complex n-plane bundle i.We shall show that a real vector bundle has a complex structure if and only if it is of the form iiw for a complex vector bundle i;furthermore, iis unique. 12.3 Lemma: I f J E GL(m, R) suris$es J 2 = - I E GL(m,R), then m is an eiien number 2n und there is a bcisis of R2" of' the form (e,, J e , ; . . . ; e,, Je,).
PROOF: For any nonzero e l E R" suppose there were a linear relation J e , = (A* + l ) r , = ( J 2 + I ) e , = 0. so that I' + 1 = 0, which is
i e , over R; then
1 74
111. Vector Bundles
not possible in R. Hence ( e , , J e , ) spans a 2-dimensional subspace I/ c R", and J induces an endomorphism J of Rm/Vsuch that J 2 = - I . The induction on dimension is clear. It follows from Lemma 12.3 that up to a change of basis in Rz" one has J = T ( i I )for the homomorphism GL(n,C)5 GL(2n, R). We shall henceforth use the basis of Rz" described in the proof of Lemma 12.3. 12.4 Lemma: I f A E GL(2n, R) satisfies AJ = J A fix J = T ( i I ) , then A lies in the image o f G L ( n , C)5 GL(2n,R).
PROOF: For the basis ( e , , J e , ; .. . ; e,,Je,) of R2" there are unique real t i x I I matrices B = (b4,) and C = (c4p) such that Ae, = ~ ~ , , ( b 4 , r GJe,,) q for p = 1 , . . . , n, and since AJ = J A , one also has A(Jr,) = J A e , = ,( - tie,, h4,JeJ; hence A = T(B iC).
+
+
+
12.5 Proposition: A real vector bundle 5 has a complex structure J if' urzd only if' it is the reulijication iw of a complex vector bundle [;furthermore iis unique.
cw,
PROOF: If 4 = then T ( i I ) is a complex structure J in 5. Conversely, suppose that J is a complex structure in 4, and let ( U i l i E I ) be an open covering of the base space X E 9 of 4 for which there are trivializations E I U i 3 U i x R" of a representation E 5 X of 4. The restriction of J to E I U iinduces a map U i 2 GL(m,R) such that (Ti J
c
'€';')(x,e) = (x, Ji(.x)e)
E
Ui x
[w"
for every(x, e) E U i x R", and since J 2 = - I , one has J,(x)' = - I E GL(m,R) for every x E U i . It follows as in Lemma 12.3 that m is an even number 2n and that there is a basis (el,Jie,; . . . ; e,,Jie,) of sections of El U i -5 U i , for each i E 1. With respect to any such basis the matrix representation ( J i ( x ) )E GL(2n, R) consists of n blocks
down the main diagonal, with zeros elsewhere; in particular, (Ji)= ( J j ) over any nonvoid intersection U i n U j . The transition function U in U j --+*1 GL(2n, R) is given by ( Y j TI: ')(x,e)
for (x, e) E (uin uj)x R2",
= (x, Il/i(x)e)
so that the identity
( T j y; 1 ) (yi,, I>
I,
J
c.
yt: 1)
= yj0 J
n
TI:1
= (yj 'I J
8'
y,: 1) . ( y jr y;
1)
175
12. Realifications and Complexifications
implies ( $ i ) ( J i )= (Jj)($fE GL(2rn,R). Since (Ji)= ( J j ) and (Ji)' = - ( I ) , Lemma 12.4 then implies that ($!) lies in the image of GL(n,C) 5 GL(2n, R). The uniqueness assertion is an easy exercise. Briefly, Proposition 12.5 asserts that any complex vector bundle i can equally well be regarded as a real vector bundle with a complex structure J . By Proposition 11.6.19, GL(n,C) and GL(2n, R) consist of one and two r components, respectively, and since the homomorphism GL(n,C) GL(2n,R) necessarily carries the neutral element into the neutral element, it follows that the image of lies in the component GL+(2n,R) c GL(2n,R). Hence Proposition 10.16 guarantees that any realification irW is orientable. In fact, there is a nuturul orientatiori X 5 O(i,) given by letting the image of each s E X be the equivalence class of e l A J r , A e, A J e , A . . . A e, A Je,, E (l\"'E,)* for any representation E X of ilW and any linearly independent elements r , , . . . , el, E E,. According to the discussion preceding Proposition 10.10,if a real m-plane bundle 4 over X E A¶ has an orientation X %O(<),then the oppositely oriented r n m-plane bundle - 5 has the orientation X O(<),where z is the canonical involution of O ( < ) .We have just defined one natural orientation of the of any complex n-plane bundle iover any X E 98.The other realification iiW naturul orientation X + O(iw)of irW is given by letting the image of each x E X be the equivalence classofe, A e , A . . . A e,, A J e , A J e , A . . . A Je, E (A2'' Ex)*. One easily verifies that the two natural orientations of the realification iW of any complex n-plane bundle idiffer by a factor ( - l)n(n-1"2. The preceding discussion suggests the converse question : is a given of a complex n-plane bundle oriented 2n-plane bundle the realification iiW i? Here is the case n = 1.
<
-+
-
<
12.6 Proposition: Any oriented 2-plune bundle unique complex line bundle 3,.
5
is the realificution of' a
PROOF: Let E 5 X represent (, with trivializations El U i 5 U ix R 2 over the sets U i in some open covering { Uil i E I ) of the base space X of 5. For each i E I the trivialization 'Pi provides a basis ( s i , t i ) of the sections Ui -+ E I U i . According to Corollary 10.17 one can reduce the structure group of ( to the rotation group Of(2),which consists of real 2 x 2 matrices cos0 -sin0 (sine cos(i)l
+
so that the bases {si,ti) can be chosen in such a way that s j = sicos 0 i i sin H and t j = - si sin 0 ticos 0 over any nonvoid U i n U j c X , for some o=o; map U i n U j + S1. For each i E I one defines El U i 2 El U i by setting
+
176
111. Vector Bundles
Jisi = ti and J i t i = -si, so that J' is multiplication by - 1 E R. Over any nonvoid U i n U j c X one has J i s j = Ji(sicos I9
+ ti sin 0) = - si sin 0 + ti cos 0 = t j = J j s j
and J i t j = J i ( - si sin 19
+ t i cos 0) = - si cos 0 - ti sin 0 = - s j = J j f j .
Thus J i l U i n U j = J j l U in U j for any nonvoid U i n U j c X , so that Ji and J j are the restrictions J I U iand JI U j of a globally defined complex structure J . It follows from Proposition 12.5 that 5 = EL, for a unique complex line bundle A over X , as asserted. The preceding result occurs only in dimension 2, a reflection of the fact that the obvious homomorphism U(1) -+ O'(2) is an isomorphism. For n > I one has monomorphisms U(n)-, 0+(2n), but not isomorphisms. Furthermore, for n > 1 one can always find an oriented 2n-plane bundle that is not a realification. For example, we shall show in Volume 2 that the tangent bundle r(S4)of the 4-sphere is not a realification. The usual inclusion R -,C of the real field into the complex field provides an obvious morphism (T,@)of the transformation group GL(m,R) x R" -, [w" into the transformation group GL(m,C) x C" C". -+
12.7 Definition: For any real m-plane bundle 4 over a base space X E d the complexijication tCis the complex m-plane bundle over X induced by the preceding morphism.
Recall from Corollary 10.18 that if 5 is a real m-plane bundle over X E g, then the Whitney sum 5 05 is orientable. In fact there is a natural orientation X -,O(5 0 5 ) in which the image of any x E X is the equivalence class of e l A . . . A emA e', A . . . A ek E (A2"Ex)*, where (e,, . . . , em)and (e',, . . . , e l ) are corresponding bases of the fibers over x of the two summands in C: 0(. 12.8 Proposition: For any r e d m-plane bundle 5 one has m ( m - 1 )/Z
50L ,forthe n~itirrnlorientations of' iCR m d < 05, respectivelj.. 5cw
= ( - 1)
PROOF: This is an exercise in definitions, using the other. natural orientation of &. Let GL(n,C)-f, GL(n,C) and C" 3 C"consist of complex conjugations, so that (I-,0)is a morphism of transformation groups, as in Definition 11.2.6. For any complex n-plane bundle the complex conjugate bundle is the bundle induced by applying (I-,@) to (, as in Definition 11.2.8. Any coordinate
r
177
12. Realifications and Cornplexifications
c,
except that scalar multibundle E 5 X representing i also represents plication in each fiber by any complex number x iy is redefined to be scalar multiplication by the complex conjugate x - iy.
+
r,
12.9 Proposition: Let irW and be the naturally oriented 2n-plane bundles ussociatrd to u complex n-plane bundle iand its conjugate respectively; then -
irW = ( - l)"iR.
r,
r
PROOF: The bundles iand trivially have the same realifications, and it remains to verify that the natural orientations differ by the factor ( - 1)". Let E 3 X represent iR, and let X O([,) be its natural orientation. Ife,, . . . , e, are any linearly independent elements of E x , for any x E X , the value of o(x) is the equivalence class of e l A J e , A e , A Je,
A.
.
'
A
e, A Je,
E
(A2"Ex)*,
by definition of 0.Similarly, if X 1,O ( t R )is the natural orientation o f t w ,the value of 7 ( x )is the equivalence class of A
(-Je,)
A e2 A ( - J e 2 ) A '
'
'A
e, A ( - Je,) E
(A2,Ex)*.
The factor ( - 1)" is clear. In the following proof we use a fixed product P Q E G L ( 2 4 C),for which as a morphism of transformation groups with T'A = ( P Q ) A (P Q ) - E G L( 2 4 C)and 0'4 = ( P Q ) e E C2' for any A E GL(2 4 C) and e E C2".If one applies (I-', @') to a given 2n-plane bundle i', then the result is again i'itself: one uses P Q to change the basis of C"in every trivialization of (See Proposition 11.2.9.) any coordinate bundle representing i'.
(r',@') is defined
12.10 Proposition:
<
ioaC = i0 lor any complex vector bundle i.
PROOF: Let (r,0)be the composition of the morphisms used in Definitions 12.1 and 12.7 to define realification and complexification, respectively; for example, the image under of (hY, icY,)E GL(n,C) is the matrix in GL(2n,C) consisting of 2 x 2 blocks
+
where h; and c; are complex numbers whose imaginary parts happen to vanish. Then ilwe is obtained by applying (T,@)to i.Let P E GL(2n,C) satisfy Pe,,- = e p and P r , , = en+, for the usual basis ( e l ,. . . , e,,) of C2", where p = 1, . . . , n, and let Q E GL(2n,C)consist of 2 x 2 blocks
,
178
111. Vector Bundles
down the main diagonal, with zeros elsewhere. If (r,@') is the morphism carrying any A E GL(2n,C)and e E C2" into ( P Q ) A ( P Q ) - E GL(2n,C)and (PQ)e E C2",respectively, then by the preceding remark the result of applying the composed morphism (l-',@') (I-,@) to i is still just i&. However, for any A E GL(n,C ) and e E C"one has
'
and (@' @ ) e = e 0 Z E C2", so that the bundle induced by applying the to iis also i 0 composition (I-',@') 0 (r,@) 0
12.11 Proposition:
& = t;@for any real vector bundle
<.
PROOF: Suppose that t; is a real m-plane bundle over X E d,represented by a coordinate bundle E X with trivializations { Y, I i E I j defined over the sets of an open covering (U,I i E I } of X . Let E' 5 X and E" 3X be the corresponding representations of the realifications SCrwand (F,),, respectively. If { e l , . . . , em} is any basis of the sections U , -+ El U , arising from a trivialization El U , 5 U , x Rm,and if J is the complex structure of tCW, then there are corresponding bases { e l , . . . ,e,,Jel, . . . ,Je,} and { e l , . . . , em,- J e , , . . . , -Je,} for E'I U , and E" I U , . It is clear that the real linear f, isomorphism E'I U , + E l ' ( U , satisfying f,(e,) = e, and f,(Je,) = - Je, for j = 1, . . . , m preserves the complex structure, and that there is a globally defined real linear isomorphism E' 5 E" with f, = fl U , for each i E I , which also and are isomorphic via an preserves the complex structure. Hence tCrw isomorphism that preserves the complex structure, so that t;@= by Proposition 12.5.
(z)@
13. Remarks and Exercises 13.1 Remark: Introductory accounts of vector bundles can be found in Atiyah [2, Chapter I], Dupont [l, Chapter I], M. W. Hirsch [4, Chapter 41, Husemoller [l, Chapter 31, Kahn [l, Chapter 41, Karoubi [2, Chapter I ] , Lang [l, Chapter 31, and Milnor and Stasheff [l, Section 21, for example. 13.2 Remark: Introductory accounts of tangent bundles in particular can be found in Auslander and MacKenzie [l], [ 2 , Chapter 41, M. W. Hirsch [4, Chapter 11, Lang [ 1, Chapter 31, and Lashof [I]. 13.3 Remark: According to the linear reduction theorem (Theorem 11.6.13) the structure group GL(m,R) of any real rn-plane bundle can be reduced to
179
13. Remarks and Exercises
the orthogonal group O(m) c GL(m,R), which maps the ( m - 1)-sphere c R" into itself; moreover, the behavior of O(m) on R" is entirely determined by its behavior on S"- l . With this observation in the background, vector bundles were first considered by Whitney [2,4,5,6] as sphere bundles. In particular, tangent bundles to n-dimensional manifolds were introduced as (n - 1)-sphere bundles, and Whitney's first consideration of Grassmann manifolds appeared within the same framework. One can also consider more general nonlinear sphere bundles that are smooth fiber bundles with fiber Sm- ' (over smooth manifolds), whose structure group is the group Diff S"- of all orientation-preserving diffeomorphisms S m - ' + S"S. P. Novikov [2] and Taniguchi [I] contain examples of such sphere bundles which are genuinely nonlinear: the structure group Diff S"- I cannot be reduced to the rotation group O'(m) ( cO(m)). Pontrjagin preferred the linear fiber R" (and structure group GL(m,R)) in his approach to tangent bundles, in Pontrjagin [l, 3,4,5]. sm- 1
'.
13.4 Remark: Both Whitney and Pontrjagin had the germ of the homotopy classification theorem (Theorem 8.9) in their very first constructions. The first explicit recognition of a homotopy classification theorem (actually an "ersatz homotopy classification theorem") occurs in Steenrod [2], and later in Chern [2, 31 and in Wu [ 5 ] . According to the linear reduction theorem (Theorem 11.6.13) one can reduce the structure groups GL(m,R) and GL(m,@)of real and complex m-plane bundles to the orthogonal group O ( m )c GL(m,R) and the unitary group U ( m )c GL(rn,C),respectively. Since the homotopy classification of such vector bundles is equivalent to the homotopy classification of the associated principal bundles, it follows that the classifying spaces Gm(Rr) and G m ( C z )of Theorems 8.9 and 11.2 are classifying spaces BG in the sense of Remark 11.8.18, for G = O ( m ) and G = U(m). For this reason, one frequently writes BO(rn) and BU(m) in place of G"(R") and G"(C"). 13.5 Remark: Other early definitions oftangent bundles appear in Steenrod [l] and in Ehresmann [5]. Stiefel [l] does not contain any vector bundles per se, in spite of its importance; instead, Stiefel works entirely with vector fields, which were later regarded as sections of tangent bundles. 13.6 Remark: The more general identification of vector bundles with corresponding modules of sections first occurs in Serre [4] and in Swan [13, with the observation that the "locally free" modules 9 of Theorem 4.6 and Proposition 5.6 are precisely the projective modules over the rings C o ( X ) and Cm(X),respectively. In the case of tangent bundles, direct descriptions of corresponding modules of differentials and their dual modules of vector
180
111. Vector Bundles
fields can be found in Osborn [2, 41 and in Sikorski [l]. One can even describe the underlying rings of smooth functions themselves algebraically, as in S. B. Myers [11 and in Nachbin [ 11, for example. Kandelaki [11 gives a sweeping generalization of the results of Serre and Swan, and Lqhsted [l] shows that if X is a finite CW space, then one can replace the ring C o ( X ) in Theorem 4.6 by a subring A c C o ( X ) that is Noetherian, with Krull dimension equal to the dimension of X : vector bundles over finite CW spaces are “algebraic” in a reasonable sense.
13.7 Remark: If E 5 X represents the tangent bundle T ( X ) of a smooth manifold X , then for each x E X the fiber E x is the corresponding tangent space. One can describe E x as the vector space of derivations of the ring of germs of smooth functions at x into the real field R, an observation due to Chevalley and Bohnenblust. (See Chevalley [l, pp. 76-78] for the analytic case, and Flanders [l, pp. 313-3141 for the smooth case.) However, if X is merely a C‘-manifold for 0 < r < 00, then the identification no longer works properly. (See Papy [1, 21, Newns and Walker [ 11, and Osborn [11.) One can impose restrictions on the derivations for which one does recover E x in the latter cases, as in Osborn [11, Sanchez Giralda [I], L. E. Taylor [11, and Ellis [13; however, the simplest procedure is to always let “smooth structure” mean “C” structure.” (Incidentally, “universal” derivations in the sense of Kahler provide further surprises, not only in the cases 0 < r < 00 but also in the cases r = 0 and r = 00; see Osborn [3,5], for example.)
13.8 Remark: Tangent bundles are also more sensitive than one might expect to the global properties of the underlying manifolds: one cannot reasonably ignore second countability (or paracompactness). For example, the tangent bundle of the “long line” is a real line bundle, as one expects; however, it is not the trivial real line bundle. (See Morrow [l].) 13.9 Remark: In spite of the preceding cautionary remarks, even topological manifolds have tangent bundles in a certain sense. The first such definition was given by Nash [11, who successfully generalized constructions we shall describe for the smooth case in Chapter VI. Nash’s definition can also be used to generalize various classical results, as in Fadell [4], R. F. Brown [11, and Brown and Fadell [l]; the smooth versions of these results will appear elsewhere in this work. Milnor later introduced (tangent) rnicrobundles, in Milnor [ l l , 141, which were not intended to be fiber bundles in the sense of Chapter 11. However, results of Kister [l, 23 show that the tangent microbundle of an n-dimensional topological manifold is a fiber bundle with fiber R“; the
181
13. Remarks and Exercises
structure group is not GL(n,R), of course, but the group of a / [ homeomorphisms R"+ [w" that leave the origin fixed. More generally, a fiber bundle with fiber R"' is a topologicul R" bundle whenever the structure group consists of all homeomorphisms R" 4 R" that leave the origin fixed; thus the results of Milnor and Kister assign a tangent topological R" bundle to any n-dimensional topological manifold. Further properties of such tangent bundles appear in Lashof and Rothenberg [l], Kuiper and Lashof [l], Kister [3], Derwent [2], and Kurogi [I], for example. Lashof [l] provides an excellent survey of tangent bundles in general. 13.10 Remark: Vector bundle sums can be constructed by other means than the one used in Definition 2.1. Let R " x R" -, R" map any pair ( ( x O , s I ,... ) , ( y o , y , , . . .)) E R" x R" into ( X ~ , J ~ ~ , X , ., .~.), E, R", and observe that if U c R ' and V c R' are linear subspaces of dimensions m and n, respectively, then the image of U x V c R" x R" is a linear subspace W c R " of dimension m n ; hence there is an induced map G m ( R a )x G"(R ') 1:G" '"(R" ) of Grassmann manifolds. If 5 and 5' are vector bundles of ranks m and n over X E & and ! I X ' E a, classified by maps X Gm(Rz) and X' 2 G"(R ), respectively, then the composition
+
xx
X'
G"l([W') x G"(R")
5 G"+"(R=)
+ 5' over X x X' E d. the preceding (m + n)-plane bundle 5 + 5'
classifies a unique (m + n)-plane bundle 5
13.11 Exercise: Verify that the (m + n)-plane bundle 5 + 5' of Definition 2.1.
is
13.12 Remark: Vector bundle products can be constructed by other means than the one used in Definition 2.2. Let N be the set {0,1,2, . . .} of natural numbers, and let N x N N be any bijection. Let R" x R" -, [W", map any pair ( (xo,xl, . . .), ( y o ,y , , . . .)) E R" x R" into that element of R" whose e(i, j)th entry is XJ, E R for each ( i ,j) E N x N, and observe that if U c R " and V c R" are linear subspaces of dimensions m and n, respectively, then the image of U x V c R' x R" is a linear subspace W t R" of dimension mn; hence there is an induced Segre map C m ( R a )x G n ( R - * ) 5 Gmn( R ') of Grassmann manifolds. If 5 and 5' are vector bundles of ranks m and n over X E d and X ' E d,classified by maps X Gm(Rm)and X ' s G ( R 7), respectively, then the composition
X x X' 5Gm(Ra) x G"(R")
Gmn(R")
classifies a unique (mn)-plane bundle 5 x 5' over X x X ' .
182
111. Vector Bundles
13.13 Exercise: Verify that the preceding (mn)-plane bundle 5 x 4’ is the (mn)-plane bundle 4 x 5‘ of Definition 2.2. 13.14 Remark: Exercises 13.11 and 13.13 provide alternative constructions of the Whitney sum ( 0 5’ and product 5 0 5’ of two vector bundles 5 and i‘ over the same base space X E 9l.If X A X x X is the diagonal map, then one simply sets 0 5’ = A!(( t’)and 0 5‘ = A!(t x 5’) for the bundles 5 5’ and 5 x 5’ over X x X E 9, as described in Remarks 13.10 and 13.12.
+
<
+
13.15 Remark: Direct sums U x V H U 0 Vand tensor products U x V H U 0 V are continuousfunctors in the sense that there are induced maps
Hom(U , U ’ ) x Hom( V , V’)
---f
Hom( U 0 V , U‘ 0 V ’ )
and Hom( U , U ‘ ) x Hom( V , V ’ )+ Hom( U 0 V , U‘ 0 V’), which are themselves continuous, for any vector spaces U , V , U‘, V’. One can therefore use the method in Atiyah [2, pp. 6-91, for example, to construct Whitney sums t 0 4’ and products 5 0 5‘ directly, without introducing 5 + 5’ and 4 x 5’. This construction also appears in Husemoller [ 1, pp. 65-67], for example. The commutative diagram GxF-F
G‘ x F’
A
F’
used in Definition 11.2.6 to define a morphism (I-,a)of transformation groups is a generalized version of the condition describing continuous functors; in particular, the morphisms (r,a) of transformation groups used to construct 5 5’ and 5 x 4’ in Definitions 2.1 and 2.2 are rephrased versions of the continuous functors U x V H U 0 V and U x V H U 0 V , respectively.
+
13.16 Exercise: Verify that the Whitney sum 5 0 (’ and product 5 0 5‘ of Definition 2.4 (or Proposition 2.5) coincide with the Whitney sum 5 0 5’ and product 5 0 5‘ described via continuous functors in Atiyah [2, pp. 6-91. 13.17 Remark: The linear reduction theorem (Theorem 11.6.13) implies that the structure group GL(rn,R) of any real rn-plane bundle 5 over any base space X E 99 can be reduced to the orthogonal subgroup O(m)c GL(rn,R). The existence of a riemannian metric on (Proposition 3.4) leads to an alternative proof of the same result, as follows.
<
183
13. Remarks and Exercises
-
Let E 5 X represent 5, let { Uil i E I } be any open covering of X with R be local trivializations El Ui 2 Ui x R", and let U i x (R" x R") the restriction to U i of a fixed riemannian metric on 5. Let (el, . . . , em)be the usual orthonormal basis of R" with respect to the usual inner product R" x R" 2R, and for each p = 1 , . . . , m and each x E U i let sp(x) = \y; '(x,e,) E E x . Then ( s l ( x ) ,. . . ,s,(x)) is a basis of E x , and one applies the Gram-Schmidt process to obtain a basis (s;(x) , . . . ,s;(x)) that is orthornor< > i R. Thus a ma1 with respect to the inner product (x) x (R" x R") given basis (sl. . . . ,s,) of local sections U j + E provides a new basis (s',, . . . , s;,) of local sections Ui + E , with sb = ;isq for a map U j 4 GL(m,R) carrying each x E U i into a triangular matrix in GL(m, R). If Ui x R" U i x R" carries each (x, e ) E U i x iw" into (x, iLi(x)e) E U j x R", Y A; I the composition El U i4 U ix R" U i x R" is a new local trivialization E I U i 2 Ui x R". There are then new transition functions U in U,i GL(m,R), defined by requiring (Yj '4'- ')(x,e ) = (x, $ii(x)e) for every (x, e ) E U in Ui x R", and one easily verifies that $:J(x)E O(m)for every .Y E U j n Ui. Hence the structure group GL(m, R) of 5 can be reduced to the subgroup O(m) c GL(m, R), as claimed.
3
CT=l
-
r
13.18 Exercise: Carry out the verification required to complete the proof of the preceding reduction theorem. 13.19 Exercise: The linear reduction theorem (Theorem 11.6.13) implies that the structure group GL(n,C)of any complex n-plane bundle 5 over any base space X E J can be reduced to the unitary group U ( n ) c GL(n,C). Prove the same result by imposing a hermitian metric on and following the pattern of Remark 13.17 and Exercise 13.18.
13.20 Remark: The C"(X)-module €(X) of differentials on a smooth manifold X was described in Definition 6.1 1 as the dual of the C"(X)-module 6*(X) of smooth vector fields on X : however, one can also construct & ( X ) directly. The ring Cr>(Xx X ) of smooth functions X x X 5 R is an algebra over the ring C " (X) of smooth functions X A R, with f F E C Y X x X ) defined by setting ( f P ) ( x , y )= f ( x ) F ( x , y )E R for any (x,y) E X x X. Let J c C x ( X x X ) be the ideal of those F E C 7 ( X x X ) that vanish on the diagonal A(X) c X x X , and let €(X) be the quotient C"(X)-module J / J 2 . There is a real linear map C m ( X -, ) J carrying any f' E Cm(X)into the function with valuef( y ) - f ( x ) E R on (x,y ) E X x X , and there is an induced real linear map C" (X) 5 € ( X ) that satisfies the classical product rule d(fy) = f 4 + Ydf.
184
111. Vector Bundles
A smooth map Y X induces an algebra homomorphism C x ( X x X) @* C * ( Y x Y), which in turn induces a module homomorphism A(X)C ’ ( Y), where 8 ( Y ) over the induced ring homomorphism Cz(X) @*f = f @ for any f E P ( X ) , so that @* df = d(@*f). -+
0
13.21 Exercise: Verify that the C“(X)-module d(X) of Remark 13.20 is canonically isomorphic to the C“(X)-module &(X)of Definition 6.1 1. 13.22 Exercise: Verify for any smooth map Y 5 X that the induced homomorphism 8(X) 3 d‘( Y ) of Remark 13.20 agrees up to canonical isomorphism with the homomorphism d‘(X) €( Y) constructed following Proposition 6.14. 13.23 Remark: There is a severe penalty for replacing Cz(X x X) by the subalgebra C r ( X )@ P ( X ) c Cm(Xx X) in the constructions of Remark 13.20. Some of the resulting pathology is described in Osborn [3]. 13.24 Exercise: Let E 5 X and E‘ 5 X represent real m-plane bundles over the same X E B, and suppose that they are described by transition functions U in U j + GL(m,R) whose values are transposed inverses of each other. Show that E 5 X and E’ % X represent the same real vector bundle. 13.25 Exercise : Replace “real” by “complex” in the preceding exercise, using conjugate transposed inverses. Show that the given coordinate bundles represent complex conjugate m-plane bundles. 13.26 Remark: A smooth n-dimensional manifold X is parallelizuble whenever the tangent bundle z(X) is the trivial bundle E” over X. For example, any Lie group G is parallelizable: the left-invariant vector fields provide a basis of sections of z(G). The spheres S’, S3, and S7 are also parallelizable; in fact S’ and S3 are the underlying manifolds of the Lie groups U(1) and U(2), respectively. However, there are no other parallelizable spheres, a result of Bott and Milnor [l] and Milnor [4], which was later simplified by Atiyah and Hirzebruch [3]; a proof will be given in Volume 3. Products of spheres are better behaved. One result of Kervaire [I] is that S P x Sq is parallelizable whenever at least one of the numbers p > 0 or q > 0 is odd. Staples [ 11 gives a simpler proof of the same result. A classical result of Stiefel [11 asserts that every orientable 3-dimensional manifold is parallelizable; more generally, Dupont [3] shows that every orientable (4k + 3)-dimensional manifold has at least three linearly inde-
185
13. Remarks and Exercises
pendent vector fields. Dediu [ I , 2,3] obtains similar results for (4k + 3)dimensional lens spaces, k 1 0 ; lens spaces are defined in Remark 1.10.20. Since parallelizability of the real projective space RP" would imply parallelizability of the corresponding sphere S", it follows that R P ' , R P 3 , and RP' are the only parallelizable real projective spaces. The question concerning more general real Grassmann manifolds is apparently still open.
+
Recall the identity T ( R P " 0 ) c 1 = (n I)?! of Proposition 7.4, where RP" is the Grassmann manifold G'(R"+l) and y: is the canonical real line bundle over G'(R"+'). Show more generally that 13.27 Exercise:
T ( G " ( R ~ + " )0 ) (1): 01.1:)
= (m
+ n)y:,
where is the canonical m-plane bundle over the Grassmann manifold G"( R"' "). :);
+
13.28 Remark: The preceding exercise is not entirely trivial; its solution can be found in Hsiang and Szczarba [11, along with corresponding results for the complex and quaternionic cases. In the latter cases the summand 0j: is replaced by 7; By:, for the conjugate bundle 7;. Different generalizations of these results are given in Bore1 and Hirzebruch [l] and in Lam [4]. ; ! :
13.29 Remark: Whitney sums my! of m copies of the canonical real line bundle 7: over RP" serve other useful purposes. For example, the immersion problem for real projective spaces RP" is equivalent to the problem of finding the largest numbers of linearly independent sections of my: for all m > 0 and I I > 0. The immersion problem for real projective spaces is approached from this point of view in Lam [3] and Yoshida [l, 21. (Recent catalogs of best-possible immersions RP" + R2"-k can be found in Gitler [I], James [ 11, and Berrick [ 11.)
If a complex vector bundle i over a polyhedron IKI has a IKI finite structure group G c GL(n,C),then there is a finite covering X such that the pullback j'!
13.31 Remark: If Y E ,# is a closed subspace of X E YB, then any trivial bundle t? over Y is the restriction to Y of a corresponding trivial bundle E" over X . In general, however, a vector bundle over Y c X is not the restriction to Y of a vector bundle over X,even when X and Y are real projective spaces or lens spaces. Counterexamples can be found in Horrocks [l],
186
111. Vector Bundles
Schwarzenberger [3], Maki [l], and Kobayashi, Maki, and Yoshida [l], for example. There is an especially easy complex counterexample in Schwarzenberger [4,pp. 65-66]. 13.32 Exercise: According to Definition 3.1, a riemannian metric on a real m-plane bundle 4 over X E 93 restricts to an inner product ( , ), for each fiber E x ; in particular, there is a matrix representation of ( , ), as a diagonal matrix with rn positive entries. One can equally well consider nondegenerate metrics of type ( p , 4): the latter requirement is replaced by requiring the existence of a diagonal representation of each ( , ), with p positive entries and 4 negative entries, where p q = m. Show that 5 possesses such a metric of type ( p , 4) if and only if it is the Whitney sum of a real p-plane bundle and a real q-plane bundle.
+
13.33 Remark: According to Theorem 9.5, any (continuous) real vector bundle over a smooth manifold can be represented by a smooth coordinate bundle. Can a (continuous) complex vector bundle over a complex manifold X be represented by a holomorphic coordinate bundle? For complex line bundles the answer is affirmative, and for complex 2-plane bundles the answer is affirmative in the special cases X = CPz (Schwarzenberger [ I , 21) and X = C P 3 (Atiyah and Rees [l]). These results, and others, are presented in detail on pages 111-138 of Okonek, Schneider, and Spindler [l], where it is shown for n 5 3 that every complex vector bundle over CP" can be represented by a holomorphic coordinate bundle. However, a construction of Rees [ 11strongly suggests that for each n > 4 there is a complex 2-plane bundle over CP" that cannot be represented by a holomorphic coordinate bun d 1e. Here is an apparently narrower problem: can a (continuous) complex vector bundle over a complex algebraic variety be represented by a coordinate bundle which is algebraic in the obvious sense? A basic result of Serre [3] implies that this question reduces to the preceding one: a complex vector bundle over a complex algebraic variety is algebraic if and only if it is holomorphic. 13.34 Remark: If E 5 X represents a real m-plane bundle 5, then n is itself a homotopy equivalence; hence the fiber homotopy equivalence relation of Remark 11.8.21 is of little direct interest. However, if E, 2 X represents the corresponding (m - 1)-sphere bundle tS,described in Remark 13.3. the projection n, is no longer a homotopy equivalence. Accordingly, two real vector bundles 5 and v] over the same base space are dejined to be jiber homotopy equivalent (or of the same jiber homotopy type) whenever the state-
187
13. Remarks and Exercises
ment is true of the corresponding sphere bundles 5, and q s ; fiber homotopy equivalence of complex vector bundles is described in terms of their realifications. It is clear that one can equally well consider fiber homotopy equivalence of vector bundles over different base spaces, provided the base spaces themselves are homotopy equivalent in the usual sense. The first major application of fiber homotopy equivalence coincided with its very definition in Thom [4]: t h e j b e r homotopy type of the tangent bundle 7 ( X ) of a smooth closed manijbld X is independent of the smooth structure assigned to X . A stronger version of Thom's result occurs in Benlian and Wagoner [ 1) : ij' two smooth closed manifolds are homotopy equivalent, then their tangent bundles are j b e r homotopy equivalent; a simplified proof of this statement is given in Dupont [2]. Incidentally, Benlian and Wagoner also show that if the given homotopy equivalent manifolds are n-dimensional, and if one of the manifolds has k linearly independent vector fields for some k S ( n - 1)/2, then the other manifold also has k linearly independent vector fields. Fiber homotopy equivalence provides a natural setting for other results. Recall that parallelizability of Lie groups was easily established in Remark 13.26, and that H-spaces are natural generalizations of Lie groups. According to Kaminker [ I], the tangent bundle of any (smooth) H-space is fiber homotopy equivalent to a trivial bundle; this is clearly the natural generalization of parallelizability.
13.35 Remark: In general one cannot strengthen the preceding remark to conclude that a homotopy equivalence X -& X ' pulls the tangent bundle T ( X ' ) back to the tangent bundle 7 ( X ) . However, one does have f ! z ( X ' )= 7 ( X )in certain special cases considered in Shiraiwa [l] and Ishimoto [l]. 13.36 Remark: Two vector bundles 4 and q over the same base space are stably equivalent if 4 0E P = q 0 for trivial vector bundles E~ and c4. For example, Proposition 7.4 asserts that the ( n 1)-fold Whitney sum (n + I)yi of the real canonical line bundle y,! over RP" is stably equivalent to the tanX ' is a homotopy gent bundle 7 ( R P " ) .In Shiraiwa [2] one learns that if X equivalence of closed even-dimensional smooth manifolds such that f ! 7 ( X ' ) is stably equivalent to 7 ( X ) , then the stronger conclusion f ! z ( X ' )= 7 ( X )is valid. There are natural homotopy classification theorems for stable equivalence classes of vector bundles. For each m > 0, let G"(R') -+ G m + ' ( R ' ) classify the Whitney sum 7" 0I;' of the universal real m-plane bundle y"' and the trivial line bundle E ' over Gm(R").In the notation of Remark 13.4, this is a map BO(m) 4 BO(m + l), and one can form the inductive limit
+
188
Ill. Vector Bundles
BO = lim, BO(m). Clearly the stable equivalence classes of real vector bundles over any X E d are classified by homotopy classes of maps X + €30. Similarly, the stable equivalence classes of complex vector bundles over any X E d are classified by homotopy classes of maps X + BU = limffiB U ( m ) , where BU(m) is also defined in Remark 13.4. Stable equivalence classes will be studied in more detail in Volume 3 : they are the very essence of K-theory. However, representations of such classes are of independent interest. For example, Fossum [13 shows that every real or complex vector bundle over any sphere S" is stably equivalent to a vector bundle represented by an algebraic coordinate bundle. In the special case n = 4k > 16 Barratt and Mahowald [l] show that any real vector bundle over S4kis either stably trivial or stably equivalent to a bundle of rank 2k 1 that is irreducible in the sense that it is not a Whitney sum of nontrivial bundles of lower rank; an alternative proof of the same result appears in Mahowald [l]. Glover, Homer, and Stong [l] show for any k > 0 that the tangent bundle r ( C P Z kof ) the complex projective space C P Z k is similarly irreducible, as a complex vector bundle; this partially strengthens a result of Tango [l], that for unj- / I > 2, there is an irreducible complex vector bundle of rank n - 1 over CP".
+
13.37 Remark: Surfaces have special properties as base spaces. Cavenaugh [ 13 shows that every nontrivial real 2-plane bundle over any orientable surface is irreducible in the sense of the preceding remark. Moore [ 13 sharpens the result of Fossum [ 13 as follows: every vector bundle over the 2-sphere Sz can itself be represented by an algebraic coordinate bundle. (The latter result is clearly related to the result of Lernsted [l], cited in Remark 13.6, that any vector bundle over any finite CW space is "algebraic" in a reasonable sense.) 13.38 Exercise: Let X be an rn-dimensional CW space as in 01.5, so that X has no n-cells for any n > rn, and for some fixed n > rn let 5 be a real nplane bundle over X . Use induction on p = 1, . . . , rn and the fact that every map Sp-' -+ S"-' is homotopic to a constant map for p < n to show that 5 has a nowhere-vanishing section. (This is an easy exercise, which is carried out in Husemoller [l, p. 991, for example; it will also be done in detail in Volume 2 of the present work, for any rn-dimensional weak simplicia1 space
x.1 13.39 Exercise: Use Exercise 13.38 and Proposition 4.8 to conclude that if > H I , then any real n-plane bundle 5 over an m-dimensional CW space X is of the form q 0 I . : " - ~ for some real m-plane bundle q over X . 11
189
13. Remarks and Exercises
13.40 Remark: The geometric dimension of a real vector bundle 5 is the least integer n such that 5 is stably equivalent to a real n-plane bundle r ] ; in case 5 is stably trivial its geometric dimension is 0. For example, according to Exercise 13.39 the geometric dimension of any vector bundle over an m-dimensional CW space is at most m. However, stronger results are possible in some cases: Remark 13.36 asserts for k > 4 that the geometric dimension of any real vector bundle over the sphere S4kis either 0 or 2k 1. There are other base spaces over which all real vector bundles have severely limited geometric dimensions; several examples can be found in Sjerve [l], Hill [l], and Davis and Mahowald [11.
+
13.41 Remark: Given an m-plane bundle 5, a stable inverse of 5 is any nplane bundle r], over the same base space and for some n 2 0, such that 5 0r] = em+''. The existence of stable inverses of real m-plane bundles over certain kinds of base spaces was established in Corollaries 8.13,9.2, and 9.4, and there were indications of upper bounds on n. In the following situation one can take n = m.
For any X E W and any point * E X the (reduced) suspension EX is the quotient of the product X x [0,1] by the subspace X x (0) u {*} x [0,1] u X x { 1), in the quotient topology. If X E W is a compact hausdorff space, and if 5 is a complex m-plane bundle over EX, then there is a complex mplane bundle r] over EX such that 5 0 r] = cZm. A proof is given in Chan and Hoffman [ 13. 13.42 Remark: The equivalence relations of Remarks 13.34 and 13.36 lead to a useful weaker equivalence relation. Two vector bundles 5 and r] over the same base space are J-equivalent (or stably jiber homotopy equivalent) if there are trivial bundles c p and cq such that 5 0cP and q 0c4 are fiber homotopy equivalent in the sense of Remark 13.34. For example, the theorems of Benlian and Wagoner [l] described in Remark 13.35 first appeared as Jequivalence theorems in Atiyah [13 and Sutherland [11, respectively. The following J-equivalence theorem is due to Atiyah and Todd [l] and to Adams and Walker [l] with a later simplification in Lam [2]: a necessary and sufficient condition that the m-fold Whitney sum my,' of the canonical complex line bundle y: over CP" be J-equivalent to a trivial bundle is that m be divisible by an integer i(m,n) defined in Atiyah and Todd [l]. An application of this result will be indicated in Remark VI.9.22. 13.43 Exercise: Let Y 5 E be the zero-section of a smooth coordinate bundle E 5 Y representing a real vector bundle r] over a smooth manifold Y. Show that & ( E ) = r] 0 T( Y ) for the homotopy equivalence r ~ .
190
111. Vector Bundles
13.44 Remark: Let X Y be the restriction of a smooth embedding X + R" of a smooth manifold X to an open tubular neighborhood Y c R" of f ( X ) c R", so that j is a homotopy equivalence. Then any vector bundle 4 over X is of the form f ' q for a vector bundle q over Y, and since T( Y ) = E" over Y the preceding exercise implies ((T f)!z(E) = 5 @ E" over X . Thus, up 0
to the homotopy equivalence (T f, any vector bundle 4 over X is stably equivalent to the tangent bundle of a smooth manifold E. 0
CHAPTER IV
Z/2 Euler Classes
0. Introduction Let H * ( X ; 2/2) be the singular 2/2 cohomology ring of a base space X E 9. In this chapter we assign a cohomology class e(() E H " ( X ; 2/2) to any real n-plane bundle 5 over X.Such classes will be used in Chapter V to obtain further 2/2 classes w ( 5 ) = 1 + w l ( l ) + . . . + w,(() E H * ( X ; L/2) for <, with w,(() = e(t), which have many useful geometric applications. Let E X represent 5, and let E* c E consist of the nonzero elements in E. The Mayer-Vietoris technique is used in $1 to construct a relative cohomology class U , E H"(E, E * ; 2/2), whose properties are further described in 92. The zero-section X 5 E and inclusion E 2 E, E* then provide a composition
H*(E, E * ; 2/2) 3 H*(E; 2/2) 5 H * ( X ; 2/2), hence a class e(5) = o*j*U, E H " ( X ; 2/2) as desired. For the universal real line bundle y1 over RP" (Definition 111.8.2) the class 4 7 ' ) E H'(RP" ; 2/2) plays a privileged role: the cohomology ring H*(RP' ; 2/2) is the polynomial ring 2/2 [e(y')] over 2/2 generated by e(7')E H ' ( R P " ; 4'2). Similarly, for the canonical real line bundle y: over RP' (Definition 111.7.2) the class e ( y i )E H'(RP' ; 2/2) generates H * ( R P ' ; 2/2), subject to the condition e(y:) u e(yi) = 0. These properties of e(y') and e(y:) lead to an axiomatic characterization of all the classes e ( 0 , given in $6. The notations H*(-,-) and H*(-) are used coitsistently in this chapter, and in the remainder of this volume, to denote singular cohomology 191
192
IV. 212 Euler Classes
H*(-,-; 2/2) and H*(-; 2/2) with 2/2 coefficients. Other coefficient rings will appear in Volumes 2 and 3.
1. The Z/2 Thorn Class U , E H"(E,E*) Let E 5 X represent a real n-plane bundle 5 over a base space X E B, and let E* consist of the nonzero elements in E . We use the local behavior of H*(E, E*) to construct a class U , E H"(E, E*). Specifically, for each x E X let E x = n - ' ( { x ) )and E: = E x n E*, and let E x , E: 2 E, E* be the inclusion of pairs; then U , will be defined by the behavior of the images of the homomorphisms H"(E,E * ) 3 H"(E,, E:). An obvious first step is to describe the 2/2 cohomology modules H*(E,, E:), each of which is isomorphic to H*(R", R"*). 1.1 Lemma: lj' n > 0, the 2/2-module H*(R",W*) vanishes except j o r single generator in H"(R", R"*).
LI
PROOF: The excision and homotopy axioms for cohomology permit one to replace the pair R",R"* by the pair D", S n - ' , where D" c R" is the unit n-disk and its boundary S"-' c R"* is the unit ( n - 1)-sphere. For the inclusions S"- 5 D" and D" 4 D", S"- the usual exact cohomology sequence
'
'
L+H4-1( D " )
Hq-l(sn-1 )
d
+ Hq(D",S"-')
a
i* Hq(D") + ...
contains many 0's; in particular, all terms with q < 0 vanish. In case y > 1 the result follows from the portion
0
-
Hq-'(S'-')
I5
4
Hq(D",S"-')
-
0
of the preceding exact sequence and the known behavior of H q - '(S"- '). For smaller values of q the exact sequence is d
0 + Ho(D",S"-') d --+
H'(D",S"-')
Ho(D")
4
5 Ho(S"-')
0.
If n > 1, then Ho(D") 2 Ho(S"- ') is an isomorphism, so that Ho(D",S"- ') = 0 = H'(D", S"- '). If n = 1, then one has H o ( D ' ) = 2/2, Ho(So)= 2/2 02/2, and i*(a) = a 0 ( - a ) for any a E H o ( D 1 ) ;hence necessarily H ' ( D ' , So)= Z/2 and 6(a 0b) = a + b for any a 0 b E Ho(So). Let E 5 X represent a real n-plane bundle over X E as before.
and define E* c E
I . The 2/2 Thorn Class
1.2 Lemma:
I/: E H"
193
( E , E* )
HP(E,E*) = 0 jbr p < n.
PROOF: Since H P ( E , E * ) depends only on the homotopy type of the base space X E -?8, one can assume that X is of finite type as in Definition 1.1.1. (This normalization is described in detail in the proof of Theorem 11.7.2, the absolute Leray-Hirsch theorem.) Let O(X ) denote the category of open sets U c X , and let E I U = n - ' ( U ) and E* 1 U = n - ' ( U ) n E* for any U E O ( X ) . There is then a Mayer-Vietoris functor {hqlq E Z)on O ( X ) to Z/2-modules, with hq( U ) = Hq(E I U , E* I U ) for any U E O ( X ) ;the connecting homomorphisms are the classical ones for relative cohomology, where E* I U is open in E I U . There is also a trivial Mayer-Vietoris functor { k4 I q E Z)on O ( X ) to 7/2-modules, with P ( U )= 0 for any U E 8 ( X ) . Finally there is a unique natural transformation 0 from {hq I 4 E Z ) to { k4 I q E Z)which annihilates everything. If U E O ( X ) is contractible to a point x E U , then hq( U ) = H q ( E x ,E:) by the homotopy axiom for cohomology, where Hq(Ex,E:) = Hq([Wn,[Wn*) = 0 for q < n by Lemma 1.1. Thus hq(U)-% P ( U )= 0 is an isomorphism whenever U is contractible and q n, so that W ( E ,E*) = hq(X) -%P ( X )= 0 is also an isomorphism whenever q < n, by Corollary 1.9.5.
-=
The main result of this section is obtained by refining the trivial MayerVietoris functor { P l q E Z} of the previous proof. For any U E O ( X ) let kq( U ) = 0 as before except in the case q = n, and let k"( V) be the Z/2-module of functions U --+ 2 / 2 which are continuous in the discrete topology of Z / 2 . For any ordered pair ( U , V ) E O ( X ) x O ( X ) the sequence 0-k"(Uu
V ) ~ k " ( U ) O k " ( V ) ~ k " ( U n l / ) - O
is trivially exact for the Mayer-Vietoris homomorphisms ",:i and j:," of 91.9, so that { k 4 ) qE Zj is still a Mayer-Vietoris functor. There is a natural transformation c) from the old {h4 q E Z)to the new { k4 q E Z>which annihilates hq(U ) for every U E O ( X )and every 4 # n. To define h"( U ) 2k"( U )
I
I
let E x , E x 5 El U , E* I U be the inclusion of pairs for any x E U , where H"(Ex,E:) = 7/2 by Lemma 1.1, and for any rx E h"(U) = H"(E1 U , E* 1 U ) let 6c,c be that function U --+ Z/2 in k"( U ) with value j,*a E Z/2 on x E U . 1.3 Proposition: I f E X represents a real n-plane bundle 5 over a base space X E J, then there is a unique class U , E H"(E,E*) such that j:U, E H"(E,, E:) is the generator of the Z/2-module H * ( E x ,E:) for each x E X .
PROOF: As in Lemma 1.2 one can assume that X is of finite type, and for any Contractible U E O ( X )one has isomorphisms h4(U ) 3 k'J(U ) whenever 4 5 n, including the case q = n; hence Corollary 1.9.5 provides isomorphisms
194
IV. 2/2 Euler Classes 0
hq(X)A kq(X) for q S n, including the case q = n. In particular the isomorphism h " ( X )-% k"(X) carries a unique cohomology class U , E h"(X) = H"(E, E*) into the constant function X -,212 with value 1 E L/2.
1.4 Definition: If E 1X represents a real n-plane bundle 5 over a base space X E a, then the 212 Thom class of 5 is the unique class U , E H"(E, E*) such that j,*U, E H"(E,,E,*)is the generator of H*(E,, E,*) for each x E X. The coordinate bundle E 5 X of Definition 1.4 induces an isomorphism H*(X) 5 H*(E) of cohomology rings, which one can combine with the relative cup product H*(E) 0 H*(E, E * ) 1;H*(E, E * ) to regard H*(E, E * ) as an H*(X)-module. Specifically,the product ofany c1 E H*(E, E * ) by any scalar p E H*(X) is n*P u a E H*(E, E*).
1.5 Proposition (A Relative Leray-Hirsch Theorem): ! f E 5 X represents 5 over X E 9,then the H*(X)-module H*(E, E*) is the free H*(X)-module generuted by the Thom class U , E H"(E, E*). u real n-plune bundle
PROOF: For each x E X the class jZU, E H"(E,,E,*) generates the 2/2module H*(E,,E:), by Proposition 1.3, and the remainder of the proof is identical to that of Theorem 11.7.2 (the absolute Leray-Hirsch theorem) with the singleton set { U , } in place of { a l , . . . , N ~ } .
2. Properties of 2 / 2 Thom Classes We now establish two properties of 212 Thom classes. In order to formulate the first property suppose that X' X is a map in the category a of base spaces and that E 1X represents a real n-plane bundle 5 over X . If E' 5 X' represents the pullback f!
X'
r
s
b
E
+x
commutes as in Lemma 11.1.3. In particular, f carries the set E'* c E' of nonzero elements into the corresponding set E* c E, so that there is an
195
2. Properties of 2/2 Thom Classes
induced map E', E'* 5 E, E* of pairs. Consequently there is also an induced map H"(E, E*) 5 H"(E', E'*) in 2/2 singular cohomology. 2.1 Proposition: Let E 5 X represent a real n-plane bundle t over a buse f ,X be any map in B, and let E' 5 X' represent the space X E a,let X ' pullback j ' t over X' E a.Then U f ! <= f*U, E H"(E',E'*), for the E / 2 Thom classes U , E H"(E, E*) and U f ! ,E H"(E', E * ) .
PROOF: For any y
E
X' there is a commutative diagram E, E*
f
E', El*
where f,, is the restriction o f f to the fiber pair over y, hence a commutative diagram H"(E, E*)
I ' H"(E', E'*)
4.
J.
H"(E/(YPE[S.(YJ
"
H"(Ey,EL*).
Since f,, is a homeomorphism of the pair E,,, Ey*, the induced isomorphism f; carries the generator j[S.cy,Urof H"(Ef(,,,,E[S.,,,)into the generator of H"(Ey, E;*); thus j;(f*U,) is the generator of H"(E;,E;) for each y E X ' . According to Proposition 1.3, however, the Thom class U !< E H"(E', E'*) is the unique class with this property of f*U, E H"(E',E'*). In order to formulate the second property of 2/2 Thom classes we recall that if X E and X ' E B, then X x X ' E 98, by Proposition 1.1.4. We also recall from Definition 111.2.1 that if E 3 X and E' 2 X ' represent real vector bundles 5 and t', then one applies a morphism (r,O) of transformation n x n' X x X ' to obtain a coordinate bundle groups to the product E x E' representing a real vector bundle 5 5' over X x X ' ; and O are the obvious maps GL(n,R) x GL(n',R) --* GL(n,R) 0 GL(n',R) c GL(n + n', R) and R" x R"' -, R"'"', respectively, which can conveniently be ignored. If E* and E'* are the sets of nonzero elements of E and E', respectively, then E* x E' u E x E'* is the set of nonzero elements of E x E', so that
-+
( E , E * ) x (E', E'*)
=(E
x E', E* x E' u E x E'*) = ( E x E', ( E x E')*).
I96
I V . 2 / 2 Euler Classes
2.2 Proposition: I f E 1X and E' 5 X ' represent real vector bundles [ and <' over X E 9 und X' E 99,respectively, then the cross product of the 212
Thom clusses U , E H"(E,E*) and U,. E H"'(E',E * ) satisfies U , x U,,
=
U,,,. E H"+"'(E x E , ( E x E ) * ) .
-
x X ' let E x , E: % E, E*, Ek,, E:* A E', E'*, and E;;,,,), E;;:,,) E', E"* be inclusions of fiber pairs, where E" = E x E'. Then j ( x , x , = , j, x j,,, so that
PROOF: For any point (x,x')E X Jtx.x')
j&xs)(uc x Ute)= ( j , x jx,)*(ut x uct) = ((j , x jx.)* x )( Ur 0 U,.) =(x
0
= j:U,
( j ; 0 j : < ) ) ( U0 , uts) x j:,U,?,
by naturality of the cross product. Since j:U, E H"(E,, E:) and j,*,U,. H"'(E,.,E:) are generators, by definition of Thom classes, and since
H*(E,, E:) 0 H * ( E : . , E ! )
E
5 H*(EY!,,,),E{;*,,))
is an isomorphism by the Kiinneth theorem, it follows for each (x,x') E X x X' that j&,xrJ U g x Ut,) is the generator of H"+"'(E{L,xj,, E;:,.+,)). However, U t + i ,E H"+" (E",E"*) is uniquely characterized by this property of U , x U,,, so that U , x U,. = Lit+,., as claimed.
3. 2/2 Euler Classes We now construct the most accessible characteristic classes of real vector bundles, establish two of their algebraic properties, and apply those properties to obtain a necessary condition for the existence of nowhere-vanishing sections of real vector bundles.
3.1 Definition: Let E X represent a real n-plane bundle ( over X E B, with 212 Thom class U , E H"(E, E*), let X 5 E be the zero section, and let E L E, E* be the inclusion. Then the H/2 Euler class e ( ( )E H " ( X ) is given by setting e(<)= o*j*U,. The trivial property that e(<)belongs to H " ( X ) c H * ( X ) will play a role in the axiomatic characterization of 212 Euler classes, given later. Here is the first nontrivial property of H/2 Euler classes.
197
3. E / 2 Euler Classes
Let X ' X be any map in the category of base .spaces, let he a real n-plane bundle over X , and let f ! t be its pullback. Then e ( j " < )= ,f*e((), f b r the 2/2 Euler classes e(5)E H " ( X ) und e ( f ! t ) E H"(X').
3.2 Proposition (Naturality):
<
PROOF: If E' 1;X ' represents f ! < , then as in 42 there is a map E' + E which is a linear isomorphism on each fiber, inducing a map E', E'* 5 E , E* of pairs such that the diagram E, E* & E', E'*
Jl
E'
E
X
C
I
X'
commutes, for the zero section 0' and inclusion j' associated to f ' t . Since U f 8 ;= f*U, E H"(E',E'*) by Proposition 2.1, it follows that e ( f ' t )= a'*j'* U,!; = a'*j'*f* U t = f*o*j* U , = f * e ( < ) ,as claimed. One easily shows the equivalence of the following two properties of 212 Euler classes, and we identify both of them by the name "Whitney product formula." There will be many further "Whitney product formulas" throughout this work.
3.3 Proposition (Whitney Product Formula): Let E X and E 5 X ' represent r e d vector bundles and <' of' ranks n und n' over base spaces X E a n x n' and X ' E d?,respectively, so that E x E' X x X ' represents the bundle + <' rank n + n' over X x X ' E B. Then the cross product of the 212 Euler clusses e(5)E H " ( X ) and e(4') E H"'(X') satisfies
<
<
-
e(5) x e(5') = e(5 + 5') E H"+"'( X x X ) . PROOF: Let X 5 E and X ' 5 E be the zero sections, and let E E , E* and E' 5 E', E'* be the inclusions, associated to ( and to t', respectively. Since ( E ,E * ) x (E',E'*) = ( E x E'), ( E x E')*, as in Proposition 2.2, it follows that X x X ' ""4 E x E' and E x E' 5( E ,E*) x ( E ,E'*) are the zero section and inclusion associated to 4 + t'. According to Proposition 2.2 the 2/2 Thom classes satisfy U , x U,.
=
UE+,E , H"(E x E', ( E x E')*).
198
IV. 212 Euler Classes
so that by naturality of the cross-product one has e(5) x e(5') = a*j*Ug x a'*j'*Ug. = ( x - (a* 0 a'*) 'J ( j * Of*))( u, 0 Ugr) = ((a x a')* 0 ( j x j')* c' x )(V, 0 U,.) = (a x a')*(j x j')*(UC x U,,) = (a x a')*(j x f)*U,+,, =4
5 + 5'1,
as claimed. Here is an equivalent formulation of Proposition 3.3.
3.4 Proposition (Whitney Product Formula): Let 5 and 5' be real vector bundles over the same base spuce X E a,and let 5 04' be their Whitney sum. Then the cup product of the El2 Euler classes e(5) E H " ( X ) and e(<')E H " ' ( X ) satisjes e(5) u e(5') = e(5 05') E H"+"'(X).
PROOF: If X 3 X x X is the diagonal map, then
u 4 5 ' ) = A*(e(t) x e(T'))= A*e(5 + 5') = e ( N 5 + 5')) = 4 5 0 5'1, by the definition of the cup product, Proposition 3.3, Proposition 3.2, and the definition 5 0 5' = A!(< <'), respectively.
+
The following lemma will be used to establish a necessary condition for the existence of nowhere-vanishing sections of real vector bundles.
3.5 Lemma: I f E" is the trivial real n-plane bundle over a base space X for any n > 0, then e(E") = 0 E H " ( X ) .
E
93,
PROOF: E" can be classified by a constant map into G"(Rm), which can be expressed as the composition X {*) 3 G"(W), where * E G"(R") is the constant value and g is the inclusion; that is, E" = f ! g ! y " for the universal nplane bundle y" over C"(W). Since H"((*}) = 0 one has e(g!y") = 0, hence e(?) = e(f!y!y") = f'*e(g!y")= 0 by Proposition 3.2.
3.6 Proposition : I f u real n-plane bundle 5 over a base space X E 93 admits a nowhere-vanishiny section, then the 212 Euler class vanishes: e(<)= 0 E H " ( X ) . PROOF: By Proposition 111.4.8,t = c1 0 q for the trivial line bundle e1 and some (n - 1)-plane bundle q ; hence Proposition 3.4 and Lemma 3.5 yield e(5) = e(c1 0r ] ) = e(E1)u e(q)= 0.
4. Gysin Sequences and H*( RP" ; 2/2)
199
According to Proposition 1.5, if E 5 X represents a real n-plane bundle 4 over X E B, then the H*(X)-module H*(E, E*) is free of rank one, the unique generator being the 212 Thom class U c E H"(E, E*). It follows that cup product by U ; induces an H*(X)-module isomorphism H*(X)+ H*+"(E,E*).
3.7 Definition: Let U , E H"(E, E*) be the 212 Thom class of a coordinate bundle E 5 X representing a real n-plane bundle 5 over a base space X E @I. The 212 Thom isomorphism H*(X) 2H*'"(E, E * ) maps any p E H*(X) into n*[j u U , E H*'"(E,E*). The L/2 Thom isomorphism provides a new characterization of 2/2 Euler classes. as follows.
3.8 Proposition: I f E : X represents a real n-plane bundle 5 over X E @I, @e with 212 Thom class U , E H"(E, E*) and 212 Thom isomorphism H*(X) 4 H*+"(E,E*), then the 212 Euler clu.s.s e ( < )E H"(X) is given by e(4) = 0;I ( u, u U , ) jiw the cup product U ; u U , E H2"(E,E*).
PROOF: Since E X E is homotopic to the identity, one can apply Definition 3.7 to e(<)= a*j*U, E H"(X) to compute
4. Gysin Sequences and H*(RP" ; Z/2) The goal of this section is to compute the 212 cohomology of real projective spaces, including RP". 4.1 Definition: Let E : X represent a real n-plane bundle 5 over X E @I, let E* be the nonzero portion of E, let H*(E*) 5 H*"(E,E*) be the connecting homomorphism in the 212 cohomology exact sequence of the pair E, E*, and let H*(X) 2H*+"(E,E*) be the 2/2 Thom isomorphism of Definition 3.7. The Gysin map H*+"(E*)3 H*"(X) is the composition 0,5'S:
H* + "(E*)5 H* + n + 1(E,E*) 2H*+ 1(XI. Trivially the Gysin map Y, is a Z/2-module homomorphism of degree 1 - n.
200
IV. 2/2 Euler Classes
4.2 Proposition (The Gysh Sequence): Let E 5 X represent a real nplane bundle 5 over X E g,let E* 5 X be the restriction of'n to the nonzero H*'"(X) be cup portion of E, let \y; be the Gysin map, and let H * ( X ) product b y the 212 Euler class e(5)E H"(X). Then the sequence
...
+
Hq+n-'(E*)
& Y p(X)
- H4+"(X)
t*
H 4 + " ( E * )+
...
is exact. PROOF: The bottom row of the accompanying diagram is the 212 cohomology exact sequence of the pair E, E*,
...
...
-
1
, H4(X) 3+ Hq+"(X)
H'?+n-I(E*)
I*
id
-
1
Hqtn(E*) id
H4fn-I (E*) 2 H4+"(E,E*) & H4+"(E)2 H q + n (E*)
__f
and the vertical maps are all isomorphisms. The left-hand square commutes by definition of Y,, and the right-hand square commutes because E* 5 X is the composition of the inclusion E * A E and the projection E X . To show that the middle square commutes, observe that if X 5 E is the zero a n section, then E E is homotopic to the identity, so that the endomoro ' n H"(E) is the identity; hence for any 4, E H q ( X )one has phism H"(E)
-
n*(B u e ( 5 ) )= n*P u n*(o*j*U,) = n*P u j * U , = j*(7C*P u U,) =
j*o,p,
as required. 4.3 Proposition: Let yl be the universal real line bundle over RP", with 212 Euler class e(y') E H'(RP"). Then the 212 cohomology ring H*(RP") is a polynomial ring in a single variable over 212, the generator being e(y').
PROOF: I f E 5 RP" represents y l , then E* = Ra*, which is contractible, as in Proposition 111.8.11. The Gysin sequence (for n = 1) is then of the form
- -
Hq(Rm*)
q.,,'
Hq(RP")
we(,,')
H q + ' ( R P " ) 5 H4"(Rz),
where Hq+'(R"*) = 0 for q 2 0 , and where Hq(Ra'*)= 0 for q > 0; in particular, u e ( y ' ) is an isomorphism for q > 0. Furthermore, the beginning of the Gysin sequence is Y
-
HO(RP") 5 H O ( R ~ *A ) HO(RP~)
UP(:l')
.
H I ( R P = )-+ 0,
where the initial homomorphism E* is the trivial isomorphism B/2 + B/2, so that we(?') is also an isomorphism for q = 0. Since H o ( R P ' ) = 2/2, this completes the proof.
.. .,
20 1
5. The Splitting Principle
A similar method applies to the 2/2 cohomology ring H*(RP") for any finite n > 0.
4.4 Proposition: For any n > 0 let y j be the canonical line bundle over RP", with 212 Euler cluss e(y:) E H ' ( R P " ) . Then the 212 cohomology ring H * ( R P " ) is a truncuted polynomial ring in a single variable over 212, the generator being e(y,l,)and the relation being e(yf)"+' = 0. PROOF: I f E: RP" represents it!, then, as in Proposition 111.8.1 1, E* is a quotient R'"+ "* x OX*/%, with (x,s) % (y,t ) if and only if sx = ty E R("+I ) * ; that is, E* = R("+')*, which is trivially homotopy equivalent to the sphere s". Since H4(S")= 0 for 0 < 4 < n, the argument used for Proposition 4.3 shows that Ho(RP") = 2 / 2 , that Hq(RP") H 4 + ' ( R P " ) is an isomorphism for 0 S 4 < n - 1, and that H"-'(RP")H"(RP") is a monomorphism; the latter conclusion implies that H"(RP") contains 2 / 2 as a submodule. Since RP" is an n-dimensional CW space, by Proposition 1.5.5, one has HY(RP")= 0 for q > n. In particular, since H " + ' ( R P " ) = 0, the Gysin sequence terminates in the fashion H"(S")2H"(RP") + 0, and since H"(S") = 2/2, the monomorphism H " - ' ( R P " ) H"(RP") is in fact an isomorphism, which completes the proof.
-
-
4.5 Corollary: For any n > 0 let RP" A RP" classify the canonical line bundle y: over RP"; then for any p 2 n the induced 212-module homomorphism H P ( R P 7 )A H P ( R P " )is an isomorphism carrying e(y')P into e(y:)". PROOF: H P ( R P " )is free on the single generator e(y')", and H P ( R P " )is free on the single generator e(y:)". Since y: = ,f!yl, it follows from the naturality of212 Euler classes (Proposition 3.2) that e(y:) = .f*e(y'), hence that e(y,!)P= j ' * e ( y ' ) P , as required. Observe that Propositions 4.3 and 4.4 establish the existence of nonzero 2/2 Euler classes, hence that the condition of Proposition 3.6 for the existence of nowhere-vanishing sections is nonvacuous. In particular, y' and y: admit no nowhere-vanishing sections; a fortiori, they are not trivial line bundles.
5. The Splitting Principle In general, an n-plane bundle 5 over a base space X E 93 is not a Whitney sum of n line bundles. Nevertheless, one can always find a map X ' 4 X in the category 9 such that the pullback 9'4 over X' E B is a Whitney sum of n line bundles, and such that the induced homomorphism H * ( X ) 5 H * ( X ' ) of 2/2 cohomology rings is a monomorphism. This result will be used often
202
IV. L / 2 Euler Classes
to verify properties of cohomology classes assigned to arbitrary vector bundles: one need only consider sums of line bundles. The construction of g will itself be used in the next chapter to define further 2/2 characteristic classes. For any n-plane bundle over a base space X there is a corresponding fiber bundle P , over X with fiber RP"- ',the group being the projective group. To construct P , let E 5 X represent and let E* c E consist of nonzero points, as usual. For each x E X one identifies two points of E: c E x if and only if one point is a real (nonzero) multiple of the other; the total space of P , is the quotient of E* by this equivalence relation, in the quotient topology. There is an obvious induced projection onto X, each fiber is homeomorphic to RP"- ',and the action of GL(n, R) on the fibers in E induces an action of the corresponding projective group on the fibers in P , .
<
<
5.1 Definition: Given any real n-plane bundle ( over X E 98,the preceding bundle P, over X is the projective bundle associated to 5.
For convenience we use the symbol P , ambiguously to denote either the projective bundle associated to or to denote its total space; in most of what follows P , will represent the total space. Since the fiber RP"-' of the projective bundle P , is a finite CW space, by Proposition 1.5.5, one easily shows that the total space P , itself belongs to the category W of base spaces. Let P, X represent the preceding projective bundle over the base space X. One can pull the original n-plane bundle back along ,f to obtain an n-plane bundle f!( over P , . One can also identify an especially attractive line subbundle 2, off!<, as follows. The elements of P , can be regarded as 1-dimensional subspaces L of the fibers E x c E, and the total space of f ! ( consists of those pairs ( L ,e) E P , x E such that L c E x and e E E x ; the total space of the line subbundle 2, is then defined to consist of those points ( L ,e ) E P , x E such that e E L , in the relative topology of the total space off!(.
<
<
5.2 Definition: For any real n-plane bundle ( over a base space X E 94, the preceding line sub-bundle 2, off!( over the base space P , E W is the splitting bundle of <; the 2/2 Euler class e(&) E H'(P,) is the splitting class of 5. The 2/2-module H*(P,) becomes an H*(X)-module with respect to the product /? * a = I*/? u a E H * ( P , ) of /? E H * ( X ) and a E H*(P,). 5.3 Proposition: For any real n-plane bundle ( over a base spuce X E 28,the H*(X)-module H*(P,) is free on the basis { l ,e(2,), . . , , e ( 2 , J - ' ) , where e(&) E H'(P,) is the splitting class of <.
PROOF: For each x E X let P,,x 5P , be the inclusion of the fiber of the projective bundle over X . Then if E L P , represents the splitting bundle 2,
203
5 . The Splitting Principle
over P;, there is a pullback diagram
1_
jx
P,,X ' pr in which Pc,x is homeomorphic to R P - ; up to this homeomorphism the map E x 5P,,x represents a bundle j& which corresponds to the canonical line bundle y:-l over RP"-'. Since H * ( R P " - ' ) is a free Z/2-module on the basis { 1, e(y:- 1), . . . , e(y,f- l ) n - l 1, by Proposition 4.4, it follows that H*(P,+,) is a free Z/2-module on the basis { 1, e(j&), . . . , e(jilt)'- >.Since e(j&) = j;e(A,) by naturality of 2/2 Euler classes (Proposition 3.2), we have therefore shown for each x E X that the ZJ2-module H*(P,,,) is free, with basis {j;l,,j;e(Q, . . . ,j;e(jLr)'-' ). Hence the absolute Leray-Hirsch theorem (Theorem 11.7.2)guarantees that H*( P r ) is a free H*(X)-module, with basis [ 1, e(E.;),. . . , e(A,J-' i, as claimed.
'
I 5.4 Corollary: Let P , + X represent the projective bundle associated to a real n-plane bundle t over a base space X E a;then the induced homomorphism H*( X) -5 H*( P r ) of 212 cohomology modules is a monomorphism.
PROOF: The image of .f'* is precisely the free H*(X)-submodule of H*(Pr) spanned by the element 1 in the basis { 1, e(J.<), . . . ,e(A,)"- ' of H*(Pr).
>
We have reached the goal of this section.
5.5 Proposition (The Splitting Principle): Let t,, . . . ,t, be finitely many real cector bundles over the same base space X E a.Then there is a map X' 3 X in J such that each of' the pullbacks g ' t , , . . . , g't, over X' E is a W i t n e y sum of line bundles, and such that H*(X) 2H*(X') is a monomorphism of 212 cohomoloyy rings. PROOF: Since pullbacks of sums of line bundles are sums of line bundles, and since compositions of monomorphisms are monomorphisms, it suffices to consider just the case r = 1. Let 5 be an n-plane bundle over X and let P: A X represent the corresponding projective bundle. Then the splitting bundle I.< is a line subbundle of the pullback j'!tover P , E a;by Proposition 111.3.6 there is an ( n - 1)-plane subbundle q of ,f!5 such that .f'!< = A, @ q, and H*(X) 2 H*(P,) is a monomorphism by Corollary 5.4. One applies the same construction to q, splitting off a line subbundle by a map P , + P, which induces a 212 cohomology monomorphism H*(P;) + H*(P,), and the induction is clear.
204
IV. 212 Euler Classes
5.6 Definition: A splitting map for finitely many real vector bundles t l , .. .,& over the same X E a is any map X ' 3 X in the category 93 such that each of g't1, . . . , g't, is a Whitney sum of line bundles, and such that H*(X)% H * ( X ' ) is a monomorphism of 2/2cohomology rings.
Proposition 5.5 asserts that splitting maps always exist, for real vector bundles and 2/2 cohomology. They are not necessarily unique, however, nor need their construction be the one given in Proposition 5.5. In particular, for any n > 0 the universal real n-plane bundle y" over G"(R"') admits an especially nice splitting map, as follows. Let yl be the universal real line bundle over RP", let (RP* )" denote the n-fold product RP" x . . . x RP", and let y' + . . . + y' be the corresponding sum of ti copies ofy' over (RP")",as in Definition 111.2.1.Then if(RP")" -% RP"' denotes the jth projection map, for j = 1,. . . , n, one has y' + . . . . , I = priy' 0 .. . 0prby', so that 'J'+ . . . + y' is itself a Whitney sum of n real line bundles over (RP")", hence an n-plane bundle. Consequently the homotopy classification theorem (Theorem 111.8.9) provides a classifying map (RPm')"1:G"(R") for yl + . . . + y ' ; that is, y 1 + . . . + 'J'= h ! f for the universal real n-plane bundle y" over G"(R").
+
5.7 Proposition: Any classifying map (RP")" 1 :G"(R"') fur the real n-plane bundle y' . . . y' over (RP")" is also a splitting map for the universal real n-plane bundle y" over G"(R").
+ +
PROOF: Since h'y" is already a Whitney sum prly' 0 . . . @ prLy' of n line bundles, it remains only to show that H*(G"(Ra)) 5 H*((RP")") is monic. Let G"(R") be any splitting map for y", as in Proposition 5.5, so that f'y" is a sum A l 0 * * .@ A, of line bundles A 1 , . . . , L,, over X and H*(G"(R")) 2H * ( X ) is monic; trivially f classifies A l 0 .. . @ A,,. If the maps X 91.RP" , . * * , X a RP" classify the line bundles A', . . . , I,,, and if g denotes the map X (RP")", then the composition X 3 (RP")" A G"(Rm) also classifies A l 0 .. . @ A,,. By the homotopy classification theorem (Theorem 111.8.9) the classifying maps f and h g for I,, 0 * * 0 A, are necessarily homotopic, and consequently there is a commutative triangle H*(G"(R")) I' b H*(X)
Xs
-
0
H*((RP")").
i
i
Since f * is monic, it follows that h* is also monic, as required.
205
6 . Axioms for L / 2 Euler Classes
A finite-dimensional version of Proposition 5.7 will be established in Proposition V.4.6.
6. Axioms for Z/2 Euler Classes The following description of 2/2 Euler classes will serve as a model for characterizing many characteristic classes of many categories of vector bundles.
<
6.1 Theorem (Axioms for 2/2 Euler Classes): For real vector bundles over htrsr spoce.s X E XI there ure unique ilomogmeous Z12 singulur cohomoloyy classes e(<)E H * ( X ) which .suti$j the j d l o w i n y axioms: ( 1 ) Naturality : a mup in the cuteyory
<
!\' is LI reul vector burzdle over X E d,and ifX ' g,then e ( j ! < = ) f * e ( t ) E H*(X').
X is
<
(2) Whitney product formula: f f und q ure r e d vector bundles over X E 39, with Whitney sum 0q o ~ e Xr , then e(<)u e(q)= e(<0q ) E H * ( X ) j o r the cup product e ( < )u e(q). (3) Normalization: is the cunonicul real line bundle over R P ' ( = S'), then is the generator o f H ' ( R P ' ) .
<
the same
!/';I;
e(;lt)
PROOF: The Euler classes of Definition 3.1 satisfy Axioms 1, 2, and 3 by virtue of Propositions 3.2, 3.4, and 4.4, respectively. Conversely, suppose that e( ) satisfies Axioms 1, 2, and 3, let y' be the universal real line bundle over RP" ,and let RP' RP" classify the canonical line bundle 7: over R P ' . Then H ' ( R P " ' ) 2H ' ( R P ' ) is an isomorphism by Corollary 4.5, and f * e ( y ' ) = e ( , f ! y ' ) = e(y:) by Axiom 1, so that e(y')E H ' ( R P m ) is necessarily the generator of H*(RP"), by Axiom 3. Now let y" be the universal real nplane bundle over G"(Ra')and let (RP")" 5 G"(R"') be the splitting map for y" described in Proposition 5.7, with h!.," = + . . . + 71 = pr!,yl 0 . . . 0priy' I / 3,'
for the n projections ( R P " ) "
R P " . Then by Axioms 1 and 2 one has
h * e ( y " ) = e(h!y") = e(pr\y' @ . . .0 prby') = e(pr\y') u . . . u e(prly') = pr?e(y') u . . . u prxe(y'), which is a unique element of H"((RP"')") since e(y') is uniquely defined, as we have just learned. Since h* is monic by Proposition 5.7, it follows that e(y")is a unique element of H"(G"(FP)).By one final appeal to the homotopy classification theorem (Theorem 111.8.9) any real n-plane bundle over any
<
206
IV. 2/2 Euler Classes
X E .9$ can be classified by a map X 5 G"(R") which is unique up to homotopy, so that by Axiom 1 one has e(<)= e(g!y")= g*e(y")E H " ( X ) for a unique homomorphism H"(G"(Rn))"*.H"(X) and the unique element e(y")E N"(G"(R")).
7. Z/2 Euler Classes of Product Line Bundles If 3, and p are real line bundles over the same base space X E a,then the product ;I@ p is also a real line bundle over X , and the real line bundles over X form an abelian group with respect to this product. Furthermore, the map carrying the real line bundle 1 over X into the 2/2 Euler class e(3.)E H ' ( X ) is an isomorphism. In this section we verify only that the latter map is a monomorphism; proofs of the stronger result are indicated later. 7.1 Proposition: For any base space X E 9? let T(X) denote the set of' real line bundles ouer X , with the product T(X) x T(X) + T(X) carrying (pi ), E T(X) x T(X) into 1. @ p E T(X). Then T(X) is an abelian group in which every element is of order 2.
PROOF : The associative and commutative properties (i@p)@v=1@(p@v)
and
1@p=p@i
were observed in Proposition 111.2.6, and the trivial line bundle E T(X) is the neutral element: I @ E' = 1 for every 1 E T(X). By the linear reduction theorem (Theorem 11.6.13) the structure group GL(R, 1) of any real line bundle can be reduced to the orthogonal group 0(1), which consists of two elements, in the discrete topology, and since the square of each of these elements is the neutral element, it follows for any real line bundle I that ;I@ i= & I . Thus each 1E T(X) is of order 2, so that each A E T(X) is its own inverse.
7.2 Definition: The abelian group T(X) of Proposition 7.1 is the real line bundle group of the base space X E a. The following trivial example will be used later. 7.3 Proposition: T ( P ) = H/2.
PROOF: There is a covering { U o ,U , } of the circle S' by two slightly lengthened open half-circles U , and U , such that the intersection U o n U , consists of two connected components V,, and V , . Since the structure group GL( 1, R) of any real line bundle over S' can be reduced to O(1) c GL(1, R)
207
7. E / 2 Euler Classes of Product Line Bundles
+
as before, where O(1) = { 1, - l), any transition functions describing 2 assume only one of the values + 1 or - 1 on each of the components V, and V , . If the values agree, then the result is A = E ' , and if the values differ, then the result is 1. = 1.1: (over the base space S' = RP').
For any real vector bundles 5 and q over the same base space X E B the product bundle 5 x q over X x X was described in Definition 111.2.2, the product 4 0 r] over X being the pullback A!(< x q ) along the diagonal map X 5 X x X , as in Definition 111.2.4. We use the former product in the following lemma. 7.4 Lemma: Let y' be the universal real line bundle over RP", and let RP" x RP" 5 R P " classify the real line bundle y' x y' over RP" x RP". I f RP" x RP' 3 RP" and RP" x RP" 3 RP" are j r s t and second projections, respectively, and if a ' , a,, and a denote the generator e(y') in each of the three copies of H'(RP"), it ,follows that h*a = .:al .!a2 E H'(RP" x RP").
+
PROOF: By the Kunneth theorem H'(RP" x RP") is isomorphic to Ho(RP" ) 0 H'(RP") + H'(RP") 0 Ho(RP"), where Ho(RP") can be regarded as the ground ring 212 in each summand. Since a l and a, generate the initial two copies of H ' ( R P " ) it follows that h*a = b1n7fa1+ b2rc?a2 for unique elements b , , h, E 2/2, and it remains to show that b , = b , = 1. Choose a base point * E RP' and let RP' 3 RP" x RP" carry x I E RP' into (x,,*) E RP' x R P " . Then n l i I is an identity map and n2i1 is a constant map, so that i:n7fa1 = (n,i,)* = a 1
and
i7fnTa2 = (n2i1)*a,= 0.
Furthermore, the pullback i\(y' x y') is clearly the bundle y1 over the first copy of RP", so that i:h*a
i:h*e(y') = e(i\h!y') = e(i\(y' x y ' ) ) = e(y') = a l . =
Hence a , = i7fh*a = b i*l n *l a l
so that h ,
=
+ b2i:nr5a2 = b l a l ,
1. A similar argument shows that b,
=
1.
7.5 Proposition: For any real line bundles 2, p over a base space X E B one has e ( i 0 p ) = e ( i ) e ( p )E H ' ( X ) ; that is, the computation of 2/2 Euler clusses of' real line bundles over X induces a homomorphism T(X) + H ' ( X ) of' abelian groups.
+
208
IV. 2/2 Euler Classes
PROOF: Let X L RP" and X 3 RP" classify A and p, respectively, let RP" x RP" 5 RP" classify y' x y', and let X 5 X x X be the diagonal map; then the composition
x
A
x x x S ~ S . R PX*R P " A
R P ~
classifies ;I@ p. Since r 1 ( f x g)A = f and n z ( f x g)A = y, the preceding lemma implies
e ( A @ p) = A * ( f x g)*h*a = A * ( f x g)*(r:a, = bl(.f' x =f*aI
+ n:a,)
s)A)*a, + (7c2(1' x g)A)*a,
+ g*a,
+ e(p),
= e(A)
as claimed. The homomorphism T ( X )-+ H ' ( X ) is in fact an isomorphism, for any X E 28.However, for the moment we prove only that it is a monomorphism; proofs that it is an isomorphism are indicated in Remarks 9.7 and 9.8.
7.6 Lemma: T(S') -+ H ' ( S ' ) is an isomorphism for the circle S'
E
a.
PROOF: By Proposition 7.3, T(S') = Z/2, the nonneutral element being the canonical real line bundle y i over RP' (=S'); but e(y:) # 0 E H ' ( R P ' ) by Proposition 4.4. 7.7 Proposition: T ( X )-+ H ' ( X ) is a monomorphism fbr any X E 93. PROOF: If e(1) = 0 for a real line bundle 1 over X , and if S' L X is any map, then by naturality of 2/2 Euler classes (Proposition 3.2) one has e ( f ' 1 ) = f*e(E,) = 0 E H1(S1);hence by Lemma 7.6, f ! A = over S'. However, since each connected component of X is pathwise connected (Exercise I.10.22), 1 is trivial if (and only if) f ' 1 is trivial for every map S' X . In Chapter V we shall derive a necessary and sufficient condition for any real n-plane bundle t to be orientable, formulated in terms of a Z/2 cohomology class. The proof will depend upon the special case that ( is a Whitney sum of real line bundles, which we now consider.
7.8 Lemma: Let ( be the Whitney sum I , 0 . . . 0 l,,of'n real line bundles A,, , . . ,1" over X E 93,let 1, Q . . . Q A,,E T ( X ) be the product ~f the sume line bundles, and let A" 5 E T ( X )be the nth exterior power of' t, as in Dejnition III.lO.l;thenk< = A 1 @ . . . @ & .
PROOF: Let S be the linear subspace of the real tensor product @"R" spanned by elements x , 0 * * * @ x, such that x , , . . , , x , are linearly de-
209
8. The L / 2 Thorn Class V: E H"(PS,, , P,)
pendent in R", and let T be the 1-dimensional linear subspace of A " R " spanned by elements e l 0. . . @ e , such that each e, E R" vanishes outside the ith summand of R". The classical computation
+ e j )0 (ei + e j ) shows that e, 0ej + e j 0e, E S, hence that @" R" = S @ T , so that A" R" = (ei 0ej + e j 0e,) + e, 0e, + ej 0ej = (e,
@" R"/S z T , by definition of
R".
7.9 Proposition: A Wliitney win i,0 , . ' . @ i fof'real , line bundles i l ,. . . ,i,, ouer a base space X E is orientable if and only if the 212 Euler classes satisjjq
e(2,) + . . . + e(&) = 0 E H ' ( X ) . PROOF: Let 5 = i1 0. . . @ Afl. Then according to Definition 111.10.4, 4 is orientable if and only if the nth exterior power A" is the trivial line bundle over X . Hence by Lemma 7 . 8 , t is orientable if and only if A l 0 .. . 0 A,, = E' E T(X). By Proposition 7.7, r(X)-+ H 1 ( X )is a monomorphism, so that 5 is orientable if and only if e(R, 0. . . @ R,) = 0 E H ' ( X ) , and by Proposition 7.5 one has e ( i l 0. . .0 i n= ) e(2,) + . . + e(lLfl). 3
The following computation will play a crucial role in the construction of other 2/2 characteristic classes, in Chapter V.
7.10 Proposition: I f ' iis u line subbundle of' a real n-plane bundle 5 over a base space X E 9, then e ( i 0 5 ) = 0 E H " ( X ) . PROOF: By Proposition 111.3.6, 5 = E. @ q for some (n - 1)-plane bundle v ] , so that jL04 = (A0A) @ (A 0 q ) = E' 0[ for the (n - 1)-plane bundle [ = i0q. Hence e ( i 0 5 ) = e(E1 @ i) = e(E1) u e([) = 0 as in Proposition 3.6.
8. The Z/2 Thorn Class Vt E H " ( P S ,1 , Pr) There are many constructions of 2/2 Euler classes other than the ones given in Definition 3.1 and Proposition 3.8. We consider an alternative in this section. Let E 5 X represent a real n-plane bundle ( over a base space X , let 1 denote the trivial bundle 6' over X, and let 5 @ 1 be the Whitney sum o f t and 1. Then the total space Pcel of the projective bundle associated to 4 01 consists of equivalence classes [e @ slz(eJ of nonzero elements e 0 .sln(elin the total space of 5 0 1, where e E E projects to x(e) E X, and where sln(elis a real multiple in the fiber over n(e) of a fixed nowherevanishing section of 1. The subset of those [e @ sln(J E PCe1with s = 0 is
210
IV. 2/2 Euler Classes
clearly the total space P, of the projective bundle associated to 5, so that there is an inclusion P , A P , , and an inclusion P,, A PCs1, P , .
8.1 Definition: The zero-section X 4 PtB1 carries each x E X into the point [0 0 l,] E PcOI . If P , , 5 X represents the preceding projective bundle associated to then H*(Pcel,P,) is clearly an H*(X)-module with respect to the product /3 * a = E*/3 u a E H*(P,, 1, P,) of /3 E H*(X) and o! E H*(P,, 1, P J . We shall show that H*(Pc,l,Pr) is isomorphic to the H*(X)-module H*(E,E*), and that if \ E H"(P,,,, P,) corresponds to the Thom class U , E H"(E,E*), then r*j* V, is the 2/2 Euler class e(4)E H"(X). In order to prove the preceding assertions, let Ps*, c P , , consist of all points of the form [e 0 sln(e)] for e # 0. In the following lemma the letters i and j each represent three distinct (but obvious) inclusions. Let E 5Pgs s P& 1, let P,, 2 P,, be carry e into [ e 0 with restriction E* 2 and the identity map, let P , 2 PFB1be an inclusion of subspaces of PYsl, set f o = ( f l , f 2 ) and g o = ( S l ?92).
5 0 1,
8.2 Lemma : There is u commutative diagram
. . bH*(E, E*)
...
T - ! s:,
x
n'o
x
J*
+
H*(E) i ' H*(E*)
s:
I
=
s:
I
>. . .
=
i* H*(P,el) i . H*(P,) d . ' H*(P,s 1 , P,) in which the four homomorphisms f T,, gT,, g:, g ; are isomorphisms. 1
PROOF: Commutativity is immediate since f 2 and g 2 are restrictions of f2 are injections of E and E* into P,,,\P, and P k l\Pc, respectively, j'T, is an isomorphism by the excision axiom, and g: is trivially an isomorphism. To show that g t is an isomorphism define P k l 1:P , by setting h , ( [ e @ = [el, so that h,g, is the identity on P , . Then g2h2 carries [ e 0 sln(e)] into [ e @ 0 (for e # 0), a map which is trivially homotopic to the identity. Thus g2 is a homotopy equivalence, so that g: is an isomorphism. Finally, since g: and g s are isomorphisms the 5-lemma implies that gT, is also an isomorphism.
f ; and g l . Since f l and
21 1
9. Remarks and Exercises
The isomorphisms ,f: and gg of Lemma 8.2 provide new Thom classes 0 U , E H"(PrO1, PFO ,) and V5 = gz W, E H"(PrO1, Pr). We are primarily interested in Vr.
w - .f. * - I
8.3 Proposition: For any real n-plane bundle 4 over a base space X the Pr), H*(X)-module H*(Prel,P,J is .free on a single generator V, E H"(P501, and T * j * V ; is the Euler class e(5)E H"(X).
PROOF: The first statement is just Proposition 1.5, combined with the isomorphism H*(E, E * ) 98J-8- ' + H*(PrO1, Pr) of Lemma 8.2. To prove the second statement one applies H"(-) to the commutative diagram
and one combines the result with the j * portion of Lemma 8.2 to conclude that the diagram H"(E,E*)
J*
' H"(E),
9. Remarks and Exercises 9.1 Remark: Thorn isomorphisms and Thorn classes (of sphere bundles over polyhedral base spaces) first appeared in Thorn [l], with sketches of elementary proofs; the constructions were extended in Thom [4], using
212
IV. 212 Euler Classes
sheaf theory. Both Thom [ 11and [4] contain the identity ace(<) = U, u U, of Proposition 3.8, ofwhich special cases had appeared in Whitney [6, p. 1191 and in Gysin [l]. Thom [4] also contains the Gysin sequence in more or less the form reported in Proposition 4.2. An independent development of the Thom isomorphism and the Gysin sequence was given by Chern and Spanier [11. The Mayer-Vietoris argument for the existence of Thom classes can be found in Milnor [3] and in Milnor and Stasheff [l]. An entirely different approach to the Thom isomorphism theorem is given in Cockroft [ 13, and a generalization of Thom classes to nonspherical fibrations appears in Schultz [4]. Also, see Bott and Tu [ I , pp. 63-64].
9.2 Remark: If E + X represents a vector bundle, the pair E, E* is the corresponding Thorn complex. The same terminology is sometimes applied to the one-point compactification of E itself, which is more often called the Thorn space of the given bundle: the cohomology of the Thom complex and the Thom space coincide except in degree 0. (The original construction of Thom [1,4], applied to sphere bundles rather than vector bundles, presents yet a third variant of Thom complexes and Thom spaces.) Thom spaces were used by Atiyah [11 to show that the J-equivalence type of the tangent bundle T ( X )of a smooth manifold X depends only on the homotopy type of X. (This result was strengthened by Benlian and Wagoner [l] and Dupont [2], as noted in Remarks 111.13.34 and 111.13.42.) Atiyah's result (applied to normal bundles rather than tangent bundles) is also valid for arbitrary closed topological manifolds: see Horvath [ 11. Other properties of Thom spaces and their relations to J-equivalence can be found in Held and Sjerve [ 1-31, for example. The most important application of Thom spaces occurs in the computation of cobordism rings, considered later in their work. 9.3 Exercise: Show for'any real vector bundle 5 over a base space X E 5? that the total space P , of the corresponding projective bundle also belongs to a,as claimed following Definition 5.1. (If one replaces the category by the category W c a of Definition 1.3.8 and Corollary 1.5.3 one obtains a stronger result, which can be found in Stasheff [l] and in Schon [l]: If E -% X is any fibration with base space X E W and fiber F E W , then E E W . ) 9.4 Remark: The cohomology ring H*(RP";2/2) was computed in Proposition 4.4. This computation and other computations of the 2 / 2 homology and cohomology of RP" can be found in Artin and Braun [2, pp. 178-1811, Feder [l], Gray [l, pp. 194-195, 285-2861, Greenberg
21 3
9. Remarks and Exercises
[l,p. 141],Hu[4,pp. 133-138],Lam[l],Massey[6,pp.241-242],Maunder [ l , pp. 140, 169-170, 348-3491, Spanier [4, pp. 264-2651, and Ta [l], for example.
9.5 Remark: The splitting principle of Proposition 5.5 was introduced in Bore1 [l]. It was used in Grothendieck [I] to provide axioms for certain characteristic classes we shall study later; Grothendieck's axioms are similar to those of Theorem 6.1. We shall use Grothendieck's procedure often, in other circumstances described in later chapters. 9.6 Exercise: Let yf be the canonical real line bundle over the real projective space RP", and let z(RP") be the tangent bundle of RP". Show that e(r(RP"))=
if n iseven if n is odd.
9.7 Remark : We constructed a Z/2-module homomorphism T(X) + H ' ( X ; Z/2) for any X E 9 in Proposition 7.5, and we showed in Proposition 7.7 that it is a monomorphism. A stronger result is valid: the Z/2-moduZe homomorphism T(X) -+ H ' ( X ; Z/2) is an isomorphism. To start the proof recall that any X E is homotopy equivalent to a metric space of finite type (Definition 1.1.2), a fortiori homotopy equivalent to a paracompact space. Lemma 11.4.5 implies that T(X) is invariant under homotopy equivalence, and since H ' ( X ; Z/2) is also homotopy invariant, one can therefore assume at the outset that X is paracompact. Clearly one may as well further assume that X is connected, hence pathwise connected by Exercise I. 10.22. Now let nl(X) be the fundamental group of X , let n , ( X ) + H , ( X ; Z) be the Hurewicz map, and let H , ( X ; Z) -+ H , ( X ; 7/2) be the coefficient homomorphism. Both homomorphisms are surjective, so that if L c nl(X) is the kernel of the composition nl(X) + H , ( X ; Z) + H , ( X ; Z/2) the quotient group n , ( X ) / Lis a Z/2-module isomorphic to H , ( X ; 4'2). Since Z/2 is a field, nI(X)/L is free, and since X is pathwise connected, one can represent each element of a basis B of n , ( X ) / Lby a map S' + X carrying a fixed base point in S' to a fixed base point in X . It follows that there is a bouquet VeS' of circles attached at a common base point, and a basepoint-preserving map V s s' X such that (i) the singular homology homomorphism H i ( V B S ' ; Z/2) H , ( X ; 2/2) is an isomorphism, and (ii) the homomorphism T(X)2 r(VB S') induced by pulling line bundles back along is an epimorphism; the second property is an easy consequence of the paracompactness of X .
214
IV. h/2 Euler Classes
Since Z/2 is a field, one has H'(X; Z/2) = Hom(H,(X; Z/2),2/2) and H'(VBS') = HOm(H,(VBS'; Z/2), Z/2) = Hom(UBHl(S1;Z/2), Z/2) = H ' ( S ' ; 2/2), for direct sums Usand direct products Clearly r(vBS') = T(S'), so that naturality ofZ/2 Euler classes (Proposition 3.2) provides a commutative diagram
nB
nB.
ns
J
J
nr(sl)tnH'(s' B
; 2/2)
B
with an epimorphism j'! and an isomorphism f*. The bottom map is also an isomorphism, by Lemma 7.6, so that T(X) -,H ' ( X ; Z/2) is an epimorphism. Since the latter map is also a monomorphism, by Proposition 7.7, this completes the proof. 9.8 Remark: Another proof of the preceding result is briefly sketched in Husemoller [I, 2nd ed., pp. 235-2361. 9.9 Remark: If one substitutes complex line bundles for real line bundles in Definition 7.2, the resulting analog of T ( X ) is the Picard group of X , which is isomorphic to HZ(X;Z); details will appear in Chapter X. Picard groups of complex algebraic varieties are frequently defined by other means; see Griffiths and Harris [l, pp. 133-1351, for example.
CHAPTER V
Stiefel-W hitney Classes
0. Introduction Let 2 / 2 [ t ] be the polynomial ring in a single variable f over the field 2 / 2 , and let j l t ) E 2 / 2 [ t ] be a polynomial 1 + a , t + . . . + aptPwith leading term 1 E 2 / 2 . Then for any real rn-plane bundle [ over a base space X E B the splitting class e(Q E H'(P,; 2 / 2 ) leads to an inhomogeneous characteristic class u,.([) E H * ( X ; 2/2). The case j l t ) = 1 + t provides the total Stieftl- Whitttey class w ( 5 ) E H * ( X ; 2 / 2 ) , which can be used to compute all the other classes u s ( [ ) . More generally, if 2 / 2 [ [ t ] ] is the formal power series ring in a single variable t over 2 / 2 , and if f ( t ) E 2 / 2 [ [ t ] ] is a formal power series 1 + a , t + a 2 t 2 + . . . with leading term 1 E 2 / 2 , there is a corresponding inhomogeneous characteristic class us(<) in the direct product H * * ( X ; 2 / 2 ) .The classes uf([) and their properties are developed in 51 and 52. For each q 2 0 the qth summand of the total Stiefel-Whitney class w([) E H * ( X ; 2 / 2 ) of a real rn-plane bundle 5: over X E B is the qth StiejelWhittiey c/u.s.s w q ( [ )E H q ( X ; 2 / 2 ) . One easily shows that wo([)= 1 E H o ( X ;2/2), that w,([) E H " ( X ; H/2) is the 2/2 Euler class e ( 5 ) of Definition IV.3.1, and that w&[) = 0 E H q ( X ;2/2)for q > rn. In particular, it is shown in $4 that if 1'"' is the universal real rn-plane bundle over the Grassmann manifold Gm(R' ), then H*(Gm(R");2 / 2 ) is the polynomial ring 2 / 2 [wl(y"), . . . , w,(;~")] over 2 / 2 generated by the Stiefel-Whitney classes wl(ym), . . . , w,(y"). This result, combined with the homotopy classification theorem, Theorem 111.8.9, provides an alternative characterization of all 2 / 2 characteristic classes of real vector bundles. 215
216
V. Stiefel-Whitney Classes
Many Stiefel-Whitney classes have a direct geometric interpretation. For example, it is shown in 53 that the first Stiefel-Whitney class w l ( < ) E H'(X; 2/2) of any real vector bundle 5 over any X E vanishes if and only if is orientable. A formal result proved in $6 will lead in the next chapter to several geometric properties of smooth manifolds X, which are proved by computing Stiefel-Whitney classes of their tangent bundles z(X). As in the previous chapter, the notations H*(-,-) and H*(-) are used consistently to indicate singular cohomology If*(-,-; 2/2) and H*(-; Z/2) with 2 / 2 coefficients.
<
1. Multiplicative Z/2 Classes Let 5 be a real m-plane bundle over a base space X E B, let P , E B be the total space of the corresponding projective bundle, as in Definition IV.5.1, and let i., be the splitting bundle and e(&) E H ' ( P S )the 2/2 splitting class, as in Definition IV.5.2. According to Proposition IV.5.3 the H*(X)-module H*(P;) is free on the basis { 1, e(A,), . . . , e(LJ"- j . It follows that the operation ue(iLt)of cup product by the splitting class is a cyclic H*(X)-module U P ( i< ) H*(P,) of degree + 1. More generally, for endomorphism H*(P;) any polynomial f ( t ) E 2 / 2 [ r ] the operation u f ( e ( R , ) ) of cup product by the class f(e(1.J) E H*(PJ is also an H*(X)-module endomorphism
-
which is in general inhomogeneous. Since H*(P;) is a free H*(X)-module of rank m < ~xone can form the determinant of the endomorphism f(ue(i.<)), with value in H*(X). 1.1 Definition: Let f ( t ) be any polynomial 1 + a,t + . . . + a# over 2/2, with leading term 1 E 2/2, and let be any real m-plane bundle over a base space X E B. The multiplicative 2/2 class u,(<) E H*(X) is given by ~ ~ (=5 detj(ue(i,)). In the special case j ( t )= 1 + t the class u,-(<) is the total Sriefel-Whitney class: w ( < ) = det(u(1 + e ( j . 0 ) )E H*(X).
<
Clearly if f ( t ) is of degree p over Z/2, then the class u,(<) E H*(X) is of the form 1 ur,,({) . . . ~ ~ , , , ~ (where 5 ) , us,,(() E H4(X)for q = 1, . . . , mp; that is, one has ~ , , ~ ( 5=) 1 E Ho(X) and uf,,(<)= 0 E Hq(X) for 4 > mp. In particular, the total Stiefel-Whitney class w(5) E H*(X) of the real m-plane writ((); bundle 5 over the base space X E !B is of the form 1 + w l ( < ) + for each q = 1, . . . ,m the class w,(O E H4(X)is the 4th Stiefel-Whitney c h s s of <.
+
+ +
- +
)
21 7
1. Multiplicative 2/2 Classes
1.2 Proposition: Let p be a real line bundle over a base space X E W,with 212 Euler class e(p) E H ' ( X ) . Then f o r any polynomial f ( r ) = 1 + a,t + . . . + apt, E 212 [ r ] with leading term 1 the multiplicative 212 class us(p) E H * ( X ) sat isfies
u r ( p ) = .f(e(p)) = 1 + a,e(p) + *
* *
+a,eW;
in purticulur, the total Stiefel-Whitney class w ( p )E H * ( X ) is gioen by w ( p ) = 1 + e(p). PROOF: Since p is of rank one, any representation P , 5 X of the corresponding projective bundle is a homeomorphism, for which ff!p is the splitting bundle A, over P , . Hence e(2,) = e(is!p)= is*e(p), and since p . a = E*fl u a for fl E H * ( X ) and a E H*(P,), by definition of scalar multiplication in the H*(X)-module H*(P,), it follows that the endomorphism f ( u e ( 2 , ) ) of H*(P,,) is scalar multiplication by f ( e ( p ) ) Since . H*(P,) is of rank one over H * ( X ) , this implies that u , ( p ) = det f'(ue(&)) = j ( r ( p ) ) as , claimed.
Now let X ' .5 X be any map of base spaces, and let be any real vector bundle over X . The pullback diagram for the bundle g!t over X ' induces a corresponding pullback diagram B
X'
p,
B
b X
of projective bundles, over which one can place the diagram for the pullback g!At of the splitting bundle A,, as indicated :
E'
X'
+ E
B
+
x.
If E denotes the total space o f t , then E consists of pairs (l,, ex) for each x E X , where I, E P , can be regarded as a 1-dimensional subspace of the fiber E x , and where ex E I,. There is a corresponding description of the total space E'
218
V. Stiefel-Whitney Classes
of the splitting bundle Ae!<, for which uniqueness of the preceding pullback diagram guarantees that I$!< = g!A<. 1.3 Proposition (Naturality): Let X ' 3 X be a map in the category a qf' base spaces, let 5 be a real m-plane bundle over X , and let g!( be the pullback over X'. Then jbr any polynomiul f ( t )E 212 [ t ] with leading term 1 E 212 the multiplicatioe 212 class u,-(g!t)E H * ( X ' ) satisfies ur(g!()= g*u,-((); in particular, the total Stiejel- Whitney class w(g!() E H * ( X ' ) satisfies w ( g ! t )= g*w(O.
PROOF: The lower portion of the preceding diagram guarantees that H*(P,) 3 H*(Ps!<)is a module homomorphism over the ground ring homomorphism H * ( X ) H * ( X ' ) ; that is, g*(p . a ) = (g*p) . (g*a) for fl E H * ( X ) and 01 E H*(Pr). Furthermore, since Ae!< = g!E.,, it follows from the naturality of 212 Euler classes (Proposition IV.3.2) that e(Ae!<)= g*e(Q, hence that g* carries the ordered basis (l,e(A<),. . . , e(AJ-l) of H*(P<)into the ordered basis (l,e(&!J, . . . , e ( A e ! J - ' ) of H*(PB!<).Since the endomorphism W A C ) + H * ( P , ) is cyclic, it has a matrix representation H*(Pc)
s
with respect to the former basis, where ,!I1, . . . , p, ue(Ae!5) + H*(Pe!<) is represented by H*(Ps!<)
E H*(X).
It follows that
with respect to the latter basis, hence that u f ( g ! < )= det j ' ( u e ( i s ! J=) g* det j { u e ( j . < )=) g*ur(<),
as claimed. We briefly reexamine the preceding matrix representation of the endomorphism ue(i,;)with respect to the basis (l,e(iLs), . . . , e(I.J'-'). By definition, ue(Rr)carries e(Qq- into e(A$ for q < m, and ue(R<)carries e(E.J"-' into fl, . 1 + p,. e(A5)+ * * ,!I1 * e(IJ-', representing e(EJ' E H"(PJ. It follows that p, E H ' ( X ) , . . . , b, E H"(X), and in particular that PI, . . . , p, are of strictly positive degree in H * ( X ) . Hence, if 9
,f(t) =
+
1
+ a l t + . . + aptP
219
2. Whitney Product Formulas
and u s ( ; ) = detj’(u4j.:)) = 1
+ us,,(<) + . . . + uJ,,,((),
then for each q 5 mp the class uf,,(<)E H 4 ( X )depends only on the coefficients 1, (I . . . , (I,, of j (t ) . Thus if one replaces polynomials f ( t ) E 2/2 [ t ] by formal power series j ( r ) = 1 + u , t + c12t2 . . . E . Z , Q [ [ t ] ] , one can compute analogous classes us(() in the direct product H * * ( X ) of the modules Hq(X).
+
+
1.4 Definition: Let f ( t ) be any formal power series 1 + a , t + a,t2 . . . E 2/2 [ [ t ] ] over 2/2, with leading term 1 E 2/2, and let 5 be any real m-plane bundle over a base space X E 9?.Then the multiplicative 212 class u,.(<) E H * * ( X ) is given by us(<) = det f(ue(j.<)),for the endomorphism f(ue(/LJ) of the free H**(X)-module H**(PJ.
One can extend Proposition 1.3 from polynomials mal power series f ( t ) E 2/2 [ [ t ] ] .
f(t) E
2/2 [ t ] to for-
1.5 Proposition (Naturality): Let X ’ 5 X be a map in the category 98 of base spuces, let he a r e d m-plane bundle over X , and let g ‘ 4 be the pullback o w r X’. Then ,for any formal power series f ( f )E 212 [ [ t ] ] with leading term 1 E 2/2 the multiplicatioe 2/2 class uJ(g!<)E H**(X’) satisfies uf(g!() = g*u&).
<
PROOF: The proof of Proposition 1.3 remains valid if one substitutes “formal power series” throughout for “polynomial.” The advantage of using formal power series f ( t ) E 2/2 [ [ t ] ] rather than polynomials, and the rationale for insisting that the leading term of f ( t ) be 1 E 2/2, lies in the existence of multiplicative inverses l/f(t) E 7/2 [ [ t ] ] , which one can compute directly from f ( t ) . For example, the multiplicative inverse of 1 + t ( = 1 - t), as an element of 2/2 [ [ t ] ] ,is the formal geometric series 1 t t2 . . . E 2/2 [ [ t ] ] . The relation between the classes u r ( 0 and u1,J() will be developed later. In most applications the cohomology H * ( X ) of the base space X E will itself be finite, so that H * ( X ) = H * * ( X ) . However, if X = R P “ , for example, then H *( R P “ ) is indeed a proper subring of H**( RP” ).
+ + +
2. Whitney Product Formulas We show for any formal power series f ( t ) E 2/2 [ [ t ] ] with leading term 1 E 212, and for any real vector bundles ( and q over the same base space X E &, with Whitney sum [ 0 q over X , that ur(< 0 q ) = uf(5) u ur(q) E H **(X).
220
V. Stiefel-Whitney Classes
Let H*(X) be the singular 2/2 cohomology ring of a base space X E g, and let H*(X)[t] be the polynomial ring in a variable t over H*(X). If 5 is a real vector bundle over X, then the free H*(X)-module H*(Pr) becomes a free H*(X)[t]-module H*(P,)[t] in the obvious way, and one can construct endomorphisms of H*(P,)[t].
2.1 Definition: For any real vector bundle 5 over a base space X E 93,the charucteristic polynomiul a,(t) E H*(X)[t] is the determinant of the endomorphism u ( t . 1 - e ( @ ) of the free H*(X)[t]-module H*(Pr)[t], cup product by t . 1 - e(A,). The Cayley-Hamilton theorem guarantees that the endomorphism H * ( P S ) q ( u e ( i < ) ) H*(P,) vanishes, and since u e & ) is a cyclic endomorphism, the kernel of the epimorphism H*(X)[t] + H*(P,) carrying t into e(A,) is precisely the principal ideal (a,(t)). Hence one can regard the H*(X)module H*(P,) as the quotient H*(X)[t]/(a,(t)). Now suppose that E -+ X represents a real m-plane bundle 5 over the base space X E 9?,and that E' + X represents the Whitney sum 5 0 q of 5 and a real n-plane bundle q over X, for any m > 0 and any n > 0. The inclusion diagram ~
induces an inclusion diagram
of corresponding projective bundles, and there is an induced H*(X)-module homomorphism H*(Pre,) H*(Pr). 2.2 Lemma: The kernel of i* is the principal ideal (a,(e(&B,)))
= H*(P,,,).
PROOF: Since the splitting bundle AS is the restriction of the splitting bundle it follows that A, = i!AteV, hence that e(A,).= i*e(Lre,,). Since H*(P,) is isomorphic to the quotient H*(X)[t]/(ar(t)), this completes the proof.
+,to P, c P g q ,
22 1
2. Whitney Product Formulas
Observe that the splitting bundle itelf of 5 @ q and the pullback &,< of the bundle 5 are both real vector bundles over the base space Pt@,E d.
2.3 Lemma: e(A,@,,0xieq<)= a,(e(A,,,)). PROOF: Since it is a line subbundle of the bundle nk< over P , , it follows from Proposition IV.7.10 that i*e(j.;@,,
Hence e(&,
0nk@,,<) = e(i!&e,, 0i!$e,<)
= e(A,
0nit) = 0.
0nke,<)E ker i*, so that Lemma 2.2 implies that e(J-'y3,,0.;e,r,
=
aq(e(A,e,))
<
for some c1 E H*(Pte,). If is of rank m, then both e(ELteq@ &,&) and r ~ : ( e ( % :lie ~ ~in~ )Hm(Pcelf), ) so that c1 E HO(PSO,,).Hence LY is of the form nTeVlj for a unique /J E H O ( X ) ;that is, e(itelf @ nke,<) = p . or(e(Ats,)). In order to verify that /J = 1, observe that for any x E X there is an inclusion diagram
(4
jx
' x,
and since any bundle over { x ) is trivial it follows that kinke,< is the trivial bundle cm over hence that k:e(&,
0I&,,<)
0k!&& = e(kiitos 0ern) = e(mk&@,) = e(k&@,,
= e(k!&,,Jm
=
k:e(ACes)"'.
The same diagram shows that H*(P,e,)
3 H*(P,@,,.)
is a module homomorphism over the ring homomorphism H * ( X ) 3 H*({x)= ) 2/2, and since j: clearly annihilates everything outside of H o ( X )c H * ( X ) , it follows that k: annihilates all coefficients of oC(t)lying outside of H o ( X )c H * ( X ) , so that k:at(e(Are,)) = k:e(Agea)m. Thus the identity e(jLtelf 0 nkelf<)= j . r ~ ~ ( e ( &has ~ , ~the ) )consequence k:e(iceV))n = (j:/l) . k.~e(j.go,f)nl.However, H*(Pte,J is the free H*({x})-module on the basis {1,k:e(itelf), . . . , k : e ( A t 8 q ) m c n - 1 } , where m + n - l z n and H*((x})a 272, so that j:/3 = 1. Hence j has the value 1 on each path component of X , which means that /l= 1 E H O ( X ) as , required.
222
V. Stiefel-Whitney Classes
Let a,(t) E H * ( X ) [ t ] and a,(t) E H * ( X ) [ t ] be the characteristic polynomials of real vector bundles 5: and q over the same base space X E 93; then a,(t) . a,(t) = ayB,(t).
2.4 Proposition:
PROOF: Two applications of Lemma 2.3, combined with the Whitney product formula for 212 Euler classes (Proposition IV.3.4) give
o n;@,,q)
o&(jL;,,l)) u a,l(e(lyeq)) = eObrellO ntB,t) u e(Ibge,, = eO.,Bv
O n:&
0 q)),
and since Lte,l is a line subbundle of n;,,(5: 0 q), Proposition IV.7.10 implies I) the that the latter expression vanishes. Thus the map t H e ( i S B Iannihilates polynomial a,(r) . a,(t), which has highest-order term P+"; but ogB,,(r) is the unique polynomial with this property. 2.5 Proposition (Whitney Product Formula): For any polynomial j ' ( t )E 2 / 2 [ t ] with leading term 1 ~ 2 1 2or , jbr any formal power seriesf(t)E 2/2 [ [ t ] ] with leading term 1 E 212, and f o r any real vector bundles 5: and q over the same base space X E 9, with Whitney sum 5: 0 q over X , it follows that Uf(5
0 q ) = Uf(5:) u U f h )
E
H*(X),
or that
0 q) = U f ( 0 u U f W E H**(X); in p a r t i c u l a r , j b r f ( t ) = l + t € 2 / 2 [ t ] , one has w ( ( @ q ) = w ( ( ) uw ( q ) ~ H * ( x ) . Uf(5
PROOF: Suppose that the characteristic polynomials of 5: and q are given by a,(t) = r" - or,P-' - . . . - CI,. 1 and a,(t) = t" - plt"-' - . . . - Bfl . respectively, so that the cyclic endomorphisms ue(i,,) and ue(i.,) have matrix representations 1 7
with respect to the ordered bases (l,e(iL,), . . . ,e(i.,)"- ' ) and (l,r(j.,,),. . . , e ( i I l ) 'I-) , respectively. The cyclic endomorphism ~ e ( l , ~has, ~a correspond) ing matrix representation with respect to the ordered bases (1, e, . . . , e m + " I ), where e = e(&,,,); however, in this case the basis (1, r , , . . , e"'-'; r~~(e),a,(e)e,. . , a,(e)e"-') is of more interest. To compute the matrix representation of ur(j.,@,) with respect to the latter basis observe that (ue(ite,,))eP-'= ep for 1 5 p < m, and that ~
.
(ue(&,,,))e'"'
=
e"'
= ct,I
+ . . . + or,P-' + a;(e).
223
2 . Whitney Product Formulas
Similarly (ut'(j.;o,l))a:(L')e4(ue(i.,o,,))a:(r)e"-
=
y < n, and
ar(e)eYfor 1
= /),,a;(e)
+ . . . + /jla;(e)ef'-I + a,(e)a,(e).
However, in the second computation one has o;(t)o,(t) = o;,,,(t) by Proposition 2.4, so that o:(e)r~,,(e) = oCo,,(e) = aco,,(ue(j.,G,l))l = 0 by the CayleyHamilton theorem: thus (Ue(i,:,,,))a;(e)e"-
I
=
+
/),pc(e) . . + [~,.,(e)e"'
It follows that the matrix representation C of ue(&,) with respect to the basis (1, e, . . . , 69'- ' ; o;(e),ac(e)e, . . . , a5(e)cn- I ) for e = e(i.seq) is given by
where D consists of 0's except for a single 1 E 2 / 2 in the upper right-hand corner. Consequently, for the given j' E 2 / 2 [ [ t ] ] with leading term 1 E 2 / 2 there is a matrix D , with formal power series entries for which
so that us(< 0 q ) = det {(C) = det j ' ( A ) u det f ( B ) = u,(<)
u u,(q),
as claimed. 2.6 Corollary: Let E, be the triuiul m-plane bundle over u base spuce X E &I. Theri jiw uny jbrrnul power series j ( t )E 212 [ [ t ] ]with leudiny term 1 E 212 it j0llow.s that u,(E,) = 1 E H * * ( X ) . PROOF: Letj'(t) = 1 + u , t
+ a 2 t 2 + . . . .Then as in Proposition 1.2 one has = I + ale(&')+ u2e(E1)2+ . . .
U,(&I) = & ( E l ) )
for the trivial line bundle c 1 over X , and since e(e') = 0 by Lemma IV.3.5, one has u , ( E ' ) = 1. The Whitney product formula then gives U J P ) = U,(d
0. . . 0c ' ) = U f ( " I ) u
.
'
.u
U,(El)
=
1,
as claimed. In the following result we use the elementary symmetric functions . . . , t,,,) = t , + . . . r,, . . . , am(t,,. . . , t , ) = t , . . . t , in the polynomial ring 2/2 [ t , . . . . , r,] in rn variables t , , . , . , t,. The elementary symmetric functions are characterized by the condition
+
a,(t,,
(1
+ t , ) . . . ( I + t,)
=
1
+ a , ( t l , . . , t,) + . . . + cm(t,>
* * * 3
tm)*
224
V. Stiefel-Whitney Classes
2.1 Corollary: Let A,, . . . , A, be m real line bundles over the same base spuce X E 8, and let I , 0 .. ' 0 I , be their Whitney sum. Then .for each q = 1, . . . , m the 4th Stiejkl-Whirney cluss w,,(A, 0 ' ' . 0 I.,) E H q ( X ) is the
qth elementary symmetric junction oq(e(lL1),. . . , e(A,)) in the 212 Euler clusses r(%,),. . . ,e(A,).
PROOF: By the Whitney product formula (Proposition 2.5) and Proposition 1.2 one has 1
+w,(I,0.*.0Am)+"'+
wm(A,0...0A,) = w ( I , 0 ' * ' 0 A,) = w(A,) u . . . u w(L,) = (1 + e(Al))u * . . u (1 + e(Am)).
2.8 Corollary: For any real m-plane bundle over any base space X has w,(() = e(<)E H"(X).
E
B one
PROOF: For any Whitney sum A, 0 .. . 0 A, of m real line bundles Corollary 2.7 and the Whitney product formula for 2 / 2 Euler classes (Proposition IV.3.4) give wm(Al0* * .0 A,) = e(Al)u . . . u e(E,,) = e(Ll 0 . . . @ A,). For any real m-plane bundle 4 over X E B o n e applies the splitting principle (Proposition IV.5.5) to find a map X ' 3 X in @ such that g!( is a Whitney sum R , 0 . . ' 0 i., of , line bundles over X ' E d and H * ( X ) 2 H * ( X ' ) is a monomorphism. By naturality of Stiefel-Whitney classes (Proposition 1.3), the preceding special case, and naturality of 2/2 Euler classes (Proposition IV.3.2), it then follows that g*wm(t) = w,(g!t) = w m ( A 1 0 .. . O A m ) = e(Al 0 . . . @ A,) = e(g!()= g*e(<) for the monomorphism g*. For any formal power series f ( t )E 2/2 [ [ t ] ] with leading term 1 E 2/2 one can use the Whitney product formula of Proposition 2.5 to compute the multiplicative 2/2 class u r ( t ) E H * * ( X ) of any real m-plane bundle ( over any X E 28 in terms of the Stiefel-Whitney classes w , ( ( ) , . . . , w,((). The computation depends upon a simple algebraic device. Let 2 / 2 [ [ u , , u , , . . .]I denote the graded formal power series algebra in denumerably many variables u l , u 2 , . . . , in which up is assigned degree p for each natural number p > 0; for example, any monomial (u,)"' . . . (u,)"~ is of degree n , 2n2 . * . pn,. If n > p , then no polynomial of degree p contains u,, so that every element of 2 / 2 [ [ u l , u z , . . . ] ] is uniquely of the form ~ n 2 0 P n ( u .1.,,.u,), where P,(ul,. . . , u,) is a degree n polynomial in u,, . . . , u,. In what follows we suppose that Po = 1 E 2/2.
+
+
+
225
2. Whitney Product Formulas
For any natural number p > 0 let Z/2 [ [ t ,, . . . , t p ] ] denote the graded formal power series algebra in which each o f t , , . . . , t, is assigned degree 1 ; for example, any monomial ( t , ) " l . . . ( r p ) " r l is of degree n , + . . . + it,. Then if f ( t ) E 212 [ [ t ] ] has leading term 1 E 2 / 2 , the product .f'(tl) * . .f'(t,) E 2 / 2 [ [ t i , . . . , t p ] ] is symmetric in t , , . . . , t,. It follows that if u ; , . . . , ub are the elementary symmetric functions a,(t,, . . . , t p ) = t , + . . . + t , , . . . , a,(t,, . . . , t,) = f l . . . t, in tl,. . . , t,,, then there are unique polynomials P,(ul,. . . , u,) of degree n in Z/2[ [ u , , . . . , u , ] ] such that .f'(tl)
. . . .f(t,)
1
=
p&;, . . . , u 2
ojnjp
+ ,I
c P,(u;, . . .
, u;,.
>p
,
Suppose that u',', . . . , u;- are the elementary symmetric functions . . . , t,- ,),.. . , ap-,(t,, . . . , t,- ,) in t , , . . . , t P - , obtained from u ; , . . . , u; by setting t , = 0; clearly u; = t , . . . t,_ . O = 0. Since j ( 0 )= 1, one has a,(t,,
,
c
. f ' ( ~ 1 ~ . ~ . . f ' ( ~ , - , ) = . f ' ( ~ , ) ~ ' ~ . f ' ( t p ~ , ) . fP"(uy,...,u;) '(0)=
+
c
ojsgp-
1
P,(u',', . . . , ui-,,O).
n>p- I
Since there are no polynomial relations among the elementary symmetric functions, it follows for each n 2 0 that there is a unique P , ( u l , . . . , u,) E 212 [ [ u , , . . . , u,]] of degree n such that P,(u;, . . . , u:) is the term of degree n in .f'(tl) . . . j'(tp),for any p 2 n. 2.9 Definition: For any formal power series f ( t ) E 2/2 [ [ t ] ] with leading term I E 212 the corresponding multiplicative sequence P,(u,, . . . , u,) E 212 [[u1,u2,. . . ] ] is determined by requiring the nth degree term of , j ' ( r l ) . * ..f'(t,) E 2 / 2 [ [ t , , . . . , t,]] to be P,(u;, . . . , ub), where u ; , . . . , u; are the elementary symmetric functions in t , , . . . , t , , .
For example, if , f ( t )= 1 + t then one clearly has P,(u,, . . . , u,) each n > 0.
= u,,
for
c,lzo
2.10 Proposition: Lrr P,,(u,, . . . , u,,) E H / 2 [ [ u , , u 2 , .. . ] ] he rhe multiplicatirc sequence o f uny jormul power series f ' ( t ) E 212 [ [ t ] ] with Ieuding term 1 E U / 2 , und let 5 be any real m-plune bundle over a base space X E a. Then the 212 singulur cohomoloyp class ~ ~ (E 5H**(X) ) i s given by
whew w,,(<) E H"(X) is the nth Stiejt.I-Whitney cluss of where w,,(<) = 0 jbr n > m.
5 for
n
5 m, und
226
V. Stiefel-Whitney Classes
PROOF: By applying a splitting map X’ + X , if necessary, one may as well assume that 5 is a Whitney sum 1,@ * . . @ Am of line bundles. Then uf(E., 0 .. . @ Am) = u r ( i l ) u . . . u uf(Am)= f ( e ( A , ) )u . . . u f ( e ( A m ) ) by Proposition 2.5 and the obvious extension of Proposition 1.2 to formal P,,(ul(<),. . . , u,(<)), where power series, so that uf(<) is of the form u,,(t) is the nth elementary symmetric functionin e(A,),. . . , e(Lm)for n 5 rn, and where u,,(<) = 0 for n > m. However, Corollary 2.7 guarantees that w,(c;’) is the nth elementary symmetric function in e(Al), . . . , e(1,) for n 5 rn.
xnrO
There is another useful version of the Whitney product formula, expressed in terms of cross products rather than cup products. Recall that if X , x X , J% X , and X I x X 2 X , are the first and second projections of a product X , x X , of topological spaces X I and X , , the cross product of any LY E H‘(X,) and any /lE H s ( X , ) is a cup product: LY x p = prTa u pr;p E H‘+’(XI x X,).Recall also that the category B is closed with respect to products (Proposition 1.1.4), and that if t1and 5, are real vector bundles over X , E 49 and X , E B, respectively, then the bundle 5 , + 5 , over X , x X , E B is the Whitney sum pr!,tl @ pr!,5, of the pullbacks pr!,(, and PI-!,[, over X , x X , . 2.11 Proposition (Whitney Product Formula): For any formal power series f ( t )E Z j 2 [ [ t ] ]with leading term 1 E 212, and for any real vector bundles 5 , and t2over X , E 9 and X , E .@ respectively, I, the sum 5 , 5 , over X , x X , E satisjks
+
+ 5 2 ) = u,(51)
H * * ( X , x X,). PROOF: Replace 4, 4‘ and X in Proposition 2.5 by pri5 1, pr!,t2, and X x X , , respectively. Then Propositions 2.5 and 1.5 imply uJt1
Uf(5l
x
uJ52)
E
+ 5 2 ) = Uf(P‘!151 0 P + 2 5 2 ) = uf(Pr!,t,) u uf(Prlt2) =
prTur(t1) u Pr:u/(t,)
= UJ(t1) x
Uf(t2h
as claimed.
3. Orientability First Stiefel-Whitney classes provide a criterion for the orientability of arbitrary real vector bundles; it is essentially a generalization of Proposition IV.7.9.
227
4. The Rings H*(G"(IW");L / 2 )
A"
3.1 Lemma : Let t he the mth exterior power of' u red m-plane bundle ooer LI husr sptrce X E J ; then e ( P = wl(<) for the 212 Euler class e ( P 5 ) E H L ( X and ) the first Stiefel-Whitney class wl(<) E H ' ( X ) .
<
c)
PROOF: Let X ' 1; X be a splitting map for 5, so that f !
f ' * r ( / y y= ) e ( P ( f ! < )=) e ( i l 0 . . @ A"') *
=
e(%,)+ . . . + e(i,)
E
H'(X').
On the other hand, j'*w(<)= w ( f ! < )= w ( i , 0. . . @ i, = , w, ( &) ) u . . . u w(n,)
+ e ( l , ) )u . . . u (1 + e(i,))
H*(X') by naturality and the Whitney product formula for (total) Stiefel-Whitney classes, and by Proposition 1.2. In H ' ( X ' ) this implies ,f*wI(<)= e ( i l )+ . . . + e(;.,")= f'*e(A"'<), and since j ' * is a monomorphism one has w l ( < ) = r(A" 2 ) E H ' ( X ) , as claimed. = (1
E
3.2 Proposition: A real m-plane bundle 5 over a buse space X E 98 is orientuhle I / ' and only if' its first Stiefel-Whitney class satisfies wl(<) = 0 E H ' ( X ) .
<
<
PROOF: By Definition 111.10.4, is orientable if and only if is the trivial line bundle over X , and by Proposition IV.7.7 one has A" = E' E T(X) if and only if the 212 Euler class satisfies e(A" 5 ) = 0 E H ' ( X ) . Hence the result follows from Lemma 3.1.
<
4. The Rings H * (Gm(R"); Z/2) We now show that the 212 cohomology ring H*(G"'(Ra))of the Grassmann manifold G"'(R" ) is the polynomial algebra generated over 212 by the Stiefel-Whitney classes wl(y"'), . . . , w,(y") of the universal bundle y"' over G"'(R' ); similarly, for any q > 0 and sufficiently large n > 0, the portion of the ring H*(Gf"(Rfn'"))in degrees 0,. . . , y is isomorphic to the corresponding portion of Z/2[w,($'), . . . , wf,,($)]. In the next lemmas (RP")" is the m-fold product RP' x . . . x RP" of copies of the real projective space R P ' , and y1 + . . . + 7' is the corresponding sum of m copies of the universal real line bundle over R P a ,
228
V. Stiefel-Whitney Classes
as in Definition 111.2.1; that is, y' + . . . + y' is the Whitney sum prly' 0 .. . 0 pray', where (RP")" J% RP" is the jth projection map, for j = 1, . . . ,m. We recall from Proposition IV.5.7 that if (RP")" 1 :G"(R") classifies t + . . . + y' then h is also a splitting map for the universal real m-plane bundle y" over Gm([WX). According to Proposition IV.4.3 the Z/2 cohomology ring H*(RP") is the polynomial ring over 2 / 2 generated by the 2/2 Euler class e(y') E H'(RP"). Hence the Kiinneth theorem guarantees that the cross product H*(RP") 0 . . . @ H*(RP") 4H*((RP")") is an isomorphism, so that H*( (RP")") is the polynomial ring over 2/2 generated by the elements pTe(y9,. . . P:e(y') E H'((RP")"). -jl
3
+
+
4.1 Lemma: Let (RP")" 1:Gm(Rm)classify the bundle y l . . . 1' ouer (RP' 1"; then the image q/ H * ( G ~ ( R ') ) II:H*((RP")") lies in the ring of symmetric functions in the elements prTe(yl), . . . , prie(y') E H'((RP")").
PROOF: Let ( R P m ) " 5 ( R P " ) " be any permutation of the m factors in RP" x . . . x RP". Then the composition (RP")" 5 (RP")" 1:Gm(R") also classifies y' + * . . + y', so that the homotopy classification theorem provides a commutative triangle
In particular, the image of h* is invariant under TL*. 4.2 Lemma: Let (RP")" 1 :G"(R") classify the bundle y1 + . . . + y' ouer (RP")"; then the image qf' H*(G"(R")) 1:H*((RP")") contains the ring of' symmetric functions in the elements prle(y'), . . . , pr:e(y') E H'((RP")"). PROOF: For each i = 1 , . . . , m let wi(y") E H'(G"([W")) be the ith StiefelWhitney class of the universal real m-plane bundle y". Then h*w,(y") = w,(h!y") = w,(y' = w,(pr!,y' 0
+ . . + y') '
. . . @ prLy')
by naturality of Stiefel-Whitney classes, where w,(pr\y' 0 .. ' 0 prky') is the ith elementary symmetric function a,(e(pr\y'), . . . , e(prLy')) by Corollary 2.7, and where e(priy') = prfe(y') for j = 1, . , . ,m by naturality of 2 / 2 Euler classes. It remains to recall that the ring of symmetric functions in a
229
4. The Rings H*(G" (R'); Z/2)
polynomial ring in m variables is generated by the m elementary symmetric functions.
4.3 Proposition: For any I N > 0 the 212 cohomology ring H*(G"(R")) of' the Grussmunn munijold G"(R"') is the polynomial ring 2/2 [w,(y"), . . . ,w,(y")] yenerared over 212 by the Stiefel-Whitney cla.s.srs wi(y") E H'(G"(R")) of the unirwrsul m-plane bundle y m o i v r C"(Rx ) jor i = 1, . . . , m. PROOF: According to Proposition IV.5.7, if (RP")" 5 G"(R") classifies . , I + . . . + . , I then h is also a splitting map for 1'". In particular, one can I use the induced monomorphism H*(Gm(Rx)) 2 H*( (RP")") to identify H * ( G m ( R a )as ) a subring H:n,((RP")") c H*((RP")"), and Lemmas 4.1 and (RP")") is precisely the subring generated by the elemen4.2 imply that HrnV( tary symmetric functions oi(prTe(yl),. . . , pr;2;e(y1)),for i = 1,. . . , m. HOWever, oi(prTe(y'), . . . ,pr;2;e(y1))= h*w,(y"), as in the proof of Lemma 4.2, and since h* is a monomorphism, H*(Gm(Rco))is therefore the polynomial ring over 2 / 2 generated by wl(y"), . . . , w,(y"), as claimed. Proposition 4.3 provides an elegant alternative characterization of the Stiefel-Whitney classes of any real vector bundle whatsoever. One temporarily ignores the provenance ofthe generators w,(y"), . . . , w,(y")ofH*(G"(R")), merely identifying H*(C"(R')) as a polynomial ring 2/2[w,,. . . , wm] over 2/2 with one generator wi E Hi(Gm(Rm')) in each degree i = 1 , . . . , m. Then if X .& G"(R") classifies a given m-plane bundle over a base space X E B, naturality of Stiefel-Whitney classes yields wi(<)= wi(f'!y") = j * w i ( y " ) = .f*wi. The abbreviated result w i ( < )= j * w i can be regarded as a dejinition of the ith Stiefel-Whitney class w i ( ( ) E H ' ( X ) of <. The computation H*(G"'(Rx')) = 2/2 [ w , , . . . , wm] of Proposition 4.3 has further importance: it shows that every 2/2 characteristic class of uny m-plane bundle over any X E B can be computed from the Stiefel-Whitney classes wl((), . . . , w,(<). For if a characteristic class u,(<) E H 4 ( X ) satisfies the naturality condition, and if X L Gm(lW") classifies <, then u,(<) = u,(f'!y") = ,f*u,(y"), where u,(y") E HY(G"(R"')) is necessarily a polynomial u q ( w I , . . . , w,) in w , , . . . , w m ;hence
<
<
. . , W m )= U,(.f*Wl,. . . , f * w m ) = v,(wl(f!y")?. . . ~ , ( f ! I J " )= ) U,(Wl(<), . . . , w,(<)).
U,(O = j*v,(w,,.
7
For example, we already know from Corollary 2.8 that the 2/2 Euler class is given by e ( < )= w,(<) E H"(X), and Proposition 2.10 gives a specific prescription u,-(<) = (w,(<),. . . , w,(<)) E H**(x) for any f ( t ) E 2/2 [ [ t ] ] with leading term 1 E 2/2, where w,(<) = 0 for n > m.
C,P,
230
V. StiefelLWhitney Classes
Proposition 4.3 can also be proved in other ways. One of the nicest alternatives is an induction which uses H*(G"-'(R")) to compute H*(G"(R")). The case m = 1 is the identity H*(C'(R")) = H*(RP") = 2 / 2 [ e ( y ' ) ] o f Proposition IV.4.3, which was obtained from a Gysin sequence. The following key to the inductive step is also obtained from a Gysin sequence. 4.4 Proposition: For any m > 1 let y"-' and y" be the universal real vector hundles over G"' I ( R ') und G"( R ' ), respectively, let ue(;~"')he cup product by theZ/2 Eulerc/u.s.se(y"')E H(G"'(R")),andlet G"'-'(R') 5 G"'(R7)clu.s.s[fj ). there ure Z/2-module the real m-plane bundle y m L @ E' over G"- '(WThen homomorphisms g for which there is an exact sequence
'
. . . 3 Hq(G"((W"))
UC'IY"'),
2, p+yGm-
H4
+
"( G"( R"' ) )
3 Hq+'(G"(R"))+. . .
~ ( ~ r n ) )
of 2/2-modules. PROOF: According to Proposition 111.8.10, if Em5 G"(R") represents y", then there is a homotopy equivalence Gm-l(Ra')>E"* such that the composition ~ m - ~ ( ~1 :mE"*) 5 G"(R")isaclassifyingmapfory"-'O&l. Since classifying maps are unique up to homotopy, we automatically have f * = h* nm* in the following diagram, whose top line is the Gysin sequence for y'", as in Proposition IV.4.2. Since h* is an isomorphism, one can set g = Y;,,,, (h*)--'to complete the proof: 0
LJ
...
"tT;."')+
\ h*l //
Hq+m(Gm(Rm))
ne*
,H q + m ( E a * )
H4
+
"( G"
-1
'f~y"'
, Hq+l(Gm(Ra))
,..
(R") ).
We omit the application of Proposition 4.4 as an inductive step in the computation of the rings H*(Gm(Rm));it is an easy exercise, which appears with a brief hint as Exercise 7.4. Here is a finite-dimensional version of Proposition 4.3.
4.5 Proposition: For any m > 0 and any q > 0 there is an N(m, q ) > 0 such that lor any n 2 N(n1,y) the 212 cohomology ring H*(C"(R"+")) ugrees in dimensions 0, 1,. . . , y with the polynomial ring Z / 2 [ w l ( y r ) ., . . , w,(yr)] generated over 212 by the Stiejel-Whitney classes w,(y;) E Hr(G"'(R"'+")) ofthe canonicul real m-plane bundle y: over Gm(R"+"). PROOF: Let 7,' be the canonical real line bundle over the projective space R P , and let y,' + * * + y,' be the sum of m copies of y,' over the m-fold product
,
231
4. The Rings H" (G" ( R ' ) ; L / 2 )
+
RPY x . . . x RPq = (RP4)";that is, 7; + . . . 7: = pr!'y; 0 . . . 0 prb,y: for thc 111 projections (RP')''' a RP', as usual. Since ( R P ) " is a smooth ym-dimensional manifold, by Proposition 1.7.3, it follows from Proposition 111.9.3 that there is an N ( m , q ) > 0 for which there is a "finite classifying map" ( R P ) " 'A G"(R"+") for i Y . . . + 11; whenever n >= N(m,q); that is, i q + . . . + 1 4 = k ' f for ti >= N ( m , q ) . If G"(R"'+'') 5 G"(R") classifies 7: (in the usual sense), then the composition j k classifies y; . . . 7:. Now let R P RP' classify the real line bundle 7 ; over R P , so that the m-fold i"' product ( R P ) " +(RP"')" satisfies + . + i q = (i")!(y' + . . . + ; ) I ) ; if (RP')"'+! - G"(R' ) classifies ;I' . . . + 7 ' . then the composition h i'" is another classifying map for 7; . . . + ;!:. By the homotopy classification theorem (Theorem 111.8.9) the two classifying maps j k and h i" for .,I + . . . + .i ,YI are homotopic, so that there is a commutative diagram
+
-1'
"1'
-1'
+
;I:
+ +
+
-
*!l
it
0
I Y
H*(G"(Rm))
h.
'*
t
I
J. (Im)*
I
H*(Gm(Rm+" ))
I
H*((RP")")
+
+
k'
H*((RP)")
of Z/Z-modules. We already know for Lemmas 4.1 and 4.2 that h* is a monomorphism whose image H,*,,(( R P " )") c H*( (RP' )"') consists of those elements invariant under the automorphisms induced by permutations of the factors R P " , as in the proof of Proposition 4.3. By Corollary IV.4.5 H P ( R P " ) 2 HP(RPY)is an isomorphism for p 5 q, so that (i")* induces isomorphisms H&,((RP")")5Hr,,((RP)") for p 5 q, for the corresponding sub-ring HZ,((RP)"')c H*((RP"))"). Hence for each p 5 q there is a commutative diagram HP(G"(R~J))
it H P ( G " ( R ~ )) +"
c
H,P,,((RP"1")
HP,,((RP)")
for any I I 2 N ( n i , q ) , with isomorphisms /I* and (i")*. Thus j * is a monomorphism (and k* is an epimorphism). Now let X & G))I(Rm+") be any splitting map for y:, so that H * ( G ( R m + " ) ) is a sum E., 0 . . . 0 E., of J'. H * ( X ) is a monomorphism and line bundles over X ; the composition j / classifies /Ll 0 * . . 0 ,Inl. If X R P " , . . . , X -kRP" classify EL,, . . . , j.", respectively, then the map
232
V. Stiefel-Whitney Classes
. . ., I
X , (RP')" induces a homomorphism H*((RP")") 1:H * ( X ) ; the composition h I also classifies A1 0 * . * 0 A,,,.Since j f ' and h I both classify i , 0 . . . @ in,the homotopy classification theorem (Theorem 111.8.9) provides a homotopy commutative diagram 1=(1,.
)
0
cm(Rm)i
G"( R"
j
T
'1)
T
hl
(RP")"
+
0
I
x,
i
hence a commutative diagram HP(G"( Rm)) j r
h*
HP(
G"( R" ") ) +
I
I
J
J
I
I
' f
HP((RPm)")1 * HP(X), whenever p 5 q and n 2 N(m,q). We already know that h* is a monomorphism with image HE,((RP")"), and that j* and f * are monomorphisms. Consequently the preceding diagram reduces to HP(G"(R"))
* HP(G"(Rm+" ) )
mono jr
I
mono f*
H f J (RP")") [ ' HP(X), where h* is an isomorphism and the compositionf* j * is a monomorphism. It follows that I* is a monomorphism, which becomes an isomorphism in the further reduced diagram 0
mono f*
J
I"
H&,((RP")") ; Im I*. Commutativity of the latter diagram guarantees that the monomorphism f * is also an epimorphism, hence an isomorphism; consequently j* is also
233
5 . Axioms for StiefelLWhitney Classes
an isomorphism, assuming as always that p 2 q and n 2 N(m,q). Since j * w r ( y m )= wr(j!$")= wr(yy)E W ( G m ( R m + "for ) ) r = 1, . . . , m, Proposition
4.3 implies the desired result.
In the preceding proof we appealed to Proposition 111.9.3 for the existence of a "finite classifying map" ( R P ) " 5 G"(R"+") for the bundle y i + . * . y: over ( R P ) " , for sufficiently large n > 0. The following property of such maps is a finite-dimensional version of Proposition IV.5.7.
+
4.6 Proposition: Given q > 0,any"jnite classifying map"(RP4)" 5 G"(R"+") f o r the reul m-plane bundle yi . ' . y: over (RP)"is also a splitting map for the canonical real m-plane bundle yy over G"'(R""), in the restricted sense that H p (G"( R" )) 5 H p (( R P ) " ) is monic for p 5 q and n 2 N(m,q).
+
+
+"
PROOF: The preceding proof contains a commutative diagram H p( Gm(R ) )
j*
* H "( Gm(R" +'))
1
HRv((RP")")
1 (imp
* Hf'tnv((Rf")")
with isomorphisms h* and (i")*, and we later verified that j * is also an isomorphism. Hence k* is an isomorphism onto the submodule H&,((RP4)") c HP( (RPq)"),for p 5 q and n 2 N(m, q).
5 . Axioms for Stiefel-Whitney Classes We now establish an axiomatic characterization of Stiefel-Whitney classes of real vector bundles over arbitrary base spaces X E B,following the pattern used in Theorem 1V.6.1 for 2/2 Euler classes. Even if one considers only smooth real vector bundles over smooth manifolds, the corresponding axioms uniquely describe Stiefel-Whitney classes, as we also show. 5.1 Theorem (Axioms for Stiefel-Whitney Classes, for the Category 3): For reul vector bundles i: over base spaces X E 9,there are unique inhomogeneous 212 cohomoloyji classes w ( ( )E H * ( X ) which satis#y the following axioms: (0) Dimension: I f ' t i s ( I real m-plane bundle over X H o ( X )0 .. . @ H"(X) c H * ( X ) .
E 2,then
w(5)E
234
V. Stiefel-Whitney Classes
( 1 ) Naturality: lj' X ' 3 X is u mup in L@,und if 5 is u r e d vector bundle over X , then w(g'5) = g*w(() E H*(X'). (2) Whitney product formula: I f ( and q ure reul vector bundles otjer the sume X E B, with Whitney sum 5 0q over X , then w(5) u w ( q ) = w(( 0q ) E H * ( X ) ,for the cup product w(() u w(n). (3) Normalization: If y i is the canonical reul line bundle over RP' ( = S ' ) , then w ( y i ) = 1 e(yi) E H o ( R P ' )0H'(RP'), where e(yf)is the generutor of H * ( R P ' ) .
+
PROOF: The total Stiefel-Whitney classes of Definition 1.1 trivially satisfy Axiom (0),and they satisfy Axioms (1)-(3) by virtue of Propositions 1.3, 2.5, and 1.2, respectively. Conversely, suppose that w( ) satisfies Axioms (0)-(3), RP" classify let y' be the universal real line bundle over RP", and let RP' the canonical line bundle 7: over RP'. Then w(y:) E Ho(RP')0 H ' ( R P ' ) by Axiom (0),and Ho(RP") 0 H ' ( R P " ) H o ( R P ' )@ H'(RP') is an isomorphism such that f * ( l e(y')) = 1 e(y:) for the generator e ( y ' ) E H ' ( R P " ) , by Corollary IV.4.5. Since one has f * w ( y ' ) = w ( j ' ! y ' ) = w(y:) = 1 + e(yi) by Axioms (1) and (3), for the same isomorphism j ' * , it follows that w(y') = 1 e(y'). Now let ym be the universal real m-plane bundle over Gm(R") and let (RP")" A Gm(Raj)be the splitting map for y" described in Proposition IV.5.7, with
+
+
+
h'y"
= y'
+ . . + y' *
% for the m projections (RP"')'" J
h*w(y")
= pr!,y'
0 .. * 0prLy'
RP". Then by Axioms (1) and (2) one has
w(h!y") = w(pr\y' 0 .. * 0 prky') = w(pr\y') u . . . u w(prky') = pr:w(y') u * . u prEw(y'), =
-
which is a unique element of H*((RP")") since w(y') is uniquely defined, as we have just learned. Since h* is monic, by Proposition IV.5.7, it follows that w(y'") is a unique element of H*(G"(Rco)). By one final appeal to the homotopy classification theorem (Theorem 111.8.9) any real m-plane bundle ( over any X E L@ can be classified by a map X 3 Gm(ROC)which is unique up to homotopy, so that by Axiom (1) one has w ( 5 ) = w(g!y") = g*w(y") E H * ( X ) for a unique homomorphism H*(G"(R")) 3 H * ( X ) and the unique element w(y") E H*(G"(R")). Now let .& denote the category of smooth manifolds and smooth maps, as before. We know from Corollary 1.6.8 that A c .@, and we know from Theorem 111.9.5 that every real vector bundle over any X E .Af is itself smooth, in the sense that it can be represented by a smooth coordinate
<
235
5 . Axioms for Stiefel-Whitney Classes
bundle E X . Since many applications of characteristic classes concern only smooth manifolds X E A' and their tangent bundles T ( X ) ,for example, it is of interest to know that the preceding characterization of Stiefel-Whitney classes applies equally well to the subcategory A' c B.
5.2 Theorem (Axioms for Stiefel-Whitney Classes, for the Category m,tt!): For smooth real vector bundles 5 over smooth manijolds X E A?,there are unique inhomogeneous 212 cohomology classes w(5) E H * ( X ) which satisjy the following axioms: (0) Dimension: If' 4 is a smooth real m-plane bundle over X
E
.%e, then
w ( 5 ) E H O ( X )0 .. . @ H " ( X ) c H * ( X ) .
( 1 ) Naturality: I f X ' 3 X is a smooth map, and if 5 is a smooth real vector bundle over X , then w(g!() = g*w(() E H*(X'). (2) Whitney product formula: I f 5 und are smooth real vector bundles over the same smooth manifold X E .A!, with Whitney sum 5 @ q over X , then w ( < ) u w(q) = w(( 0 q ) E H * ( X ) ,,for the cup product w(() u w(q). (3) Normalization: l f y i is thecanonical real line bundleover RP' ( =S1), then ~ ( 7 ;=) 1 + ~ ( 7 :E)H o ( R P ' )0 H ' ( R P ' ) , where e(yi) is the generator of H*(RP').
PROOF: The existence of such classes follows automatically from the inclusion K c d and Theorem 5.1. Conversely, suppose that w( ) satisfies Axioms (0)-(3). For any q > 0 one easily obtains w(y:) = 1 e(y,') E H o ( R P )0 H ' ( R P ) from Corollary IV.4.5 and Axioms (0),(l), and (3), exactly as in the proof of Theorem 5.1, where y,' is the canonical real line bundle over R P , and where e(y,') E H ' ( R P ) is the uniquely defined generator of H * ( R P ) . Now, for any rn > 0, let q = m, so that there is a finite classifying map (RP")"' Gm(Rm+")for the real m-plane bundle y,!, + . . . + y,!, over (RP")" whenever n 2 N(m, 4 ) = N(m, m), as in Proposition 4.5; that is, k'y; = y ,I + . . . + y; = pr\y,!, 0 .. . @ prLy;
+
for the canonical real m-plane bundle y; over G"(R"'+"). By Axioms (1) and (2) one has k*w(y;)
= w(pr\y,!, 0 .. prky,!,) w(pr\yi,) u . . . u w(prLy,!,) = pr:w(yA) u . . . u pr:w(y,!,), = w(k!y;) =
and since we already know that w(y,!,) = 1 + e(yf), it follows that k*w(yT) is a uniquely defined element of H*( (RP")"). However, Axiom (0) guarantees that w(y;) vanishes in degrees above m, and since q = m, Proposition 4.6
236
V . Stiefel-Whitney Classes
guarantees that HP(G"'(R"'+n)) 5 HP((RPm)"')is monic for p 5 rn; hence w(y;) is uniquely defined in H*(G"'(R"'+" )), for any n 2 N(m,m). Finally, if 5 is any smooth real m-plane bundle over any X E ~then , Proposition 111.9.3 provides a finite classifying map X 5 G"'(R"'+") for sufficiently large n. By Theorem 1.6.19 one can assume that f is itself smooth; furthermore, one may as well suppose n 2 N(m,m), so that Axiom (1) implies w(5) = w(,f!y;) = f*w(y;) E H * ( X ) for the preceding class w(y;) E H*(Gm(R"'+"1). The final step of the preceding proof clearly does not depend on the choice of the finite classifying map f ;for if w( ) and It() both satisfy the axioms, the earlier steps of the proof give w(y;) = It(yy) E H*(G"(R""")), hence
w(5) = w(f!f) = f*w(y;) = f*It(yY) = I t ( f ! y ; )
= It(S)E
H*(X).
However, there is also an explicit demonstration that the choice o f f is immaterial. Suppose that X % G"'([W"+"') is another smooth finite classifying map for the smooth real rn-plane bundle t over X E 4, for some n' 2 N(m, m). The ersatz homotopy uniqueness theorem (Proposition 111.8.14) then provides (smooth) finite classifying extensions gn,n..and gn.,n..for which the compositions Gm(,m+n) g n * * ~ ' ' Gm(,rn+n") + and 3 p(Rrn+n') !/*l'm'' G"'(R"'+"'')
x
x
-
are homotopic. The axioms then imply f*w(Y;)
= f*w(gb,,*,r;4 = f*g:,,4y;,4
= (gn,,,,
f)*w(y;4
= g*g:,,,..w(y;?.)
= (gn.,d, g)*w(y7,)
= g*w(g~~,,~~y;~.)
= g*w(y3
as expected. Since the Grassmann manifolds Gm(Rrn+")are both closed and smooth (Proposition I.7.3), they are a fortiori compact and triangulable. One therefore has analogs of Theorem 5.2 in which the category 4 of smooth manifolds is replaced by any one of several categories of topological spaces. For example, Theorem 5.2 is valid for real vector bundles over spaces in the category of smooth closed manifolds. Theorem 5.2 is also valid if one replaces A' by the category of those compact spaces which happen to lie in 99,or by the category of finite-dimensional simplicia1 spaces (which automatically lie in by Corollary 1.2.2 and Proposition 1.4.6). In each case one of Propositions 111.8.12,III.9.1, or 111.9.3 furnishes the required finite classifying maps.
231
6 . Dual Classes
6 . Dual Classes Since 2/2 is a field, and since we work exclusively with formal power series f ( t )E 2/2 [[t]] with leading term 1 E 2/2, it follows for each such f ( t ) that there is a unique formal power series (l/f)(t) E 2/2 [[t]] with leading term 1 E 2/2 for which , f ( t ) . (l/f)(t) = 1 E 212 [[t]]. For example, if f ( t ) = 1 t , then ( l / . f ) ( t = ) 1 t tZ .. . .
+
+ + +
6.1 Definition: For any formal power series f ( t ) E 2/2[[t]]
with leading term 1 E 2/2, and for any real vector bundle over a base space X €98,the multiplicative 2/2 class u,,,-(<) E H * * ( X ) is the dual class of the multiplicative 2/2 class us(<) E H**(X). In particular, for f ( t ) = 1 + t and ( l / f ) ( t )= 1 t + tZ + . . . , the class u,,,-(<) E H * * ( X ) is the dual Stiefel-Whitney cluss of (, denoted @(<).
<
+
6.2 Lemma: For uny real line bundle p over a base space X E 99,and for any ,formu1 power series ,f(t)E 2/2 [ [ t ] ] with leading term 1 E 2/2, it follows that U / ( P ) u Ul//(P) = 1 E H**(X). PROOF: There is a unique multiplicative homomorphism 2/2 [[t]] + H * * ( X ) carrying t into e(p),and since p is a line bundle, the relation f ( t ) . ( l / . f ) ( t )= 1 defining (l/f)(t) and Proposition 1.2 imply that u,(p) u ul/f(p) = f M 4 ) u ( l / f ) ( e W = 1 E H**(X).
<
6.3 Lemma: For any real vector bundle over a base space X E 9, and for any formal power series f ( t ) E 212 [[t]] with leading term 1 E 2/2, it follows that u,,,-(<) is the unique class such that us(<) u u,,,-(<) = 1 E H**(X). PROOF: Any splitting map X ' 3 X provides a sum g!( = p I 0 .. .0 p,, of line bundles p,, . . . , p,,over X' and a monomorphism H * * ( X ) H**(X'), and one uses Lemma 6.2 to complete the proof.
<
6.4 Proposition: Suppose that and q are real vector bundles over the same base spuce X E d,and suppose that there ure trivial bundles E~ and E~ over X such [hut 0 q 0 E P = E". Then for any formal power series f ( t )E 212 [[t]] with leuding term 1 E 2/2 it follows that u,,,(<) = u,-(q) E H**(X).
<
PROOF: Since the leading coefficient of f ( t ) is 1 E 2/2, it follows that the coefficient of us(() in H o ( X )is also 1, and since 2/2 is a field, there is a unique cohomology class c1 E H * * ( X ) with us(<) u c1 = 1. However, us(<) u u,,,-(<) = 1 by Lemma 6.3, so that u,,,-(<) = a. On the other hand, according to Corollary 2.6 one has us(&") = 1 and u,(P) = 1, so that one can apply the Whitney
238
V. Stiefel-Whitney Classes
<
product formula to the identity 0q 0,cP = c4 to conclude that uf(<) u us(q) = I , so that u,-(q) = u. Hence u , , ~ ( <=) c( = uf(q), as claimed. 6.5 Corollary: Suppose that 5 and q are real vector bundles over the same base space X E 3?, and suppose that there are trivial bundles ,cP and i? such thut 5 0 q 0z p = c4. Then the Srief&Whitney class w ( q ) E H * ( X ) und dull1 Stiejel-Whitney cluss a(<) E H * * ( X ) satisfj- w ( q ) = W(5) E H * * ( X ) ; ajbrtiori
W(<)E H * ( X ) .
We recall once more that in most applications the base space X will have a finite-dimensional cohomology ring H * ( X ) , so that H * ( X ) = H * * ( X ) .
7. Remarks and Exercises 7.1 Remark: Stiefel-Whitney classes were first constructed in Stiefel [ I ] and in Whitney [2], independently written papers of 1935. Stiefel considered the existence of k linearly independent vector fields on a smooth n-dimensional manifold X . Suppose for convenience that X is oriented and that it has a fixed triangulation K ; the (n - k)-skeleton IKn-kl c JKI = X consists of all geometric p-simplexes in with p 2 n - k . There are always k linearly independent vector fields on IKn-kl. However, Stiefel found an obstruction class in H n - k + l ( X )whose vanishing is a condition for the existence of k linearly independent vector fields on IKn-k+ I ), where the coefficient group of H n - k + l ( X )is Z or Z/2 according as n - k is even or odd. Since vector fields can equally well be regarded as sections of the tangent bundle z ( X ) , Stiefel's obstruction classes can be regarded as characteristic classes of r ( X ) ; furthermore, if one reduces the coefficient ring Z to 2/2, then Stiefel's obstruction classes are precisely the Stiefel-Whitney classes w , - k f l ( ? ( X ) ) E H n - k + ' ( X ;Z/2). We already noted in Remark 111.13.3 that the first definition of fiber bundles is in Whitney [2]; the very same paper contains a sketch which generalizes Stiefel's construction to arbitrary oriented sphere bundles over a polyhedron IKI. In either paper one can ignore the orientation by reducing the coefficient ring Z to Z/2; however, one then loses part of the obstruction information. A more recent account of Stiefel [13 and Whitney [2] is given in Steenrod [4, pp. 190-1991, and a summary of the same material is given in E. Thomas [2, pp. 159-1611; the homotopy computation leading to the coefficient groups Z and Z/2 is presented in modern dress on pages 202-203 of G. W. Whitehead [11 for example. The special case k = 1 will be treated in detail in Chapter VIII of the present work, and the cases 1 < k I n will be summarized in the Remarks at the end of that chapter.
IKI
7. Remarks and Exercises
239
There are many other constructions of Stiefel-Whitney classes, in addition to that of Definition 1.1. One of these (suggested by Proposition 4.3) is discussed in the following remark, an entirely different one of Thom [l, 41 and Wu [3] is presented in Exercise 7.17 and Remark 7.18, and de Carvalho [I] contains yet another construction, for example. However, the original methods of Stiefel [I] and Whitney [2] are still of current interest: Stern [l, 21 assigns Stiefel-Whitney classes to the topological R" bundles of Remark 111.13.9, with an interpretation as obstruction classes in the sense of Stiefel [I] and Whitney [2] (over suitably restricted base spaces). 7.2 Remark: Following Proposition 4.3 it was observed that the Stiefel. . . , w m ( [ )of a real m-plane bundle 5 over any X E 93 Whitney classes w,((), could be defined by setting wi(()= f * w i for the generators w,,. . . , w, E H*(G"(R"); Z/2); over appropriate base spaces one can replace G"(R") by Gm(Rm+") for suitably large n and appeal to the ersatz homotopy classification theorem for a similar result. This technique was first introduced in Pontrjagin [l], the details appearing in Pontrjagin [5]. The technique also appears in Chern [2,3], in Wu [2, 51, in Takizawa [l], and in Bore1 and Hirzebruch [I, pp. 483-4871, for example; a group-theoretic interpretation of the construction is given on pp. 496-497 of the latter paper. Hodge [3] contains one of the many early complex analogs of the same technique.
7.3 Remark : The preceding characterization of Stiefel-Whitney classes requires the structure of the ring H*(G"(R'); 2/2), which we computed in Proposition 4.3 in terms of the particular Stiefel-Whitney classes wl(y"), . . . , w,(;'"'). However, the study of H*(G"(R');Z/2) and H*(G"(R"+"); Z/2) antedates the very definition of cohomology. In order to trace the early history of H*(G"(Rm+"); Z/2) we begin with the work which led to the first computation of the complex analog, H*(G"(@"+");Z), considered in Volume 2. During the 1870s Hermann Cisar Hannibal Schubert developed a technique for solving certain geometric intersection problems in R3, the enutnerutiur culculus of Schubert [I]. The technique was applied during 1886 to similar problems in (complex) vector spaces of arbitrary finite dimension, for which the Grassmann manifolds Gm(C"'+'')provided a natural setting. In particular, Schubert uarieties were constructed as submanifolds of G"'(@"''") in Schubert [2], and it was shown (with the rigor of 1886) that certain Schubert varieties represent cycles whose dual cohomology classes form a convenient basis of the cohomology modules H*(G"(@"+"); Z). Intersections of such Schubert cycles were computed in Pieri [I] and in Giambelli [l], furnishing 1894 and 1902 versions, respectively, of the cup ); Z).Todd [I] later used the original product structure of H*(Gm(@"+"
240
V. Stiefel-Whitney Classes
methods of Schubert [2] to derive Giambelli's results from those of Pieri. In 1934 Ehresmann [I] used topological intersection theory to give the first rigorous statement and proof of the Schubert "basis theorem" for G"'(C"l+"). Schubert varieties and Schubert cycles are constructed for G"'(R"'+") as they are for G"'(C"'+''), and in 1937 Ehresmann [2] used topological intersection theory to establish an analogous "basis theorem" for G"'(R"'+"). In 1942 (and 1947) Pontrjagin [ 1, 51 used Ehresmann's methods to give a corresponding "basis theorem" for the oriented double covering @'(Rm+") of G"'(R"'+"). (The cohomology of G"'(C"'+") and @""Rrn+")will be considered in Volume 2.) Meanwhile, Hodge [l] developed a rigorous and entirely algebruic justification for the Schubert "basis theorem", in 1941, followed by a further investigation of intersections of Schubert cycles in Hodge [2]. The Schubert calculus computation of H*(G"(C"+");Z) is presented in Hodge and Pedoe [l, Chap. XIV] and in Griffiths and Harris [l, pp. 193-2061; as in all the work just cited, there is no explicit mention of cohomology. General expositions of Schubert calculus are given in Zeuthen [11, Pieri and Zeuthen [11, Severi [l, 21, Kleiman and Laksov [I], and Kleiman [l]. In 1947 Chern observed that the closest real analog of H*(G"'(C"'+"1; 2) is H*(G"'(Rm+"); 2/2), with the coefficient ring 2/2, and he used Schubert calculus in Chern [2,.3] to provide the first proof of Proposition 4.5. Chern's results inspired entirely different computations of H*(G"'(R"); Z/2) and H*(G"'(R"'+"); Z/2), such as the ones used for Propositions 4.3 and 4.5, respectively. Other computations of H*(G"'(R"); 2/2) are sketched in Exercise 7.4 and Remark 7.5, and similar computations of H*(G"'(R"); 2/2) are given in Milnor [3, pp. 26-31] and in Milnor and Stasheff [l, pp. 83881; an alternative using spectral sequences is given in Liulevicius [2, 117-1221. 7.4 Exercise: Use Proposition 4.4 to compute H*(G"'(Rm); 2/2) by induction on m. (Hint: The initial step is in Proposition IV.4.3. To establish the inductive step observe that the homomorphisms f * in the exact sequence of Proposition 4.4 are surjective, hence that there are short exact sequences
0 + Hq(G"(Rm))
=
H"+"(G"(R"))
-5Hq+"(G"-'(R"))
+ 0.)
7.5 Remark: Since H*(RP"'; Z/2) is the polynomial ring generated over 2/2 by the 2/2 Euler class e(y') = w,(y') E H'(RP'), one can use Remark 7.2 to define the Stiefel-Whitney class w,(A) E H ' ( X ; 2/2)of any line bundle 3, over any X E B, hence the total Stiefel-Whitney class 1 + w,(A) E H * ( X ; 2/2). The Whitney product formula (Proposition 2.5) and the splitting principle (Proposition IV.5.5) then provide yet another characterization of StiefelWhitney classes, essentially the uniqueness theorem proved earlier. This
24 1
7 . Remarks and Exercises
construction originated in Grothendieck [ 11, in a slightly different setting; it can also be found in Braemer [11. 7.6 Exercise : Some of the preceding remarks indicate alternative constructions of Stiefel-Whitney classes, and there are more to come. For example, show that the special case f ( t ) = 1 t of Definition 1.1 is equivalent to the following jormulu of'Hirsch for any real m-plane bundle 5 over any X E g :
+
e(i.;)"
+ w,(c)e(l;)"- + . . . + w,_
,(&(A,)
+ W,(()l
=0
in the free H*(X)-module H*(Ps), where i., is the splitting bundle over P, E a. This identity was apparently known to Chern and Wu before its publication (in a slightly different setting) in G. Hirsch [7]. The language "formula of Hirsch" appears in Hirzebruch [3, p. 751. 7.7 Remark : The axiomatic characterization of Stiefel-Whitney classes reported in Theorems 5.1 and 5.2 was first given in Hirzebruch [2]; it can be found in Hirzebruch [3, p. 731. 7.8 Remark: A special case of the Whitney product formula of Proposition 2.5 was first announced as a "duality theorem" in Whitney [6], without proof. The proof is implicit in Chern's computation of the ring structure of H*(Grn(Rrn+" ); 2/2) in Chern [2,3], for example, and an explicit proofappears in Wu [l]. More recent product formulae in H*(G"(Rrn+"); 2/2) can be found in Oproiu [I]. 7.9 Exercise: Use the identity e(A 0 p) = e(A) + e ( p ) of Proposition IV.7.5 to prove the following property of Stiefel-Whitney classes: For any natural numbers m 2 0 and n 2 0 there is a unique polynomial P,,,nin m + n variables over 2/2 such that
~ ( 05V ) = Prn,n(wl(
*
9
same base space X E J.(This is a relatively easy exercise; however, alternative approaches to its solution can be found in Bore1 and Hirzebruch [ 13 and in E. Thomas [11.) 7.10 Remark: In Thom [4] one learns that the total Stiefel-Whitney class w(() of a vector bundle 5 depends only on the J-equivalence class of 5. This suggests that Stiefel-Whitney classes should be assigned directly to J equivalence classes; it also suggests that one should try to construct StiefelWhitney classes directly from less information than a vector bundle carries.
242
V. Stiefel-Whitney Classes
For example, since Thom [4] also shows that the J-equivalence class of the tangent bundle z ( X ) of a smooth manifold X is independent of the smooth structure assigned to X , it follows that w ( r ( X ) )is independent of the smooth structure of X. Nash [l] proves the latter result directly; in fact, Nash's construction assigns total Stiefel-Whitney classes to arbitrary topologicul manifolds X , the results agreeing with w ( r ( X ) )for smooth manifolds X . (Nash's construction also leads to generalizations of several classical results about tangent bundles of smooth manifolds, as noted earlier in Remark 111.13.9, in Fadell [4], R. F. Brown [l], and Brown and Fadell [l].) Stiefel-Whitney classes of more general fibrations, including the topological R"' bundles of Remark 111.13.9, have been constructed more recently in Teleman [l, 2,3] and Bordoni [l]. According to Teleman [2], the specialization to the tangent topological R" bundle of an n-dimensional manifold X provides an alternative computation of Nash's total Stiefel-Whitney class of X . It has already been observed in Remark 7.1 that the individual StiefelWhitney classes arising in such constructions can be interpreted as obstruction classes, as in Stern [1,2]. An earlier construction of the Stiefel-Whitney classes of the preceding paragraph can be found in Vazquez [l], using generalizations of methods described in some of the following remarks; an application of the result appears in Vazquez [2]. 7.11 Exercise: Let A be any commutative ring with unit, and let Po = 1, P,(u,), P,(u,, u,), . . . be a sequence of polynomials P,(ul,. . . , u,) E A[u,, . . . ,u,] such that if uy is assigned degree q, then P,(ul, . . . u,) is of degree n. The sequence can be regarded as an element Cn2 P,(u, ,. . . , u,,) of the formal power series ring A[[u,,u,, . . .]I, as in Definition 2.9; for example, if u,, = 1, then u, is itself such an element. One can compute the formal product
where w,
=
and
c,120 P,(ul, . . . , u,) is multiplicative whenever -
, P,(u,, . . . , u,) is the multiplicative sequence assigned to a Show that if formal power seiiesf'(r) E A[[t]] as in Definition 2.9, then it is multiplicative in the present sense. 7.12 Exercise: Conversely, let sense of Exercise 7.1 1; show that
P,(ul,. . . , u,) be multiplicative in the
if is the multiplicative sequence assigned
243
7. Remarks and Exercises
as in Definition 2.9 to the formal power series . f ( t )=
1 P,,(r,O.O,
. . . ,O) E A[[r]].
n>o
7.13 Remark: Multiplicative sequences were introduced in Hirzebruch [I]. Other expositions can be found in of Hirzebruch [3, pp. 9-16], Milnor [3, pp. 111-1 161, and Milnor and Stasheff [ 1, pp. 219-2221. 7.14 Remark : Thom provided a useful alternative construction of StiefelWhitney classes in Thom [l, 41. In order to discuss it we first describe a classical 2 / 2 cohomology operation which is not ordinarily presented in beginning algebraic topology courses. Let H * ( - , -) (and H*(-)) denote singular cohomology with 2/2 coefficients, as before, and for each pair ( X , Y ) of topological spaces with Y c X let Sq' denote a Z/2-module homomorphism H * ( X , Y ) --* H * ( X , Y ) of degree i 2 0, carrying each summand H " ( X , Y ) into the summand H""(X, Y). The direct sum Sq = Sqo @ Sq' @ Sq2. . . is the Steerirod square if it satisfies the following axioms: (0) Dimension:
lj' a E H o ( X , Y ) 0 . . . @ H " ( X , Y), rhen Sq'a
=0
fbr
i > I?.
(I) Naturality: V.f is any nmp f h n a pair (A",Y') to a pair ( X , Y), f/JPlJ sq f * X = ,f'* sq 3! E H * ( X ' , Y') ,for U H y E f f * ( X ,Y). (2) Cartan formula : The cup producr a u /j E H*( X , Y ) of' o! E H*( X , Y ) L I /j ~ E H * ( X , Y ) .sclri.sfie.s
Sq(a v
p, = s q a v s q p.
(3) Normalization: !f' a E H"(X , Y ) ,for. c r n y
Sqo
= CI
U I I ~
Sq" ~1 = a u
IJ
2 0, fhen
E
H 2 " ( X ,Y),
The Steenrod square first appeared in Steenrod [3], and the preceding axiomatic characterization (including the Cartan formula) first appeared in Cartan [I]. Early general expositions of the Steenrod square can be found in Exposes 14 and 15 ofcartan [3], in Thom [5] and Wu [6], and in Steenrod [5]; the latter papers contain generalizations for other primes p than the prime p = 2, independently presented in Bott [I]. Somewhat more recent expositions of the Steenrod square (and its generalizations) can be found in Steenrod and Epstein [I, pp. 112-113, 124-1321, Spanier [4, pp. 269-2761, Mosher and Tangora [ I , pp. 12-32], and Steenrod [9], for example. R. J. Milgram developed a simplified construction of the Steenrod square, first published in Gray [I, pp. 310-3231. 7.15 Remark: Among the consequences of the preceding axioms are the Adem relariotis, which will figure in later remarks and exercises, and which
244
V. Stiefel-Whitney Classes
can be found in most of the preceding references; they were discovered independently by Adem [l, 21 and Cartan [2].
7.16 Remark: The Cartan formula of Remark 7.14 can also be expressed in terms of cross products. Recall (from Dold [8, pp. 221-2221, Greenberg [ l , p. 2011, or Spanier [4, p. 2531, for example) that if X and Y are any : topological spaces whatsoever, then the cross product H*(X) 0H * ( Y ) 1 H*(X x Y ) can be obtained as the composition H*(X)0H*( Y )
pr?@ p r t
H*(X x Y ) 0 H*(X x Y ) 2 H*(X x Y )
for the projections X x Y 3 X and X x Y 3 Y. Naturality and the preceding Cartan formula for Sq then imply Sq(a x p) = Sq a x Sq fi E H*(X x Y ) for any a E H*(X) and fi E H * ( Y ) . There is an obvious relative version of the same result, and both versions are regarded as Cartan formulae. Conversely, if X 3 X x X is the diagonal map one uses the composition
H * ( X ) 0 H*(X) 1:H*(X x X) 5 H*(X) to define cup products in H*(X), so that naturality and the latter Cartan formula for Sq imply the absolute version of the Cartan formula of Remark 7.14; the relative version of the same construction is clear.
<
7.17 Exercise: Let be a real n-plane bundle over X E a,with 2/2 Thom class U , E H"(E,E*) and 7/2 Thom isomorphism H*(X) 2H*+"(E, E*), and let Sq be the Steenrod square of Remark 7.14. Use the axiomatic characterization of Stiefel-Whitney classes (Theorem 5.1) to show that w(<)= @'r1(SqU r ) =
Sq@,(l) E H*(X)
for the element 1 E Ho(X).
7.18 Remark: The identity w(<)= @; Sq@,(l) can also be written in the concrete form n*w(<)u U , = Sq U , E H*(E, E*). Since H*(E, E*) is the free H*(X)-module generated by the Thom class U,, this clearly characterizes
45). 7.19 Remark: If 5 is a real n-plane bundle over X E 98,then Corollary 2.8 u U,) E and Proposition IV.3.8 provide an identity w,(<)= e(<)= @i '(Ur H"(X). Since U , u U , = Sq"U , , this identity can be regarded as the dimension n portion of the identity w(<)= '(Sq U , ) of Exercise 7.17; in fact, one uses this observation as a part of Exercise 7.17. 7.20 Remark: Since 212 is a field, one can regard the Steenrod square as an automorphism of the 2/2 cohomology rings H*(X, Y), satisfying the obvious
245
7. Remarks and Exercises
'
naturality condition; in particular, there is an inverse Steenrod square Sqwhich is also natural. The same remarks apply if one substitutes the group of units in H**(X, Y ) for H*(X, Y ) ; that is, 0th entries are required to be 1 E H o ( X , Y), and one considers products only. One can therefore define the rota/ Wu class of any real vector bundle 5 over any base space X E B by setting Wu(5) = Sq-'w(()= Sq-'o,5'Sqo;(l)EH**(X) for the total Stiefel-Whitney class w(5) E H * ( X ) (cH**(X)) and the Thom isomorphism at; that is, Wu(() E H**(X) is the unique class such that s q W u ( 0 = w(5) E H * ( X ) . 7.21 Exercise: Show that there is a formal power series f ( t )E 2/2 [[t]] with leading term 1 E 2/2 such that Wu(5) = us(() E H**(X), as in Definition 1.4, for any vector bundle 5 over any X E 98.
7.22 Exercise: Show that the "Wu series" of Exercise 7.21 is given by . f ( t ) =1 r Z 4 E 2/2 [[t]]. Show furthermore that f ( t ) 2 = f ( t 2 ) = f ( t ) + t , and that (l/f)(r) = Cq20rzs-1 E 2/2[[t]].
+Iqzo
7.23 Remark: Wu classes of tangent bundles first appeared in Wu [3]. The definition was extended to arbitrary real vector bundles by Adams [3], who observed that Wu classes could be computed in terms of Stiefel-Whitney classes. The specific recipe Wu(5) = Sq-' (D; Sq(Dt(l)was given by Atiyah and Hirzebruch [2]. Exercises 7.21 and 7.22 provide an alternative construction of Wu classes, without Steenrod squares.
'
7.24 Exercise: Let (f) denote the 2/2 image of the usual binomial coefficient (f) whenever a 2 h 2 0, let (-;) = 1 E Z/2, and let (f) = 0 E B/2 otherwise. Show that the Stiefel-Whitney classes wo = 1, wI, w2, . . . of any real vector bundle satisfy the following Wu relation:
7.25 Remark: The Wu relation of Exercise 7.24 first appeared in Wu [4], the complete proof appearing in Wu [7]; a proof related to that of Wu [7] is given in Bore1 [2]. One can also use the Adem relations of Remark 7.15 to obtain the same result, as in Hsiang [l]. The best way to prove the Wu relations involves a technique developed in Brown and Peterson [l]. 7.26 Remark: Any real rn-plane bundle 5 over a contractible base space Y is the trivial bundle E ~ by , Proposition 11.3.5, so that Corollary 2.6 gives
246
V. Stiefel-Whitney Classes
u f ( ( ) = uf(cm)= 1 E H * * ( Y ) for any f(t) E 2 / 2 [[t]]
with leading term 1 E 2/2; equivalently, all the Stiefel-Whitney classes wl(c), . . . , w,,,(c) vanish. There are other base spaces Y with the latter property. For example, let X be any CW space and choose a point * E X . As in Remark 111.13.41 the (reduced) suspension C X is the quotient of the product X x [0,1] by the subspace X x (0) u {*} x [0,1] u X x (1). in the quotient topology; by iteration one obtains the n-fold (reduced) suspension C”X for any n 2 0. One of the consequences of Atiyah and Hirzebruch [3] is that all the StiefelWhitney classes of any real vector bundle over C”X vanish whenever n 2 9; that is, such suspensions have the preceding property of contractible base spaces Y . According to Sutherland [2], weaker results of a similar nature apply to the Stiefel-Whitney classes of real vector bundles over suspensions C 2 X and C 5 X .
7.27 Remark : The Stiefel-Whitney classes of various specialized real vector bundles satisfy certain polynomial relations, some of which will be developed for geometric applications in the next chapter. For example, we shall learn in Corollary VI.8.6 that if t ( X ) is the tangent bundle of any smooth closed n-dimensional manifold X , then the Wu class W u ( t ( X ) )= 1 Wu,(t(X)) . . . Wu,(t(X)) satisfies Wu,,(z(X)) = 0 E H P ( X ) whenever 2 p > n ; since the individual Wu classes are polynomials in the StiefelWhitney classes w l ( t ( X ) ) ., . . , w , ( t ( X ) ) , the result can be regarded as a set of polynomial relations among the latter classes. In Brown and Peterson [ 1,2] one finds all the polynomial relations satisfied by w , ( z ( X ) ) ,. . . , w , ( T ( X ) ) for every smooth closed n-dimensional manifold X . Incidentally, E. H. Brown [2] and Stong [l] provide independent proofs that if 2p 5 n, then there are no such “universal” polynomial relations P ( w l ( z ( X ) ) ., . . , w,(T(X)) = 0 E H P ( X )in degree p . In sufficiently high degrees there are “universal” polynomial relations satisfied by the dual Stiefel-Whitney classes W l ( t ( X ) ) ., . . , W,,(t(X))of every smooth closed n-dimensional manifold X , where 1 + w l ( t ( X ) )+ . . . + W , ( t ( X ) ) = w ( r ( X ) )= u f ( 7 ( X ) )for f ( t ) = 1 + t + t 2 + * . . E Z/2[[t]]. Such relations were investigated in Massey [l, 31 and in Massey and Peterson [ 11; the latter paper shows that if 0 S p < cc(n), where a(n) is the number of 1’s in the dyadic expansion of n, then W,-,,(t(X)) = 0 for every smooth closed n-dimensional manifold X . The set of all polynomial relations satisfied by iVl(r(X)),. . . , W,(t(X)) for every smooth closed n-dimensional manifold X is implicitly described in Brown and Peterson [l, 21: one merely replaces w by W throughout their original argument (and result); related results are given in Bendersky [l] and in Papastavridis [2]. The result itself is an essential ingredient of our later Remark VI.9.14.
+
+ +
7. Remarks and Exercises
247
7.28 Remark: According to Proposition 3.2 one can reasonably define orientability of a smooth manifold X by the requirement that wl(z(X))= 0 E H ' ( X ) . An orientable manifold which satisfies the additional relation W ~ ( T ( X=) 0) E H 2 ( X )is called a spin munifbld. (Alternative characterizations of spin manifolds are given in Bore1 and Hirzebruch [2, p. 3501, the end of Milnor [ 121, and at the beginning of Milnor [ 161.)The total Stiefel-Whitney class W ( T ( X ) ) of a spin manifold automatically satisfies further relations, some of which are discussed in Wilson [ 13, for example. 7.29 Remark: Since realifications of complex vector bundles are somewhat specialized, one expects their Stiefel-Whitney classes to satisfy universal polynomial relations. For example, the realification laof a con3plex vector bundle ( is naturally oriented, so that w , ( i a ) = 0 by Proposition 3.2; other such relations will appear in the second volume of this work. There are also universal polynomial relations satisfied by the Stiefel-Whitney classes of realifications of "quaternionic bundles," some of which appear in Marchiafava and Romani [I, 2, 31. 7.30 Remark: One of the classical results of differential topology is that S', S3,and S7 are the only standard spheres which are parallelizable in the sense of Remark 111.13.26; that is, if S" has n linearly independent vector fields, then n = 1, 3, or 7. This result was proved in Bott and Milnor [l] and Milnor [4], using the following property of Stiefel-Whitney classes : If S"is the base space of a real vector bundle 4 with w,(() # 0, then n = 1,2,4, or 8. (Incidentally, the result of Barratt and Mahowald [l] reported in Remark 111.13.36 instantly implies that if n = 4k > 16, then w,(() = 0 for any real vector bundle ( over S",) I n the third volume of this work we shall formulate and prove a result of Adams [l, 21, one of whose principal corollaries is that S', S3, and S7 are the only parallelizable standard spheres. Adams's original proof, which also appears in Cartan [ 5 ] , uses many of the techniques introduced in this chapter, including Steenrod squares; however, the proof given later will be based on an entirely different technique of Adams and Atiyah [ 13. 7.31 Remark: There is a more general result than the one just described: the maximum number of linearly independent vector fields on the standard sphere S" is known for any n > 0. One of the early guidelines for the computation was provided by Stiefel [2], who used Stiefel-Whitney classes to show that if n = 2 k -~ 1 for an odd number u, then the projective space RP" cannot have 2k linearly independent vector fields. Steenrod and Whitehead [ 13 used Steenrod squares to obtain the same result with S" substituted for RP", and their work eventually led to the complete solution of the problem,
248
V. Stiefel- Wh itney Classes
in Adams [4]. (We shall not attempt to present a proof of this result. Adams’s original proof is outlined in Eilenberg [l] and in Husemoller [l], and some simplifications by Karoubi are reported in Gordon [l], for example; Adams himself streamlined the proof in many ways, and a major simplification appeared in Woodward [l]. A complete proof is given in Karoubi [2], incorporating all improvements through 1978; a complete proof is also given in Mahammed et al. [I], incorporating all improvements through 1980.) 7.32 Remark: To any finite-dimensional real representation I/ of a finite group G one can associate a real vector bundle 5“ over the classifying space BG of Remark 11.8.18; the total Stiefel-Whitney class w(tv) E H*(BG) is then the total StieJel- Whitney class w(V ) of the representation V . Segal and Stretch [11 present an alternative construction of w(V ) ,which leads to new results about group representations. 7.33 Remark: There is an algebraic construction of Delzant [l], which assigns Stiefel-Whitney classes to certain “quadratic modules” rather than to real vector bundles; the construction was further developed in Milnor [ 191. A related construction exists for real vector bundles, given in Patterson [l]; however, this variant produces only the terms 1 + wl(t) + w2(t) of the total Stiefel-Whitney class w(() of a real vector bundle 5. A recent exposition of the constructions of Delzant and Milnor is in Chapter 4 of Micali and Revoy [l]. 7.34 Remark: Lemma 6.3 is a special case of a more general result, with essentially the same proof. Let f ( t ) and g(t) be any formal power series in 212 [[t]] with leading terms 1 E 212, so that the product (.fg)(t)is also such a formal power series. Then for any real vector bundle 5 over a base space X E 2 one has u,-(<) u u,(<) = ur,(<) E H**(X). Thus t induces a homomorphism from the multiplicative group of formal power series in 212 [[t]] with leading terms 1 E 2/2 to a corresponding multiplicative group in H * * ( X ; 212).
CHAPTER Vl
Unoriented Manifolds
0. Introduction Let z ( X )be the tangent bundle of a smooth manifold X that is not necessarily orientable. Several geometric properties of such manifolds X can be described in terms of the multiplicative 212 classes u , ( z ( X ) ) E H * ( X ; 2 / 2 ) of the bundles t ( X ) . For example, if X is n-dimensional one can use the dual Stiefel-Whitney class i i j ( t ( X ) )E H * ( X ; 2 / 2 ) to provide necessary conditions for the existence of immersions or embeddings X -, R 2 n - pfor a given p > 0. As another example, if X is a closed such manifold the Stiefel-Whitney classes w l ( t ( X ) ) ., . . , w , ( t ( X ) ) provide elements in 212 whose vanishing is a necessary and sufficient condition for X to be the boundary of a smooth compact ( n + 1)-dimensional manifold Y . The chapter begins with some standard properties of the 212 homology module H * ( X , X ; 2/2) and 2 / 2 cohomology module H * ( X ; 212) of any smooth compact n-dimensional manifold X with boundary 8. The bestknown feature of H J X , X ; 2/2) is the existence of a findamentul2/2 homo/ogy d a i s p x E H,(X, X ;2 / 2 ) , which is used to provide the 212 PoincariLefschet; duality isomorphisms H 4 ( X ;212) 5 H,_,(X, X ;212) for any y E 2.The classes ccx and isomorphisms nli, are introduced in $51 and 2, respectively. The classes lcx and isomorphisms n p x are temporarily ignored in 993 and 4, which treat the immersion and embedding problems mentioned earlier. Specifically, 43 consists entirely of the computation of the dual Stiefel-Whitney classes W(?(RP"))of real projective spaces RP". Then immersibility and embeddability criteria for arbitrary smooth manifolds X are 249
250
V1. Unoriented Manifolds
established in 54,in terms of the dual Stiefel-Whitney classes W ( z ( X ) ) ;the computation of 93 provides explicit examples of n-dimensional manifolds X which can neither be embedded nor immersed in IW2"-p for p >= 0 unless p is sufficiently small. If 1 + w l ( r ( X ) ) . . . w , ( t ( X ) ) is the total Stiefel-Whitney class w ( T ( X )E) H * ( X ;2/21 of a smooth closed n-dimensional manifold X , then for any ordered n-tuple ( r l , . . . , r,) of natural numbers such that r l + 2r2 . . . + nr, = n the Kronecker product ( w l ( ~ ( X ) Yu' . . . u w,(t(X))'", px) E E / 2 is the ( r l , . . . , r,)th Stiefel-Whitney number of X . In 95 it is shown that if X is the boundary Y of a smooth compact (n 1)-dimensional manifold Y , then all the Stiefel-Whitney numbers of X vanish. The converse assertion is also true, although not proved here, and the combined results can be used to compute the unoriented cobordism ring 'Jl, whose elements are equivalence classes [XI of smooth closed manifolds X . The set of diffeomorphism classes of smooth closed manifolds X itself forms a semi-ring +2, and there is a natural epimorphism U i -+ 91 whose kernel corresponds to those manifolds which are boundaries. For any formal power series f ( t ) E 212 [ [ t ] ] with leading term 1 E 212 there is a homomorphism 4!l% 212 whose construction uses the multiplicative 212 classes u f ( z ( X ) )of the tangent bundles z ( X ) of manifolds representing elements of @.The intersection ker G(f)properly contains the kernel of the epimorphism %! + 'Jl, so that the homomorphisms G ( f ) can equally well be regarded as homomorphisms '3 212. Such Stiefel-Whitney genera C ( f ) , in either interpretation, are considered in 96. The boundary X of any smooth closed manifold X is void. Consequently if X E JU is n-dimensional its fundamental class p x is an element of H , ( X ; 2 / 2 ) , and the 212 Poincark-Lefschetz duality isomorphism npx reduces to the Poimar6 duulity isomorphism H * ( X ; 212) H , ( X ; 212). Since E / 2 is a field one has H J X ; E/2) = H o m Z i 2 ( H * ( X ;2 / 2 ) , 2 / 2 ) , so that D, can equally well be treated as a nondegenerate bilinear form
+
+
+
+
of
a
H * ( X ; 212) Q H * ( X ; 212)
a212,
which happens to be symmetric. The dual 212 Tkom form
H J X ; 212) 0 H J X ; 212)
212
is constructed geometrically in 97 as an element
j * T x E H " ( X x X ; 212) ( zH * ( X ; 2/2) @ H * ( X ; L / 2 ) ) ,
H * ( X ; 2 / 2 ) to the thereby providing a specific inverse H J X ; 2 / 2 ) 212 Poincart: duality isomorphism D,. One of the most useful properties of j*T, is that if X A X x X is the diagonal map, then A*j*Tx E H " ( X ; 2/2) is the 2 / 2 Euler class e ( r ( X ) )of the tangent bundle z ( X ) .
25 1
I , 212 Fundamental Classes
For any smooth closed manifold X E 92,and for any formal power series f ( r ) E Z/2 [ [ t ] ] with leading term 1 E Z/2, the Z/2 multiplicative class u / . ( z ( X ) )E H * ( X ; 2/2) depends only on the homotopy type of X ; in particular, u / . ( T ( X ) )is independent of the smooth structure assigned to X . The volume closes with a proof of this classical result, due to Thom and Wu.
As in the previous two chapters, the notations H*(-,-) and H*(-) are used to indicate singular cohomology H*(-,-; Z/2) and H*(-; 212) with 2/2 coefficients. A corresponding convention applies to direct products H**(-) and to singular homology H*(-, -) and H*(-).
1. Z/2 Fundamental Classes Let X be a smooth compact n-dimensional manifold with boundary
2 = X \ i , where 8 is the interior of X . We shall show that the 212 homology module H,(X, X ) is free, with one basis element for each connected component of X ; in case X is closed, in the sense that X is void, the result applies to H,(X). For example, for any n > 0 let D" be the n-disk, whose boundary dnis the (n - 1)-sphere S"'; in case n = 1, So consists ofjust two points. A portion of the exact homology sequence of the pair D", S"- ' is H,(D") - Hj e, ( D " , S " - ' ) - H , , _ , ( S " - ' ) ~ H (D"), n-
1
where H,(D") = 0. If n > 1, then H,- ' ( 0 " )= 0, so that H,(D",S"-') = H,- ' ( S n - I ) = Z/2, and if n = 1, then the preceding exact sequence is 0
2 H'(D',SO)
I-
Z / 2 @ 2 / 2 i'-H/2,
which implies H , ( D ' , So)= 212. Thus H,(D", of rank 1 for every n > 0.
Sn-l)
is the free ZJ2-module
1.1 Lemma: Let X be any n-dimensional munijold with interior 8. Then for any x E the Z/2-module H,(X, X \ { x ) ) is,free of rank 1, and H,(X, X \ { x ) ) = 0 for 4 # n.
PROOF: Since x lies in the interior 2 c X , it also lies in the interior of some n-cell D" c 2, and since the inclusion D",D"\{x] -+ X , X\{x) is an excision, it suffices to consider just the case X = D". The boundary s"- ' of D" is a strong deformation retract of D"\{x), so that the 5-lemma provides isomorphisms H,(D", S"- ') 5 H,(D",D"\{x)) for all q E Z;however, we have just just observed that H,(D",S"-') = Z/2, so that H,(D",D"\{xJ) = 212. A similar argument gives H,(D", D"\{x}) = 0 for q # n.
252
VI. Unoriented Manifolds
One feature of Lemma 1.1 deserves special mention. Since there is one and only one nonzero element in Z/2, there is also one and only one generator of the Z/2-module H,(X, X \ { x ) ) for any x E k:there are no choices to make. In later chapters, where H,( - ) will have coefficients other than 2 / 2 , corresponding results will require choices. In Proposition 1.8.4 we learned that if X is a smooth compact manifold with interior k, then there is a finite covering { U l , . . . , U , } of by open cells such that each nonvoid intersection U , n U , is a cell U , in the covering, and such that the closures in X satisfy U , n U , = 0, n 0,. Then in Proposition 1.9.7 we let 9(k)be the category whose objects are unions of the sets U , , . . . , U , , morphisms being inclusions, and we learned that there is a Mayer-Vietoris functor {h41qE 2) on 9(8)with hq(U) = H,-,(X,X\U) for every u E 2(&. There is another Mayer-Vietoris functor { k4 I q E Z)on S(@, defined as follows. For any U E 2(k)let k o ( U ) be the Z/2-module of continuous functions U + 2/2, in the discrete topology of 2/2, and let P ( U )= 0 for q # 0. If U -+ I/ is a morphism (inclusion) in 9(&, then the restriction to U of any element of k o ( V ) is clearly an element of ko(U),so that ko is indeed a contravariant functor from S ( X ) to the category 93 of Z/2-modules. The remainder of the verification that { k4 1 q E Z} is a Mayer-Vietoris functor on d ( i )is trivial. For any U E 9(k) and any a E ho(V)( = H , ( X , X \ U ) ) let 0,a be the function U + Z/2 whose value on any x E U is the image of a under the homomorphism H,(X, X\U) + H,(X, X \ { x } ) = Z/2 induced by the inclusion ( X ,X\U) + ( X ,X\{x)). A function U + 2/2 is continuous if and only if it is constant on each connected component of U , and since U is an open set in a manifold one can take “connected component” to mean “arcwise connected component.” Now suppose that x and y are distinct points of the same arcwise connected component of U , so that they are end points of an embedded arc. One can fatten t$e arc into a closed n-cell D” c U containing both x and y in its interior D”,for which the image of a under the homomorphism H , ( X , X\U) + H , ( X , X\b”)= H,(D”, S”- ’) = 2/2 is the common value of U,a(x) and O,a(y). Hence 0,a is continuous, so that the map a H 0,a is a Z/Z-rnodule homomorphism ho( U ) 2ko( U ) . Naturality of the homomorphisms (0,) for U E 9(8)is trivial, and since k q ( U ) = 0 for q # 0, the con0 struction automatically extends to a natural transformation h + k of MayerVietoris functors {h4}and { P }on 242). For any smooth n-dimensional compact manijold X with interior k the 2212-module homomorphisms h4(@ k 4 ( 8 ) are isomorphisms .for 4 0. 1.2 Lemma:
s
a
253
I . E / 2 Fundamental Classes
PROOF: Let [cl) be the family of open n-cells U . . . , U , used to construct d ( X ) .Then h4(D3= H,-&X,X\b;) = H,-,(D;,S;-') for a closed n-cell Dl with boundary S y - ' , so that ho(by) = 212 and h4(@) = 0 for 4 < 0. Since kY(D;) = 0 for y < 0, the homomorphisms h4(D;)% k4(@) are the trivial isomorphisms 0 + 0 for 4 < 0. For any c1 E hO(&) ( = H,(DY, S y - ' ) ) the function D ! %2/2 carries x E D; into the image of ct under H,(DY,S;- ') + H,,(DY,Dy\ [xi)= E/2, so that the homomorphisms ho(D;) %ko(D;)are also isomorphisms. Hence the Mayer-Vietoris comparison theorem (Theorem 1.9.8) implies the desired result. 1.3 Proposition : For any smooth n-dimensional compact manifold X with boundury X = X\X, the Z/2-module H , ( X , X ) is free, with one basis element jor each connected component q f k.
PROOF: For y = 0, Lemma 1.2 provides an isomorphism H , ( X , X ) = H , ( X , X\X) k O ( X ) ,where the Z/2-module k0(& consists of the continuous functions X + 2/2. Clearly k O ( X )is free, with one basis element for each connected component of X . Since there is only one nonzero element in 2 / 2 , the basis elements assigned by Proposition 1.3 are unique, so that their sum is also unique. 1.4 Definition: For any smooth n-dimensional compact manifold X with boundary X , the fundamental 212 homology class px E H,(X, X ) is the sum of the basis elements described in Proposition 1.3. 1.5 Corollary : Let X be any smooth n-dimensional compact manijold with interior i und houndury X = X \ i , and for any x E J? let j , be the inclusion ( X ,2 )= ( X ,X \ i ) + ( X ,X \ ( x ) ) . Then the fundamental class p x E H,(X, X ) is uniyurl\, churuc~erizedhj, the property thut ( jr)*/ix E H,(X, X\ ( x )) generates for euch x E X .
PROOF: This is just a reformulation of the case 4
=0
of Lemma 1.2.
1.6 Corollary: Let X he a smooth cornpact n-dimension$ manifold, with interior X und (smooth)boundary X = X\X us usual, let 2 ( X ) be the category qj'open sets in described just before Proposition 1.9.6, and f o r any U E 2 ( i ) let 0 he the closure of U in X , with boundary = 0\U. Then there is a unique class p" E H,( 0,f ) with the property that i f (0, 6)A (0, U\(x}) is o\(xi). the inclusion jiw uny .Y E U the element (j,.)*pU generates H,( 0,
PROOF: The metbod of Lemmao1.2 aad Proposition 1.3 applies equally well to any U E 9 ( X ) ,not just to X E 3 ( X )itself.
254
VI. Unoriented Manifolds
The class pu is the fundamental Z/2 homology class of 0, for any U E a(2). It will appear along with the following constructions as part of a MayerVietoris argument in the next section. 1.7 Corollary: Zf X is a smooth compact manijold as in Corollary 1.6, then for any inclusion U V of U E 2 ( i )into V E 2(k)there is an induced L/2module homomorphism H , - , ( 8 , V* ) S' H , - & r f , for every q E Z,such that for q = 0 one has f*pv = pu E H,,(rf, 0).
-
f)
c)
PROOF: Let 9 and h be the inclusions ( V , = ( V , V\V) + ( V , V\U) and (0, r f ) = (0, rf\V) + ( V , v\U), respectively. Then h is an excision, so that H,-,(rf, 6) H,-&V, V\U) is an isomorphism, and one can define f* as
'
the composition (h.Jg*. The property ,f *pv = pu is a consequence of the characterization of fundamental classes given in Corollaries 1.5 and 1.6. 0
1.8 Corollary: Zf X is a smooth compact n-dimensional manijold, thenathere 9(2) with k4( U ) = H , -,( 0, r f ) for and every q E Z,such that the homomorphisms k4( U v V ) it.&every U E 9(2) P(U)0 k q ( V )and P ( U )0 k4( V ) kq(U n V ) are obtained as in $1.9 by applying Corollary 1.7 to the inclusions U 2 U v V , V 2 U u V , U n V j k l , u , a n d U n V*V.
-
I
is a Mayer-Vietoris functor { k4 q E Z}on
PROOF: According to Proposition 1.9.7 there is a Mayer-Vietoris functor 9(k)with k 4 ( U )= H , - , ( X , X \ U ) for each U E 2 ( i ) .However, the excision ( 0 , c )= (0, o\U) + ( X , X\ U ) induces isomorphism H,-,,(rf, f ) H , _ , ( X , X\U), and one uses further such excision isomorphisms to verify that the homomorphisms it," andjt." implicit in Proposition 1.9.7 correspond to the homomorphisms it," and jt," obtained from Corollary 1.7. (k41q E 12) on
In case X is closed in the usual sense that its boundary X is empty, then the fundamental class of Definition 1.4 is an element of H , ( X ) . In case X does have a nonempty boundary X , then X is itself an (n - 1)dimensional manifold which is closed and smooth. Furthermore, the connecting homomorphism d in the exact 212 homology sequence
H,(X) A H,(X,X)
d +
Hn-l(X)
5 H,-,(X)
carries the fundamental class p x E H , ( X , 2 )into a class dpx E H,- , ( X ) . 1.9 Proposition: Let X be a smooth compact n-dimensional manifold with a nonempty boundary X , and with fundamental Z/2 homology class px E H , ( X , X ) . Then the class 8px E H , - l(X) is the fundamental 212 homology class p i E H , _ ' ( X )of the boundary X .
255
I , 2/2 Fundamental Classes
PROOF: As in Corollary 1.5 the class p i E H , - l(i) is uniquely characterized X , X \ [ y ) is the inclusion induced by any by the property that if 2 1%E X , then (j,,)*pi generates H,- l(X, X \ [ y ) ); we shall show that L7px has the same property. Let x E i lie in the same arcwise connected component of X as y E k,let I be an arc connecting x and y , with interior in i ,and let ( X , X ) A ( X , X \ ( x ) ) be the inclusion. Then / can be fattened to an n-cell D"
containing / \ ( y ) in its interior (see the accompanying figure). One easily verifies that the diagram Cj,h
P
H , , ( X ,2) A H , -
i(X)
A
H n -i ( X 9 X \ { Y ) 1
I-
commutes, all maps except the three connecting homomorphisms 13 arising from inclusions. Three of the vertical maps are excision isomorphisms, as indicated, and the two lower maps are isomorphisms because they are portions of long exact sequences H,,(D")
-
and H,]- l ( D " \ / )
-&+ H,]- ,(D"\(x))
H,,(D",D"\(xj)
H I ] -, ( D " \ ( x ) )
+
-
H f l -~ ( D " \ { x ) , D " \ I )
Hn- I(D")
P
H , AD"\/) -
whose outer terms vanish because D" is an n-cell and D"\/ is homotopy equivalent to an n-cell. (For n = 2 the second sequence still produces the desired isomorphism because H,(D"\/) + H,(D"\ jx) ) is an isomorphism; the adjustments needed for the degenerate case n = 1 are trivial.) Since j x * p x generates H , , ( X ,X \ [.Y!), it follows that jy*i?px generates H n _l(X,* i ; ( y ) ) , as required.
256
VI. Unoriented Manifolds
2. Z/2 Poincare-Lefschetz Duality Let X be a smooth compact n-dimensional manifold with boundary X , and let p x E H,(X, X ) be the fundamental Z/2 homology class of Definition 1.4. One of the classical products of singular homology and cohomology is the cap product H 4 ( X )0 H,(X, X ) 1H , - , ( X , X ) for q E Z,some of whose properties will be recalled as needed. (Further details about cap products are readily available on pages 238-245 of Dold [8] and on pages 254-255 of Spanier [4], for example.) In particular, cap product npx by the fundamental Z/2 homology class p x is a Z/2-module homomorphism W ( X ) H , - , ( X , X ) for any 4 E Z.We show in this section that n p x is an isomorphism for any q E Z. We continue to use the category 2($) of open sets described just before Proposition 1.9.6. Specifically, 9(k)consists of unions of certain open n-cells U 1 , . . . , U , covering $, such that each nonvoid intersection U i n U j of two such open n-cells is itself one of the open n-cells U,, and such that the closures in X satisfy U i n U j = U i n Oj in addition to the usual identity U i u U j = Oiu Oj. One Mayer-Vietoris functor {hqlq E Z)on 2 ( i )is given in Prop osition 1.9.6 itself: one has h4(U)= Hq(D) for every U E 2 ? ( i ) .Another Mayer-Vietoris functor {PIq E 2 ) on 2 ( X ) is given in Proposition 1.9.7, and modified via excision isomorphisms in Corollary 1.8: one has k4(U ) = H , -¶( O,O) for every U E J?(k). The following lemma will be used to construct a natural transformation from {h4I q E ZJ to { k4 1 q E Z). 2.1 Lemma: Let U L V be a morphism (inclusion of one open set into another) in 2?(2), inducing a n inclusion U- f+ V- , hence a Z/2-module homorphism H q ( V ) z H 4 ( 0 )for any q E Z,and let H , - J V , 8)fl. H,,-¶(O, 6)be the induced Z/2-module homomorphism of Corollary 1.7. Then cap products by the 212 fundamental classes pp E H,( V , ?) and p~gE H,( 0 , O ) provide commutative diagrams H4(
V)
f'
PROOF: The lower Z/2-module homomorphism .f* was described in Corollary 1.7 as a composition, which appears in the bottom line of the
257
2. hi2 Poincare Lefschetz Duality
following diagram W ( V ) id* H " V )
* H4(D)
s*
H,-,(P, V\U) ++ . Hn-,(D, 0). Each of the two squares commutes by naturality of the cap product (as on page 239 of Dold [S] or on page 254 of Spanier [4]). The ambiguity of the definition of the middle vertical arrow is quickly resolved: each of the classes g * p v E H,( P \ U ) and hirpu E H,( V , V\ U ) is uniquely defined by the property that if ( V , V\U) 5( V , v \ [ x } )is the inclusion for any x E U , then each of ( , j x ) * g * p v and (,jx)*h,pcris the unique generator of HI,(P, V\ ( x )).
Hn-,(V, V )
v,
2.2 Lemma : For any smooth compact n-dimensional manifold X with houndarjl X there is a natural transformation 8 ,from the Mayer-Vietoris ,functor ( hYI y E Zj on 2(R) to the Mayer- Vietoris junctor {/@I q E L j on Y(X ) . 0u the hom~morphisms h4(U ) ky(U ) bring cap products H4(a) H,,-,(iJ, a) by 212 jundumentul classes p o E Hn(O, for each U E S(2) and each y E R.
-
-
c),
PROOF: One must show that the diagram h4(U u V )
!fhu..
it.v
Ilifl I
h4(U)0 h 4 ( V ) % h4(U n V ) % h 4 + l ( U u V ) O l i r , 1'
P ( U u V ) 9P ( U ) O P ( V )
F(U n V )
!
0L'"I.
P + ' ( uu V )
commutes for every U E d ( i ) , V E d(k),and q E Z.However, both the left-hand and middle squares break into two copies of Lemma 2.1, using the inclusions U 2 U u V and V 5 U u V for the left-hand square, and using the inclusions U n V U and U n V V for the middle square. The right-hand square commutes by the stability property of the cap product (as on pages 239-240 of Dold [S]), of which a special case asserts that the diagram
a d
258
VI. Unoriented Manifolds
commutes, for the Mayer-Vietoris connecting homomorphisms S and 3 in cohomology and homology, respectively.
2.3 Theorem ( 2 / 2 Poincark-Lefschetz Duality): Let X be u smooth compact n-dimensionul manifold, with boundary X und 212 fundamental class p x E H,(X, X ) ; then jor each 4 E Z the cup product Hq(X) 3 H,_,(X, X ) is u ZI2-module isomorphism. PROOF: Let 0 be the natural transformation of Lemma 2.2, from the Mayer-Vietoris functor (hq14 E Z)on 2 ( X ) to the Mayer-Vietoris functor ( k 414 E Z}on 2(k).Let 6" be any one of the open n-cells U , , . . . , U , used to construct the category A?(&, the closure of 8.being an n-disk D" with boundary S"-'. Both HY(D")= 0 and H,-,(D",S"-') = 0 for q # 0, and since 1 n pD,,= pD,,forthe generator 1 E Ho(D")and the fundamental 212 class pDnE H,(D", S"- ')(as on page 239 of Dold [8] or on page 254 of Spanier [4]) it follows that each hq(Ui) kq(Ui) is an isomorphism. Hence each hq(& A kg(& is an isomorphism by the Mayer-Vietoris comparison H,-,(X, 2 )is an isomorphism theorem (Theorem 1.9.8); that is, H 4 ( X ) for any 4 E Z,as asserted.
a
Recall that a compact manifold is closed if its boundary is empty. 2.4 Corollary ( 2 / 2 Poincark Duality): Let X be u smooth closed ndimensional mani$old with B/2 fundamental class p,y E H,(X); then for any 4 E Z the cap product H 4 ( X )npX. H,-,(X) is a Z/2-module isomorphism. PROOF: This is the special case k =
of Theorem 2.3.
There are also Zj2 Poincare-Lefschetz duality isomorphisms H"-,(X, X ) -, H,(X) that specialize to Poincare duality isomorphisms; see Exercise 9.41. Explicit inverses to Poincare duality isomorphisms will appear in Corollary 7.9.
3. Multiplicative 2/2 Classes of RP" As indicated in the Introduction on this chapter, we now put Z / 2 fundamental classes px and 212 Poincare-Lefschetz duality isomorphisms npx aside, temporarily. The classes px will reappear in $5, and the isomorphisms n p x will reappear in $7. In this section we compute the Stiefel-Whitney classes w ( z ( R P " ) )and the dual Stiefel-Whitney class % ( t ( R P " ) of ) the (tangent bundle of the) real
3. Multiplicative L / 2 Classes of R P
259
projective space RP". The former computation will show that w , ( z ( R P " ) )= 0, hence that RP" is orientable, if and only if n is odd. The latter computation will be used in the next section to obtain necessary conditions for the existence of immersions RP" C_ [wztl-p and embeddings RP" c R 2 n - q . 3.1 Proposition: Let s(RP") be the tangent bundle of RP", and let y: be the canonical red line bundle over RP". Then
w(z(RP"))= (1 + e(y,")"" for the total Stii$&Whitney e(y:) E H'(RP").
class w ( t ( R P " ) E) H*(RP") and Euler class
PROOF: In Proposition 111.7.4 we learned that t ( R P " )0 E' = ( n + l)y,!, so that the Whitney product formula and the normalization of Stiefel-Whitney classes give w(r(RP"))= w(r(RP"))u w ( E ' ) = w(z(RP")0 E ' ) = w((n I)?:) = w(y:)"+' = (1 e(y,'))"+
+
+
',
as claimed, since w ( E ' )= 1 by Corollary V.2.6.
+
In computing the product (1 e(y:))"+' it is understood that the highest term e(g:)"+' vanishes, since the ( n + 1)st cohomology module H"+'(RP") of the n-dimensional CW space RP" vanishes. It is also understood that if one expands (1 + e(y:))"+' by the binomial theorem, then the coefficient of e(y;)" is the 2 / 2 image of the integral binomial coefficient ("i ').
3.2 Corollary: RP" is orientable if n is odd; RP" is nonorientable if n is even. PROOF: The 2 / 2 image of (": ') is 0 E 2/2if n is odd, and 1 E 2 / 2 if n is even. However, the product of this coefficient and e(y:) E H'(RP") is the first Stiefel-Whitney class w l ( z (R P " ) ) , to which one then applies Proposition v.3.2. We need the following lemma to compute the dual Stiefel-Whitney class %(z(RP"))E H*(RP").
3.3 Lemma: For any element c( of' a Z/2-algebra, and for any natural number r 2 0, one has (1 + a)''' = 1 + a'*. PROOF: This is trivial for r = 0, and the computation (1
+ a ) 2 ' + ' = ( ( 1 + U ) 2 * ) 2 = ( I + $')2
is the inductive step.
=
1 + 2azr + $ ' '
= 1
+ cJZr+l
260
VI. Unoriented Manifolds
'
3.4 Proposition: I f 2' 5 n < 2" the dual Stiefel-Whitney class of RP" is qiuen bji W(r(RP"))= (1 + e(y,!))2'tl-n-' E H*(RP"),where e(y,!)E H'(RP") is rlie 212 Euler clu.ss oj' the cunonicul line bundle y,!, PROOF: Since H2'"(RP") and Lemma 3.3 imply that
=0
by the hypothesis n < 2'+', Proposition 3.1
w ( r ( R P " ) ) ( l+ e(y;)yrt1-"-
+
hence that ( 1 e(yf))2'"-"-' of Lemma V.6.3.
+ e ( y f ) ) " + ' ( l+ e(y,!))2'+1-n-1 = ( 1 + e(yi))2"+1 = 1, -(1
is the unique class satisfying the conclusion
4. Nonimmersions and Nonembeddings According to the Whitney immersion theorem (Theorem 1.6.9) there is at least one immersion X -+ S2"-' of any smooth n-dimensional manifold X into the (2n - 1)-sphere S2"- ; if n > 1, then there is at least one immersion X A R2"-' of X into the euclidean space RZn-l.Similarly, according to the Whitney embedding theorem (Theorem 1.6.6) there is at least one proper embedding X 3 R2"of X into the euclidean space R2".We shall show for some dimensions n > 0 that these results are best-possible: for certain n > 0 there are smooth n-dimensional manifolds which cannot be immersed in ~ 2 n - 2nor embedded in R2"- The proof gives a suggestion of best-possible results for any dimension n > 0. In view of the Whitney immersion and embedding theorems, it is reasonable to try to find the largest natural number p 2 0 such that a given smooth n-dimensional manifold X immerses in R 2 n - p or embeds in R 2 n - p ,The tangent bundle r(R2"-") is the trivial bundle c 2 " - P over [ W 2 " - p , and c 2 " - P has a riemannian metric induced by the usual inner product R2"-P x RZn-p S R .Then if X 5 R 2 n - pis an immersion, the pullback ~ ! T ( I W ~ " - ~is) the trivial bundle E ~ over ~ X, - and ~ according to Lemma 111.3.3 there is an induced riemannian metric ( , ) on f ! r ( R 2 n - p )Since . f is an immersion, the tangent bundle T ( X )of X itself is a subbundle of ~ ' ! T ( R ~ " of - ~rank ) , n, and according to Proposition 111.3.6 the riemannian metric on f ! Z ( R ~ " - ~ ) provides a well-defined subbundle vs of rank n - p such that r ( X )0 vs = , f ! t ( R 2 " - p ) = c 2 " - P over X.
'.
4.1 Definition: For any immersion X A R 2 n - pof a smooth n-dimensional manifold X into the euclidean space R2n-p, the preceding (n - p)-plane bundle v,. over X is the normal bundle of the immersion f .
26 1
4. Nonimmersions and Nonembeddings
Since r ( X )0 = c2"-", it follows from Corollary V.6.5 that the total Stiefel-Whitney class w(vf) E H * ( X ) of the normal bundle vf agress with the dual Stiefel-Whitney class W ( s ( X ) E ) H * ( X ) of the tangent bundle: w(v,-) = W ( T ( X ) ) Furthermore, . since W ( r ( X ) )is independent off', one can use its properties to establish properties of unj' immersions X + [ W 2 " - p , including their nonexistence for sufficiently large values of p. We now carry out such a program. \is
4.2 Lemma: Let R ( s ( X ) )be the dual Stiefkl-Whitney class of' the tangent bundle of' u smooth n-dimensional manifold X , and suppose that X immerses in [W2"-" jbr some p > 0 ; then W,,(z(X)) = 0 E H q ( X ) for any q > n - p. PROOF: The normal bundle vf of any immersion X 5 ( W 2 " - P is of rank n - p. so that wy(vf) = 0 for q > n - p by the dimension axiom of Theorem V.5.2. However, we havejust noted that w(v/.) = W ( s ( X ) )so , that W , ( r ( X ) )= 0 for y > n - p, as claimed. Thus if W,(z(X)) # 0 for some q > n - p then X cannot possibly immerse in OX2"- ". Here is the simplest example of such a result.
4.3 Proposition: If' n = 2' for r 2 0, then the real projective space RP" does not immerse in R2"-'. PROOF: According to Proposition 3.4 the dual Stiefel-Whitney class of RP" is given by W(?(RP"))= (1
+ e(y,!))2"'-2'-'
= (1
+ e(y,!))'-',
so that Wn-,(s(RP"))is the nonzero element e(y,!)'-'
E H"-'(RP").
There is a natural generalization of Proposition 4.3, which will be given in the following notation. 4.4 Definition: For any natural number n > 0 let a(n) > 0 be the number of 1's in the dyadic expansion of n; that is, if n = a02' a121 . . . ar2', where the natural numbers u 0 , a , , . . . , u, are either 0 or 1, then a(n) =
+
+
+
C;=oUj.
4.5 Proposition: For any natural number n > 0 there is a smooth closed l. n-dimensional manifdd X that does not immerse in R2n-a(n)-
PROOF: For the set J of those indicesj with aj = 1 in the expansion n = + . . . + ar2', let X be the product nj,,RP2'. Then, for the projections X RP2', the tangent bundle of X is a Whitney sum t ( X ) = @jEJprjs(RP2'), so that the dual Stiefel-Whitney class of r ( X ) is a cup uo2O
262
VI. Unoriented Manifolds
nJEJ
nJEJ
product w ( r ( X ) )= w(prir(RP2'))= prTW(z(RP2')).The highestorder nonvanishing class in iij(z(RP2'))is m2'- l(z(RP2')) = e ( ~ : , ) ~ ' -E H 2 ' - l ( R P 2 ' ) , as in Proposition 4.3. Since the Kunneth isomorphism OJGJ H*(RP2') H * ( X ) carries the highest-order nonvanishing tensor prj*W2,-l ( t ( R P 2 ' ) ) , product @ , E J w 2 J - I ( r ( R P 2 ' ) )into the cup product and since CJEJ(ZJ - 1) = n - a(n), it follows that w ~ - ~ ( , , ) ( T (#X )0) E H" - a(")( X ) .Hence by Lemma 4.2 the product X cannot immerse in R2"-""- '.
nJEJ
One cannot improve Proposition 4.5 without further specialization, for the following reason: for any natural number n > 1, every smooth n-dimensional munifdd X immerses in R2n-a(n). This "best-possible'' immersion theorem of Cohen [l] is briefly discussed in Remark 9.14. Proper embeddings are specialized immersions, for which there are improved versions of Propositions 4.3 and 4.5. Specifically, an immersion X R2"-P is an embedding if it is injective ( . f ( x ) = f ( y ) E R 2 n - p implies x = y E X ) , and it is proper if inverse images of compact sets in R 2 " - pare compact in X . For example, any embedding of a compact manifold is necessarily proper. For any proper embedding X 5 R2"-P of a smooth n-dimensional manifold X, for any E > 0, and for any connected open set V c X whose closure V is diffeomorphic to a closed disk D" c R", let C,( V ) denote the set of points y E R2"-p such that l f ( x ) - yl < E for some x E V . Since the closure is compact, the inverse image consists of at most finitely many connected components, so that for sufficiently small E > 0 the set j'-'(W)) contains no points outside the connected component of B c X . In what follows, a cocoon C ( V ) c R2"-P of the image j'( V ) R 2 n - pIS any set of the form C,( V ) ,where E > 0 is chosen in such a way that j ' - '(C3,(V ) ) contains no points outside the connected component of V c X .
<
f-'(v))
'
4.6 Lemma : Let E X represent the normal bundle v f of a proper embedding X R 2 n - p of a smooth n-dimensional manifold X . Then there is a dgfJeo-
morphism of some open neighborhood of' the zero-section of E onto an open neighborhood c R 2 n - pof' the image f ( X ) c R 2 n - p .
PROOF: Let x = (x', . . . , x") be local coordinate functions on some open set in X , so that the restriction off is given by f ( x ) = ( f ' ( x ) ,. . . ,f 2 " - p ( ~ ) ) for smooth functions f', . . . , f 2 n - p on an open set U c R", for which the jacobian matrix (df'ldxj) = (1:)has rank n. By restriction to a connected open subset V c U , with closure diffeomorphic to D" c R", and by a
4. Nonimmersions and Nonembeddings
263
relabeling of coordinates, if necessary, one can suppose that
f'; . . . f ; d e t (j!' ; . . . f'; !]#O. It follows for each x E V and each y = ( y " " , . . . , y 2 " - p )E R"-p that there is a unique z = z(x, y ) = (z', . . . , z2"-P) E R2n-p such that
f;"
...
fit1
...
1
0
0 hence a well-defined smooth map V x R"-P (x,y) E V x KYPinto '
(f'(x)
+ z'(x, y), . .
*
,fZ"-P(x)
5 R2"-p carrying
+ zZn-p(x, y ) ) E
any point
[WZn-p.
Clearly F is affine in y, so that the restriction of F to V x (0) is ,f itself, and the restriction of the jacobian determinant of F to V x (0) is
Since j ' is an embedding, the inverse function theorem guarantees that the restriction ofFtosomeopen neighborhood Wof V x ( 0 )is a diffeomorphism. We let ?!(, V ) c R 2 " - pbe the (open)intersection F( W ) n C(V ) ,for any cocoon C ( V )c RZn-pofj'(V). The n linear conditions f j z l + . . . + f ? " - p z 2 " - p = 0 appearing in the the vector definition of z(x, y ) guarantee that for each (x, y ) E I/ x z(x, y) E R Z n - p is orthogonal to the tangent space at f(x) of the embedded f ' ( X ) , in the usual inner product of R 2 " - p ;hence z(x, y ) can be identified as a point in the fiber over x E V of the normal bundle v f of the embedding X A R Z n - p . Specifically, one can regard V x R"-P as a trivialization of the restriction of the normal bundle to V c X , an open neighborhood of V x (0) being identified with the neighborhood E( V ) off( V ) .Since i(V ) lies in the cocoon C ( V ) ,it is clear for any coveTing [ V } of X by open sets V of the type described earlier that the union E( V )is a diffeomorphic image i c R2n-p of an open neighborhood of the zero-section of E , as required.
uv
264
.
VI. Unoriented Manifolds
R2"-P 4.7 Lemma: Let vs be the normal bundle of a proper embedding X of' a smooth manifold X of dimension n > 0. Then the 212 Euler class vanishes: e(vs) = 0 E H " - P ( X ) .
FROOF: By definition, e(vs) = cr*j*U,, for the 212 Thom class U , , E H"-P(E,E*),where E 5 X represents v,., where E A E, E* is the inclusion, and where X 5 E the zero-section. By the excision axiom one may as well replace E by any open neighborhood of the image of cr, which we also denote E, with a corresponding E* c E. Then Lemma 4.6 provides the isomorphisms f ; ,f : ,and j'* in the accompanying commutative diagram, and the remain-
+
j+
H" - "( E, E*)
T
a*
H""(X)
- -
2
H"- Y( [WZ"- P ,
-
1; =
H"-P($E\f(X)) YX
' H"-p(E)
'i
I.'*
H"-P(i)
)]I
H"-P(f(X))
p"-P\f(x))j * H" -P( R2n-P)
ing homomorphisms are induced by the obvious inclusions. The inclusion [ W 2 " - p induces an excision i,i \ f ( X ) 3 ( W 2 n - p , Rzn-p\ f ( X ) , it follows that gg is also an isomorphism. Since H"-P([W2"-p) = 0, it then follows that
2%
e(vs) = o*j*U,,
= j'*r*j*(gg)- ' ( j ' : ) -
U,,
= 0 E H"-p(X),
as asserted.
4.8 Theorem: Let W ( r ( X ) )be the dual Stiefel-Whitney class of the tangent bundle r ( X ) of a smooth n-dimensional manifold X , and suppose that there is a proper embedding X R z " - p ; then m,(r(X)) = 0 E H 4 ( X )for q 2 n - p. PROOF: The normal bundle v f is an ( n - p)-plane bundle and one has W ( r ( X ) )= w(v,-),where wq(vI)= 0 for q > n - p as in Lemma 4.2. However, according to Corollary V.2.8 w,-,(vs) is just the 212 Euler class e ( v f ) E H " - p ( X ) ,which vanishes by Lemma 4.7.
4.9 Proposition: I f n = 2' for r 2 0, then the real projective space RP" does not embed in R2"-'. PROOF: Since RP" is closed, a fortiori compact, any embedding is proper. However, W n -,(r(RP")) is the nonzero element e(y,!)"- E H"-'(RP") as in
265
5 . Stiefel-Whitney Numbers
Proposition 4.3, so that Theorem 4.8 forbids an embedding of RP" into [W2,- I
For any n > 0 the number a(n) > 0 is described in Definition 4.4. 4.10 Proposition: For any natural number n > 0 there is a smooth closed n-diniensional nzun@ld X which does not embed in [W2" '("). PROOF: In the proof of Proposition 4.5 it was shown that iij,-m(,J7(X)) # 0 E H"-"("'(X)for the closed manifold X = RP". Since any embedding of a closed manifold is necessarily proper, Theorem 4.8 therefore forbids any embedding of X into R2n-a(n'.
n,,,
5. Stiefel-W hi tney Numbers Let X be any smooth closed n-dimensional manifold, and let ( r l , . . . , r,) be any ordered set of natural numbers such that r l + 2r2 . . . nr, = n. We shall use the total Stiefel-Whitney class w ( t ( X ) )of the tangent bundle r ( X ) to assign an element of 2/2 to each ( r l , . . . , r,). One obtains the value 0 E 2/2 for each ( r l , . . . , r,) if (and only if) X is the (smooth) boundary Y of a smooth compact (n 1)-dimensional manifold Y.
+
+
+ 1 + w l ( t ( X ) )+ . . . + w , ( t ( X ) )
If is the total Stiefel-Whitney class w ( T ( X ) E) H * ( X ) of the tangent bundle T ( X )of the smooth closed n-dimensional manifold X , and if(rl, . . . , r,) is an ordered n-tuple of natural numbers such that r l Zr, . . . nr,, = ti, the product w , ( ~ ( X ) ) lu' . . . u w,,(T(X)P in H*( X ) belongs to H"( X ) . The boundary X of X is void since X is closed, so that the fundamental 2/2 homology class of Definition 1.4 is an element p x E H , ( X ) . Consequently the Kronecker product H " ( X ) 0 H , ( X ) 2/2 assigns an element of the ground ring 2/2 to the manifold X and the n-tuple ( r l , . . . , r,), as follows.
+
+
+
-
5.1 Definition: Let X be a smooth closed n-dimensional manifold, and let ( r l , . . . , r,) be any natural numbers such that r l + 2r2 + * * . + nr, = n. Then the (rl, . . . , r,) th Stieid- Whitney number of X is the element (Wl(T(X))ll
U
.
'
*
U
W , , ( T ( X ) P , p x )E 2/2.
5.2 Proposition: Suppose that Y is a smooth compact (n + 1)-dimensional manifold with boundary Y , so that Y is a smooth closed n-dimensionalmanijold; then all the Stiefel-Whitney numbers of Y vanish. PROOF: According to Proposition 1.9 the fundamental 272 homology class Y , Y ) under
pu E H,( Y ) is the image of the fundamental 2/2 class pLy E H,,
266
VI. Unoriented Manifolds
the connecting homomorphism d of the exact homology sequence
H , + 1( Y ) -L H , ,
1(
Y , Y ) PH,( Y ) i tH,( Y )
for the pair Y , Y ; that is, p p = d p y . Furthermore, the connecting homomorphism S of the corresponding exact cohomology sequence H " ( Y ) ' I , H " ( Y ) "-H"+'(Y,Y) j t .n+l(y) is the adjoint of d with respect to the Kronecker product ( , ); that is, (cc,dv) = (6cr,v) for any o! E H"(Y) and any v E H,, l ( Y , Y ) . Consequently the ( r l , . . . , r,)th Stiefel-Whitney number of Y satisfies (w;l
...w?,py)
= (w;'
. ' . w',",dpy)= ( S ( W ; l . . . w ? ) , p y ) ,
where W?
' ' '
W',"
= W l ( T ( f)rlU *
' *
U W,(5(
Y)y";
We shall show that S(w;l . . . w',")= 0. Let T ( Y ) be the tangent bundle of Y itself. Then for the inclusion Y f Y of the boundary Y , the restriction i ! z ( Y )of t ( Y ) to Y is the Whitney sum z( Y ) 0 c1 of the tangent bundle z( Y ) and the trivial line bundle c 1 spanned by the inward-pointing unit vectors normal to Y , with respect to any riemannian metric on t( Y ) . Since W(T(Y
) )= W ( T ( Y ) @ E l )
= W ( i ! T (Y
) ) = i*W(T( Y ) )
for the total Stiefel-Whitney classes W ( T ( Y ) )E H*( Y ) and it follows that
W ( T (Y
) )E H*( Y ) ,
6 ( w l ( t ( Y ) ~u1 . . . u w n ( 7 ( Y ) p = ) 6i*(wl(T(Y)ylu * * . u W , ( T ( Y ) P )
=
o
as desired, since Si* annihilates H"( Y ) in the exact cohomology sequence of the pair Y , Y .
5.3 Remark: Proposition 5.2 is due to Pontrjagin [ 5 ] . The converse assertion is also true, due to Thom [3, 6-81. Thus a smooth closed manifold is a boundary if and only i f all its Stiefel-Whitney numbers vanish. We shall not prove the converse of Proposition 5.2. The following result depends only on Proposition 5.2. 5.4 Corollary: If n is even, then the real projective space RP" is not the boundary of any ( n 1)-dimensional manifold.
+
PROOF: By Proposition 3.1, w ( T ( R P " )= ) (1 + e(y,!))"+',with binomial coefficients computed in 2/2, where according to Proposition IV.4.4, H*(RP")is the polynomial ring over 2/2 generated by the 2/2 Euler class e(y,!), modulo
267
5 . Stiefel -Whitney Numbers
("L
the relation e(y,')"+' = 0. In particular, if n is even, then ') = 1 E Z/2, so that w,(z(RP")) is the generator e(y:)" E H"(RP"). Hence the (0,. . . , 0,l)th Stiefel-Whitney number of RP" satisfies (w,(z(RP")),p R p n )= 1 # 0, so that Proposition 5.2 prevents RP" from being a boundary. 5.5 Proposition: I f n is odd, then all Stiefel-Whitney numbers of' the real projective space RP" vanish.
PROOF: If n = 2m - 1 for m > 0, then Proposition 3.1 and the technique of Lemma 3.3 give w ( t ( R P " ) )= ( 1 + e(~;))~"' = ( 1 + e(+jj,!)2)"'; in particular, all the odd Stiefel-Whitney classes of RP" vanish. However, ifr, + 2r2 +. . . + nr, is to equal the odd number n, then rp # 0 for at least one odd number y n, so that w,(t(RP"))" u . . . u w,,(r(RP"))'"= 0. 5.6 Corollary : If' n is odd, then the real projective space RP" is the boundary of some ( n
+ 1)-dimensionalmanifold.
PROOF: Remark 5.3 and Proposition 5.5. 5.7 Definition: Two smooth closed n-dimensional manifolds X and X ' are cobordant if their disjoint union X X ' is the boundary of a smooth compact (n 1)-dimensional manifold Y .
+
+
+
If X is any smooth closed n-dimensional manifold, then X X is the boundary of X x [0,13, so that cobordism is a reflexive relation. Since the disjoint union X + X ' is assigned no order cobordism is a symmetric relation. Finally, if X + X ' is the boundary of Y and x' x'' is the boundary of Y', then one can identify the common portion X ' of Y and Y' to create a smooth compact (n + 1)-dimensional manifold Y" with boundary X X " ; hence cobordism is also a transitive relation. In summary, cobordism is an equivalence relation. Let [ X I and [ X ' ] denote the cobordism classes of smooth closed manifolds X and X ' , respectively. If X and X' are of the same dimension, then one easily verifies that the class [ X + X ' ] of the disjoint union X + X ' depends only on [ X I and [ X ' ] , and for any dimensions one easily verifies that the class [ X x X ' ] of the smooth product X x X ' also depends only on [ X I and [ X ' ] . Hence the set 'Jz of cobordism classes of smooth closed manifolds admits a sum '.Jz x (Jz: 'Jz and a product 'Jz x (Jz i'Jz. Furthermore 'Jz is graded by dimension: if 'Jz, c (Jz consists of classes represented by smooth closed n-dimensional manifolds, then (Jz is the disjoint union of the subsets 9tn, and sums and products satisfy 91, x ' J z l l ~ Y 1 ,and , J' z,, x 91, i'ill,,,+,,.
+
+
268
VI. Unoriented Manifolds
5.8 Lemma: The set % of' cobordism classes is a graded commutative ring with respect to the preceding operations
a,, x 3,; a,,
and
%, x %,, i %,+,.
PROOF: This is a trivial verification. In particular, since X + X is the boundary of X x [0,1],it follws that - [ X I = [ X I , hence that every element of % is of order 2 with respect to addition. 5.9 Definition: The preceding graded commutative ring % of cobordism classes of smooth closed manifolds is the unoriented cobordism ring. 5.10 Proposition: If [ X I = [ X ' ] in a,,for two smooth closed n-dimensional manifolds X and X ' , then X and X ' have the same Stiefel-Whitney numbers; conversely, if X and X' have the same Stiefel-Whitney numbers, then [XI = [X']. PROOF: The first assertion is a rephrasing of Proposition 5.2, and the second assertion is a rephrasing of Remark 5.3. There is an alternative way to construct the unoriented cobordism ring %. Observe that if DP and D4 are disks of dimensions p and 4 with boundaries
+
S p - ' and S q - ' , respectively, then each of the ( p q - 1)-dimensional manifolds DP x S4-' and S P - x Dq has S P - x S4- as its boundary. Suppose that DP x S q - is smoothly embedded in a smooth closed manifold X o of dimension p + 4 - 1. One can then remove DP x S4-' from X , , leaving a boundary S p - x Sq- ; since S p - ' x Sq- I is also the boundary of S p - I x Dy, one can (smoothly) insert S p - ' x D4 into- X,\DP x S - ' , identifying common points on the common boundary S p - ' x S 4 - ' , to obtain a new q - 1. Two such manismooth closed manifold X , , also of dimension p ifolds X , and X , are surgically related; in general, surgically related manifolds are not diffeomorphic to each other.
'
5.11 Definition: Two smooth closed manifolds X o and X , of the same dimension are surgically equivalent if and if there are smooth closed manifolds X , , . . . , X , - such that X i - is surgically related to X i for each i = 1, . . . , 4 .
,
Clearly surgical equivalence is an equivalence relation, and the set of equivalence classes [ X I of smooth closed manifolds X forms a commutative semi-ring %' with respect to disjoint union and smooth product: [ X I + [ X ' ] = [ X + X ' ] and [ X I . [ X ' ] = [ X x X ' ] . The semi-ring %' is graded by dimension. We shall show that (JL' is canonically isomorphic to the unoriented cobordism ring %. Let 11 ( I p and 11 It, denote the usual euclidean norms on R P and W. The product Rp x Ry then carries a norm 11 I( with Il(x,y)l(= max{IIxllp,llyllp),
2 69
5. Stiefel- Whi tney N urn bers
and D'' x S'-'u S"-' x DY is the unit sphere in R p + qwith respect to 11 11. By stretching the unit sphere D p x S Y - ' u S"-' x Dq at the "equator" S - I x S Y - ' it follows that D p
x S 4 - ' x {O)
u Sp-'
x Sq-' x [0,1]
u S p - ' x D4 x (1)
is the boundary of a ( p + 61)-cell in Rpfy, hence homeomorphic to the sphere SPfq-1
c
RP+4.
5.12 Lemma: Any two surgically related smooth closed manifolds are cohordunt : con.sequentlji any two surgically equivalent smooth closed manifolds ure cohordant. PROOF: Suppose that X , and X I are smooth closed manifolds of dimension p 4 - 1, where one replaces some Dp x S4- c X , by S P - ' x 0 4 to obtain X I ,as before. Then Xo\Dp x S 4 - and X \Sp- x 0 4 are each diffeomorphic to a smooth compact manifold X with boundary S P - ' x S - ' .Let Y be the union of the product X x [0,1] and the ( p q)-cell with boundary
'
+
+
Dp x S4-' x {0)
usp-'
x SY-' x [0,1] u S p - ' x
0 4
x 11)
described earlier, with the obvious boundary identifications: the two boundaries DP x S4-I x (0) are identified with each other, the two boundaries Spx S4- I x [0,1] are identified with each other, and the two boundaries Spx Dq x { 1 are identified with each other. Then Y is a smooth compact ( p + 4)-dimensional manifold with boundary Xo x (0) + X I x (l}, as required.
' '
The converse of the second assertion of Lemma 5.12 requires some elementary Morse theory, which we sketch. According to Definition 111.6.11 the djflerential of a smooth function Y R on a smooth manifold Y is the C5( Y)-linear map €*( Y ) % Cm'( Y ) carrying any smooth vector field L E &*( Y ) into the smooth function Lf' E Cm(Y). A critical point off is any point J' E Y at which df vanishes; that is, ( L f ' ) ( y )= 0 for every L E Q*( Y).
5.13 Lemma (Regular Interval Lemma): Let Y be a smooth compact ndimensional munijdd whose boundary is the disjoint union of two smooth closed ( n - 1)-dimensional manijolds X , and X , , and let YL [a, b] c R be a smooth ,function with no critical points, such that X , = f - ' ( { a ) ) and X , = j - ' ( { b ) ) . Then there is a diffeomorphism Y 3 X , x [a, h] whose composition with the second projection X , x [a, b] 3 [a, b] is 1' itself'; u jbrtiori X , is difleomorphic to X,.
-
PROOF: There is a smooth riemannian metric Q*( Y ) x €*( Y ) Cr'(Y ) by Proposition 111.5.8, which induces an isomorphism b(Y ) --* €*( Y ) carrying
270
VI. Unoriented Manifolds
a(Y ) into a vector field grad j ' E a*(Y ) such that ( M , grad,/') = M f E Ca( Y ) for any M E a*(Y ) . By hypothesis df is nowhere-vanishing; hence grad f is also nowhere-vanishing, so that (grad f , grad f ) is everywhere positive on Y. It follows that there is a unique vector field L E &*( Y ) such that (grad f', grad f ' ) L = grad f ' , for which one trivially has &f = ( L , gradj') = 1 E Cq-(Y). The definition of L also yields the identity (grad f , grad f ' ) ( L , L ) = 1, and since Y is compact, it follows that there are positive constants A and B such that 0 c A 5 ( L , L ) B < 00 uniformly on Y . In order to construct Y 3 X u x [a, b] we shall first construct its inverse X u x [a,b]+! - Y by solving ordinary differential equations. In outline, observe that partial differentiation with respect to the coordinate t E [a, b] provides a vector field d / d t on Xu x [u, b]. We shall show that there is a uniquely defined h such that h,(d/dt) = L on Y and h(x,u) = x for each
dj' E
xE
x,.
For any c E [a,b] let X , be the closed set f - '({c}) c Y . Since Y is compact X , is also compact, so that it is contained in the union of finitely many open coordinate neighborhoods Y' in Y. As in Proposition 1.6.10 one can shrink each of the open coordinate neighborhoods Y' to a smaller open coordinate neighborhood Y" whose (necessarily compact) closure satisfies P " c Y ' , in such a way that X, is also contained in the union of the finitely many open sets Y"; one may as well assume that X , actually intersects each Y". If ( y ' , . . . ,y") are coordinate functions on one of the former coordinate neighborhoods Y ' , the restriction of L to Y' is of the form I' d/dy' + . . . . + Ind/dynfor smooth functions I', . . . , I" on Y', as in Lemma 111.6.6.Suppose temporarily that c lies in the interior (a,b) of the closed interval [a, b]. Then according to the classical existence and uniqueness theorem for ordinary differential equations there are positive constants 6 > 0 and E > 0 and a unique n-tuple ( h ' , . . . , h") of smooth real-valued functions on the intersection X , n P" x [ c - 6, c E ] such that
+
ah' (x; t ) = Il(h*(x;t), . . . , h"(x; t ) ) , at
-
ah" at
-(x; t ) =
l"(h'(x; t), . . . , h"(x; t ) ) ,
such that each x E X , n Y" has coordinates ( h ' ( x ; c), . . . ,h"(x;c)),and such that each n-tuple (h'(x;t ) , . . . , h"(x; t ) ) is the coordinate description of a point in the larger neighborhood Y'. The constants 6 > 0 and E > 0 depend on the geometry (in R") of the inclusion 7'' c Y' and on the constants A and
27 I
5. Stiefel-Whitney Numbers
B described earlier, with 0 < A 5 ( L ,L ) I B < 00; however, A and B themselves depend only on the given function Y s [a,b] and the choice of the riemannian metric ( ,),so that 6 and E effectively depend only on the inclusion Y" c Y'. In the special cases c = a or c = b one has 6 = 0 or E = 0, respectively. One can rewrite the preceding system of differential equations in the form
'
?/I
(7t
s
(7 h " (7
(7
--+...+--
24'1
d t dy"
=I' -+... dyl
a + PF, UY"
or equally well in the coordinate-free form h,(?r/dt) = L. Since X, is contained in the union offinitely many of the coordinate neighborhoods Y", the uniqueness of the n-tuples (A', , . . , 11") implies for L' E (a, b) that there are positive h Y with constants 6, > 0 and E , > 0 and a unique map X, x [c - b,, c EJ h,(?/?r) = L over the image of h, and h(x,c ) = s for each s E X , . Since
+
-
one has f(/z(x, t ) ) = t for each t E [c - b,, c + E,]. Hence for any closed sub) interval [ t , t'] c [c - 6,, c 4 there is a bijection from X, (=f-' ( { t j , )to X,. ( = j ' - ' ({ t i ) which ) identifies the X,-end-point of each integral curve of L with the X,,-end-point of the same integral curve; one easily verifies that the preceding bijection is a diffeomorphism from X, onto X,. . In the special cases = ( I or c = h one has 6, = 0 or cb = 0, respectively. To complete the proof of the regular interval lemma it suffices to note that finitely many of the intervals [c - S,, c + E,] cover [a, b], so that there is a unique diffeomorphism X, x [a, b] 1 :Y such that h,(d/dt) = L on all of Y and h(x,a) = x for x E X,. It follows as in the local case that f ( h ( x ,t ) ) = t for any (x,t ) E Xu x [a, 61. Furthermore, if the inverse diffeomorphism YA X, x [u,b]carriesy~Y i n t o ( . x , t ) ~ Xx, [u,b],thenf(y)= t =pr,(g(y)), so that j ' = pr, y as claimed.
+
(8
19
Now let Y A R be a smooth function on a smooth manifold Y of dimension n = p q, and suppose that y E Y is a critical point o f f . If there are coordinate functions u ' , . . . , u p , o', . . . , ziq on some neighborhood of y, all vanishing at y, such that
+
j ' = f ( y ) - [(u')' f . . .
+
(UP),']
+ [(u'),' + . . . + (u",']
in that neighborhood, then y is a nondegenrrate critical point o f f , with criricul ualue ,f(y ) E R. (There is a less restrictive-appearing characterization of nondegenerate critical points; however, for the moment we merely observe that d j ' does indeed vanish at y, so that y is at least a critical point off in the earlier sense.)
272
V1. Unoriented Manifolds
For notational convenience let u = ( u ’ , . . . , u p ) , u = ( u ’ , . . . , u4), lu11’ = + . . . + (up)’, and llull’ = (u’)’ + . . . + (uq)’. For any E > 0 let c Rp+4be the disk of radius 28 about 0 E Rpfq, consisting of those (u, u ) E RP++“ such that IIu11’ + l(u)1’ S ( 2 ~ ) ’and , for any 6 > 0 let S : + q - l c Rpf4 be the sphere of radius 6 about 0 E Rp+4, consisting of those points (u, u) E Rp+q such that lu11’ + lu11’ = 6’. The notations DP, D4, SP- S q - without subscripts will denote standard disks and spheres of the indicated dimensions, identified only up to diffeomorphism. (u’)’
’,
+
5.14Lemma: ForanyE > O l e t f = -IIuI12 Ilull’onDP,r4;thenj’-’((-~’)) is dfleomorphic to S p - ’ x 0 4 and f - ’ ( { + E ’ ) ) is diffeomorphic to DP x S Q -
’.
PROOF: If 0 5 6 5 2 ~then , the intersection f - ’ ( { - ~ ’ ) n ) SgfY-l consists of those (u,u) E 01;:“ such that -IIuII’ + ( 1 011’ = -E’ and llu11’ + llu11’ = 6’; that is, l)u11’ = i(6’ E ’ ) and llu11’ = $(a’ - c’). Clearlyf-’({ - 8 ’ ) ) n S : + 4 - - ’ is void for 0 S 6 < E , of the form S P - ’ x (*) for 6 = E , and of the form S p - ’ x S - ’ for E < 6 S 2.5, so that f - ’ ( { -e’J) is diffeomorphic to S p - ’ x D4. Similarly f - ’( { + 8 ’ ) ) is diffeomorphic to DP x S 4 -
+
’.
5.15 Lemma: Let Y be a smooth compact n-dimensional manifold whose boundary is the disjoint union of two smooth closed ( n - 1)-dimensional manifolds X u and xb,and let Y [a, b] c R be a smooth function with a single critical point y E Y, which is nondegenerate, whose critical wlue c E R satisjies a < c < b, and for which X u = f - ’ ( { a j ) and X b = f - l ( ( b ) ) .Then X , is surgically related to xb.
s
PROOF: One has j’ = c - lu11’ + llull’ in some neighborhood of the critical point y, as in Lemma 5.14, with (u,u) = (0,O) at y itself. One can choose c > 0 sufficiently small that a < c - E’ and c + E’ < b, and also sufficiently small that lies in the preceding neighborhood of y. By the regular interval lemma (Lemma 5.13) X , is diffeomorphic to X C - - E and L X C + Eis2 diffeomorphic to X,, so that it remains to show that X c - E 2is surgically related to X,,,,. Let Y, = f - ‘([c - E’, c + E’]) n Dp2:4, and let Y, be the closure in Y ofthe complement f-l([c-&’, C+E’])\Y~. Then X,-,, n Y, and X,+,* n Y, are diffeomorphic to S P - ’ x 0 4 and DP x S 4 - respectively, by Lemma 5.14, and since X c - - c zn Y2 and X,,,, n Y, are mutually diffeomorphic, by the regular interval lemma, it follows that X,- tl is surgically related to X C + E 2as, required.
’,
5.16 Lemma: Let Y be a smooth compact n-dimensional manifold whose boundary is the disjoint union of two smooth closed (n - 1)-dimensional manifolds X , and X , . Then there is a smooth function Y [0,1] with X , = f - ’ ( { O ) ) and X , = f - ’ ( { l ) ) , and with only jnitely many critical points, all
s
273
5. Stiefel ~WhitneyNumbers
qf which are nondegenerate, whose criticul levels ure mutuully distinct real numbers in the open interval(0, I).
PROOF: We omit the proof of this classical result, which is Lemma 1 on pages 41-42 of Milnor [9] and Theorem 2.5 on page 9 of Milnor [18]. It also follows from Corollary 6.8 on page 37 of Milnor [131 and from Theorem 1.2 on pages 147-148 of M. W. Hirsch [4]. A function Y [0,1] satisfying the conclusion of Lemma 5.16 is an udmissible Morse function. 5.17 Theorem: Two smooth closed manifolds X, and XI ofthe same dimension ure surgicully equivalent if and only if they are cobordant.
PROOF: According to Lemma 5.12 any two surgically equivalent smooth closed manifolds are cobordant. Conversely, if the disjoint union of two smooth closed ( n - 1)-dimensional manifolds X , and X I is the boundary of a smooth compact n-dimensional manifold Y, Lemma 5.16 provides an s [0,1] with critical levels c l , . . . , c, satisfying admissible Morse function Y + 0 < c1 < . . . < c, < 1, where r is the number of (nondegenerate) critical ~ ) 4 = 1 , . . . , r - 1, let a, = I, points. Let a, = 0, let uq = +(c, c ~ + for and let X,,q = j - ' ( { a , , ; ) for 4 = 0,. . . , r. Then Lemma 5.15 asserts that X u qis surgically related to X u q +I for 4 = 0, . . . , r - 1, so that X , is surgically equivalent to X I , as claimed.
+
5.18 Corollary : The semi-ring Y1' of surgical equivalence clusses of smooth closed muniJolds is canonically isomorphic to the unoriented cobordism ring fl.
PROOF: Since addition and multiplication in each of 3' and fl are induced by disjoint unions and smooth products of smooth closed manifolds, this follows immediately from Theorem 5.17. 5.19 Corollary: Two smooth closed manifolds represent the same surgical equivcilence cluss (f and only if they have the same Stiefel-Whitney numbers.
PROOF:This follows immediately from Proposition 5.10 and Theorem 5.17.
One can extend Definition 5.1 in an obvious way without altering the preceding results. Let 2/2 [o1, . . . , a,] be the polynomial ring over 2/2 in ~ assigned degree p , for p = 1,. . . , n; that is, each monomial which each ( T is 0;'. . . T;(: E 2/2 [ol,. . . , (T,,] is assigned degree r 1 + 2r2 + . . . + nr,. A polynomial p(o,,. . . , (T") E 2/2[0,, . . . ,o,,]is homogeneous of' degree n if it is a sum of monomials of degree 17. For any smooth closed n-dimensional manifold X , there is then an element p ( w l ( ~ ( X ). ). ,. , w , ( r ( X ) ) )E H " ( X )
274
VI. Unoriented Manifolds
providing a sum ( p ( w l ( r ( X ) ) ., . . , w , ( r ( X ) ) ) , p x )E 2/2 of the StiefelWhitney numbers of Definition 5.1, also called a Stiefftl- Wliitney number. For later convenience, we define particular such Stiefel-Whitney numbers. Let 2/2 [ t , , . . . , t,] be the polynomial ring over 2/2 in which each t, is assigned degree 1, for p = 1,. . . ,n. According to the fundamental property of elementary symmetric polynomials, there is a unique s,(~~,. . . , G,) E 2 / 2 [ G ~ ,. . . , GJ, which becomes t; + . . + t: E 2/2 [ t , , . . . , t,] when one replaces G,,. . . , G, by the elementary symmetric polynomials t l + * * . t , , . . . , t , * * * t , ; clearly s,,(o1, . . . , a,) is homogeneous of degree n in the sense of the preceding paragraph. The following definition will be used in Remark 9.23.
+
5.20 Definition: For any n > 0 and any smooth closed n-dimensional manifold X , the number ( s n ( w l ( r ( X ) ) ., .
. ?
w , , ( . ( X ) ) ) , ~ xE) 2/2
is the basic Stiefel-Whitney number s , ( X ) of X .
6 . Stiefel-Whitney Genera The set 42 of diffeomorphism classes of smooth, not necessarily orientable, closed manifolds X is a commutative semi-ring with respect to disjoint unions and smooth products. The subset 9 c 42 of sums of two copies of any X E +2 is an ideal in 42, in the obvious sense, and there is an equivalence relation -in % with X , X , ifand only ifthere are elements X , + X , and X , + X , in 9 with X , X , + X , = X 1 X , X , . The algebraic operations in 0, induce corresponding operations in the quotient %/9, and if [ X I E %/Sis the equivalence class of X E 42, then the relation [ X I + [ X I = 0 E %/9 implies that 42/9 is a commutative ring 4 in which every element is of order two. The quotient of 42 by the ideal represented by smooth closed boundaries is precisely the unoriented cobordism ring !Ti of Definition 5.9: the natural epimorphism @ -+ !Ti factors out boundaries. The purpose of this section is to introduce certain other homomorphisms 42 -+ 2/2 which will serve as models for later constructions.
-
+
+
+
For convenience we use the same notation X to denote either a smooth closed manifold in 42 or its equivalence class [ X I E 08. Since every element of 4 is of order two, 4 can be regarded as a (graded) 2/2-algebra. For any n-dimensional X E % the 2/2 cohomology module H 4 ( X ) vanishes for q > n, so that H * * ( X ) = H * ( X ) . Let r ( X ) be the tangent bundle
275
6 . Stiefel-Whitney Genera
of X, and for any formal power seriesf(t) E 2/2[[t]] with leading term 1 E Z/2 let u,(z(X)) E H*(X) be the resulting multiplicative Z/2 class, as in Definition V.1.4. Since X is closed, in the usual sense that its boundary X is empty, the fundamental Z/2 homology class of Definition 1.4 is an element p x E H,(X). The Kronecker product (u,(z(X)),pX) E Z/2, in which the homogeneous element px of degree n annihilates all cohomology classes except in degree n, clearly depends only on X as an element of 42. 6.1 Definition: For any formal power series f ( t )E Z/2[[t]] with leading term 1 E 2/2,and for any X E 42, the Kronecker product (uf(r(X)),p x ) E 712 is the flyenus G(f)(X) of X.
-
6.2 Proposition: For any formal power series f ( t ) E 212 [[t]] with leading W) 212 carrying each term 1 E 212, there is a semi-ring homomorphism g# X E u& into the flyenus G(f)(X) E 212.
PROOF:Since addition in 42 is induced by disjoint union of manifolds, the additivity of G ( f ) is trivial. For any X, E 42 and X , E 42 the tangent bundle ofthe product X, x X, E 42 is given by z(X, x X,) = pr\t(X,) 0 pr\.r(X,) = t ( X , ) + z(Xz),for the projections X I x X2 -% X , and X , x X, X,, so that the cross-product version of the Whitney product formula (Proposition V.2.11) gives u/(r(X, x XZ)) = U f ( T W 1 ) + G,)) = Uf(.r(Xl)) x UJ(dX2)) E H*(X, x X2). The characterization of fundamental 212 homology classes appearing in Corollary 1.5 immediately implies that the fundamental Z/2 homology class p x I x X 2 ~ H * (xXX,)ofX, 1 ~ X ~ i s t h e c r o s s p r o d u c txpp~x, , ~ H , ( X 1xX,) of the fundamental 212 homology classes p X , E H,(X,) and p x z E H,(X,). Since pxl and p x 2are homogeneous, their degrees being dim X , and dim X,, the classical relation between cohomology cross-products and homology cross-products implies G(.f)(Xl x X,)
(u,(.r(X, x X2)L PX] X X J = (U,(W,)) x Uf(.r(X2)),pxl x P x , ) = (Uf(.r(X,))?PX1)(~f(4x2))9 Px,) = G ( f ) ( X l ) .G ( f ) ( X , ) , =
as desired. (See page 217 of Dold [8], e.g., for the cross-product relation; there are no & signs in the present computation because the coefficient ring is 212.)
276
VI. Unoriented Manifolds
One can equally well apply the term figenus to the 212-algebra homomorphism $2 % 212 induced by Proposition 6.2. Any formal power seriesf'(t)E 2/2 [ [ t ] ] with leading term 1 E 2 / 2 induces P,(ul,. . . , u,) E 212 [ [ u , , u 2 , . . .]] as in a multiplicative sequence Definition V.2.9. Furthermore; if X is of dimension n, then Proposition V.2.10 guarantees that G ( , f ) ( X )= ( P n ( w l ( T ( X ) ).,. . , w n ( t ( X ) ) ) , p x )Thus . G ( f ) ( x )is a Z/Z-linear combination of the Stiefel-Whitney numbers of X ; it is therefore reasonable to call G ( f ) a Stiefel-Whitney genus, as in the title of this section. According to Proposition 5.2 this remark implies that G ( f ) ( X )= 0 whenever X is the boundary Y of a smooth compact (n 1)dimensional manifold Y . Consequently 42 212 annihilates the kernel of the natural epimorphism % + 3 onto the unoriented cobordism ring \n, so that G ( f ) can equally well be regarded as a HJ2-algebra homomorphism 3 -+ 212, which we shall also denote G ( f ) . However, there are many nonzero elements of 3 which lie in the intersection of the kernels of all Z/2-algebra homomorphisms 3 --t H/2, as we shall learn in Remark 9.33. The simplest Stiefel-Whitney genus O 2 212 is induced by the polynomial f ( t ) = 1 t E 2/2 [ t ] (c212 [ [ t ] ] ) .In this case the corresponding multiplicative sequence C n 2 0 ~ n ( u.l.,. , u,) satisfies p,(ul,. . . , u,) = u, for each n > 0, as we observed following Definition V.2.9. Hence if X is a smooth closed n-dimensional manifold one has u J , , ( t ( X ) )= w,(T(X))E H " ( X ) , and by Corollary V.2.8 w , ( t ( X ) ) is the 212 Euler class e ( z ( X ) )of z ( X ) . Consequently
+
+
G ( f ) ( X )= ( u f ( G ) ) , P x ) = ( u f , , ( W ) , Px) = ( W f l ( d X ) ) , P x )= ( e ( 4 v ) ? P x ) E 212. 6.3 Definition: For any smooth closed manifold X characteristic x 2 ( X )is the element ( e ( t ( X ) )px) , E 212.
E
42 the 212 Euler
Since we have just observed that the H/2 Euler characteristic is merely a specialized Stiefel-Whitney genus, Proposition 6.2 guarantees that it too can be regarded as a ZJ2-algebra homomorphism @ --%212 or as a 212algebra homomorphism 3 3 212; the kernel of the latter homomorphism is geometrically characterized in Remark 9.25.
7. 2/2 Thorn Forms For any smooth closed manifold X E 02 let H * ( X ) 2H J X ) be the 2/2 Poincare duality map npxof Corollary 2.4and let H * ( X ) @ H * ( X ) 212 be the corresponding bilinear symmetric H/2 PoincarP form, with
277
7. 212 Thorn Forms
E 272 for any (a,8)E H * ( X ) x H * ( X ) ;since is nondegenerate. We shall construct a bilinear symmetric 2/2 form H J X ) 0 H J X ) 22/2 which is dual to ( , )p, in the obvious sense, hence a specific inverse H , ( X ) 5 H * ( X ) of the 2/2 Poincare duality map. The form ( ,)T is an element j*T, E H"(X x X ) , where n is the dimension of X ; its behavior will provide the major result of the next section. In this section itself we shall furthermore show that if X 4 X x X is the diagonal map, then A*j*T, is the 2/2 Euler class e ( s ( X ) )E H " ( X ) of the tangent bundle r ( X ) . As in the rest of the chapter, all coefficients lie in 2/2.
(a,/?Ip = ( a . DPP) = ( a
u /?,px)
D, is an isomorphism
(,)p
Let E : X represent the tangent bundle r ( X ) of a smooth closed manifold X E &. One can identify X with the image a ( X ) c E of the zero section X E , and since one can also identify X with the image A ( X ) c X x X of the diagonal embedding X 5 X x X there is a canonical diffeomorphism a ( X ) .+ A ( X ) . One can extend o ( X ) .+ A ( X ) in many ways to a diffeomorphism of open neighborhoods of a ( X ) c E and A ( X ) c X x X , as in the following lemma. Recall that two maps i A X x X and ,6 3X x X are homotopic relative to a subset a ( X ) c E if they are the restrictions to i x {O) and 6, x { 1 respectively, of a map x [0,1] 5 X x X , such that F(x, t )E X x X is independent of t E [0,1] whenever x E a ( X ) . In the following lemma X x X % X and X x X 3 X are the first and second projections of the product X x X .
I,
7.1 Lemma: Let E 1 :X represent the tangent bundle T ( X ) of a smooth closed manijbld X E 4Y. Then there are embeddings E 3 X x X and 2 X x X of' some open neighborhood E c E of the image a ( X ) c E of the zerosection X 5 E, which restrict to the d?fSeomorphism a ( X ) + A ( X ) , and such that each 01' the compositions
X ~ E F " ' X X X - = + X and X-E'X
n
-
F
x X-X
is the identity on X . Furthermore, there is a homotopy F ,from F , to F , , relative t o t ( X ) c E, which induces a homotopy from the restriction
, 6 \ a ( X ) A X x X\A(X) to the restriction i\o(X)
AX x
X\A(X).
278
VI. Unoriented Manifolds
PROOF: The initial stage of the construction can be found on pages 108-109 of Bishop and Crittenden [13, on pages 32-40 of Helgason [13, or on pages 32-40 of Helgason [2]. One introduces an afJine connection in T ( X ) ,such as the Leoi-Ciuita connection associated to a riemannian metric on z ( X ) . This provides an exppential map Ex X for each x E X , defined on an open neighborhood Ex c Ex of 0 E Ex,with the following properties: each exp, is a diffeomorphism such that expxO = x E X , and there is an open neighborhood c E of a ( X ) c E such that each exp, is the restriction to E, c ?I of a map -+ X . One can therefore define E x [0,1] 5 X x X , as required, by setting
a
F(e9 t ) = (exp,,,,(te), exp,,,,(
- (1 -
04 ).
For any X E %! the El2 cohomology module H * ( X x X , X x X \ A ( X ) ) is an H*(X)-module with respect to the composition
-
H*(X)0 H*(X x X , X x X\A(X)) pr: Q id H*(X x X ) 0 H*(X x X , X x X \ A ( X ) ) U H*(X x X , X x X \ A ( X ) ) carrying any o! 0 T E H * ( X ) 0 H*(X x X , X x X \ A ( X ) ) into ( a x 1) LJ T E H*(X x X , X x X \ A ( X ) ) . The 212 cohomology module H * ( E , E * ) is also an H*(X)-module, as usual, with respect to the product
H * ( X ) 0 H*(E, E * ) carrying o! 0 U x E X let X , X\{x)
-
E H * ( X )0
=
n*@id
H * ( E ) 0 H * ( E ,E * )
H*(E, E*) into n*a
LJ
uH * ( E , E * )
U E H*(E, E*). For any
{ x ) x X , { x ) x X \ A ( X ) A X x X , X x X\A(X)
be the lefi inclusion, and let E x , E: 5 E, E* be the inclusion of the fiber pair over x, where E 5 X represents the tangent bundle r ( X )as usual.
7.2 Lemma: For each X E & and each X E X there is an H*(X)-module isomorphism G, and a ZI2-module isomorphism G,,, such that the diagram H*(X x X , X x X\A(X))
of U/2-module homomorphisms commutes.
H*(E,E*)
279
7. HI2 Thom Forms
PROOF: Let i c X x X be the image F , ( i ) of the embedding F , of Lemma 7.1, define a neighborhood Y c X of x E X by setting {x) x Y = ({.XIx X ) n d, and observe that the inclusion I, induces a commutative diagram - * H*(b,D\A(X)) H * ( X x X , X x X\A(X))
H*({x) x
I': I-
1
I:
X,{x)x X\A(X)) 5 H * ( ( x ) x Y,{x) x Y\A(X))
-
=.
H * ( X , X\ ( x )1
* H*( Y, y \ (x)1
with horizontal excision isomorphisms. If Ex = E n E x , = E n E,*, and E* = E n E*, the inclusion E x , E: 5 E, E* also induces a commutative diagram H*(E,E*) H*(&E*)
with horizontal excision isomorphisms. Since the composition
is the identity map, F , induces a commutative diagram
- -
E,,E,* A { x ) x YJx) x Y\A(X) = Y, Y \ { x ~
1
1
E,E*
b,b\A(X)
with vertical inclusions, hence a commutative diagram
H*(b,d\A(X))
7 -
J
H*( Y, Y\{x) )
* H*(E,E*)
4.
F!x
* H*(E,,
g,*)
280
VI. Unoriented Manifolds
with isomorphisms F: and F:,x. To complete the proof one defines G, and to be the obvious compositions of excision isomorphisms with Fg and F:,x, respectively. The proof that GI is an H*(X)-module isomorphism is a direct verification. One can interchange the two factors X x X in Lemma 7.2, in which case H*(X x X, X x X\A(X)) is an H*(X)-module with respect to the composition carrying
p 0 T E H*(X) 0 H*(X into (1 x
p) u T E H*(X
x
X, X x X\A(X))
x X, X x X\A(X));
for any x E X the right inclusion is X, X\(X) = X x {x), X x {xf\A(X) -% X x X, X x X\A(X). As in Lemma 7.2 one then has an H*(X)-module isomorphism G, and Z/2-module isomorphisms Gr,xsuch that the diagrams H*(X x X,X x X\A(X))
H*(X,X\{x).)
H*(E,E*)
* -
H*(E,,E,*)
of Z/Zmodule homomorphisms commute.
In the following lemma we temporarily ignore the two H*(X)-module structures of H*(X x X,X x X\A(X)). 7.3 Lemma: The isomorphisms GI and G , from H*(X x X, X x X\A(X)) to H*(E, E*) are the same H/2-module isomorphisms. PROOF: GI is defined up to excision isomorphisms as the Z/Zrnodule isomorphism H*(b,d\A(X))5 H*(E, B*), and G, is defined up to the same F' excision isomorphisms as the Z/2-module isomorphism H * ( b , b\A(X)) -A H*($ I?*). By Lemma 7.1 there is a homotopy F from I? 3 b c X x X to k 3 b c X x X relative to 4x1 c i?, which also induces a homotopy from the restriction i \ a ( X ) 2d\A(X) c X x X\A(X)
to the restriction E\a(X) 3d\A(X) c X x X\A(.X). Hence F:
=
FT, so that GI = G,.
28 1
7. 212 Thorn Forms
7.4 Lemma: For uny E H * ( X ) and any T E H*(X x X , X x X \ A ( X ) ) one = T u (1 x p) E H * ( X x X , X x X \ A ( X ) ) ; hence
hu.5 T u ( / I x I )
j*Tu(px l)=j*Tu(l xfl)~H*(XxX)
for the inclusion X x X
X x X , X x X\A(X).
PROOF: Since the Z/Zalgebra H * ( X x X , X x X \ A ( X ) ) is commutative, it suffices to show that ( p x 1) u T = (1 x p) u T. However, Lemma 7.3 gives G , ( ( P x 1) u T )= .*[) u G,T =
.*P
u G,T = G,((l x
for the common 2/2-module isomorphism G,
=
p) u T )
G,
Lemmas 7.3 and 7.4 together imply that G, and G, are the same H * ( X ) module isomorphism, which we henceforth denote
H*(X x
x,x x
x\A(x))~H*(E,E*).
7.5 Definition: For any smooth closed n-dimensional manifold X E 42 the diagonal 2/2 Thom class Tx E H"(X x X , X x X \ A ( X ) ) is the inverse image G - Ur(,) of the 2/2 Thom class CJrcx, E H"(E, E*) of the tangent bundle z ( X ) ; the 2 / 2 Thom form of X E 42 is the image j*Tx E H"(X x X ) of T, under the inclusion-induced homomorphism H"(X x X , X x X \ A ( X ) ) 5 H"(X x X ) . There are useful alternative characterizations of the diagonal 212 Thom class T,. Recall from Lemma 1.1 and Corollary 1.5 that for each x E X the 2/2 homology module H,(X, X \ ( x } ) is free on the single generator jx,*px, where X 5 X , X \ ( x ) is an inclusion and p, E H,(X) is the fundamental 212 homology class. (The boundary X of any X E 42 is empty, so that H,(X, 8)= H,(X).) Hence the 212 cohomology module H " ( X , X \ { x } ) is also free on a single generator, which we denote a,.
7.6 Proposition: For any smooth closed n-dimensional manifold X uny x E x , let
X , X\(x)
= {x)
X , X\[xj
=
x X , { x ) x X\A(X)
AX
E& !
and
x X , X x X\A(X)
and
X x [x), X x ( x ) \ A ( X ) A X x X , X x X\A(X)
be left and right inclusions, respectively. Then the diagonal 212 Thom class Tx E H"(X x X , X x X \ A ( X ) ) is uniquely characterized by the property that l:T, = w, E H " ( X , X\{x}) for each x E X ; similarly, T, is also uniquely characterized by the property that r,*T, = a, E H " ( X , X\{x>) for each x E X .
282
V1. Unoriented Manifolds
PROOF: To prove the first assertion it suffices to reexamine the diagram H*(E, E * )
H*(X x X , X x X \ A ( X ) )
-
J
GI
J
x
H * ( X ,X\{XS) H * ( L E,*) of Lemma 7.2, where GT, = Ll,(,), and to recall from Definition IV.1.4 that E H"(E, €*) is the unique class such that j,*U,,,, is the 212 Thom class UT(X, the generator of H*(E,, E:) for each x E X . The proof of the second assertion is similar. The following result is a partial rephrasing of Proposition 7.6. 7.7 Lemma: Let p x E H , ( X ) be the fundamental 212 homology class of' a smooth closed n-dimensional manifold X E (42, with H/2 Thom form j*Tx E H"(X x X ) ; then ( j * T x ,1 x p x ) = 1 E 212 and similarly ( j * T x , p x x 1) = 1 E 212.
PROOF: One starts with an inclusion diagram X
=
jX
{x}x X
'-I
x x x
j
+ {x} x
+
X , { x ) x X\A(X)
=
X,X\(x)
X x X , X x X\A(X).
By applying H*( ) and H,( ) and computing Kronecker products, it follows that ( j * T x ,1 x pX) = (j*Tx71,,,pX)= (l,*j*Tx,px) = ( j X T x 9 p x )= (CTxLix)*px) = (w,,(j,)*px)= 1
for the unique generators w, E H"(X, X\{x} and ( j x ) * p xE H , ( X , X \ { x ) ) . A similar proof yields the second assertion. Here is the main result of this section. 7.8 Proposition : For any smooth closed n-dimensional manifold X E 02,let j*Tx E H"(X x X ) he the 212 Thom form of Definition 7.5, and let H * ( X ) H J X ) he the PoincarP duality isomorphism n p , of Corollary 2.4. Then ( j * T , , a x DpP) = ( P , a ) E Z/2 jbr any a E H J X ) and any P E H * ( X ) , and (j * T x ,D,a x b ) = (a, b ) E 212 for any a E H * ( X ) and b E H J X ) .
283
7. L/2 Thom Forms
PROOF: Observe that if u and fl are homogeneous of different degrees then ( j * T x , u x DpP) and ( P , u ) both vanish; hence one may as well assume that u and are homogeneous of the same degree, in which case the cap product [I n u is just (p, a ) 1 E H,(X). Since the coefficient ring is 2/2, cup products commute, and one also has the classical identity (0 x cp) n ( p x v) = (0 n p) x (cp n v) relating cap and cross products (as in Spanier [4,p. 2551, for example). Consequently (j*Tx
x DpB) = ( j * T x , (1 n 4 x (b' n pX)) = ( j * T x , (1 x PI n ( a x pX)> = ( j * T x u (1 x PI, a x p x ) = ( j * T x u ( B x 11, a x px) =
( j * T x , ( P x 1) n (a x PX)) = (.i*Tx, ( B n a) x (1 PX)) px} = ( B , a ) ( j * T x ,1 x pX) = (P,Q>,
= ( j * T x , (B,u>1 x
as claimed, by Lemmas 7.4 and 7.7. Similarly, if a and b are homogeneous of the same degree one has ( j * T x , Dpa x h ) = (j*T,,(a n px) x (1 n b ) ) = ( j * T x , (a x 1) n ( p x x b ) ) = ( j * T , u ( a x l ) , p x ~ h ) = ( j * T X u x( la ) , p , x b ) ( j * T x , (1 x a) n ( p x x b ) ) = ( j * T x , ( 1 n px) x (a n b)) = ( j * T x , p x x ( a , b ) l ) = ( a , b ) ( j * T x , p x x 1) =
Observe that the preceding proof does not use the property that the 2/2-module homomorphism H * ( X ) 5 HJX) is actually an isomorphism. Recall that the 212 Poincare form is a symmetric bilinear form H*(X) x H*(X)*2/2 defined by setting (a, ,O)p = ( a , Dp[j) = ( a u f l , p x ) for any (a,p ) E H * ( X ) x H * ( X ) ,and observe that the 2/2 Thom form j*Tx E H*(X x X ) of Definition 7.5 can be regarded as a bilinear form
H * ( X ) x H * ( X )32 / 2 by setting ( ~ l , h=) (~j * T x , u x b ) for any (u,b)E H J X ) x H,(X). Since H*( X ) is canonically isomorphic to Hom,,,(H,(X), 2/2),the 2/2 Thom form induces a unique 2/2-module homomorphism H,( X ) L H * ( X ) such that ( D T L I , ~= ) ( ~ , hfor ) ~any (a,b)E H J X ) x H,(X). 7.9 Corollary: For utiy X E %Y, the compositions DpDT und DTDp ure the identity isomorphisms on H , ( X ) und H * ( X ) , respectively; thut is, DT = Dp so [hut the Z/2 Poincurl jorm ( , )p and 212 Thom jorm ( , )T are inverse nondegenerate bilineur jorms.
',
284
VI. Unoriented Manifolds
PROOF: For any (P, a) E H * ( X ) x H J X ) one has ( P , DpDTa) = (P, DTa)p = (DTu, P ) p = (&a, D p P ) = (a, DpPh = (j * T x ,a x D p P ) = ( P , a> by ProPosition 7.8, so that D p D T is an isomorphism. Similarly, for any (a,b ) E H * ( X ) x H , ( X ) , one has (DTDptl,b ) = (Dptl,6 ) =~ (j * T x ,Dptl x 6 ) = (tl, b ) by Proposition 7.8, so that D T D p is an isomorphism. For later convenience we exhibit the duality between ( , )p and ( , )T in terms of Dp and D T , as it occurs in the preceding proof: jbr uny (S, a) E H * ( X ) x H J X ) . Here is one of the most useful properties of 2/2 Thom forms. (&a,
DpP)
( P , a ) E 4'2
=
7.10 Proposition: Let X E 02 be any smooth closed n-dimensional manifold with 212 Thom jbrm j*Tx E H"(X x X ) , and let X 4 X x X be the diagonal map. Then the 2 / 2 Euler class of'the tangent bundle z ( X ) i s gioen by e ( z ( X ) )= A*(j * T x )E H"(X ) .
PROOF: Let E 5 X represent z ( X ) .Then the vertical arrows on the left-hand side of the clearly commutative diagram H"(X x X , X x X \ A ( X ) ) & H"(X x X )
=I I
I\
H"(B,d \ A ( X ) ) z.r
jb
* H"(b)A H " ( X )
Ff,
H"(E,E * )
j*
+
H"(E)'
are precisely the isomorphisms used to construct the isomorphism Gl ( = G) of Lemma 7.2; hence one has a commutative diagram
H"(X x X , X x X \ A ( X ) ) i '
I
Since the 2/2 Euler class of t ( X ) is given in Definition IV.3.1 by setting e ( z ( X ) )= o*j*U,(x,for the 2/2 Thom class U r ( XE) H"(E,E*), it follows that j*A*Tx = cr*j*GTx = o*j*U,,,, = e ( z ( X ) ) ,
as claimed.
8. The Thorn Wu Theorem
285
8. The Thorn-Wu Theorem We shall show for any formal power series f(t) E 2/2 [ [ t ] ] with leading term 1 E 2/2 that the multiplicative 2/2 class u r ( z ( X ) )E H * ( X )( c H * * ( X ) ) of the tangent bundle z ( X ) of any smooth closed manifold X E 42 depends only on the 2/2 homology and cohomology of X itself. Hence u s ( z ( X ) )is homotopy invariant (up to canonical isomorphisms); in particular, u r ( t ( X ) ) is independent of the smooth structure assigned to X . The proof uses the Steenrod square Sq of Remark V.7.14, which is not part of elementary algebraic topology; nevertheless, the result is too important to omit. 8.1 Lemma : For any smooth closed n-dimensional manifold X E 42 let j * Tx E H"(X x X ) he the 2/2 Thorn jbrm, and let w E H * ( X ) be the total StiejelWhitney class w ( s ( X ) )of the tangent bundle r ( X ) ;then j*Tx u (w x 1) = j*Tx u (1 x w ) = Sqj*Tx E H * ( X x X ) ,
lor the Steenrod square Sq of Remark V.7.14.
PROOF: Let E 5 X represent t ( X ) and let UT(x)E H"(E,E*) be the usual L/2 Thom class. By Remark V.7.18 one then has an identity n*w u Ur(X)= Sq Ur,x, E H*(E, E*) to which one applies the 2/2-module isomorphism G-'
H * ( E , E * ) --LH * ( X x X , X x X \ A ( X ) ) of Lemma 7.2 to obtain ( W x 1) u Tx = Sq Tx E H*(X x X , X x X \ A ( X ) ) . Since Sq is natural, and since H * ( X x X ) is commutative with respect to cup product, one can then apply the inclusion-induced homomorphism H*(X x X , X x X \ A ( X ) ) H * ( X x X ) and Lemma 7.4 to complete the proof.
For any X E 02 the Steenrod square Sq induces a transpose 2/2-module homomorphism H J X ) % H , ( X ) , defined by the requirement that ( p , Sqx a ) = (Sq p, u ) for any a E H J X ) and any p E H * ( X ) . 8.2 Lemma: For any X T E H*(X x X ) one has
E 02
(Sq T,a x b )
and any elements a E H , ( X ) , b E H , ( X ) , and =
(T,Sqxa x Sqxb)
E
Z/2.
PROOF: Since T is a 212 linear combination of cross products a x B E H*(X x X ) , and since Sq(a x p) = Sq a x Sq p by Remark V.7.16, it suffices to observe that (Sq(a x PI, a x b )
(Sq a x Sq fl, a x b ) = <Sq a, a>(Sq A b ) = (a,Sqxa>
286
VI. Unoriented Manifolds
In the following result H J X ) 3 H * ( X ) is the inverse of the 212 DP=nlrx Poincare duality isomorphism H * ( X ) H,(X), as in Corollary 7.9.
8.3 Proposition (Thom and Wu): For any smooth closed manifold X E O/L the total Stiefil-Whitney class w ( z ( X ) )of the tunyent bundle s ( X ) suti~fies the identity w ( z ( X ) )= Sq D T S q X p x E H * ( X ) f o r the fundamental 212 cluss Px E H * W . PROOF: Set w ( z ( X ) )= w E H * ( X )as before. Then for any a E H J X ) one has ( w , a> = ( D T D P w , a> = ( D p w , ~ )= , (j*Tx, Dpw x a ) = (j*Tx, ( w n px) x (1 n a ) ) = (j*T,, ( w x 1) n (p, x a ) = (j*T, u ( w x I), p, x a ) = (Sqj*Tx,Px x a> = (j*TxlSqxPx x = (%X =
sqX a)T = ( D T sqX P X > sqX SqX P X a> E
pX,
( s q DT
7
by Lemmas 8.1 and 8.2. Proposition 8.3 is a variant of the original result of Thom [2] and Wu [3]. Another formulation is given in Corollary 8.5.
For any smooth closed manijold X E 42 and any formal power series f ( t )E 212 [ [ t ] ] with leading term 1 E H/2, the 212 multiplicative class u f ( z ( X ) )E H * ( X ) depends only on the algebraic structure of H , ( X ) and H * ( X ) ; in particular, u f ( r ( X ) )is independent of the smooth structure assigned to X .
8.4 Theorem (Thom and Wu):
PROOF: Since the fundamental H/2 class px E H , ( X ) and the operators Sq,, D, Sq depend only on algebraic properties of H J X ) and H * ( X ) , the identity w ( z ( X ) )= Sq DT Sq, px of Proposition 8.3 implies the result for the total Stiefel-Whitney class, which is the special case f ( t ) = 1 + t. However, Proposition V.2.10 provides polynomial computations for any multiplicative 212 class u f ( z ( X ) )in terms of w(.(X)), which are also independent of
Go.
Since H,(X) and H * ( X ) depend only on the homotopy type of X E 42, the preceding result can be rephrased as follows: For any X E 42 the 212 multiplicative classes u , - ( r ( X ) )depend only on the homotopy type of X . Here are some more corollaries of Proposition 8.3. 8.5 Corollary (Wu): For any smooth closed manijold X E 42 the W u class Wu(z(X)) of the tangenr bundle z ( X ) is the unique class v E H * ( X ) such that ( u u a,p,) = (Sq a , ~ , ) for every a E H * ( X ) .
287
9. Remarks and Exercises
PROOF: Since ( Du a,px) = ( 0 , ~ )for ~ the nondegenerate 2/2 Poincare form ( , )p, uniqueness is clear. It remains to observe that (Wu(r(X)) u 4 P x ) = (Sq-' W(dX)) u %Px) sqX PX > cl)P = (DT sqX PX Dpa) = (a, sqx Px) = (Sq % Px) = (DT
9
by Remark V.7.20 and the reformulation of Corollary 7.9. 8.6 Corollary: !f X E "11 is of' dimension n, then Wu(r(X))E H * ( X ) vanishes in all dimensions p such that 2p > n.
PROOF: If u E H,(X), then DTu E H"-"(X) and (Wu(r(X)),u)= (DTSqXPX,a) = (DTa,SqXPX)
=
(SqDTa,PX).
By the dimension axiom for Steenrod squares (Remark V.7.14) SqqD,a = 0 E H"-p'4(X) for q > n - p. Hence, if p > n - p one has Sqp DTu = 0 E H " ( X ) , so that ( w u ( ~ ( X ) ) , u=) 0, for every a E H,(X).
8.7 Corollary: For any X E OiY qf'dimension n > 0 one has (Wu(r(X)),px) 0 E 212.
=
PROOF: Since 2n > n, this is a consequence of Corollary 8.6.
9. Remarks and Exercises 9.1 Remark: The 2/2 fundamental class px E H,(X, X ; Z/2) of Definition 1.4 was defined only for smooth n-dimensional compact manifolds X with boundary X because the given proof of its existence required results (Propositions 1.8.4 and 1.9.7) which were established only in the triangulable case. However, with additional effort one can extend Definition 1.4 to n-dimensional compact topological manifolds. If an n-dimensional compact topological manifold X with boundary X is oriented in the sense described in the next volume, then for any commutative ring A with unit there is a unique fundamental class px E H,(X, X ; A); the case A = Z suffices. We shall obtain this result for the smooth case in Volume 2 as an oriented analog of Proposition 1.3. As before, smoothness is not really required; proofs of the topological case can be found in Dold [8, pp. 259-2671, Massey [6, pp. 200-2051, Milnor and Stasheff [l, pp. 2732741, Spanier [4, pp. 299-3061, and (for closed topological manifolds) in Samelson [11.
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9.2 Remark: An alternative proof of the identity d p , = p i of Proposition 1.9 is given in Spanier [4, p. 3041.
-
9.3 Remark: Here is an alternative proof that the Z/2 Poincare duality Dp'npx H , ( X ) of Corollary 2.4 is an isomorphism. The fact map H * ( X ) that D , is an isomorphism is not used in the proof of Proposition 7.8; furthermore, the Thom form j*T, E H*(X x X ) and induced homomorphism H J X ) 5 H * ( X ) were constructed independently of D,. However, according to Corollary 7.9 the compositions DTDp and D,D, are identity isomorphisms on H * ( X ) and H , ( X ) , respectively; hence D, is an isomorphism with inverse D T .
9.4 Remark: If X is an oriented compact smooth manifold with boundary X , then one can replace the coefficient ring 2/2 of Theorem 2.3 by any commutative ring A with unit to obtain an oriented Poincare-Lefschetz duality theorem. We shall do so in the next volume, using the obvious analog of Lemma 2.2; a different algebraic format for the proof is given on pages 1-17 of Browder [l]. PoincarbLefschetz duality is also valid for topological manifolds, using the coefficient ring 2/2 except in the oriented case. One of the standard proofs is similar to the proof of Theorem 2.3; it can be found in Dold [8, pp. 291-2981, Greenberg [l, pp. 162-1891, Milnor and Stasheff [I, pp. 276-2801, and Spanier [4,pp. 296-297.1. There are also proofs of the same nature in Bore1 [3] and in Griffiths [I], using Alexander-Spanier and Cech cohomology, respectively, rather than singular cohomology. Given a smooth closed oriented manifold X , one can establish Poincare duality for arbitrary coefficients by a procedure analogous to that of Remark 9.3; the details appear in Milnor [3, pp. 51-52] and Milnor and Stasheff [l, pp. 127-1281. A similar technique applies to closed topological manifolds; the details appear in Spanier [3] and in Samelson [l]. Finally, there are older proofs of the Poincare-Lefschetz duality theorem for triangulable manifolds (a fortiori for smooth manifolds) in Mayer [2], Lefschetz [l, pp. 188-2041, and Maunder [l, pp. 170-1991. A survey of Poincare-Lefschetz duality and related topics appears in Dold [7], and a 1965 bibliography of proofs of Poincark duality appears in Samelson [l]. 9.5 Remark : Stiefel-Whitney (2/2 cohomology) classes of tangent bundles of smooth manifolds were first constructed in Stiefel [l] and in Whitney [2] in 1935, using obstruction theory, as noted in Remark V.7.1. Whitney presented the details of his original construction in Whitney [3,4, 61, and some related work appeared in Rohlin [3], also using obstruction theory. In
9. Remarks and Exercises
289
Pontrjagin [ 11 classifying space techniques were applied exclusively to tangent bundles to extend the results of Stiefel [I]; the same techniques were applied in the detailed exposition of Pontrjagin [5]. Chern [ I ] and Pontrjagin [2] used integral formulas to express StiefelWhitney classes of tangent bundles. Classifying space techniques like those of Pontrjagin were then developed in Chern [2] for tangent bundles; they were extended to vector bundles in general in Chern [3] and in Wu [2, 51. 9.6 Remark : The general computation of Stiefel-Whitney classes via Steenrod squares (Exercise V.7.17) appeared in Thom [l], which concerns arbitrary real vector bundles. However, the method was instantly applied to tangent bundles in Thom [2], Wu [31, and Thom [4]. There are other methods for computing Stiefel-Whitney classes of tangent bundles of smooth manifolds. For example, Bucur and Lascu [l] compute Stiefel-Whitney classes of smooth closed manifolds by a method originally applied in B. Segre [l] to algebraic varieties. The alternate computations of Nash [ 11 and of Teleman [1-31 form part of the next remark. 9.7 Remark: We already know from Remark 111.13.34 that Thom [4] shows that the fiber homotopy type of the tangent bundle z ( X ) of a smooth closed manifold X is independent of the smooth structure assigned to X ; a fortiori the same conclusion applies to the J-equivalence class of z ( X ) . (These results were subsequently strengthened in Atiyah [13 and in Benlian and Wagoner [l].) We also know from Remark V.7.10 that Thom [4] shows that Stiefel-Whitney classes of vector bundles depend only on the J-equivalence classes of the bundles. Consequently for any smooth (closed) manifold X the Stiefel-Whitney class w ( z ( X ) )is independent of the smooth structure assigned to X . (The same result was established in Theorem 8.4by methods of Thom [2] and Wu [3].) This is perhaps the best place to recall from the remainder of Remark V.7.10 that one can most easily explain the preceding result by assigning Stiefel-Whitney classes directly to topological manifolds X in a way which produces w ( t ( X ) )whenever X happens to be smooth. The first such construction is that of Nash [l]. A later construction begins with the microbundles of Milnor [l 1, 141, reinterpreted as tangent topological R" bundles in Kister [ l , 21, to which one assigns total Stiefel-Whitney classes as in Teleman [I, 2, 31, for example; the latter Stiefel-Whitney classes agree with those of Nash.
9.8 Remark: The combinatorial construction on page 342 of Stiefel [l] and a later sketch in Whitney [5] assign 2/2 homology classes (rather than 2/2 cohomology classes) to smooth manifolds, using barycentric subdivisions of
290
VI. Unoriented Manifolds
given triangulations. These constructions were revitalized by Cheeger [11, Sullivan [13, Halperin and Toledo [11, and Latour [13, with the observation that the H/2 homology classes so assigned to a smooth closed manifold X are the 2/2 Poincare duals of the Stiefel-Whitney classes wi(z(X)) of the tangent bundle t ( X ) . Halperin and Toledo [l] provided the first complete proof of the duality assertion; later proofs were given by L. R. Taylor [ 11 and Blanton and McCrory [l]. Goldstein and Turner [l] give a variant of Whitney's construction which is explicitly independent of barycentric subdivision, Banchoff [l] gives a visually appealing interpretation of the construction, and McCrory [l] and Porter [l] give an intepretation which is based upon singularities of mappings of the manifold X into euclidean spaces. Related constructions can be found in Akin [l] and in Goldstein and Turner
PI
*
9.9 Remark : The preceding remark suggests that Stiefel-Whitney cohomology classes of tangent bundles also lead a life of their own, without the machinery of vector bundles in general. In fact, an axiomatic characterization of Stiefel-Whitney classes of tangent bundles is given in Blanton and Schweitzer [13, and another axiomatic characterization of such classes is given in Stong [7]. The value of the Blanton-Schweitzer axioms is demonstrated in L. R. Taylor [l] and in Blanton and McCrory [l], where they are used to prove the duality theorem mentioned in the preceding remark. The key to Blanton and Schweitzer [l] lies in Remark 111.13.44. 9.10 Remark: For any smooth manifold X the classical Whitney duulity theorem in H * ( X ; H/2) is merely the Whitney product formula for the Whitney sum z ( X ) 0 vJ of the tangent bundle z ( X ) and the normal bundle vf of any immersion X R2"-k; the result was first announced without proof in Whitney [6], as already noted in Remark V.7.8. An analogous duality theorem in H , ( X ; 2/2) was established in Halperin and Toledo [2], helping to verify that the H / 2 homology classes of Remark 9.8 are Poincark duals of the corresponding Stiefel-Whitney (cohomology) classes; the homology duality is visually interpreted in Banchoff and McCrory [l]. Finally, the suggestion formulated in Remark 9.9 is further dramatized by a direct combinatorial construction in Banchoff and McCrory [2] of the dual Stiefel-Whitney (cohomology) classes wi(r(X))of any triangulated manifold X .
29 1
9. Remarks and Exercises
of Exercise 111.1 3.27 to compute the total Stiefel-Whitney class w ( r ( G " ( R m + " ) ) E H*(G m(Rm+"of )) the tangent bundle T(G"'(R~+"))of the real Grassmann
manifold Gm(R*+").[ H i n t : Do Exercise V.7.9 first.] 9.12 Exercise: For any smooth closed manifold X the tangent bundle T ( X x X ) of the product X x X is trivially the sum t ( X ) t ( X ) . For any riemannian metric ( , ) on t ( X )one can therefore define a riemannian metric ( e ' , f ' ) for elements on t ( X x X ) by setting ( e e',f + f ' ) = ( e , f ) e, e', j ' ,J" in the total space of t ( X ) such that Tce = ~ fand ' ne' = nj', where 7c is the projection onto X . Hence one can define the normal bundle vA of the diagonal map X X x X by requiring t ( X )0 vA = A ! t ( X x X ) as in Definition 4.1. Show that vA is 7 ( X )itself.
+
+
+
>
9.13 Remark:
Some general immersion theorems were briefly described in Remark 1.10.13,and a classical nonimmersion theorem was proved in Proposition 4.5. Here is a very general necessary and sufficient immersion criterion. If X -$ ( W 2 n - p is any immersion of a smooth n-dimensional manifold X, then the normal bundle vf is of rank n - p and t ( X )0 vf = E ~ over ~ X, as in Definition 4.1. Conversely, according to M. W. Hirsch [l], if there is any real vector bundle 5 of rank n - p over X such that T ( X )@ 4 = . z 2 " - P , then there is an immersion X 5 RZn-pwhenever X is compact. The "easy" Whitney immersion theorem of Remark 1.10.13 trivially provides an immersion X 1:R2"+' for any smooth n-dimensional manifold X, and according to another result of M. W. Hirsch [l] there is a smooth homotopy x x [O, 11 + R 2 " + 'relating any two such immersions, ho and h such that each X x ( t )+ Rz"+ is also an immersion. It follows that the classifying maps X + G"+' (Ra')for the normal bundles vhuand v h , are homotopic, hence that vho = v h l ; that is, the normal bundle v h is uniquely defined by X itself. (According to Kervaire [2] this result was already known for embeddings X 1:[ W 2 " + l . In fact, for n > 1 Wu [8] had proved an even stronger result: any two smooth embetidings X 2 RZni I and X 2 [W2"+' of a smooth ndimensional manifold X are isotopic in the sense that there is a smooth map x x [O, 13.5 [WZf1' such that each X R2"+' is itself a smooth embedding.) One can always weaken any two immersions X L R 2 n - pand X 3 Rz'8-q with normal bundles vf and vg, using inclusion maps R2"-p+ R 2 " + ' and [WZn-q [ W a n + 1 to obtain immersions X R2"-p + R 2 " + 'and X 3 R2"-q + R2"+ with normal bundles vf 0 E P + ' and vg 0 .zq+ respectively. According to the preceding paragraph one then has vf 0 E P + = vg 0 E ~ + ; that is, in
',
~
'
', '
'
~
29 2
VI. Unoriented Manifolds
the terminology of Remark 111.13.36,any two normal bundles of X determine the same stable equivalence class, the stable normal bundle of X . By combining the preceding results it follows that the question of a R 2 n - p of a given smooth n-dimensional "best-possible'' immersion X manifold X is equivalent to the determination of the geometric dimension of the stable normal bundle of X , as in Remark 111.13.40. Specifically, if the geometric dimension of the stable normal bundle of X is n - p or less then X immerses in RZn-p,at least when X is compact.
s
9.14 Remark : The "best-possible'' immersion conjecture of Remark 1.1 0.13 was proved by Cohen [I], using results of R. L. W. Brown [2,3] and Brown and Peterson [4,5] : any smooth n-dimensional compact manifold X admits a smooth immersion X + [WZn-'("), where a(n) is the number of 1's in the dyadic expansion of the dimension n. According to Proposition 4.5 one can do no better. Here is an outline of the proof. Let BO(m) denote the classifying space G"([W") for real m-plane bundles, as in Remark 111.13.4, and let BO denote the classifying space lim, BO(m)for stable real vector bundles, as in Remark 111.1 3.36. Since the 212 cohomology rings H*(BO(m))are polynomial rings with one generator w,(y"), . . . , w,(y") in each degree 1, . . . , rn, the identities w i ( y m 0 E ' ) = wi(y") and the naturality of Stiefel-Whitney classes imply that the 2 / 2 cohomology ring H*(BO) is a polynomial ring with one generator wi in each degree i > 0. Now let X be any smooth n-dimensional compact manifold, and let vx be its stable normal bundle; as in Remark 111.13.36, there is a stable homoBO for v x . According to Remark 9.13, one topy classifying map X must show that the geometric dimension of vx is at most n - a(n);that is, one must factor j" in the form X + BO(n - a ( n ) )+ BO, up to homotopy. (According to the result of Wu [8] and M. W. Hirsch [I] cited in Remark 9.13, one can regard v x more concretely as an ( n + 1)-plane bundle \I,, whose classifying map X + BO(n + 1) is to be factored in the form X -+ BO(n - a ( n ) )+ BO(n l), up to homotopy. However, the stable classifying space BO is more convenient than BO(n + 1). Let 1, c H*(BO) be the ideal of all polynomials in w l , w z , , . . lying in the kernel of all the homomorphisms H*(BO) * H * ( X ) induced by the stable homotopy classifying maps ix,for all smooth n-dimensional manifolds X . According to Brown and Peterson [4,5] there is a topological space B o i l , and a universal map BO/I, 3 BO with the following two properties: (i) each .fx can be factored in the form X + BO/I, 3 BO, and (ii) the inclusion I , c H*(BO) induces a short exact sequence
*
+
0
-- I,
H*(BO)
f
H*(BO/I,)
-
0.
293
9. Remarks and Exercises
Briefly, BO/I,, is itself a classifying space for stable normal bundles of smooth n-dimensional manifolds, and its 2/2 cohomology ring is as small as possible. Brown and Peterson also obtain a weakened form M O I I , + M O ( n - a ( n ) )+ M O of a possible factorization BO/I, -+ BO(n - a ( n ) )--* BO of,j,,. Recall from Remark 1.10.13 that R. L. W. Brown [2,3] showed that every smooth closed n-dimensional manifold is cobordant to a manifold that immerses in [W"-'(") . Cohen [ 11 uses this result to construct a space X , and maps j;,and g n for which the compositions X , 5BO(n - a ( n ) )+ BO and X I , 3 BO/I,, -% BO are homotopic maps from X , to BO; the corresponding weakened compositions T X , 3 MO(n - a($) + M O and T X , 2MOII, % M O are then homotopic maps from T X , to M O . Cohen also constructs a map (T, such that MOII, 3 T X , 3 MOII, is homotopic to the identity and TX,,
Moil,,5TX,,
IJ,,
M O ( H- ~ ( n ) )
T g,,. on
is homotopic to Tj". Consequently the composition M O / I , MO(rr - cr(n))+ M O is a new weak version of a possible factorization of j,,, homotopic to the version of the preceding paragraph; moreover, Cohen strengthens the new version toan actual factorization BOII, + BO(n-a(n))+ BO ofj,. It follows that the stable homotopy classifying map fx of the stable normal bundle v x of any smooth compact n-dimensional manifold X E A' can be factored in the form X + BO/I, + BO(n - a ( n ) )-+ BO, and hence that \fx is ofgeometric dimension at most n - a(n);consequently X immerses in [WZ"- ? ( , I ) as desired, by Remark 9.1 3.
9.15 Remark: Theorem 4.8 first appeared in its present form in Chern and and Spanier [2], following unpublished work of Hopf. The result can also be found in Milnor [3, pp. 43-44], Husemoller [I, pp. 261-2621, and Milnor and Stasheff [I, p. 1201. There is an important partial converse in Haefliger and Hirsch [I], as follows. Let X be a smooth closed k-connected n-dimensional manifold with 0 5 2k < ti - 4; then X embeds in R 2 n - k - ' if and only if m n - k - l ( T ( X ) ) = 0E ' ( X ) .(A manifold is k-connected whenever it is connected and the homotopy groups n l ( X ) ,. . . , n,(X) vanish.) Thus, in this special case, the embedding criterion of Theorem 4.8 is both necessary and sufficient. The case k = 0 of the Haefliger-Hirsch theorem will be used in the next remark. Let X be uny smooth closed manifold of dimension n > 4; then X embeds in R2"- if and only if R,- ' ( ~ ( x=)0)E H " - ' ( X ) . Observe that the connectedness condition is entirely deleted; one simply applies the k = 0 version of the Haefliger-Hirsch theorem separately to each connected component of X . '
294
V1. Unoriented Manifolds
9.16 Remark : Some general embedding theorems were briefly suggested
in Remark 1.10.11, and a classical nonembedding theorem was proved in Proposition 4.10. This is an appropriate place for more details about the strongest known general embedding theorem. According to M. W. Hirsch [2], any smooth open n-dimensional manifold whatsoever embeds in R2"- Accordingly we henceforth consider only smooth closed n-dimensional manifolds, for n = 1, 2, 3, . . . . The "hard" Whitney embedding X -, Rz"is itselfbest-possible for smooth closed manifolds of dimensions n = 1 and n = 2; for according to Proposition 4.9 the real projective spaces RP' and RP2 do not embed in Iw' and R3, respectively. On the other hand, every orientable closed manifold of dimension n = 2 is one of the familiar orientable surfaces of some genus p , which visibly embed in R3. According to Wall [11 every smooth 3-dimensional manifold X embeds in R5. The "hard" Whitney embedding X --+ R2" is best-possible for smooth closed manifolds X of dimension n = 4; for according to Proposition 4.9 the projective space RP4 does not embed in R7. However, there is a conjecture that every smooth closed orientable 4-dimensional manifold does embed in R7, and many special cases of this conjecture have been verified in M. W. Hirsch [3], Watabe [l, 21, and Boechat and Haefliger [l, 21, for example. At the end of the preceding remark we learned that a smooth closed manifold X of dimension n > 4 embeds in R2"-' if and only if the dual Stiefel-Whitney class W,- I(z(X))E H " - ' ( X ) vanishes. However, according to Massey [l, 31 and Massey and Peterson [l] one has W n - ' ( T ( X ) = ) 0 for all orientable manifolds of any dimension n > 1 and for all nonorientable manifolds whose dimension n is not of the form 2'. It follows that every smooth closed orientable manifold of dimension n > 4 embeds in R2"-'; the same result is true for smooth closed nonorientable manifolds of dimension n # 2'. Incidentally, the proviso n # 2' is necessary in the nonorientable case; for according to Proposition 4.9 the projective space RP" does not embed in RZn-' for n = 2'. The preceding results justify the conclusion of Remark 1.10.11 that, except for easily identifiable exceptions, every smooth n-dimensional manifold can be embedded in RZn-'. The case of orientable 4-dimensional manifolds is probably not an exception to the general rule; however, at the moment it is not known whether every smooth closed orientable 4-dimensional manifold embeds in R'. Portions of the preceding results were obtained independently by other authors. S. P. Novikov [l] showed that every simply connected smooth manifold ofodd dimension n > 6 can be embedded in R 2 " - ' . Wu [9] showed
'.
29 5
9. Remarks and Exercises
that every smooth closed orientable manifold of dimension n > 4 can be embedded in Rz"- Rohlin [6] showed that every smooth notiorientable 3dimensional manifold embeds in R5. 9.17 Remark: In view of the preceding abundance of smooth embeddings X RzfI- 1 of smooth n-dimensional manifolds X , it is of interest to examine the normal bundles ti,. of such embeddings. According to Massey [2], vf has a nonvanishing section if and only if w 2 ( v f )u w , - , ( v f ) = 0 E H " ( X ; 2/2); since w ( v f ) = E(T(X)),as in Lemma 4.2, the latter condition is just W 2 ( . r ( X ) u ) \ t ' f l - 2 ( T ( x ) ) = 0 E ff"(X; 2/2). 9.18 Remark: The "best-possible" embedding conjecture of Remark 1.10.11 states that every smooth n-dimensional manifold X embeds in R2"-'(")+ I . One justification for the conjecture is the following result of R. L. W. Brown [2,3]: If X is closed, then it is cobordant to a smooth closed n-dimensional manifold which does embed in R Z f I - ' ( ' ' ) + l . There is a related result in R. L. W. Brown [4]: a necessary and sufficient condition for X to be cobordant to a smooth closed n-dimensional manifold which embeds in S n t k is that all Stiefel-Whitney numbers involving % i ( ~ ( Xfor ) ) i 2 k vanish. According to M. W. Hirsch [13, if a smooth closed n-dimensional manifold X admits a smooth embedding X R 2 n - k t 1for which the normal bundle vf has a nowhere-vanishing section, then there is an immersion X + R 2 n - k . The condition on vf is necessary since Mahowald and Peterson [l] construct a smooth closed n-dimensional manifold X with a smooth R 2 " - k t 1, for which there is no immersion X + R 2 n - k embedding X whatsoever; since k > a(n) in this example, there is no contradiction with the "best-possible" embedding conjecture. 9.19 Remark: Immersions and embeddings of real projective spaces RP" are of special interest, partly because the nonimmersion and nonembedding techniques of Propositions 4.3 and 4.9 can easily be extended to other dimensions. However, the following Exercise does not in general represent best-possible results.
+
9.20 Exercise: Show that if n = 2' q for q 2 0, then RP" does not immerse in [ w 2 * * ' - 2 and does not embed in R2" -
'.
9.21 Remark : Many best-possible immersions and embeddings are known for real projective spaces RP", and there is now a large literature on the subject. Most of the results known through 1979 are catalogued in Berrick [l]; there are also earlier catalogs and large bibliographies in Gitler [l] and in James [ 1, 21.
296
VI. Unoriented Manifolds
The problem of enumerating immersions and embeddings (up to isotopy) of real projective spaces is considered in Larmore and Rigdon [l, 21 and in Yasui [2, 31, for example.
9.22 Remark : Best-possible immersions and embeddings of complex projective spaces CP" are also of interest, although most of the known results involve techniques which will be introduced only in later chapters of this work. For example, for any n > 0 Atiyah and Hirzebruch [ 11 show that the complex projective space CP", of real dimension 2n, does not embed in R4,- 2a(n). Sanderson and Schwarzenberger [11 use this nonembedding theorem to show for certain values of n that CP" does not immerse in [ W 4 n - 2 n ( n ) and Sigrist and Suter [I] find necessary conditions for which CP" does immerse in R4"-2a(n)-'. A s of 1977 all known nonimmersion results for CP" were consequences of a general technique of Davis and Mahowald [2]; these results depend on the geometric dimensions of Whitney sums my: of the canonical complex line bundle y i over CP", and on the Atiyah-Todd number i(m, n), briefly mentioned in Remark 111.13.42. (According to Remark 111.13.29 the corresponding immersion problem for real projective spaces RP" is equivalent to finding the geometric dimensions of Whitney sums my: of the canonical real line bundle y i over RP", for all m > 0 and n > 0.) Yasui [11 enumerates certain embeddings of complex projective spaces CP" (up to isotopy). Oproiu [2] generalizes some of the nonembedding results for real projective spaces RP" ( = G'(R"' ')) to the particular Grassmann manifolds G2(Rn+')and G3([W"+3),and Opriou [3] contains further such generalizations; immersions of Grassmann manifolds are considered in Hiller and Stong [l]. Kobayashi [l] considers certain immersions and embeddings of lens spaces.
',
9.23 Remark: Following some initial work of Rohlin [I, 2,4], the unoriented cobordism ring % was completely computed in Thom [3,6-81. Specifically, YI is a graded polynomial algebra 2/2 [x2, x4, xs,x6,x 8 , . . .] with one generator x, in each degree n not of the form 24 - 1. Furthermore, each generator x, can be represented by any smooth closed n-dimensional manifold X , such that the basic Stiefel-Whitney number of Definition 5.20 satisfies s,(X,) = 1 E 2/2. Specific manifolds representing the generators xs and x Z mfor all m > 0 were constructed in Thom [8], and manifolds representing the remaining generators of 91 were constructed in Dold [3]. Alternative sets of manifolds X, representing the generators x, of % are given in Milnor [17], in Stong [ 6 ] , and in Royster [l], for example. Complete expositions of Thom's computation of the unoriented cobordism ring YI are given in Liulevicius [l, 31 and Stong [2]. There are also
9. Remarks and Exercises
297
surveys containing some of the computation in Milnor [lo], Rohlin [S], Poenaru [I], and Gray [ 11. Quillen [1,2] applied jorrnal groups to compute the unoriented cobordism ring 91 via Steenrod squares, and Brown and Peterson [3] used Steenrod squares in an entirely different way to compute '32. Quillen's approach is given in detail in Brocker and tom Dieck [l], and surveys of formal groups and their applications to the computation of '32 appear in Schochet [l], Buhitaber et al. [13, and Karoubi [ 13.
9.24 Remark: Unoriented cobordism is a relatively weak equivalence relation, so that one expects each class in '32 to be represented by at least one manifold with special properties. For example, it has already been noted that according to R. L. W. Brown [2,3] each class in '32 contains at least one representative satisfying the "best-possible" immersion conjecture and at least one representative satisfying the "best-possible'' embedding conjecture. According to Stong [4], each class of positive degree in '32 can also be represented by a fibration over the real projective plane R P 2 . 9.25 Remark: Certain classes in 91 contain representatives with further special properties, especially those classes in the kernel of the 2/2 Euler characteristic ' 3 2 3 2/2 of Definition 6.3. According to Conner and Floyd [l], an unoriented cobordism class lies in the kernel of x 2 if and only if it can be represented by the total space of a fiber bundle over S' with structure group 2/2. According to Stong and Winkelnkemper [I], such classes are characterized by the property that there is a representative which admits a locally free action by the product group S' x S'. According to Iberkleid [l], such classes are also characterized by the property that they can be represented by at least one smooth closed manifold X whose tangent bundle s ( X ) splits into a Whitney sum of real line bundles. This implies that every smooth closed manifold is cobordant to a smooth closed manifold X such that the Whitney sum r ( X ) @ c 1 splits into a Whitney sum ofreal line bundles; furthermore, according to Stong [ S ] , if 2 2 2k c n, then every smooth closed n-dimensional manifold is cobordant to a smooth closed manifold X such that t ( X ) = 5 @ q for a 2k-plane bundle 5 and an (n - 2k)-plane bundle q. Finally, according to R. L. W. Brown [ 13, every even degree cobordism class in the kernel of !It 52/2 can be represented by a fibration over the 2sphere S2. 9.26 Remark: One can obtain results similar to those of the preceding remark by imposing restrictions on other Stiefel-Whitney numbers than the one { w " ( r ( X ) )p, x ) which defines the 2/2 Euler characteristic. For example,
29 8
VI. Unoriented Manifolds
the result of R. L. W. Brown [l] continues as follows: If X is a smooth closed manifold of odd dimension n, then X is cobordant to a fibration over S2 ,whenever (w2(z(X))u W , - ~ ( T ( X ) ) = , ~0. ~) According to Milnor [17] the unoriented cobordism class of a given manifold X contains a complex manifold if and only if all Stiefel-Whitney numbers constructed from at least one odd-dimensional Stiefel-Whitney class w,(t(X)) vanish; this happens if and only if the unoriented cobordism class of X also contains the square Y x Y of a smooth closed manifold Y. Finally, Yoshida [3] shows that if X is a smooth closed n-dimensional manifold such that all Stiefel-Whitney numbers other than those constructed from wl(z(X)), . . . , W , _ ~ ( Z ( Xvanish, )) for some k 5 6,then X is cobordant to a smooth closed manifold with at least k linearly independent vector fields. (The special case k = 1 is an instant corollary of the result of Conner and Floyd [l] described in Remark 9.25.)
9.27 Remark: In some cases the vanishing of certain Stiefel-Whitney numbers of a smooth closed manifold X implies that X is cobordant to a manifold for which related Stiefel-Whitney classes vanish. Here are three such results. (i) Recall from Remark V.7.28 that a (smooth closed) spin manifold X is characterized by the conditions wl(7(X))= 0 and w2(z(X))= 0.Suppose that one knows only that X is a smooth closed manifold such that every Stiefel-Whitney number containing one of the factors wl(z(X))or w2(t(X)) vanishes. Then, according to Anderson, Brown, and Peterson [l, 21, X is cobordant to a spin manifold. (ii) Milnor [17], already cited in Remark 9.26, has the following further consequence. If X is a smooth closed manifold such that every StiefelWhitney number containing any odd factor wZp+,(z(X)) vanishes, then X is cobordant to a smooth closed manifold X' such that w2,,+,(5(X')) = 0 for every p 2 0. (iii) Let X be a smooth closed n-dimensional manifold, let i,, . . . , is satisfy 2i, 2 n + 1, . . . , 2i, 2 n 1, and suppose that every Stiefel-Whitney number containing one of the factors wi,(z(X)),. . . , wi,(t(X))vanishes. Then X is cobordant to a smooth closed manifold X' such that wi,(z(X')) = 0, . . . , wi,(t(X')) = 0.This result was established under the more stringent conditions 2i, > n 1,. . . ,2i, > n 1 by Reed [l], and the same version is in Wall [3, p. 171; the present version is due to Papastavridis [l, 31.
+
+
+
9.28 Exercise: Show for any n > 0 that the complex projective space CP" has the same Stiefel-Whitney numbers as the square RP" x RP" of the real projective space RP", hence that CP" is cobordant to RP" x RP". Given this
9. Remarks and Exercises
299
result, construct a specific cobordism from CP" to RP" x RP". (The second part of this problem is nontrivial; it is solved in Stong [3].) 9.29 Exercise: According to Corollary 5.6, for each n > 0 the real projective space RP2"-' is the boundary of a smooth compact 2n-dimensional manifold find such manifolds X 2 " . (See Husemoller [l, pp. 262-2631 for example.) 9.30 Remark: The definition of surgical equivalence and the proof of Theorem 5.17, that surgical equivalence is cobordism, appeared independently in Milnor [9] and in Wallace [l-41. Wallace uses the terminology "spherical modification" in place of "surgery." 9.31 Remark: The existence of the admissible Morse functions required to complete the proof ofTheorem 5.17 is easily established in Milnor [13, 181 and in M. W. Hirsch [4], as already indicated in Lemma 5.16. Other general accounts of Morse theory are given in Pitcher [11 and Morse and Cairns [I], for example. Admissible Morse functions are also easily constructed for any smooth manifold X by means of the following technique, in Morse [11. Given any embedding X + R", there is a dense set of linear functionals R"' + R whose restrictions to X are smooth functions all of whose critical points are nondegenerate. It remains only to separate critical values, as in Smale [l, 21. 9.32 Remark: Given a smooth manifold X , it is of interest to construct an admissible Morse function X 3 R with as few critical points as possible. According to M. W. Hirsch [2], if X is not closed, then there is a nonconstant smooth function X R with no critical points whatsoever; hence the question centers on closed manifolds. According to Reeb [I] and Milnor [I], any smooth closed n-dimensional manifold X with only two critical points is homeomorphic (but not necessarily diffeomorphic) to the sphere s". A related set of conditions, involving an entire family of Morse functions, is used in Rayner [l] to characterize those manifolds X which are djfeomorphic to S". One easily constructs an admissible Morse function RP2 -5 R with only three critical points. Eells and Kuiper [l, 21 and Banchoff and Takens [11 study other manifolds with the same property. Specific admissible Morse functions are constructed for Grassmann manifolds in Hangdn [I] and in Alexander [I], for lens spaces in Vrhceanu [11, and for other special cases in Vrgnceanu [2] and Masuda [11, for example.
300
VI. Unoriented Manifolds
Aside from the intrinsic interest of constructing admissible Morse functions with a minimal number of critical points on a given smooth manifold X , one is interested in relating the latter number to the Ljusternik-Schnirelmann category of X (Remark 1.10.2),a relation which is studied in Threlfall [ 11 and Takens [ 11, for example. A similar question for cobordism classes of smooth closed manifolds is studied in Mielke [l-41. 9.33 Remark: Since any Z/Zalgebra homomorphism '31 % 2/2 whatsoever can only assume one of the values 0 E 2/2 or 1 E 2/2 on each of the generators x2 E '31 and x4 E '31, for example, it follows that the element (x2)3x4+ xZ(x4)' E '31 of degree 10 lies in the kernel of every such cp. Consequently any manifold X E 02 representing the element (X$X4 + x2(x4)2E % lies in the kernel of every Stiefel-Whitney genus 42 3 2/2. Thus one cannot compute all Stiefel-Whitney numbers in terms of Stiefel-Whitney genera: there are smooth closed manifolds that are not boundaries all of whose Stiefel-Whitney genera vanish. It is probably not difficult to compute the ideal n,kerG(f) c of those unoriented cobordism classes lying in the kernel of every StiefelWhitney genus '31 --% 2/2, or to compute the quotient 91/(), ker C ( f ) . However, the geometric significance of such computations is not clear to the author.
9.34 Exercise: Observe that the structure theorem '31 = Z/2[x2, x4, x5,x 6 , x8,.. .] of Remark 9.23 implies that every smooth closed 3-dimensional manifold is the boundary of a smooth compact 4-dimensional manifold, a result first announced in Rohlin [l]. It follows from Proposition 5.2 that all the Stiefel-Whitney numbers of any smooth closed 3-dimensional manifold vanish. Prove the latter statement directly, without using Remark 9.23. 9.35 Exercise: Show that the 2/2 Thom formj*T,,,, E H"(RP" x RP") of any real projective space RP" is the sum x p + q = n e ( y ! )xP e(~!)~,for the generator e(y,') E H'(RP"). 9.36 Exercise: Show that the 272 Thom form j*T,, E H"(S" x S")of any sphere s" is the sum 1 x w(Sn) w(S") x 1, for the generator w(Sn)E H"(S").
+
9.37 Remark: The conclusion of Theorem 8.4 can be reformulated as follows: For any smooth closed n-dimensional manijold X the Stiejhl- Whitney classes w ~ ( T ( X ).) ., . , w , ( T ( X ) ) ure independent uj' the smooth structure assigned to X . This result was also formulated in Remark V.7.10, with indications of the alternative proof given in Thom [4].
30 1
9. Remarks and Exercises
9.38 Remark: According to Remark V.7.27 there are no “universal” polynomial relations P ( w l ( t ( X ) )., . . , w , ( z ( X ) ) )= 0 E H P ( X )of weighted degree p which are valid for every smooth closed manifold X of dimension n 2 2p; this was proved independently in E. H. Brown [2] and in Stong [l]. However, the total W u class W u ( z ( X ) )= 1 + W u , ( . c ( X ) )+ . . . + W u , ( z ( X ) ) of the tangent bundle T ( X ) of an n-dimensional manifold X is of the form P p ( w l ( z ( X ) ). ,. . , w,,(T(X)), Pp(ul,. . . , up) being the multiplicative sequence associated to the “ W u series” of Exercises V.7.21 and V.7.22, where each P p ( u l ,. . . , up) is a nontrivial polynomial of weighted degree p . Since W U , ( T ( X ) )= 0 whenever 2p > n, by Corollary 8.6, the Wu polynomials P p ( u l , .. . , up) do provide nontrivial relations P p ( w l ( z ( X ) )., . . , w , ( z ( X ) ) ) = 0 E H P ( X ) which are valid for every smooth closed manifold X of given dimension n < 2p. It follows that the ideal generated by the Wu polynomials P p ( u l ,. . . , up)such that 2p > n is contained in the ideal of all polynomials P such that P ( w , ( T ( X ) ) ., . . , w , ( z ( X ) ) )= 0 for every smooth closed manifold X of dimension n. The latter ideal was completely determined in Brown and Peterson [ l , 21, as indicated in Remark V.7.27.
cp20
9.39 Remark: For any given n > 0 let P ( u , , . . . , u,) be a polynomial over 2/2 of weighted degree n such that P ( w l ( z ( X ) )., . . , w,(.c(X))) = 0 E H ” ( X ) for every smooth closed manifold X of dimension n. According to Dold [2], in this special case the polynomial P(ul, . . . , u,) then lies in the ideal generated by the W u polynomials P,(u,, . . . ,up)for 2p > n. This result does not generalize to polynomials of weighted degree less than n. 9.40 Remark: Let X Y be any map of smooth closed manifolds X , Y of dimensions m and n, respectively, and let H J X ) H,( Y) be the induced 2/2 homology homomorphism. Since X and Y are closed, one also has Poincare duality isomorphisms H * ( X ) -%H J X ) and H * ( Y ) % H,( Y), as in Corollary 2.4 and Remark 9.3, with specific inverses H J X ) 3 H * ( X ) and H,( Y ) 3 H*( Y ), as in Corollary 7.9. The composition H*(X)
aH * ( X )
H*(Y)
*
DT=
Dp”
H*(Y)
is the Gysin homomorphism H * ( X ) H*( Y). Since fH proceeds in the reverse direction ( = Umkehr) of the usual cohomology homomorphism H*( Y ) I*,H * ( X ) , the Gysin homomorphism is an example of an Umkehr homomorphism; more general Umkehr homomorphisms will be constructed in the third volume of this work, along with an explanation of the notation f H and a proof of a general result which includes the following special case.
302
VI. Unoriented Manifolds
Recall from Remark V.7.14 that the Steenrod square Sq is natural in the sense that H*(Y) I
I
*
* H*(Y) I
I
J.
J.
H*(X)
f*
sq H * ( X )
s
commutes for any map X Y , and that Sq is multiplicative in the sense that it satisfies the Cartan formula Sq(a u p) = Sq a u Sq p. It is reasonable to ask whether Sq is in some sense natural with respect to the Gysin homomorphism H * ( X ) & H*( Y ) induced by any map X Y . The answer is affirmative only if one introduces a correction consisting of cup products by the Wu classes Wu(z(X))E H * ( X ) and Wu(z(Y ) )E H*( Y ) of the tangent bundles z(X) and z( Y ) . Specifically, the 2 / 2 Riemann-Roch theorem asserts that
s
H*(X)
j*l
H*(Y)
*
b
H*(X)-
uWu(r(XII
ss H * ( Y ) - H * ( Y )
uWu(r(Y1)
H*(X)
I*
commutes. This result underscores the importance of Wu classes. The preceding 2/2 Riemann-Roch theorem is a slightly specialized version of the main result of Atiyah and Hirzebruch [2], later generalized by Spanier [2]. However, it is also a special case of a much broader generalized Riemann-Roch theorem, which will be proved in Volume 3. In the most useful special case the Steenrod square H*(X; Z/2) 5 H*(X; 2/2) will be replaced by the Chern character K @ ( X ) H*(X; Q) and the correction factor Wu(z(X)) E H * ( X ; 2/2) will be replaced by the Todd class td(r(X))E H * ( X ; Q), along with corresponding changes for Y .
a
9.41 Exercise: Let X be a smooth compact n-dimensional manifold with boundary X and 2/2 fundamental class px E H,(X, X ) , Use couariant MayerVietoris functors on L?(@to show for each q E 2 that the cap product H"-4i(X,X)-HI(X) is a 2/2-module isomorphism. (This is also a 2/2 Poincare-Lefschetz duality theorem.)
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This Page Intentionally Left Blank
Glossary of Notation
L
C" C P" CP'
C ( V )c R2"-p CW complex CW space C W structure CW structure of projective spaces @'
C"(X)-modules .F C"(X)-modules .F*
any element of the complex general linear group the adjoint of A E GL(n,C) the modulus of A E GL(n,C) the category of 72-graded abelian groups classifying space for a topological group G barycenter of a simplex I category of base spaces complex field, in its usual topology standard n-dimensional complex vector space, in its usual topology complex projective space ( = Gl(C"+1 ) ) complex projective space (=Gl(C")) cocoon of V c X
87 87 89
45 101 9
2, Sff, 7 34 23, 34 23.34
262 20 19ff, 20 20 22.23 the complex vector space limn@" 34 in the weak topology the ring of continuous functions 116 X+R 117, 118, 119 1I9
35 1
352
Glossary of Notation the ring of X+Iw
C'(X)-module d ( X ) of smooth differentials on X C'(X)-module 6 * ( X ) of smooth vector fields on X C"(X)-module 9 of smooth sections X + E C'(X)-module 9* C"(X)-module .F** M(G, F )
EG
9
.F
Y* Y** f-genus
f!5 G G x F+F
I I9 I 34, I 35. in3 127, 129, 130, 131 I22
category of families of fibers, with respect to a group action G x F+F Poincare duality map n p x inverse Poincart duality map D; ' the closed unit n-disk the differential of g E C r ( X ) the total space of a family E X of fibers (especially a coordinate bundle E : x) the total space of a universal Gbundle ya the fiber n - ' ( ( x } ) over X E X of a fiber bundle represented by E:X the exponential of A E End C" the C'(X)-module of smooth differentials on X the C" (X)-module of smooth vector fields o n X the 2/2 Euler class of a real m-plane bundle over X E S? the fiber of any family of fibers (especially a coordinate bundle), also applied to fibre bundles (and especially to fiber bundles) a C"(X)-module of (continuous) sections X + E a C"(X)-module ofsmooth sections X-+E the first conjugate of 9 the second conjugate of 9 a homomorphism 4 %2/2 or 'JL G U I 7 / 2 assigned to a forma1 power series f(t) E 2 / 2 [ [ t ] ] pullback of 5 along X X ' structure group action of a transformation group G (structure group) on a fiber F
r
F
smooth functions
-
I23 I33 61
250, 258,216 217,283 19 134, 183, 184 57, 61, 61
I02 67
87 134, 135, 183 127, 129, 130. 131 196 3, 51, 60
117, 118, 119
122 118, 119. 123 I33 214tT, 275,216
3, 63 3, 58, 60
2, 60,65
Glossary of Notation
G-related isomorphisms G-related set of homomorphisms G(j) GIK G U n , C) G U m , R) G L + ( m R) , G"(C" ") +
G"(Q3") G*( R" '" ) G"(R') G"(R") GI(@"+I )
GI(@') G1(R"' ')
G'(k-8")
s'5 H*(Gm(R"'" ); 6/21 H*(G"(R"); Zj2) H*(RP"; Z/2) H*(RP', 212) I I , , . . . , I , and I
i(m,n ) J j* T ,
K K'
IKl*
353
Stiefel- Whitney j-genus homogeneous space complex general linear group real general linear group the component of the identity in G L h k-8) complex Grassmann manifold complex Grassmann manifold real Grassmann manifold real Grassmann manifold the total space O(ym)of o(y") complex projective space CP" complex projective space CP" real projective space RP" real projective space RP" pullback of ( along X 5 X'
a two-sided ideal in @ V with @ V / I = AV simplexes in a simplicial complex K . which also constitute the vertices of the barycentric subdivision K' Atiyah-Todd number a complex structure in a real vector bundle a Thom form ( , )T an abstract simplicial complex the first barycentric subdivision of K the vertex set of K a family of functions K -+ [O, I ] a family of functions KO+ [O, I] the simplicial space of K the simplicial space of K' metric simplicial space of K weak simplicial space of K the metric telescope of IKI
K-theory
x lim,, X , , (in the weak topology) In A E GL(n,C)
Steenrod's convenient category of compactly generated spaces the logarithm of a positive element A
E
60 60 2 1 4 6 275.276 83 59.87 I, 5n 59.90 34 34 34 34 I66 34 34 34 34 63 227tT, 230 221fT, 229 199ff. 201 199fr, 200 35 9
I 89,296 I13
n 9
n n 9 17, in 17, i n 12 170 48 19.20 n7
GL(n,C)
left inclusion
278.281
Glossary of Notation any real or complex vector bundle of rank m the category of smooth manifolds and smooth maps a category of modules over a fixed commutative ring the category of Z-graded modules over the commutative ring R the natural numbers (0. I. 2,. . .;
m-plane bundle
w1: N n-cell (for n = 0. I, 2, . . . ) n-disk D" (for n = I , 2, 3 , . . . ) n-plane bundle
2, 23ff, 25 41 45
19 19
any real or complex vector bundle of rank n
( n - 1)-sphere s"-' (for ii = I , 2, 3 , . . .) nth type of topological space (for n = 0, I , 2,. . . ) n-skeleton (for n = 0, 1, 2 , . . . ) n-sphere S" (for 11 = 0, 1,2, . . .) 91
a'
O(1) c GL(1,R) O(0
p-simplex (for p = 0, 1,2,. . . ) q-dimensional abstract simplicia1 complex K (for q = 0, I , 2,. . . ) y-dimensional simplicia1 space K (for y = 0, 1. 2,. . .)
106. 110 19
6 20 19 250. 268.213 268, 273
the unoriented cobordism ring the unoriented surgical equivalence ring the orthogonal group, usually act- 3, 59, 81. 86,90 ing on [w" or on S"-I the rotation group, usually acting 59, 81, 86. 90 on [w" or on S"- I the orthogonal group (acting on S o ) 160 the total space of the unique coordi- 162, 163 nate bundle representing an orientation bundle o ( 5 ) [i? particular, O ( y r ) is denoted Gm([wm+") and O(y") is denoted (7'""R")] the orientation bundle of a real 160, 161, 162, 163 vector bundle 5 over X E &3 the category of open sets in V c X 43 for X E the category of open sets in X E D 41 Poincare form 276, 217,283,284 the projective bundle of a real vec- 3,202 tor bundle 5. or the total space of the projective bundle 8 9
1 I
v
2JV) t e(v)
a specific family of open sets in
2(k,
a category of open sets in the interior X of a manifold X real field, in its usual topology
R
1, 106, 170
43
vcx 46, 252tf. 256f
355
Glossary of Notation R"
R P" RP" Iw"
T x E H"(X x X , X x X \ A ( X ) ) ( 9 )T
u1
,(O
standard n-dimensional real vector space, in its usual topology real projective space ( =GI(&!"' I ) ) real projective space ( =G1(R')) the real vector space limnR" in the weak topology right inclusion the Steenrod square the(n- I)-sphere(forn= 1,2,3,. . .) the n-sphere (for n = 0, I , 2, . . . ) the 0-sphere (consisting of two points) the diagonal Thom class of X E .A Thom form j*Tx the unitary group(usually acting on @" or on S 2 " - ' ) L/2 Thom class of a real n-plane bundle 5 multiplicative Z/2 class of a real vector bundle 5 with respect to a formal power series f ( t ) E 212 "[I1 dual of a 2/2 multiplicative class
34 22. 23. 34 22, 23, 34 22 280,281 243R, 285ff 19
19 19, 162, 167
281tl' 216K 211,283, 284 59, 81, 86, 89 3. 191, 192K 194ff, 21 1 2, 3, 215tl', 216
231,238
U,(O
Y Y
V
the semi-ring of smooth closed unoriented manifolds the category of spaces homotopy equivalent to spaces in Steenrods category x' any of the real or complex vector spaces R" ",R" , C" ",@ (used to generate the tensor algebra @V and the exterior algebra AV = @V/I 2/2 Thom class of a real n-plane bundle 5 2/2 Thom class of a real n-plane bundle 5 the qth Stiefel-Whitney class of a real vector bundle 5 the total Stiefel-Whitney class of a real vector bundle 5 the dual Stiefel-Whitney class of a real vector bundle 5 the category of spaces homotopy equivalent to metric simplicia1 spaces the category of spaces homotopy equivalent to countable metric simplicial spaces +
#a
214 49
34
+
20tl', 21 I 21 I 4,216 2, 4, 215, 216 2,231 2, 13, 21
14
356
Glossary of Notation
x
x Y
e
H 212
the boundary of a manifold X the interior of a manifold X the boundary of a manifold Y the interior of a manifold Y the ring of integers the field of integers modulo 2
24 25 24 25
number of 1's in the dyadic expansion of n real line bundle group over X E
2, 50, 52, 246, 261.292 206lT, 207,208, 213,214 70, 71, 72, 81, 109 110,111,115, 161, 162, 172, 173. 176. 177, 178 170 3, 149 164
2
Greek iilphabet
u(n)
morphisms groups
of
transformation
universal complex m-plane bundle universal real m-plane bundle universal oriented m-plane bundle
n(v")!r"
.. .. (and also 5, <', . )
canonical complex m-plane bundle canonical real m-plane bundle canonical oriented m-plane bundle n(Y:)'Y:: diagonal map X -,X x X image of a diagonal map A trivial real or complex vector bundles of ranks m, n, . . , complex vector bundles (usually) realification of a complex vector bundle i real vector bundles the linear map A" V + V induced by a linear functional 0 on
I70 144 166
real or complex line bundles splitting bundle (over P s ) of a real vector bundle 5 real or complex line bundles (such as A, I.', . . .) H/2 fundamental class of a compact manifold X with boundary k cap product by px normal bundle of an immersion
106,202,204, 206 3.202
256.258, 302 260
arbitrary fiber bundle real vector bundles principal G-bundle
3, 63, 64 I , 106, 107 99
I10 277 I12 170 172, 173 106, 107 35
A"-' v
x L, ~ 2 n - p
.)
24Y. 253
Glossary of Notation
A
a
r
357 restriction of 5 to a subspace U , given as the pullback i!( along the inclusion i of U the projection E + X of a family of fibers (especially the projection of a coordinate bundle) the universal oriented rn-plane bundle g"' a canonical oriented rn-plane bundle 3; section X + E of a coordinate bundle E : X image of the zero section X E the characteristic polynomial of a real vector bundle 5 zero-section X + P s , , for a real vector bundle 5 tangent bundle of RP" tangent bundle of any smooth manifold X E .X L / 2 Thom isomorphism of a real vector bundle 5 the 2 / 2 Euler characteristic & + 212 local triviahation El Ii, + Ii, x F transition function U ,n U, + G
63
1, 57. 61, 64, 67
164
I66 3. 82
3,211 220,222 210 I38 2, 4, 130 1Y9
276,297 51.61 58.67
This Page Intentionally Left Blank
Index
A Absolute Leray-Hirsch theorem, 94 Abstract simplicia1 complex, 8 Abstract star of a vertex, 10 Additive functors O(X)+!Ul, 41 Adem relations, 243, 245 Adjoint of a matrix, 87, 90 Adjunction space, 19 Admissible Morse function, 273, 299, 300 Admissible topology of a structure group, 60 Affine connection, 278 Algebraic coordinate bundles, 180, 186, 188 Algebraic vector bundles. 180, 186, 188 Arcwise connectedness, 252 Associated bundle, 83 Associated principal G-bundle, 100 Associative H-space, 102 Associative laws for vector bundles, I I I Atiyah-Todd number, 189, 296 Atlas, 23, 130 Attaching maps, 19 Axioms for characteristic classes Stiefel-Whitney classes, base spaces in 1, 233 Stiefel-Whitney classes. base spaces in 91, 235 H / 2 Euler classes, base spaces in 9, 205 Axioms for Steenrod squares, 243
B Barycenter E'. 9 Barycentric coordinate functions, 16 Barycentric coordinates, 8 Barycentric subdivisions, 9 Base space of a bundle, 1, 2, 5, 57, 60, 61, 64 the category 4, 2, 5-7, 13, 14, 19, 21, 29, 37, 38,47, 59, 64 of a family of fibers, 61 Basic Stiefel- Whitney numbers, 274, 296 Basis theorem of Schubert, for Gm(Cm+") and Gm(Rm+"),239, 240 Best-possible embedding conjecture, 50, 295, 297 Best-possible immersion theorem (Cohen), 52, 262, 292, 293, 297 Bohnenblust, H. F., 180 Boundary k of a compact manifold X,24 Bouquet of circles, 2 I3 Brown, E. H., 47, 104 Bundle associated bundles, 83 associated principal G-bundles, 100 over contractible spaces, 59, 73-75 coordinate bundles, 1, 66-68, 97 complex vector bundles, 169ff fiber bundles, 3, 57ff, 64
359
360
Index
fibre bundles, 63 homomorphisms, 108 induced bundles, 72 line bundles, 3, 106, 170 microbundles, I80 m-plane bundles, 1, 106, 170 numerable bundles, 2, 98, 102 n-plane bundles, 1, 106, 170 principal coordinate bundles, 100 principal C-bundles, 99, 100, 102 projective bundles, 3, 202 real vector bundles, I , 105ff, 106 smooth vector bundles, I19ff, 158ff sphere bundles, 162, 179 splitting bundles, 202 tangent bundles, 2, 4, 105, 126ff, 130, 134, 137, 146, 178, 180, 185,259 topological R" bundles, 181, 242, 289 trivial bundles, 57, 63, 73, 74, 75, 112 universal bundles, 3, 101, 105, 149, 153, 170 vector bundles, 1, l05ff, 178ff
C
Cairns- Whitehead triangulation theorem, 28, 52, 159 170 Canonical complex line bundle Canonical complex m-plane bundle ff, 170 Canonical involution of a double covering, 165 Canonical orientation of n(<)'t,163 Canonical oriented m-plane bundle 7;. 166 Canonical real line bundle y:, 144 Canonical real m-plane bundle ff, 140-144, 155-1 59 Canonical vector bundles, 140ff Cap products, 256, 257 Cartan formula, 243, 244, 302 Category, Ljusternik-Schirelmann category, 47,48, 104, 300 Steenrod's convenient category X , 48 Vogt's convenient categories, 48 Category dd' of Z-graded abelian groups, 45 Category B of base spaces, 2, 5-7, 13, 14, 19, 21, 29, 37,47, 59, 64 Category %'(G,F) of families of fibers, 61 Category X , Steenrod's convenient category of compactly generated hausdorff spaces, 48 Category 4 of smooth manifolds, 25, 29, 50, 235
?A,
Category W ofmodules over a fixed commutative ring, 41 Category W of Z-graded R-modules, 45 Category O ( T ) of all open sets in X E O ,41 Category 9 ( X ) of some open sets in k t X,46 Category 'y1 of homotopy types ofspaces in X , 49 Category W of spaces homotopy equivalent to simplicia1 spaces, 13, 21,48 Category W o of spaces homotopy equivalent to countable simplicial spaces, 14, 47 Cayley-Hamilton theorem, 220 Cell, n-cell, 0-cell, 19 Cell complex, 19 Characteristic classes, 2, 4, see also Dual Stiefel-Whitney classes; Multiplicative 2/2 classes; Stiefel-Whitney classes; Wu classes; H/2 Euler classes Characteristic polynomial of a real vector bundle, 220,222 Chern character, 302 Chevalley, C., 103, 180 Classification of fiber homotopy types, 103 Classification of manifolds, 52, 53 Classification theorems ersatz homotopy classification of complex bundles, 17 I of oriented bundles, 166 of real bundles, 155, 157-159 fiber homotopy types, 103 homotopy classification of complex bundles, 170, I7 I of oriented bundles, 164 of principal bundles, 101 of real bundles, 153, 155-157 Classifying extensions, 157, 171 Classifying maps. 3, 4, 156 for complex bundles, 170, 17I for oriented bundles, 164, 166 for principal bundles, 101, 102 for real bundles, 153, 156- 159 for the real bundle y1 + . . . + 7'. 204, 228, 231 Classifying spaces, 3 for complex bundles, 170, 17I for oriented bundles, 164 for principal bundles, 101, 102 for real bundles, 3, 153 Closed manifolds, 23,24,249ff, 250,254,258, 267,268, 273ff Closed star, 10 Closure finite cell complex, 19
Index
36 1
Coarse topology, 16. 17 Cobordism (unoriented), 50, 52, 267, 268 Cobordism ring 91. 250, 268,296,297 Cocoon C( V ) c R 2 " - p of V c X,262 Coefficient ring A, 59 Coefficient ring 2/2, 191, 192, 216, 251 Cohen immersion theorem, 52,292,293 Combinztorial construction of StiefelWhitney classes, 289, 290 Commutative laws for vector bundles, 1 1 1 Compact Lie groups, 3,8 1,86,91 Compact manifolds, 24,258, 302 Compact manifolds with boundary, 24 Compactly generated hausdorff spaces, 48 Comparison theorem, Mayer-Vietoris, 46,47 Complex abstract simplicial complex, 8 cell complex, 19 closure finite cell complex, 19 CW complex, 20 Complex conjugate bundle, 176ff Complex dimension ofa complex manifold, 29 Complex ersatz homotopy uniqueness theorem, 171 Complex line bundle, 170 canonical complex line bundle, 170 universal complex line bundle, I70 Complex manifold, 29 Complex n-plane bundle, 170 Complex projective space CP",23, 296 Complex projective space CP", 23 Complex structures J in real vector bundles, I73ff Complex vector bundles. 169ff canonical complex vector bundles, I70 universal complex vector bundles, 170 Complexifications of real vector bundles 169, 172, 176ff Conjugate complex vector bundles, 176 Conjugates 9*,P**, . . . of modules 9 , 1 1 8 , 119, 123, 133, 134 Connectedness arcwise connectedness, 252, 255 k-connectedness, 293 pathwise connectedness, 56, 208, 213 Connections in tangent bundles, 278 affiine connections, 278 Levi-Civita connections, 278 Connectivity in 9,56 Continuous functors, 182 Continuous sections, 82, 124 nowhere-vanishing continuous sections, 126
rc
r,
Continuous vector fields, I37 Contractibility of EG, 102 Contractible spaces, 6, 59 bundles over contractible spaces, 59.73-75 Convenient category X of Steenrod, 48 Convenient categories of Vogt, 48 Convex set, 10 Coordinates barycentric coordinates, 8 Plucker coordinates, 36 projective coordinates. 24. 37 Coordinate bundles, 1.66-68,97,100,121,159 algebraic coordinate bundles, 180, 186, 188 holomorphic coordinate bundles, I86 principal coordinate bundles, 100 smooth coordinate bundles, 121, 158-159, 172, 186 Coordinate covering of a manifold, 23 Coordinate neighborhood, 23 Countable simplicia1 spaces, 14 Covering, double covering associated to a real vector bundle, 162 Covering homotopy property, 98 Covering homotopy theorem, 98,99 Coverings by open sets, 6, 23, 25, 30, 39, 40 Critical point of a map X + W. 269 nondegenerate, 271 Critical value, 271 Cross product and cup product, 244 Cross product relation between homology and cohomology, 275 Cup product and cross product, 244 CW complexes, 20 CW spaces, 19-20,49,50 CW structures, 20
D Decompositions polar decompositions, 86ff polar decompositions of GL(n, C), GL(m, R), and G L + ( m ,W), 89,90 Decomposition theorem of Iwasawa and Mal'cev, 81.86, 103, 104 Diagonal map X A X x X, 110,277 Diagonal 2/2 Thom class T, E H"(X x X,X x WA(X)), 28 1 Diffeomorphisms, 24, 110 Differential, modules of differentials, 134, 135, 183, 184
362
Index
Differential forms, 134, 135, 183, 184, 269 Differential of a real-valued smooth function, 269 Differential topology, I , 105 Dimension of a CW complex, 20 of a CW space, 20 geometricdimension of avector bundle, 189 of a manifold, 23, 25 of a simplex, I 1 of a simplicial complex, 9 of a simplicial space, 9 Dimension axiom for Stiefel-Whitney classes, 233 for Steenrod squares, 243 Direct sum, 8, 41 Disk, 19 Distributive law for vector bundles, 1 I I Dold-Lashof construction, 102 Double covering associated to a real vector bundle, 162, 167 Double coverings, the canonical involution, I65 Dowker homotopy equivalence, 18,53,54,158 Dual classes, 237ff dual Stiefel-Whitney classes, 2, 237 of RP",258,260 Dual .F*of a locally free module 9,118, I 19. 123, 133, 134 Duality isomorphism over 2/2 (Poincare and Poincare-Lefschetz duality), 249, 250, 256-258,302 Duality theorem of Whitney, 241, 290 Dyadic expansions, 2, 50, 52, 246, 261, 292
E Easy Whitney embedding theorem, 50 Easy Whitney immersion theorem, 51, 291 Effective action of a transformation group, 60, 100 Ehresmann-Feldbau fibrations, 97, 100 Em bedding of complex projective spaces, 296 conjecture for smooth manifolds, 50, 295, 297 of real projective spaces, 295, 296 of smooth manifolds, 2,27,28, 50.51.260262, 264,265,294-291
.
Embedding theorems easy Whitney embedding theorem, 50 embeddings in R", 27 embeddings of separable metric spaces, 51 embeddings of simplicial spaces, 5 I embeddings of smooth compact manifolds, 28 embeddings of smooth manifolds into R", 27 hard Whitney embedding theorems, 28,50, 260, 294 isotopies of embeddings, 5 I , 29 1 strongest known general embedding theorems, 294ff Enumeration of immersions and embeddings, 296 Enumerative calculus of Schubert, 239, 240 Ersatz homotopy uniqueness theorem complex version, 171 real version, 147, 157, 179, 236 Euclidean norm, 19, 124 Euler characteristic x 2 ( X )E 2/2, 276, 297 Euler classes 2 / 2 Euler classes, 3,4, 191ff, 196-198,205ff alternative construction of e(r(X)) E H'"(X; 2/2), 284 axioms for 2/2 Euler classes. 205 naturality of 2/2 Euler classes, 197, 205 normalization of 2/2 Euler classes, 20 I , 205 of products of real line bundles, 206-208, 213, 214 the relation e(5) = w m ( < ) ~ H m (2/2), X ; 4, 224 Whitney product formula for 2 / 2 Euler classes, 197, 205 Exotic free involutions of manifolds, 54, 55 Exotic spheres, 55 Exponential map of an affine connection, 278 Exponential of a matrix, 87, 90 Extensions, finite classifying extensions, 157, 171 Exterior algebra, 35 Exterior power Apt of a vector bundle 5 , 161 Exterior products, 35, 55
F Families of fibers, 60,6 I , 62 $genus G O , 275
363
Index Fiber, I , 3, 57, 67 over a point, 67 Fiber bundles, 3, 57& 6Off. 63, 64, 97 Fiber homotopy equivalence, 103, 186, 187 stable fiber homotopy equivalence (J-equivalence), 189 Fiber homotopy type, 103, 186, 187, 289 Fiber spaces, 97ff Fibrations, 97ff Ehresmann-Feldbau fibrations, 97, 100 Hurewicz fibrations, 99 Serre fibrations, 99 Fibre bundles, 60ff, 63. 64, 68, 97, 98 Fine topology, 19 Finite classifying extensions, 157, 171 Finite classifying maps, 155- 159, 171, 172, 230,233 Finite homotopy classification theorems, 155159, 171, 172, 231,233 Finite simplicial complex, 9 Finite simplicial space, 9 Finite type (spaces of finite type), 6, 10, 13.47, 59 Finite-dimensional CW complex, 20 Finite-dimensional CW space, 20 Finite-dimensional simplicial complex, 9 Finite-dimensional simplicial space, 9 Finitely presented groups, 52 First barycentric subdivision, 9 Five-lemma, 43 Formal groups, 297 Formula of Hirsch, 241 Free involutions of spheres, 54, 55 Fundamental homology classes p x E Hn(X,?; 2/2) and p x E H , ( X ; 2/2), 249, 251 -254, 275 pij E H , ( o . U ; 212). 254
General linear group GL(n,C), 59,8 1,85,86ff, 89, 91, 93, 170 General linear subgroup GL+(m,R) c GL(m,R), 59,81,85,86ff, 90,91,93, 168 Geometric dimension of a vector bundle, 189, 292 Geometric simplex, 8 Gram-Schmidt process, 115, 183 Grassmann, H., 55, 56 Grassmann manifolds G"( V ) , Gm(R'"+"), Gm(Roo),G'"(C"+"), G"(Cm), 3, 34-38, 55, 56 Grassmann manifolds and canonical vector bundles, 140ff as objects in 1, 37, 38 as objects in A, 36 and universal vector bundles, 3, 147ff Groups compact Lie groups, 3, 81,86,91 general linear groups GL(m,R), I , 58, 59, 81, 85, 86ff, 90, 91, 93, 105, 106 general linear groups GL(n,C), 59, 81, 85, 86ff, 8 9 , 9 l , 93, 170 general linear subgroups G L + ( m , R ) c C L ( m , R ) ,59,81,85,86ff,90,91,93,168 Lie groups, 3, 81, 85, 86, 91, 120, 184, 187 linear Lie groups, 59, 81, 85, 86, 91 nonlinear Lie groups, 91 orthogonal groups O ( m ) c GL(m,R), 3,59, 81, 86ff, 90-92 rotation groups O+(m) c GL+(m), 59, 81, 86ff. 90-92, 168 structure groups, 3, 58, 60, 61, 85, 91, 97 transformation groups (used as structure groups), 60, 70-72 unitary groups U(n) c GL(n,@),59,81,86ff, 89, 91, 92 Gysin homomorphism, 301 Gysin map, 199 Gysin sequence, 199-201, 212,230
G
G-bundle principal, 99- 102 universal, 101, 102 G-related isomorphism, 60 G-related set of homeomorphisms, 60 Genera G o . 274-275,300 Generalized Riemann-Roch theorem, 302 General linear group GL(m,R), I , 58, 59, 81, 85ff, 90. 91, 93, 105, 106
H Haefliger-Hirsch theorem, 293 Hard Whitney embedding theorem, 28, 50, 260, 294 Hard Whitney immersion theorem, 29, 51,52, 260 Hausdorff spaces, 6, 7 paracompact hausdorff spaces, 25
3 64
Index
Hermitian inner product, 87 Hermitian metric, 170, 183 Hirsch's formula, 241 Holomorphic coordinate bundle, 186 Homogeneous space, 83 Homomorphisms of vector bundles, 108 Homotopic maps, 5, 6 pullbacks along homotopic maps, 75-79 Homotopy, relative, 31 Homotopy classification theorems. 147ff complex bundles, 170, 17I fiber homotopy types, 103 oriented bundles, 164, I66 principal G-bundles, 101, 102 real bundles, 153, 155-157 Homotopy equivalence, 6 fiber homotopy equivalence, 103, 186. 187 J-equivalence (stable fiber homotopy equivalence), 189, 241, 242, 289 proper homotopy equivalence, 54 stable fiber homotopy equivalence (J-equivalence), 189, 241, 242, 289 weak homotopy equivalence, 21,49 Homotopy groups, 293 Homotopy lifting property, 98, 99 Homotopy type, 6 fiber homotopy type, 103, 186, 187,289 stable fiber homotopy type, 189, 241, 242, 289 Homotopy uniqueness theorem (homotopy classification theorem), ersatz versions, 147, 157, 171, 179, 236 H-spaces, 102, 187 Hurewicz fibrations, 99 Hurewicz map, 213
Immersions of smooth manifolds, 27, 29, 5 1, 52,26Off, 291, 292. 293, 297 Induced bundles, 72 Inner proffucts, hermitian, 87 Interior X of a compact manifold X,25 Intersection theory, 240 Inverse bundle (stable inverse of a vector bundle), 189 Involutions canonical involution of a double covering, 165 smooth free involutions of manifolds, 54,55 Irreducible vector bundles, 188 Isomorphism problem for finitely presented groups, 52 Isomorphisms in a category V ( G , F ) of families of fibers, 62,63 of coordinate bundles, 68 G-related, 60 of smooth coordinate bundles, 121 Isotopy of simplicia1 embeddings, 5 1 of smooth embeddings, 291 Iwasawa, K., 103, 104 Iwasawa-Mal'cev decomposition theorem, 81,86, 103, 104
J
J-equivalence (stable fiber homotopy equivalence), 189, 241, 242, 289 Jacobian matrices, 2, I30ff
K I Immersion conjecture (Cohen immersion theorem), 52, 292, 293 Immersion criterion of Hirsch, 291 Immersion theorem of Cohen, 52, 292,293 Immersion theorems of Whitney easy, 5 1, 29 1 hard, 29, 51, 52,260 Immersions in R", 27 Immersions of complex projective spaces, 296 Immersions of real projective spaces, 185,261, 295, 296
Kahler derivations, 180 k-connected manifolds. 293
L Left inclusions I,, 278, 281 Lens spaces, 55, 185, 296 Leray-Hirsch theorem, 3,41,93ff, 97,104,194 absolute case, 93ff. 104 relative cases, 97, 194 Lev-Civita connection, 278
365
Index Lie groups, 3, 81, 85, 86, 91, 120, 184, I87 compact, 3, 81, 86. 91 linear, 59, 81, 85, 86, 91 nonlinear, 91 Lifting property (homotopy lifting property), 98.99 Line bundle group r ( X ) over X E 9,206-208, 213, 214 Line bundles (real and complex), 3, 106, 170 canonical, line bundles y:, 144, 170 products of, 206-208,213, 214 universal line bundles y ' , 149, 155, 170 Linear Lie groups, 59, 81, 85, 86, 91 Linear reduction theorem, 91, 116, 178, 179, 182 Ljusternik-Schnirelmann category, 47, 48, 104, 300 Local trivializations, 67, 121 Locally trivial projections, 57, 99 Locally finite open coverings, 16 Locally finite simplicial spaces, 53 Locally free Co(X)-modules, I 18, 119, I79 Locally free C"(X)-modules, 122, 123, 129, 134 Logarithm of a positive matrix, 87, 90 Long line, 180
M Mal'cev, A., 103, 104 Manifolds, 23ff category .X of smooth manifolds, 25 classification problem for manifolds, 52, 53 closed smooth manifolds, 23,24,249ff, 254, 258,267, 268, 273R complex manifolds, 29 Grassman manifolds, 3, 34ff, 36, 37, 38, 147. 153, 154, 157, 164, 166, 170, 171 nonorientable manifolds, 51, 160, 169, 259, 295 open manifolds, 25, 294 orientable and oriented manifolds, 50, 52, 259, 287, 288, 294 smooth manifolds, 23ff, 25 smooth unoriented manifolds, 249ff, 267, 268, 273ff topological manifolds, 29, 53, 54, 180, 242, 288
unoriented smooth manifolds, 249& 267, 268, 273ff triangulable manifolds, 54 Map, 5 proper, 27,54, 262 smooth, 24, 25, 33 Matrix jacobian, 2, 130ff orthogonal, 90 rotation, 90 self-adjoint, 87, 90 unitary, 89 Maximal compact subgroup of a Lie group, 3, 81, 86, 91 Mayer-Vietoris cohomology sequence, 46,56 Mayer-Vietoris comparison theorem, 46,253, 258 Mayer-Vietoris functor, 42, 95 natural transformations of, 42-46, 96, 193, 252 on O(<), 42,44,45,95,96, 193 on 2 ( X ) , 46,47,252,254,256-258 Mayer-Vietoris sequences, 42, 56 Mayer-Vietoris technique, 5, 41R, 44-47, 96, 253, 258 Metric, 6 metric of type @,q). 186 metric simplicial space, 8-19, 21, 49 riemannian metric, 112-1 14, 116, 182, 183 smooth riemannian metric, 123 Metric topology, 6,9, 15-19 metric topology of a simplicial space, 8ff Metrizable spaces, 6 m-fold exterior product, 35, 55 Microbundles, 180 Milgram, R. J., 243 Milgram-Steenrod construction, 102 Miyazaki's theorem, 50 Module of differentials, 134, 135, 183, 184 Module, locally free, 118, 119, 179 of smooth sections, 122, 123, 129, 134 Module of sections of a vector bundle, 1 18,122 module of smooth sections of a smooth vector bundle, 122 Module of vector fields, 126ff, 129ff Modulus IA ( E GL(n,C) of A E GL(n,C), 89 Mobius band, 58, 59, 160, 169 Morphism (I-,@) of transformation groups, 70-72, 81, 109, 110, 111, 161, 162, 172, 173, 176, 177
366
Index
Morphism of families of fibers, 60,61 Morse functions, 273, 299 Morse theory, 269 Multiplicative refinements, 48 Multiplicative sequence of A f ) E 2/2 [ [ f ] ] . 4, 225, 242, 243 Multiplicative Z/2 classes naturality, 218, 219 of RP,258-260 U , ( ~ ) E H * * ( X2/2), ; 216ff, 219, 225 Whitney product formula, 219-222, 226
N Natural orientation of certain real vector bundles realification CR of a complex bundle [, 175 the other orientation of C R , 175, 176 the sum @ t, for a real bundle <,168, 176 realification Tn of a complex conjugate bundle 177 Naturality of characteristic classes multiplicative 2/2 classes u,(t), 218, 219 Stiefel-Whitney classes, 218, 234, 235 2/2 Euler classes, 197, 205 Naturality of Steenrod squares, 243, 302 Natural transformations of Mayer-Vietoris functors, 42-46, 96, 193, 252 n-cell, 19 ndimensional CW complex, 20 ndimensional CW space, 20 ndimensional manifold, 25 n-dimensional simplex, 8 n-dimensional simplicial complex, 9 ndimensional simplicial space, 9 ndisk, 19 Neighborhoods (coordinate neighborhoods),
c,
L5
n-fold reduced suspension of a pointed space, 246 (n-I)-sphere, 19 Nondegenerate critical points, 271 Nonembeddings, 260ff, 264,265,294,296 Nonembeddings of R P in R2"- 264 Nonimmersions, 260-262, 295, 296 Nonimmersions of R P in R'"-2, 261, 295 Nonlinear Lie groups, 9 I Nonlinear sphere bundles, 179 Nonorientability of the Mobius band, 160, 169
',
Nonorientable manifolds, 51, 160, 169, 259, 295 Nonspherical fibrations, 2 I2 Normal bundle of an immersion, 260 stable normal bundles, 292, 293 Normal spaces, 7 Normalization of characteristic classes Stiefel-Whitney classes, 217, 234, 235 2/2 Euler classes, 20 I , 205 Normalization of Steenrod squares, 243 Nowhere-vanishing sections, 119, 126, 198 n-plane bundle, I , 106, 170 canonical, 144, 170 universal, 3, 105, 149, 153, 170 n-skeleton of a CW complex, 20 n-skeleton of a CW space, 20 Numerable bundles, 2, 98, 102
0
Obstruction classes, 238. 239, 242 Obstruction theory, 288 Open convex sets, 10 Open manifolds, 25, 294 Open star of a vertex, 10 Opposite orientation, 165, 175 Orientability of real vector bundles, 160- 164, 168, 169, 209,216, 226ff Orientability of RP",259 Orientable and oriented manifolds, 50, 52, 259, 288, 294 Orientable vector bundles, l60ff, .we also Orientability Orientation bundle o(5)of a real vector bundle 6, 160-163 lrientations of some real vector bundles canonical orientation of the pullback n(<)'(. 163 canonical orientation of canonical oriented bundles jj;, I66 canonical orientation of universal oriented bundles 8'". I64 natural orientation of a realification in,175 natural orientation of a sum 5 @ 5 of two copies of 5 , 168, 176 opposite orientations, 165, 175 the other natural orientation of a realification in.175, 176 Oriented classifying map, 164
367
Index Oriented Poincare duality theorem, 288 Oriented Poincart-Lefschetz duality theorem, 288 Oriented 2-plane bundles, 175 Oriented vector bundles, 163, 164, 166 canonical oriented m-plane bundles 7;. 166 homotopy classification theorem for oriented bundles, 164 realifications of complex bundles i,175 sums 5 @ 5 of two copies of 5 , 168, 176 universal oriented m-plane bundle y"', 164 Orthogonal group O ( m ) c GL(m,R),3,59,81, 86ff: 90-92 other natural orientation of a realification jR, 175, 176
[,,
P Paracompact hausdorff space, 25 Parallelizable manifolds, 184, 185, 187. 247 Parallelizable spheres, 247 Partial derivatives, 127 Partitions of unity, 16, 25 Pathwise connectedness in a, 56, 208, 213 Picard group, 214 Plucker coordinates. 36 Plucker relations, 36, 55, 142 Poincare duality over Zj2. 250, 258, 276, 277, 282, 288 Poincare form ( , )p over 2/2, 276ff. 283 Poincare -Lefschetz duality over 212, 41, 249, 250.256-258, 288 Poincare Lefschetz duality for oriented manifolds, 288 Polar decompositions of GL(n,C), GL(m,R), and GL+(m.R), 86-90 Polyhedra ( K ( ,I 1 Polynomial relations for Stiefel-Whitney classes, 246, 301 of Wu, 287. 301 Positive element A E GL(n,C),87 Principal coordinate bundles, 100 Principal G-bundles, 99, 100, 102 Product family of fibers, 61, 62 Products of complex line bundles, 214 of real line bundles, 206-208,213. 214 of vector bundles, 109, 110, 111, 170, 181, 182, 241
t x 5'
of bundles over products X x X', 109, 181 Projection of a coordinate bundle, I , 57, 64, 67 of a family of fibers, 61 of a fibre bundle, 64 Projective bundle P, of a real vector bundle 5 , 3,202 Projective group PGL(rn,R), 3, 202 Projective modules, 179 Projective spaces RP",RP", CP",CP",22,23, 24, 29, 34 Proper embeddings, 21, 262 Proper homotopy equivalences, 54 Proper maps, 27, 54,262 p-simplex, 8 Pullbacks, 3, 4, 59, 61, 63, 64,66, 113, 121, 153, 155, 156, 158, 159, 162-164, 166, 170-172 along homotopic maps, 3, 4, 75-80
Q Quadratic modules, 248 Quaternionic bundles, 247 q-dimensional simplicia1 complex, 9 Quotient topology, 19
R Rank of a vector bundle, 1, 106, 170 Real ersatz homotopy uniqueness theorem, 147, 157, 179, 236 Real line bundle, 3, 106 canonical, 3, 144 universal, 149, 155 Real line bundle group T ( X ) , 206-208, 213, 214 Real line bundle products, 206-208, 213, 214 Real m-plane bundles, I , 106 canonical, 140ff, 144, 155-159 trivial, 112 universal, 3, 105, 149, 153 Real projective space RP",22,23,34,264,266, 267, 295, 296 Real projective space RP", 22, 23, 34
368
Index
Real vector bundles, 1, 105ff canonical, 140ff, 144, 155-159 orientable, 160ff, 163, 226ff smooth, 119-121, 158ff tirvial, I12 universal, 149, 153 Realification CR of a complex vector bundle [, 173-175, 177 Reduced suspensions, 189,246 Reducing maps, 84 Reduction of structure groups, 3, 59, 80ff, 85, 86,91, 116, 178, 179, 182, 183 Reduction theorem, 80, 81, 85, 86, 91 linear cases, 86, 87, 91, I 16, 178, 179, 182, 183 Refinement of an open covering, 25,48 multiplicative, 48 Regular interval lemma, 269 Relative homotopy, 31 Relative Leray-Hirsch theorem, 97, 194 Relative topology, 9, 11 Restrictions of families of fibers, 63 Riemann-Roch theorem over hi2, 302 Riemannian metric, 112-1 14, 116, 123, 182, 183 Right inclusion r,, 280, 281 Rotation group Of(rn) c GLt(m,R), 59, 81, 86ff, 90-92, 168 S
Schubert, H. C. H., 56,239 Schubert basis theorem, 239, 240 Schubert calculus, 239, 240 Schubert cycles, 239, 240 Schubert varieties, 239, 240 Second conjugate F** of a module 9,133 Second countability of manifolds, 25, 180 Sections of coordinate bundles, 82 continuous sections, 124- 126 modulus of sections, 117-119, 122, 123, 129ff, 134ff nowhere-vanishing sections, 119, 126, 198 sections of cotangent bundles (differentials), 134, 135 sections of tangent bundles (vector fields), 126ff, 129ff sections of vector bundles, 116-1 19, 198 smooth sections, 122, 124-126 zero sections, 3, 117, 210, 277
Segre map, I8 1 Self-adjoint matrices, 87, 90 Semiring of vector bundles, I 1 1, 1 I2 Semiring Q of smooth closed unoriented manifolds, 274, 275 Serre fibrations, 99 Shrinking lemma, 30 Simple elements in AmV, 35 Simplex, 8 Simplicial complex K, 8, 49 Simplicial embedding theorem, 5 I Simplicia1space IKI, 8, 49 countable simplicial space, 14 finite simplicial space, 9 finite-dimensional simplicial space, 9, I0 locally finite simplicial space, 53, 54 8-19, 49 metric simplicia1space IKI = [Kim. polyhedron ( K ( ,1 1 weak simplicial space IKI = I KI,, I Sff, 1721 Singular cohomology, 2, 4, 42, 250ff. 256ff. 258, 302 Singular homology, 249,25Off, 256ff, 258,302 Skeleton of a C W complex or CW space, 20 Smooth closed manifolds, 23, 249ff Smooth compact manifolds, 24, 258, 302 Smooth coordinate bundles, 121, I 58ff, 172, I86 Smooth involutions, 54, 55 Smooth manifolds, 23ff, 33, 50ff category A of smooth manifolds, 25 classification of smooth manifolds, 52, 53 Smooth maps, 25, 33 Smooth maps into R", 27 Smooth nowhere-vanishing sections, 126 Smooth partitions of unity, 25, 26 Smooth riemannian metrics, 123 Smooth sections, 124- 126 Smooth structure of Grassmann manifolds, 36, 37 of projective spaces, 24, 29 of a smooth manifold, 24 Smooth triangulations, 28 Smooth vector bundles, 119-121, 122, I58ff, 172. 186 Smooth vector fields, 126ff, 129, 130, 137 Smoothness of a manifold, 24 Spaces of finite type, 6, 10, 13.47,59 finitedimensional metric simplicial spaces, 10
369
Index metric simplicial spaces, 11-13 telescopes of metric simplicial spaces, 12, 13 Sphere bundles, 162. 179 nonlinear, 179 Spheres S"- I , 19 0-sphere So. 19, 162, 167 I -sphere S' (circle), 167, 205, 234, sc'r also RP1 Spherical modification, 299 Spin manifolds, 247. 298 Splitting bundle I, of a real vector bundle (, 3, 202 Splittingclass e(A;) of a real vector bundle (, 3, 201 Splitting map for real vector bundles, 204 for universal real n-plane bundle 7". 204 Splitting principal for real vector bundles, 201-203, 213 Square root of a positive matrix, 89 Stable equivalence of vector bundles, 187 Stable fiber homotopy equivalence (J-equivalence), 189, 241. 242, 289 Stable inverse of a vector bundle, 189 Stable normal bundle, 292, 293 Stability of the cap product, 257 Star of a vertex (abstract, closed open), 10 Steenrod's convenient category X , 48 Steenrod squares, 243, 245.285-287, 289 Stiefel-Whitney classes, 2, 4, 215ff, 224, 225, 227,229,230,238-248,288-290,298 axioms for base spaces in B , 233,234, 241 for base spaces in A, 235, 241 combinatorial construction, 289, 290 history, 238-240 naturality, 218, 234, 235 normalization, 217, 234, 235 of quadratic modules, 248 of representations of a finite group, 248 of RP", 259 of tangent bundles, 242, 289 of topological manifolds. 242, 289 Whitney product formula, 222, 234, 235 Stiefel- Whitney genera Ccf'),250, 274ff; 300 Stiefel- Whitney homology classes, 289. 290 Stiefel-Whitney numbers, 250. 265-268, 273, 274. 296 basic Stiefel-Whitney numbers, 274, 296 Stone's lemma, 39, 79. 114
Stone- Weierstrass theorem, 30, 31, 124 Strongest known embedding theorems, 294, 295 Structure groups (transformation groups), 3, 58. 60, 61, 85, 91, 97 admissible topology, 60 effective action, 60 reduction of structure groups, 3, 59. SOK, 81, 85. 86, 91, 116, 178, 179, 182, 183 Subbundles of vector bundles, 114, 115, 170, 186 Subdivision of simplicial complexes, 9 first barycentric subdivision, 9 Subgroups maximal compact, 3, 81, 86, 91 orthogonal, 3, 59, 81, 86ff, 90-92 rotation, 59, 81, 86ff, 90-92, 168 unitary, 59, 81, 86ff, 89, 91, 92 Sums of vector bundles, 108-110, 170, 181, 182 Sums 5 + 5' of bundles over products X x X , 109, 181 Surgical equivalence, 268, 273, 299. see also Cobordism (unoriented) Surgical equivalence ring, 268, 273, see also Cobordism ring Suspensions, reduced suspensions of pointed spaces, 189, 246 Tangent bundles r(X), 2, 4, 105, 126ff, 130, 134, 137, 146, 178, 180, 185,259 r(G"(R"'+ ")) for the real Grassmann manifold G'"(R"'+"), 185 r(RP")for the real projective space RP", 146 r(X) for topological manifolds X , 180, 181 Tangent spaces, I80 Telescope IKI* of a metric simplicial space IKI, 12, 13, 49 Telescope function, 1 I Tensor algebra @I V generated by a vector space V , 35 Thomclasses. 3. 41, 191, 192ff. 211 for real n-plane bundles 5 . 212 coefficients U cE W ( E , E * ) , 3, 192ff V<E H"(P,, I Pc),209ff W c ~ H " ( P C , PFal). I. 211 for real tangent bundles r(X), the diagonal Thom class T, E H"(X x X , X x y A ( X ) ; 2/2), 281ff Thom complexes, 212 Thom forms ( , )T =yT,, 276ff. 281-284
.
370
Index
Thom isomorphisms a<,199. 21 I , 212 Thom spaces, 21 2 Thom-Wu theorems, 285ff Todd classes, 302 Top Stiefel-Whitney class, 4, 224 Topological intersection theory, 240 Topological manifolds, 29, 53, 54, 180, 242, 288 Poincark duality, 288 Poincare-Lefschetz duality, 288 Stiefel-Whitney classes, 242, 289 tangent bundles, 180 triangulation, 53 Topological R" bundles, 18I , 242, 289 Topologies of simplicia1 spaces, 8, 10, 16- 18 weak, 17, 18, 20, 22 Total space E of a coordinate bundle, I , 57, 64, 67 of a family of fibers, 61 of a fiber bundle, 57,64,67 of a fibre bundle, 64, 67 Total Stiefel-Whitney class of a representation of a finite group, 248 Total Stiefel-Whitney class w ( Q E H * ( X ;h/2) of a real vector bundle over X E B ,2.4, 215ff Total Wu classes, 245 Transformation groups (structure groups), 58, 60, 61, 70 morphisms (I-,@) of transformation groups, 70 Transition functions t/j/ of coordinate bundles, 58, 67, 120, 121 of vector bundles, 116, 118, 120, 121 Triangulable manifolds, 53 Triangulation theorem of Cairns and Whitehead, 28, 52, 159 Trivial fibre bundles, 57, 63, 73-75, 112 Trivial vector bundles E", c", . . ., 1 I2 Trivializations of a coordinate bundle, 67, 121
U Umkehr homomorphisms, 301 Uniqueness theorems (ersatz homotopy uniqueness theorems), 147, 157, 171, 179, 236
Unitary group U ( n ) c GL(n,C),59,81,86-89, 91.92 Universal complex line bundle yl. 170 Universal complex n-plane bundle y". I70 Universal oriented m-plane bundle 7". 164 Universal polynomial relations for normal bundles, 246, 292 for tangent bundles, 246, 301 Universal principal G-bundles, 101, 102 Universal real line bundle y ' , 149, 155 Universal real m-plane bundle y". 3, 105, 149, 153 Unoriented cobordism, 50, 52, 267, 268 Unoriented cobordism ring 'JL, 250, 268, 296, 297 Unoriented smooth manifolds, 23ff. 50ff. 249tT, 258, 273ff
V
Vector bundles, 1, l05ff, 178ff algebraic vector bundles, 180, 186, 188 canonical complex bundles y;, 170 canonical oriented bundles j j ; , 166 canonical real bundles y:, I40ff. 144, 155159 complex bundles, 169ff holomorphic bundles, 186 introductory accounts, 178 irreducible bundles, 188 normal bundles vI of immersionsf; 260 nowhere-vanishing sections, I 19, 126, I98 orientable and oriented bundles, 160ff, 163 real bundles, I , 105ff sections of vector bundles, 116-119, 198 smooth bundles, 119-122, l58ff, 172, 186 stable normal bundles, 292, 293 tangent bundles r ( X ) , 2, 4, 105, 126ff, 130, 134, 137, 146, 178, 180, 185, 259 trivial bundles E", E", . . . , 1 12 universal complex bundles y". 170 universal oriented bundles j", 164 universal real bundles y", 3, 105, 149, 153 Vector fields, 126ff, 129, 137, I79 smooth and continuous, 137 nowhere-vanishing, 137 Vertex, 8 Vogt's convenient categories, 48
37 1
Index W
2
Weak homotopy equivalence, 21, 49 Weak simplicial spaces, 15-21 Weak topology of a cell complex limn X,,20 of R", 22 of a simplicial space IKI, 17, 18 Whitney duality theorem. 241, 290 Whitney embedding theorems, 28,50,260,294 analogs of, 5 I Whitney homology classes, 289, 290 Whitney immersion theorems, 29,51, 52.260, 29 1 Whitney products of bundles, 108-1 1 I , 170 of complex vector bundles, 170 of real vector bundles, 108-1 I 1 Whitney product formulas for multiplicative 2 / 2 classes of real bundles, 219-222, 226 applied to sums 5 , t2,226 applied to Whitney sums 5 0 q , 222 for Stiefel- Whitney classes of real bundles, 219-222.234. 235 for 212 Euler classes of real bundles, 197ff, 205 applied to sums 5 , + t 2 ,197 applied to Whitney sums 5 0 q , 198, 205 Whitney sums, 108-111, 168-170 of complex bundles, 170 of orientable bundles, 168, I69 of real bundles, 108- 1 1 1 Word problem for finitely presented groups, 52 Wu classes, 245 Wu polynomial relations for tangent bundles, 287,301 Wu relations for Steenrod squares, 245 Wu series, 245, 301
h / 2 characteristic classes, 2, 4, see also Dual Stiefel- W hitney classes; Stiefel-Whitney classes; Wu classes; 2 / 2 multiplicative classes; 2 / 2 Euler classes 212 diagonal Thom classes & E H"(X x X,X x wA(X)), 281ff Hi2 Euler characteristic, 276, 297 2 / 2 Euler classes, 3, 4, 191ff, 196ff, 205ff axioms for, 205 naturality of, 197, 205 normalization of. 201, 205 of product line bundles, 206ff. 213, 214 Whitney product formula for, 197ff, 205 2 / 2 Euler classes ~ ( T ( X )of ) tangent bundles r(X), 284 2 / 2 fundamental homology classes, 249, 251254, 275 H/2 multiplicative classes, 216-219, 225 naturality, 218, 219 Whitney product formula, 219ff, 226 212 Poincark duality, 250, 258,276, 277, 282, 288 2 / 2 Poincark forms ( , ) p , 276ff, 283 2 / 2 Poincare-Lefschetz duality, 41, 249, 250, 256ff, 288 2j2 Riemann-Roch theorem, 302 h/2 Thom classes CJc~HII(E,E*), 3, 41, 191ff ~ , 209ff 2/2 Thom classes % E H " ( P ~, PJ. 2 / 2 Thom classes W<EH"(PS,, , P:,,), 21 I 2 / 2 Thom diagonal classes T X € H"(X x X,X x X\A(X)), 281ff 2/2 Thom forms ( , h = J*T,E H"(X x X ) , 276ff, 28 1-284 212 Thom isomorphisms mZ,199.21 1 Zero section X 5 E , 3, 1 17, 277 Zero section X + P s B 1 210 , Zero sphere So, 19, 162, 167
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