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„(')· Since F u( 0) = 0, F u(l) = γ , and γ is an integral multiple of In, we get f u(t) = /0( 0) + u[f
x
= 0()O-0(O) = O; therefore, = /(0), for each u e l . Next we show that f u(t) # 0 , μ € [0, 1]. We have f u(t) = h u{t)-
Jo
f h u(s)ds.
h (t) e K.
If y ψ 0, then /„' h u(s)ds lies in the interior of Κ , because by Schwarz's inequality rt f h u(s)ds\ < [ \\h u(s)\f ds. Jo Jo But h u(s) is not a constant number, and hence the above inequality must be a strict inequality. Moreover, = L, implies that 2
/ hl(S)dS
Hence, /(/) φ 0. Thus, we have shown that f u is a regular closed curve, and so the { f u } is a regular homotopy connecting f 0 and /, . (ii) Case γ = 0. Suppose we can change F u(l) so that for each u e [0, 1], F u(t) is not a constant map. Then f u(t) Φ 0 for each u and we have the proof. To make such a change take a point t 0 with /-", (/0) Φ 0 and deform FJt) to F,(/) in a sufficiently small neighborhood of t 0 . Denoting by F u the newly obtained deformation of F 0 to F, we repeat the above process. We then see that F u is not a constant map for each tt. • Lemma 0.1 suggests a later development of our subject.
CHAPTER I
C r Manifolds, C r Maps, and Fiber Bundles In this chapter we shall collect together the fundamental facts about C' manifolds, C' maps, andfiber bundles as well as other preparatory items necessary in the later chapters. §1. C\n itfy manifolds and
c x
maps
Here we give a brief summary of C\n itfy manifolds. A. C\n itfy manifolds. First we define a C\n itfy manifold. Let R" be η dimensional Euclidean space with afixed coordinate system. Then a poinl ν of R" is represented by the «-tuple -v =
· ·ν2
VJ'
Consider a function defined on an open subset i' of R /:(.' — R1 . Let r be a natural number or oo. We say that / is differentiahlc of class C' , / is of class C', or simply f is C' if at each point ν of V all partial derivatives of / of the form dx. Ox, ...Ox, exist and arc continuous. Consider a map / : U — > R'' from an open subset U of R" to R''. Writing f(x) = (f x(x) /,,(*)) e R'', we say that / is differential^· of class C r if for each 1 < / < ρ, f t : U — R is a differentiate function of class C' . The definition of a c\infty map is similar. A real analytic map f is sometimes called C'" . DEFINITION 1.1. A topological space M" iscallcdan «-dimensional topological manifol·' if it satisfies Ihe following: (i) M" is a HausdorfT space. (ii) For each point ,v of M" there exists a neighborhood ί'(.ν) which is homeomoφhic to R" . ( i i i ) M" satisfies the second axiom of countabilitv. 7
I. C' MANIFOLDS. C"' MAPS, AND FIHLK MINIM I S
Η
Now we define a differentiable structure on a topological manifold. Let M" be a topological manifold of dimension η. By a C\n itfy coordinate system or an C\n itfy atlas for M" we mean a family S" = {( V ., φ ) | j e J } pairs (F^, ψ.) of open sets Κ in M" and homeomorphisms : Κ — R" of Vj in R" satisfying the following: (i) M" = U J e J V j . (ii) If V, η V j Φ 0 , then the map φ )0φ- χ
^ . ( ^ n l ' . ) - 9j{V,nVj)
from an open subset of R" to an open subset of R" is of class c\infty (Figure 1.1).
The pair (V , φ j) is a chart or a system of local coordinates and V coordinate neighborhood. Two c\infty atlases & = {(V. , V j)\ j e J } and f = { < p ' k ) \ k e Κ ) are equivalent, ~ S^ , if the combined family 5?' of the two systems is also a C\n itfy atlas for M n . Evidently the relation ~ is an equivalence relation. An equivalence class 2! = [S^] in M" is a differentiable structu or a C°° structure for Μ" , and the pair (Μ" , 3) is a differentiable manifold with the underlying topological manifold M" . The above definition is known as Whitney's definition. More generally if the maps φ. ο φ~ 1 in the definition of a c\infty manifold are of class C' . 0 < r < ω , we say that (M" ,3>) is a c f manifold. A c 0 manifold is a topological manifold. Often a differentiable manifold is understood to be a c l manifold; however, in this book we agree for simplicity that a differentiable manifold is a C\n itfy manifold, which is also called a smooth manifold. Next we discuss orientations of a c\infty manifold (Μ" , 3). Let 3 = [./"]. = {(V-, φj)\ j € J } . For χ e V Γ V let α ·,·(*) be the Jacobian matrix of φ )οφΤ ι at ψ :(χ): a, i(x) = D{
. R p : k(x) - k we must have f k\V k = f k_ t\ V k . But the {V-) IS locallyfinite and so for some neighborhood U(x) of .v, U(x) Π Vj Φ 0 for onlyfinitely many j . Denoting by jH(.v) the largest such j wc get jtf(.v) = ./) |(0 ( v). The definition of g(x) readily implies that g is a c x map which is an immersion as well. It is also clear that ,ι»|Λ' = /|Λ'. It remains to show that g is one-to-one. Suppose now g(x) = g{x u) and .ν φ .v„. Then from what was discussed above we have f k_ t(x) - f k_|(.v 0), ^t.(.v) = (.v„) for all integers k > 0. From the former equation wc get f(x) = ./(.v0). Therefore. χ and XQ do not belong to the same . But from the latter equation wc see that if χ e V k for k > 0, then ,v0 must also be in I\ . which cannot happen; hence, both χ and ,v0 must be in V (the I' were reindcxed iliis way). This is a contradiction, however, as f is one-to-one on Γ Finally, a close look at the definition of f k reveals that j k and t\ , are regularly homotopic relative to Ν . and hence the same is true for ί and g. • DEFINITION 2.8. Let M" bean /i-dimcnsional C x manifold and lei I : M" —> R'' be a continuous map. Wc say that the set 0\OV. , ,, °01 fo (iii) Cor.n(Q
ψ : G —· G , vAg) = g '
arc smooth. G L ( / J , R ) and SO(ti) arc Lie groups. Here SO(n) is the group of «-dimensional orthogonal matrices whose determinants are of the value one, which is called the n-dimensional rotation group. There are natural inclusions SO(/<)\subsetGL(n . R) c R" . With the relative topology of GL(tt,R), SO(/i) is a topological group. DEFINITION 1.19. Consider a coordinate bundle .'Ί8 = {II. p. V. )'. ii) satisfying: (i) Β, Χ, Y are C\n itfy manifolds. (ii) G is a Lie group and its action on )' is C^ . (iii) The maps ρ, φ ). . g n are all c x . EXAMPLE.
We say that 38 is a smooth coordinate is a smooth fiber bundle.
bundle and its equivalence class {·%)
20
C' MANIFOLDS. C
MAS AND FIBER BUNDLES
61
io
FIGURE 1.7
If Χ , Y arc C\n itfy manifolds and if G is a Lie group acting smoothly on Υ , then we can make {,3?) = {Β, ρ, X, Υ, (!) into a smoothfiber bundle by introducing a suitable C\n ifty structure on Β. REMARK. In the above definition we often take G to be a subgroup, not necessarilyfinite dimensional, of the group Diff)' of diffeomorphisms of Y . DEFINITION 1.20. Let 38 = {Β , /;, Χ , Υ , G} be a coordinate bundle. By a cross section of 98 we mean a continuous map / : X —» Β with po f = 1 . A smooth map / which is a cross section of a smooth coordinate bundle 38 is called a smooth cross section. EXAMPLE. The center line of the Mobius strip {Β , ρ , X, )'", G} gives a cross section. See Figure 1:7. DEFINITION 1.21. Let 38 = {Β , ρ, Χ , Υ , G} be a coordinate bundle, and let /I be a subset of A". Then the restriction p\p-\A):p~\A)
— A
inherits naturally a coordinate bundle structure from 38 , which is called the restriction of 38 to the portion of A and denoted by 38\A . C. Bundle maps. DEFINITION 1.22. Let 38 = [B,p ,
X, Y, G } and <2' = {B\p.
X',
Y',
C}
be coordinate bundles such that Υ = Y' and G = G'. A continuous map Ii : Β —> B' is called a bundle map and is denoted by h : 38 —> 38' if it satsfies the following: (i) h maps a fiber Υ χ of Β homeomorphically onto a fiber Y^ over χ e X' ; thus, putting h(x ) = χ we have a continuous map h : Β —> Β and the commutative diagram Β - ί — Ιί
Ί χ (ii) For a point χ of defined by (i), the map
l< χ' and the homeomorphism 1/γ : >'v -> Υ χ·
— >
is a homeomorphism of Υ, which coincides with the action of an element of G.
2. FIBER BUNDLES
21
(iii) The above correspondence defines a continuous map g k J:V,nh-\v
k)-.
G.
We say that h is the map induced by h . Evidently the identity map 1 H : Β -> Β defines a bundle map S8 — .'5?'. Further the composition of bundle maps is also a bundle map. The maps {g k j) defined in (iii) are called the mapping transformations It and satisfy I lie following: A',;(-v)i';,(.v) = ^,(.v), g' l kCHx))§
k l(x)
=
,v6 i ; n i - n r ' ( < ) ,
.ί',,(-ν).
|
of
(i)
ν e ι; ηΓ'(ΐ;'η κ/), j
These relations follow easily from Definition 1.22. LEMMA 1.5. Let 3S = (Β,ρ,
Χ , Υ , (7) and .<%' = {Β',
ρ'.
X' .
>''.
G')
be coordinate bundles with Υ = Y' and G = G' . Let h : Λ' — V be a continuous map. Suppose continuous maps
satisfy the relations (1). Then there exists a unique bundle map Ii such that (i) h induces h , and (ii) the gj k are the mapping transformations PROOF.
of h .
Existence. Put p(b) = χ 6 V C\h~\V k) and h k j(b)=4>' k(h(x).g k l(x)-p im·
Then h k J is continuous in b. Here p j(b) =
P(b) = χ € Γ ( . and
.ν € ν η Vj η Λ ~1 (v'k η vj); that is, b is contained in each of the domains of h k j and h h. Then by Lemma 1.4, h k J(b) = φ[(χ', = ^ k(x'.g = Φ',(χ\
g k j(x) • gjx) k i(x)-P
i(b))
• ρ,(!>)). = h k,(b)
g' l k(x') • g k l(x) • P,0>))
Hence, we have
fl = U(Kn/r'(i'/)),
χ = l>(x)
22
. C' MANIFOLDS. C' MAS, AND FIBER BUNDLES
η=χχγ γ
P x~v FIGURE 1.8
and the family {hj k } of continuous maps h J k : p~ι ( V j η /Γ1 ( V k)) —> B' . j e J, k e /, which agree on the intersection on any two domains, defines a continuous map Β —» Β' . Further by the definition of h- k we have ρ' ο h = h ο ρ , that is, h induces h . We also note that p'k °
h
° Φί.χΨ) = p'k ° Ί>Λ Χ' - s k J(x) • Pj °
Hence, Λ is a bundle map and the {g
k j}
are its mapping transformations.
Here p' k(b') = 4>' k% x~\b') and p'(b') = x' e V' k . Uniqueness. Suppose h : Β —< Β' is a bundle map satisling (i) and (ii). Then if p(b) — χ e Κ Π Λ~1 ( V k ), we have h(b) =
23
2. FIBER BUNDLES
LEMMA 1.6. Let 38 = (Β. ρ, Χ, Υ, (7} and = {//,//. X' , >''. be coordinate bundles with .V = X' , )' = )'' and G = Cr' . Then and only if there exist continuous maps
Skj •
v
j
n
38 ~
Κ — G
such that 5,,(-v) = ^;.(.v)g;,.(.v).
Λ-e i ; n ' ; n i ( ' .
i'/,(-v) = 4(.v)g A . y (.v),
AG rnr/m;\
where the primed terms belong to the bundle 38'. PROOF. To prove the "only if pan suppose .Ct? ~ . Then there exists a bundle map h : 38 — 38' with Ji = 1 y ; so the maps g k i defined b\
v ^ S k j =
satisfying
the relations: ,?;,(.v) = /V(.vf'S >((.Y);.,(.Y).
ν ei;m;.
(3)
PROOF. TO prove the 'only if part suppose 38 ~ 38' . then by Lemma 1.6 we have maps g k / which satisfy requirement (2); therefore, we define continuous maps λ / by A (.V) = (g /J(x)) ' . which satisfy requirement (3). To prove the 'if part, assume conversely that there exists a family | λ ( : j 6 J } of maps satisfying condition (3). Putting =
A-E
I'/H·.
we gel requirement (2) from (3). Hence, by Lemma 1.6 we get 38 ~ •·/}'. Γ)
D. Steenrod's structure theorem. DEFINITON 1.25. Let A' be a topological spacc and let G be a topological group. By a system of coordinate transfomations in X with values in mean a pair of families ( { Κ } , which satisfy the following: (i) The { Κ \jeJ) is an open covering of .V . (ii) The maps g μ : Γ, Π Γ —· G arc conlinuous and satisfy
24
I. C' MANIFOLDS C' MAS AND FIBER BUNDLES
It follows immediately that the pair ( { Κ } , {#,,}) of coordinate neighborhoods and coordinate transformations of a coordinate bundle iJS is a system of coordinate transformations in its base spacc with values in its structure group. The converse also holds. THEOREM 1.1 (STEENROD [ A 7 ] ) .
be a topological transformation of coordinate transformations
Let A", >' be topological spaces, let ( group of Y and let ({ V '.}, [g^ }) of X with values in (!. Then
(i) There exists a coordinate bundle fi8 with base space A' coordinate neighborhoods { V. }, and coordinate transformations (ii) Two coordinate bundles satisfying requirement (i) are equi PROOF, (i) Introduce the discrete topology on the index set ./ in { Γ | / € J ) and set Τ = {(α-, y, j)€X χΥ χ y|-v e V }.
Then Τ is a topological space which is the union of pairwisc disjoint open sets Vj χ Υ χ j . Define a relation ~ in Τ as follows: (.ν , .v, j ) ~ (χ', y' , k)
if and only if .v = .v' and g k l(x)y
= r',
(,v, y, j) , (χ', y . k) e Ί. By (4), ~ is an equivalence relation. Let Β denote the quotient spacc of Τ by ~ ; Β = Τ/ Let q : Τ — Β be the natural projection map; then q is continuous (B is given the quotient topology: U \subsetΒ is open if q~\U) is open in Τ ). Define a map ρ : Β — X by //({(.ν , ρ , /)}) = .ν . Then the following commutative diagram shows that ρ is continuous:
where p t is the projection onto thefirst component. We next define coordinate functions φ. : Κ χ Υ — ρ ' ( ) by 0(.(.γ . Γ) = q(x , y, j) . As q is continuous so is each φ.. Further, ρ ο q(x . y. j) = .v implies ρ οφ^χ , y) = χ , and so φ ) : V. χ Υ —> p~ l (V. ). Further, each φmaps Vj χ Y onto p " ' ( K ) , for if b = {(.ν, y, A)) e />"'(!' ( ) i h e n .γ € I ' n i ; and (x,y,k)~(x, g j k{x)-y , j) ; hence, we can write b = φ,(.χ , g j k (. Y)·.!'). To show that is one-to-one, suppose φ^χ , y) = φ^χ' ,y')\ that is, ( x > y , j ) ~ (x • y , j ) · Then λ- = χ and g tj(x) • y = y • But g n(x) = e , and so y = y . Hence, φ. is one-to-one. We now prove that φ~ ι is continuous. Suppose that H' is an open subset of Vj χ Y ; we want to show that φ t(W) is open in Β . To do so it is enough
2. FIBER BUNDLES
25
to show that q~ is open in T. But '/' is a pairwisc disjoint union of open sets Vk χ Υ χ k; hence, it sulfices to show that the intersection of Q~ {
\
VjxY-Up
(I·' ) C B =·/·/-
υ
υ
»•'
Here r(.Y, t>. k) = (.ν , g j k(x)-y) . Since r is continuous, t' '(M7) is an open set. Thus. "' is continuous. Now the map φ~\.οφ ι v , ν e l^n V , is a homeomorphism of )'. Putting y = Φ~\. ° Φ, ,.(.!'), we get φ^χ,ν') = ·) = g„(x)·y . y e )' We have constructed the desired coordinate bundle whose coordinate transformations arc the -
= {/?,/», V . >', (7}
(ii) Putting A;(.v) - e, .v € I7 in Lemma 1.7. we sec that two coordinate bundles with equal coordinate transformations are equivalent. Thus, the coordinate bundle constructed above is unique up to equivalence classes. • E. Tangent bundles of dilferentiable manifolds. We take an «-dimensional manifold M" for the X, Euclidean «-space R" for the Υ, and the general linear group GL(«, R) for the G in Stcenrod's structure theoremTheorem 1.1. In this case G acts smoothly on >'. Choose a C\n itfy atlas S" = { (t/ .
υ
,
η υ
ι
— *GL(«,
R),
a t/(x) = the Jacobian matrix of φ l ο φ ι
1
at φ ((χ).
Thefiber bundle constructed from this system via Stcenrod's structure theorem is called the tangent bundle of M" and is denoted by τ(Μ") = { T(M
n),
ρ, Μ" , R" , GL(« . R)).
It turns out that T ( M " ) = U T X (M") ΛΈΛΓ
26
I C' MANIFOLDS C' MAS, AND FIBER BUNDLES
with suitable differentiable structures. DEFINITION 1.26. Let M n be a C\n itfy manifold and let τ (Μ") be the tangent bundle of M" . A cross section of τ (Μ") is called a vector field 011 M\ DEFINITION 1.27. Let M" be a C\n itfy manifold. We say that M" is parallelizable if the tangent bundle τ(Μ") is trivial. EXAMPLE 1. A Lie group is parallelizable. EXAMPLE 2. The //-sphere S" is parallelizable if and only if 11 = 1 , 2 , or 7 (cf. J. Adams, On the nonexistence of elements of Hopf invariant one, Ann. of Math., 72 (I960)). F. Reductions of structure groups. DEFINITION 1.28. Let C be a topological group, let // be a subgroup of G, and let 1 : Η —» G be the inclusion map. Consider the coordinate bundle 38 = {B,p,X,Y,H\ {Κ.}, whose structural group, coordinate neighborhoods, and coordinate transformations are //, { ' , } . and { gjj }. We assume that G acts on )' extending the given action of // on Y. Using these Υ , G , X , {K.},and {/' ο g }, we construct a coordinate bundle 38 according to Theorem 1.1, which is called a G-image of or a coordinate bundle obtained by enlarging the structural group of 33 to G . Conversely if coordinate bundles 38 and 38' arc related as above we say that 38 is a coordinate bundle whose structural group is a reduction of ihe structural group G Xo Η. The structural group of the tangent bundle τ(Μ") of a C v manifold M" is GL(/i, R); θ(/ι) is a closed subgroup of GL(/i, R). DEFINITION 1.29. A reduction of the structural group of the tangent bundle r(M") of a C\n itfy manifold M" to O(n) is called a Riemannian metric on M". When a Riemannian metric is defined on a C\n itfy manifold M" we may assume a Euclidean metric ( , ) v on the fiber 7\(M") over each \ c- Μ" , of r(M"). Note that this metric varies smoothly in χ . The converse is also true. THEOREM 1.2.
A C°°
manifold
M" admits a Riemannian
metric.
a,&),
PROOF. Let M" = { M 3 = \S?\, and .5" = ( ( C ,
( « , »>,., = (ΦΐΐΜ,φ-^ν)). where φ ; : V i χ R" -»
u.v e TJM").
, φ [ V(.v) = φ^χ,ν),
and ( . ) on the right-
hand side of the equation is the usual inner product in R" . The desired
27
2 FIBER BUNDLES
Euclidean metric may be defined by
<"·">Λ = Σ ( · * ) ( » . •">,.,.
». " e /,(.»/").
I We can readily check thai { , ) is symmetric and positive definite. • REMARK. WC may also carry out our proof using the reducibilitv of the quotient GL(ti, R)/0(/z). The //-dimensional unitary group U(«) sits naturally in the rotation group SO(2//): consider the map ρ : U(i?) — SO(2n) defined by P(C) C = (c 0),
=
JQ ,
c ( j = a ;j + \T- \ bjj ,
CeU(n). A = (a it).
/* = (/>,).
Then ρ is a continuous isomorphism of U(//) in SO(2//). DEFINITION 1.30. Let M~" be a 2//-dimensional C X manifold. By Theorem 1.2 wc may take 0(2n) for the structural group of the tangent bundle τ(Μ") of M" . A reduction of the structural group 0(2n) of M" to U(//) is called an almost complex structure of M" . An almost complex manifold Μ " is a manifold with an almost complex structure. A complex manifold is almost complex. It is also evident that an almost complex manifold is orientable. Since SO(2) = U(l), a two-dimensional smooth orientable manifold lias an almost complex structure. G. Induced bundles. DEFINITION 1.31. Lei ·1ΰ' - {IS' , // . A"', ) ' . (i\ be a coordinate bundle. Let A" be a topological space, and let ι; : X — .V' be a continuous map. For a system of coordinate neighborhoods { Vj \j'ej'}. the family { i/~ (I'') \ j e j ' } is an open covcr of X . Setting J?„(A-) = jfV(/;(jr)). ve r , n r . we obtain a system of coordinate transformations ({I |.{&' /( }) in A with values in G. We define the pullhack or the induced bundle tf( /> ) of" ./) over X by /; to be the coordinate bundle as constructed in Theorem I.I from { r , ( 7 ; A \ { K } , { g „ } } . The following is an alternative definition of an induced bundle over A Let £6' = {/Λ/Λ X', >', G) be a coordinate bundle and let η : X — V' be a continuous map. Consider the following subspace II of Α χ It : Β = {(χ, b') e Χ χ Β' I //(λ ) = p'(b')}. We then have the commutative diagram Β
B'
X
X'
28
I. C' MANIFOLDS C' MAI'S, AND FIBER BUNDLES
where π, : Χ χ Β 1 —< X and π 2 : Χ χ Β' —· Β' are the projections onto Χ and Bl respectively, ρ = and h = π2|Β. Putting Vj = >f\Vj) and defining φ ] : V χ Υ -> p~'(Vj) by
gg[ ~ gg'
2
bundles over
X' . and
==> η\<%[ ~ η'.<% 2.
(ii) For a coordinate bundle 38 over X we have 1 ~ . whe 1 x : X -» X is the identity map. (iii) For a coordinate bundle 38' over X' and a constant map c : X the induced bundle c 38' is trivial. (iv) For a coordinate bundle 38" over X" and continuous maps /; : X' . η': X' -» X" , (ν) For continuous maps f , g : X —> X' . which arc homotopic. f38' ~ g'38'. (vi) If 38' is trivial so is η~.3?'. Η. Associated bundles, principal bundles. DEFINITION 1.32. A bundle {Β,ρ, Χ, Y . G) is called a principal bundle if y = G and G acts on Y by left translations. EXAMPLE. Suppose Β is a Lie group and G is a closed subgroup of Β . Then the natural projection ρ : Β —> B/G is a principal bundle (cf. Stecnrod[A7]). DEFINITION 1.33. Let 38 = {Β , ρ , Χ , Υ , G} be a coordinate bundle with coordinate neighborhoods { K } and coordinate transformations {g y / }.The associated principal bundle 38 of 38 is the bundle given by Theorem 1.1 using (G, G; Χ , { Κ } , }) where G acts on G by left translations. In short, we obtain 38 from 38 by replacing Y by G. DEFINITION 1.34. Suppose bundles 38 = {Β, ρ , X . Y . G) and .0' = {Β' , ρ , X , y \ G} have the same base spacc and the same structural group. We say that 38 and 38' are associated if the associated principal bundles 38 and 38' of 38 and 38' respectively are equivalent. We also say that 38' is the associated bundle of 38 with thefiber Y' . In particular, a bundle 38 and its associated principal bundle Cj are associated.
!i2. ΙΊΜ:Κ IIIINDI I s
In otltcr words 38' is associated with 38 if ·ίβ' is a replacement of 03 in which the new fiber is Y'. EXAMPLE. The Mobius strip and the Klein bottle arc associated as bundles cl over Λ . I. Quotient spaces of Lie groups. An ordered A-tuplc ν =(//,,/t, li k) of linearly independent vectors in //-dimensional Euclidcan space R" is called an orthonormal k-fi vine. The set of all orthonormal A-lramcs is denoted by Vn k . Since V k is a subset of R χ · χ R . we give I n ; the k relative topology of R" χ · · • χ R" . Wc say that I n k is the Sticfcl manifold k of orthonormal A-frames in //-dimensional Euclidcan spacc R" Evidently 0(/i) acts transitively on l'n k . Now pick an orthonormal Aframe Vg . We may think of Ο(/; - A) as the isotropy group of i\t . Hence. In particular, we have Κ The natural map
V, k «θ(//)/θ(//-Α). = θ(/ί) and Γ , = S"~ ] .
K.k = 0(/z)/0(/z - A) — I'„ t._, = O(n)/O(n
- A + I)
is a bundle whosefiber and structure group are S"~ 1' and 0(/i - A + I). respectively (cf. Steenrod [A7]); thus, wc sec thai \'N k is a C x manifold. Let R(l, n denote the set of «-dimensional subspaccs of /// + //-dimensional Euclidean spacc R"' + " . Consider the map /) : Γ ) ι Μ η m — R m n which associates to cach element v" of the Sticfcl manifold l' m - J 1 N the //-dimensional subspace spanned by v" in r " " " This map is clearly surjectivc. Wc give R„, n the quotient topology; a subset U of R m n is open if //"'((.') is open. We say that R m n is the Grassmann manifold of /t-dimcnsional subspaccs in R'" + " . The standard action of the orthogonal group Ο (iii + n) on R m n makes 0(m + n) a transitive topological transformation group of R ;ll ( J . Denote by R[j the following subset of R"' + " : Ro
= {
x «m.i) e
R
Km,
=
· ν ιιt:
=
=
v
= 0»·
The isotropy group of R," is O(w) χ Ο'(///), where ()'(//!) is a subspace of 0(m + //),
0'(m)= | ( E «
e 0(»i» + «)|,lg0(/H) . /;(l is the //-dimensional identity matrix |.
Hence. R w ,, ~ Ο(/// + η)/0(η) χ Ο'(/»).
30
I. C' MANIFOLDS C' M A S AND FIBER BUNDLES
Further, the natural map Ρ :V
m+n n
— Ο(/// + η)/0(η) χ Ο'(///) = R,„ „
= 0(m + n)/0'(m)
is a principal bundle whose structural group is 0(/i) (cf. Steenrod [A7]). Thus, we have the following PROPOSITION 1.3.
dimension
The Grassmann
manifold
RM
is a
C\N I FTY
manifo
mn .
J. Classifying spaces. With a fixed η wc have the following natural map between Grassmann manifolds: Ρ nι ' R„ , π ' II θ(/ζι + n)IO(n) χ O'(m)
^/H11. η II Ο(m + I + η)/0(η) χ θ'(ιη + I )
which give us an inductive system { R m n , p m : /// € Ν }. The inductive limit of this system denoted by , B o(«)
= |ΪΠ}θ(/η + ιι)/0(η) χ O'(m), m is called the classifying space for Ο (//). In general we think of a compact Lie group G as a closed subgroup of θ(λ) for sufficiently large k , and define the classifying spacc ll i; for G in a similar manner as above. If G and Η are compact Lie groups it readily follows that a continuous homomorphism ρ : G —> Η naturally induces a map ρ : li t; — . Κ. Vector bundles. DEFINITION 1.35. A bundle = {//,//. A", V,<7} in which )' = R" and G = GL(/i, R) acts on )' by the usual linear transformations is called an //-dimensional vector bundle. EXAMPLE. The tangent bundle τ(Μ~') of a smooth //-manifold is an ndimensional vector bundle. The natural map Ρ : K„ +n.m = 0(m + n)/0'(m)
— Ο(/// + η)/0(η) χ O'(m) = R„, „
is a principal bundle whosefiber is Ο (η) , and wc have the following commutative diagram: 0(m + n)/0'(m)
Ί Ο(m + η)/0(η) χ Ο\m)
0(m + 1 + n)/0'(m
+ 1)
Ί 0(m + 1 + η)/0(η) χ 0'(m + 1)
§2. FIBER BUNDLES
31
FIGURE. 1.9
whose inductive limit with respect to m
Ii οι») is also a principal bundle with fiber Ο(/;). The latter is called the universal bundle for Ο(//) and its associated bundle γ Η with thefiber R" is called the universal vector bundle. Now we define simplicial complexes and polyhcdra. In what follows wc shall carry out our discussions in Λ'-dimcnsional Euclidean spacc R A for a sufficiently large natural number Λ'. The points /'0 l' m in R x are linearly independent or in general position if the vectors l\ ll\ .•• . I' linearly independent. It is routine to sec that this definition does not depend on the order of the points l' { ) Pm Ρ be linearly independent points in R x . DEFINITION 1.36. Let I o· Wc call the set + >.„OPn. ] | X f R v OX = λ„ΟΡ„ t t ( /.,. - ι . / > ο / (I an n-simplex. The η is the dimension of the simplex \I' 0I\ • /'J . A zero-simplex |/-"0| is the point P0 , a one-simplex |/'01\ \ is the line 7'()l\ • a two-simplex {PqP^P,] is the triangle with vertices P Q. !\ . I\ . and a threesimplex \PQP\ Ρ-,Ρ^ is the tetrahedron with vertices 1\ . I\ , and l\ See Figure 1.9. /' among the DEFINITION 1.37. Any set of q + I points Ρ ι , P t 0 < q < η . are ... , P of an //-simplex σ = \ l' 1\ •••!'„] n 0 vertices P,0 ' again linearly independent; hencc, they define a (/-simplex •''I
τ
called a q-face
,
= ΙΛ p
· ρ,
I-
of σ . If τ is a face of a . wc write τ χ tr or σ y τ.
(
32
I. C
MANIFOLDS. C
MAI'S, AND FIBER BUNDLES
DEFINITION 1.38. Afinite set Κ of simplices ir. Ν dimensional Euclidean space R v is called a simplicial complex if it satisfies the following: (i) If a e Κ and ay τ , then τ e Κ . (ii) If σ, τ 6 Κ and σ Π τ Φ 0 , then σ Π τ -< σ and σ η τ -< τ .
The dimension of a simplicial complex Κ is the maximum value among the dimensions of simplices belonging to Κ and is denoted by dim Κ . DEFINITION 1.39. Let AT be a simplicial complex. The union of all simplices belonging ίο Κ is a polyhedron of Κ denoted by |A'|:
<τ€λ {classification of vector bundles). Suppose thai I' is a polyhedron. Then there is a one-to-one correspondence between the lence classes of n-dimensiona! vector bundles over Ρ and the set of homotopy classes of continuous maps from Ρ to B Q. n), the corres being {/} >— {/*'/„}. THEOREM 1.3
AN OUTLINE OF THE PROOF. The classification of equivalence classes of //-dimensiorial vector bundles over Ρ reduces to the classification of equivalence classes of principal θ(/ι) bundles over Ρ which is readily seen to be in one-to-one correspondence with [Z5, 5 0 ( H | ] ; this follows from B n.„. = lim θ(/ζ + m)/0(n) χ Ο (in) "1 m—«oo
π,(θ(/ζ + m)fO(m)) = 0. 0< i < m . where π;(Α') is the homotopy group of X in dimension 1. For a more detailed discussion see Steenrod [A7], Let ξ = {£(£), P (, X , R", θ(/ζ)} and η = (/:(//), Ρ η. Υ . R'" , θ(//ζ)} be vector bundles of respective dimensions n and m . Suppose that a continuous map h : Ε(ξ) — » £(/;) maps each fiber R" of ξ , .v e A', homomorphically into afiber R"' of η . We then say that h is a homomorphism of ζ in η and write h : ζ — » η . Such an h is called a vector bundle homomorphis EXAMPLE. Let Μ" , V p be smooth manifolds and let / : Μ" — Γ'' be a C°° map. Then the differential df of / defines a homomorphism of τ {Si") in r ( V ) . §3. Jet bundles A. Jets. Denote by C'(n , p) the set of all C' maps from R" to R'' sending the origin 0 to the origin 0, r > I . Wc introduce the following relation in C r(n,p ): / and g in C'(n , p) arc r-cquivalcnt at 0 if the partial derivatives of / and g agree at 0 in each order up to r . 5=1,2 r. 9 f, ι = 1,2 p. Ox, dX: • dX: Ox j OXj • •• O.Xj h h Λ ι <;,<·"
1.1
53. JET BUNDLES
We write / ~ g if f and g are /•-equivalent. It is clear that ~ is an r
'
r
equivalence relation. DEFINITION 1.40.
in C r(n,p)
under
We denote by J'(n, p) the sel of all equivalence classes An clement of J'(n,p) is called an r-jct. Wc write r
J'(f) , /(/). or p r ) for the equivalence class which contains / e C'(n . p). In the case where / is a C\n itfy map the r-jet J r(f) of / is just the truncated Taylor expansion of order r . We notice, in particular, that ./'(/'./') = M{p, n \ R), where M(/z, n) denotes the set of (/>, //)-matriccs over Κ The set ./'(//, p) is in onc-to-onc correspondence with R' . where Λ' = („//, + · • · + nH r) χ ρ , n Η s = (n + s - I )!/.s!(ti - I)!: we associate to cacli element of J r its partial derivatives of order up to r . This correspondence defines a natural topology in J'(n . p). Now consider C' maps f: (R", 0) — (R', 0), g \ (R'\ 0) —· (R v . 0). We define a map J r(p, q) χ J'(n . p) .l'(n , q) by (g' r ) , / l r | ) i— (g ° f ) ( r > • This definition is independent of the choice of representatives. Further as any partial derivative of go f can be written as a polynomial in partial derivatives of / and g . the above map is algebraic. In particular for // = ρ = q , the above map defines a product in .1' (ii . n) . and the r-jct (l K *) l ' i of the identity map !„., of R" is both the right and left unit of J'(n , n) with respect to this product. Denote by L'(n) a subset of J r(n,n) consisting of all invcrtiblc elements. For 1 < s < r , define a map : J'(" • P) —* •/'("· P) by *,.,(/"') = / 1 " · PROPOSITION 1.4. ( i )
l' (n) = GL(T/ , R ) .
(ii) Z/(//) = (7Tr t)-\L\n)). (iii) L'(n) is a Lie group for each r. I < /· < x . (iv) L r(n) has the homotopv ivpe of GL(ZJ. R) (cf. Thom and Lcvinc [Bll]). B. Singular sets. Wc write L'(n,p) for //(/>) χ //(;/) and define an action of L'(n , p) on J'(n , p) as follows: L r(n , ρ) χ ./'(//, p) —* j'(n , ρ). ,, , ('K),/ /-(<•), , ,(0 . ((α (rl , Λ )>—· ,(a-ι ο ,·Job) DEFINITION 1.41. A jet f >r ) e J'(n. p) is regular if its representative J in C'(n, p) has the maximal rank at 0. Wc shall denote by ''.] r(n. p) the set of all regular jets in J'(n . p).
34
I. C' MANIFOLDS. C' MAS, AND FIBER BUNDLES
Let q — min(//, p). We identify j\n , p) with M(p , η ; R) and set S k(n , p) = {A e M(p,
η ; R)|rank of A is q - k },
0 < k < q.
In what follows we simply write S k = S k(n , p). is dense in J'(n,p). (i) S 0 = "j\n,p). (ii) J 1 (//, p) = S Q U S, U · · · U S , pairwise disjoint
PROPOSITION 1.5. ( 0 )
pJ r(n,p)
union.
(iii) S k is the orbit space of /_'(/'. p) • (v) S k is a codimension
(η - q + k)(p - q + k ) submanifold
of ./'(//
PROOF, (i), (ii), and (iii) are immediate. We want to show (iv). We shallfirst prove S k \SUBSETS k . U - • - U S . Let A E S\ . In an arbitrary neighborhood U(A) of A there is a Π whose rank is q - k . Hence, the ."ank of Λ must be less than or equal to q — k . Next show S k U · • · US q C S k . Let A € S k U · · · US q . Then the rank of A is q - k — i, 0 < 1 < q - k . Hence, we can transform A by an element of L l(n , ρ) to
o
"i-k-i
0J ·
where / . stands for the (q - k - /') χ (q - k - 1) identity mat. ix. Hence, it is enough to show that E q_ k_ j is in . But this is obvious, (v) follows readily from the following
LEMMA 1.8. Let M(n,p\ R ) be the set of (ρ, n)-matriccs over R Sinc Μ (ρ , η ; R) is in one-to-one correspondence with R'"'. we give the usu ogy of R^" to M(p , //; R); thus M(p , η ; R) becomes a smooth manifo Let M(p , n \ k) be the set of all (ρ, n)-mat rices of rank k . If k < min(p , η). M(p , η ; k) is a k(p + η - k)-dimensional submanifo of M(p, η• R). PROOF. Let E 0 be an element of M(p,n\k). Without loss of generality we may assume that E 0 = (*<> £»), A0 is a (k , A:)-matrix and |. l0| # 0. Then there exists an ε > 0 such that \A\ / 0 if the absolute value of each entry of A - A0 is less than ε . Now let U C M(p , η ; R) consist of all [ρ , /z)-matrices of the form Ε = (a o) • where A is a (k , /c)-matrix such that the absolute value of each entry of A - A0 is less than ε . Then we have Ε e M(p , n \ k) if and only if D = CA~ lB.
because the rank of (h
0 \(A
B\ =(
\ A - I p_ k)\C D)
Α
Β
\
νΛ'Λ + C xb + D)
.
BUNDLES
35
is equal t o the rank of Ε for an arbitrary (p - A , k )-matrix X . Now setting X = -C.4 - 1 we see that the above matrix becomes (Α Β \ \0 -CA~ lB + D)' The rank of this matrix is k if D = CA Β. The converse also holds, since if -CA~'B + D Φ 0. then the rank of the above matrix bccomcs greater than k . Let ΙΓ be an open subset of />//-(p-K )(//-A ) - A(/H II - A (-dimensional Euclidean space: "
=
{ (
o)
c
e Λ1{ 1'·
"•
R) H
}
(*): the absolute value of each entry in A - l„ does not exceed v.. Then the correspondence (A B\ \C 0J
(Α \C
Β \ CABJ
defines a dilfcomorphism between ΙΓ and the neighborhood ί.'ηΛ/(/>. //: A ) of E 0 in λί(ρ , η: A). Therefore, Μ (p. ir. A) is a k(p + n- A (-submanifold of M(p. n: R). • C. Jet bundles. Let I'" and M'' be CA manifolds of respective dimensions η and ρ , s > I . For .v e V" and ρ € Λ/'1. I < »' < .ν. set cr
,.(('". M") = (./': I'" - M'\ c'map|/(.v) = r }.
Elements /' and g of
,.(!'", Λ/'') are r-cquiralcni
αι v. / ~ ,tr, if V.τ the partial derivatives of / and g at .v in some local coordinate system agree up to order r. The relation ~ is well defined and is an equivalence relation. Set ./; ,.(!·". A/") = c : ,.('•". M")/ - • We write J[(f) for the cquivalcncc class containing f and we say that J[( f) is the r-jci of f at χ . Set c r
'1
/ ( ! ' " , A/")=
(J
J[
,.(! ". A/").
A€I " .yetf
Using the atlases of I'" and M''. wc turn .!'(Γ",Α/'') into the total space of a bundle over V" χ M n withfiber J'(n.p) and structure group L'(n.p): J[
,.(l'". A/'')\subsetj'(V
n.
A/'') ,— J r(n. p)
i
J
(x.y)
e ν" χ A/''
36
I. C' MANIFOLDS, C' MAPS, AND FlttUR BUNDLES
This is called a jet bundle. An alternative definition of a jet bundle comes from the structure theorem of Steenrod (Theorem 1.1). Set Y = J r(n , p) and G = L r(n,p)\ G acts on Y. Set X = V"xM p . Take a C' atlas ^ = {((/„, φ η) \ a e A ) of V" and an atlas S* = { (W k , ψ λ) \λ e A } of M" . Then with" X t λ = U it χ Η\ , the family {X a e Α, λ e A] is an open cover of X . For X n i Π X fi μ Φ 0 , we define a map Λβ.μ)'· χ„.λ ηΧρ. μ
—
L'("'P)
by ε (.,Α)Αβ. μΜ> y) = ( C · Ό where
(r)
rr
/
^
6
l(0
rr
x /
-l\
°«.β = ^(Λ,^,. ). 6;..;, = ° ψ„ >• Then the { X a λ, g ( ( t ^ j |α, β 6 Λ , Α, μ e Λ } is a system of coordinate transformations in V n χ M p with values in G . Hence, we construct, by Theorem 1.1, afiber bundle, which turns out to be the above jet bundle. The total space J r(V" , M p) may be regarded as a C s~' manifold when r < oo. Let / : V" —» M p be a C s map. Then we call the map J\f):
V" — J'(V"
, M"),
A" — f x(.f) the r-extension of /. The r-extcnsion J r(f) following diagram commute: J'(V"
•/
, M") <— J'(n. j
is a C'~ r map making the p)
V" —ί V" χ M". In the following we take r = 1 . The submanifold S k(n , //)\subset· / ' ( / ' . P) is invariant under the action of L l(n , p). Hence, wc consider ihc associated bundle (this is a subbundle) of the jet bundle (J K(V\ Λ/"), p. V" χ M p), whosefiber is S k(n , p): J\V"
, M") DS k(V
i
n,
M") <— S k(n , /;)
I
V" xM" = V" χ M" Since S k(n, p) is a codimension (n - q + k)(p - q + k) submanifold of y'(n,p),the set S k(V" , M") is also a codimension (// - q + k)(p - q + k) submanifold of j \ V , M").
54. MORSE FUNCTIONS
Let / : V" —. M 1' be a S k(f)
c l
.17
map and set
= {.v e V"\ rank of / at .v is q-k
}.
We have the following routine PROPOSITION 1.6.
D. Mapping spaces. Let V" and At 1' be C' manifolds of respective dimensions η and p. Let c > ( r " , M p) be the set of C* maps from Γ" to M p, and introduce the C r topology in c l ( l " . A'''). I < /· < ,v . DEFINITION 1.42. Let / E C ' ( R " , M p ) . Let A be a compact subsei of V" and let Ο be an open subset of M'' such that f(K) \subsetΟ. For ρ € I'" . Q — f(p) ε Λ-/'', choose a chart ((.', φ) about ν such that A' C U and a chart (U', ψ) about q . Let c > 0 and 0 < / ' < . < . Write φ(/>) = (A, , ... , A'(J) and define the set N'(f : .Ν . Ρ : Κ . ()\r.) as follows: N r(f\
Λ", ρ: Α. Ο; £) = (.?€
c ,
( Γ" , ,1/'')|(i).(ii).(iii) }.
where (i) 8 ( K ) c O , (ii) V, ° f(p) ~ Ψ, ° ·?(/>)! < f- · for each // 6 A'. I
The C' topology of C' r(
Γ"
. M 1') is the weakest topolo
making the map J r:C r(\\M'
,)—^c"(y".J'(V".M"))
continuous with respect to the compact-open
topology in
cl l(
Γ", ./'(!".
§4. Morse functions 1.43. Let Μ be an ti-dimensional C x manifold, and let f : Μ — R be a c\infty function. Further let ρ e M 1' be a critical point of f . Choose a local coordinate system (U , φ), φ : U — R" . φ(ρ) = 0 . The square //-matrix o\f Ο φ) OXjOXj v=n DEFINITION
38
I. C' MANIFOLDS C
M A S AND FIBER BUNDLES
is called the Hessian of / at the critical point //. If the Hessian //(/) is a regular matrix we say that ρ is a nondegenerate critical point. If ρ is nondegenerate critical point we define the index of ρ to be the index (the number of the negative eigenvalues) of H(f) p • It is readily seen that these definitions do not depend on the choice of local coordinate systems.
THEOREM 1.4 (the lemma of Morse). Let Μ bean η dimensional manifold and let f : Μ —· R be a C\n iy tf function. Suppose //„ 6 Μ nondegenerate critical point of f . Then there exists a chart ( U satisfying the following : (i)
f(x ) = f(P
0)
-x 2
Xr + xL + •••+ xl .
where r is the index of p 0 .
LEMMA 1.9. Let Μ be a C°° manifold and let (U , φ) be a chart p 0 € Μ, where ψ{ρ) = (χ,, ... , χ η) .peU. Let f : U — R be a C'^ function. Then there exist a neighborhood W of p n in W and a C\n iy tf functions h . : IV R such that on IF , / has the following ex
+ Σ U-lPo )Κ· - - V M , » + Σ hij • (·ν, ,))<·»•/ - · ν ,(λ."· /=1 ' ' ι. I- I We leave the proof to the reader. Hint: a Taylor expansion. PROOF OF THEOREM 1.4. Using / - f(p Q) in placc of f wc may assume without loss of generality that f{p o)=0. Choose a chart (U .φ) about //(l such that
0)
By Lemma 1.9 there exists a sufficiently small neighborhood II of /;(1. on which / has the following expression:
t=l
·'
/ ,7=1
where φ(ρ) = (y,, ... , y n), ρ € U . Since p 0 is a critical point of /, thai is, ifw-
0'
/ =
1
the expression for / reduces to /= Σ
V,
"·
54. MORSE FUNCTIONS
Define a tj : IV — R by α ί( = 1 /2(li
= V
+
:j
f=
Σ
, then we have
(6)
w v
The coificient matrix A = (a :j ) is regular at the point />g . In fact, by differentiating (6), we obtain Of
A
{) a„
-A ;=ι
i.j= I
dv,.<>v, *
^ (. y= I
0-a. Ov..Ov/ '
*·
^
^
Oa,
+
' '
/= 1
On,
,+ - y , .
0 v
' *
_/=. 1
',
'
+
' '
thus, by (5) al the point p 0 wc have O'f But p Q is a nondegenerate critical point and so the matrix °2-r
t
ι
is regular. Hence, the matrix A(p 0) = (α(-·(ρ0)) is also regular. Now as the functions a ( ; arc continuous, we can find a small enough neighborhood Γ, of p a on which the matrix A = (a ) is regular. Hence, by selecting a sufficiently small neighborhood U { \subset\\ of the point p n wc have the following:
\
0 ' Ρ (p)A{p)P(p)
ι
=
Ρ e L\
V
I /
Denote by Q = ( q ) the inverse matrix of Ρ , and define functions .v( : U t R , 1 = 1,... , η . by n
= Σ
;
then (.*! x n) is local coordinates about //„ : (l/;.v, by differentiating (7), we get
(7)
v (1 ). In fact.
I. C' MANIFOLDS. C MAPS, AND FIBER BUNDLES
40
which gives us Ox
and hence we have D{x x ,. -y η) Further, by (5) we have
= det[
= det(<7,.,.(/>„))/0.
"a
χλΡ 0)
= · " = X„(P
0)
=°-
and by (6), (7), we have
f =Σ W>
=
'y/ly = 'x/,/iiPx fx t\
= (*
xn)
\ XJ I7
\
2 2 A +X"V+l r J_. "V, ^
2 +--+X., -r
*l τ (8) Finally we show that the index of / at p 0 is r. By differentiating equation (8) twice, we obtain 'ί - 2 , 1=1,. 2, / = /' + ax? I o 2f ϋχβχ;
= 0,
Hence, the Hessian of / at p Q is 1-2
0 =
0 2/ and thus its index is r. We now have proved the theorem. • DEFINITION 1.44. Let Μ be a J°° manifold. We say that a C\n ifty function / : Μ —> R is a Morse function if it satisfies the following: (1) The critical points of / are nondegenerate. (2) If ρ and q are critical points of / such thai ρ # q, then f(p) Φ /(
§4. MORSE FUNCTIONS
41
Μ,
THEOREM 1.5. Let Μ be a C\n itfy manifold. Then there exists a Morse function f : Μ — R . Further, if Μ is open, we can choose f to be with the index of each critical point less than η .
A continuous map / : X >' from a topological spacc ,V to a topological space Υ is proper if for each compact set Κ in >'. f~\K) is compact in A'. The ease of an open manifold is due to Phillips [C 14], PROOF OF THEOREM 1.5. Wc know that the set .// of Morse lunctions defined on Μ is dense in the set C\infty(M, R) of c v functions of Μ (cf. Milnor [A2] for a proof). Hence, a Morse function of Μ exists. For the case of an open manifold Μ ,first express Μ as a union of an expanding sequence of compact manifolds with boundary: oc M = (jM,. Λ/\subsetint + ( , /=l (see Figure 1.10). Next wc denote by M': the union of ,1/ and compact components of Μ - intM ( . Then Μ is again a union of an expanding sequence of compact manifolds with boundary: cxi Μ = (J M[. A/'\subsetml Λ/'.,. 1=1 To see thisfirst notice that MI c intA/j+|\subsetiniA/' , . Next wc note that if A is a compact connected component of Μ - int A/(. 0 A is a union of connected components of i)M i . Hence, either /lcintM + . or .1 contains some connected component of 'M/ 1+| . In the second case A - (A Hint Μ t l ) is a union of compact components of Μ - int Λί. , . Hence. A\subsetint Μ' , Consider now the compact manifold with boundary A/'+| - int Λ/'. Since M\ C intA/('+| , d(M' +{ - int A/;') falls into two disjoint parts corresponding to 9M' i+l and ΰM[ (Figure 1.11). Then there exists a Morse function f : M' i+l - intA/(' — R such that (0) the range of f c [0, 1]. (1) f~\0) = dM ,'. f->(\) = 0M' (ii) f has no critical points in 0(M'
i+v j+ ]
and - int A/')
42
I. C' MANIFOLDS C
M A S AND FIBER BUNDLES
(Milnor [A4], Theorem 2.5; this might be thought of as a relative case of what is described above). Fitting the / together we obtain a proper Morse function / on Μ . We next eliminate the critical points of index η . By the construction of M\ we have H 0 (M' j + l -intM;,cM/ ( ' +| ) = 0. In fact, if a connected component of M ( ' +| - int"M(' cannot be joined to yM / + | by a curve, it must be a compact component of int M' m l - int Λ/;', and hence it must be a compcact component of Μ - intM\. But this is a contradiction. Thus, we can remove critical points of index η (cf. Milnor [A4], Theorem 8.1). §5. The transversality theorem of Thom This section presents the transversality theorem of Thom, one of the most fundamental theorems in the theory of differential topology, which will be used in Chapter VII. DEFINITION 1.45. Let Μ" , N1' be C x manifolds of dimensions η and p. Let f : M" —> N p be a continuous map and let H' p ~ (l be a // - qdimensional submanifold of N p . We say that / is l-regular on ΙΓ'' - '' or that / is transverse to ]V P~' 1 if for an arbitrary point ρ of li' p~' 1 and an arbitrary point χ of J~ (y) the composite map Τ χ(Μ")
^
T y{N p)
T y(N p)IT r(W'"
q)
is a surjection, where π is the natural projection. The concept of /-regularity wasfirst established by R.Tliom.
LEMMA 1.10. Let M" and N" be C\ n itfy manifolds, and let f : M " - N be a smooth map. Let W p~q be a (p - q)-dimensional submanifold If f is t-regular on W p~q , f~\w p~(') is either an (// - q)-submanifo M" or the empty set.
The lemma routinely follows from the definition.
43
55. THE TRANSVERSALITY THEOREM Ol-THOM
THEOREM 1.6 (the transversality theorem of Thorn). Let M" be an ndimensional compact C x manifold, let N p be a p-dimensional C'^ m ifold, and let \V p~q be a {p - q)-dimensiona! submanifold of Ν 1'. L r p p C {M",N ) be the space of C°° maps from M" to N with the C r to ogy. r > 1 . Then the set
T w = { / e c ' ( M " , Ν") I f is t -regular is open and dense in C r(M"
, N p).
We omit the proof (cf. Thorn and Lcvinc [Bl I]).
on II'""" }
Ιί*
CHAPTER II
Embeddings of C x Manifolds Our discussions in the present chapter will revolve around theorems of Wliitney, which were later generalized by Haefiiger. We shall save Haelliger's works for Chapter VII. The main problem here is the existence of embeddings of Γ in Μ and their classification up to isotopy for a given pair of manifolds Γ and Μ . Throughout this chapter we shall work in the ( " v category. §1. Embeddings and isotopies We first define isotopies of embeddings and then state the theorems of Whitney and Haefiiger. Let V be an /(-dimensional c x manifold and lei Μ be an /(/-dimensional manifold. DEFINITION 2.2. Let / and g be embeddings. We say that /' and g are isotopic and write f ~ g if there exists a C map /·' : I ' χ / — · M .
/ = [0.
I]
which satisfies the following, with the notation /,(.v) = /'(.v. t): ( ' ) ./;, = / ·
/, = i ? ·
(ii) f t : I' — Μ is an embedding for each / € [0. 1). The map /·" or the family of maps { ./ } is called an isotopy between / and g . It is a routine that ~ is an equivalence relation (the transitivity requires a bit of thinking). The basic problem concerning embeddings is whether given I and .'·/ there arc embeddings of I' in Μ . and if so can we classify these embeddings by isotopies? We stated in Chapter I lhal if / : !' — Μ is an embedding, then the image f(V) of Γ under f is a submanifold of Μ (the converse is false). Hence, when Μ is a Euclidcan space of low dimension we can fairly readily grasp the /(Γ) and so V . Hencc. wc always try to embed Γ in Euclidean space of the lowest possible dimension. The question of isotopic classifications is called the I'roblcm dcr luxe. that of classifying topological spaces, polyhcdra. manifolds and so on by 45
46
II. EMBEDDINGS OF C
ο As')
MANIFOLDS
5 (S 1 )
FIGURE 2.1
homeomorphisms (or diffeomorphisms) is called the Problem der gestalt. There has been considerable progress on the Problem der gestalt in the last half century but progress on the Problem der lage seems to have stagnaied somewhat. EXAMPLE. Consider two embeddings / and g of the circle S 1 in Euclidean space R 3 as shown in Figure 2.1. Here f(S') and g{S l) are homeomorphic but not isotopic. The problem of classifying embeddings of J?1 in R3 is called knot theory and forms one of many activefields of research in topology (for the interested reader we recommend Introduction to knot theory , by Crowell and Fox(') strictly speaking isotopies in knot theory differ slightly from our isotopies). Another expression for isotopies comes in the following THEOREM 2.1. Let V be a closed manifold and let f, g : I ' — Μ embeddings. Then f and g are isotopic if and only if there exis topy Λ,: Μ — Μ, t € [0, 1 ] such that (i) for each t e [0, 1], /;, is a diffeomorphism of Μ , and ('*) ho = ' « · 8 = h tcf. As we have no occasion to use this theorem in our book wc omit the proof (cf. Thom [BIO]). In 1944 Whitney showed the following THEOREM 2.2 (Whitney's embedding theorem). Let V"bean n-dimensional C°° manifold, η > 3. Then we can embed V" in R2" , i.e., there ex embbedding f : V" —> R 2 " . DEFINITION 2.2. We say that a topological space X is k-connected if it is arcwise connected and n i(X)=0, 0 < t < k. Later Haefiiger discussed and honed Whitney's work and in 1961 obtained the following THEOREM 2.3 (Haefliger's embedding theorem). Let V"be an n-dimensional C\infty manifold. (') Introduction York, 1977.
to knot theory.
Ginn, Needham Heights, MA. 1963: Springer-Verlag, New
52. TWO APPROXIMATION THEOREMS
47
I
Ir, -+
- +I I
1 2r C'(x„ ',)
"I FIGURE 2.2
R
2
(a) Suppose V"
2k + 3 < η . Then we can embed Γ
is k connected,
(b) Let V" be k-connected. in R are isotopic.
2k + 2 < //. Then
two embeddings
of Γ
Taking k = 0 in Haefliger's theorem we obtain ihe embedding theorem of Whitney. In this chapter we discuss Whitney's theorem and in Chapter VII we will focus on Haefliger's work. §2. Two approximation theorems In this section as well as in the next we follow closely Milnor's lecture notes [B6]. We present two approximation theorems fundamenlal for the proof of Whitney's embedding theorem. First some preparations. DEFINITION 2.3. We say that a subset A of Euclidcan ti-space R " has measure zero if for an arbitrary f > 0 there exists an open covcr of I such that acΰΓ"(·νο·
£ ' · ; < ' · 1=1
i'=l
where C"(.v(, r ) is the //-cube centered at x j in R" , whose edges are each of length 2r(. (Figure 2.2); when no confusion arises we might abbreviate C" (*,,/·,) by C(x, , r,.).
PROPOSITION 2.1. If a subset A of R " has measure zero, then the com ment R" - A of A is everywhere dense.
Let U be an open subset of R " . and let f : U — R " be a map. If a subset A of U has measure zero, so does f(A) .
LEMMA 2.1.
C°°
PROOF.
Let C bean //-cube whose closure C is contained in U. Put b -- max xtC.i.j
'Of Ox.
then by the mean value theorem we get \\f(x)-f(y)\\
χ .veC.
II. EMBEDDINGS
48
MANIFOLDS
C
As A has measure zero by assumption, AnC also has measure zero, i.e., for an arbitrary e > 0, we can choose an open cover of Α Π C : OO
CO
AnC C jj c"(x,, r,.), 1=1 But the above inequality implies that
i=l
/( c n(x,. >r,.))c c"(/(A-,),b//r). Hence, we get f(A
Π C) C / ( Q C"(x, , R)J = U./K'"(-V, , -·,)) OO
cU C n(Ax,),lmr,). (=1 But the sum of volumes of cubes in the above open cover of A η C is oo oo -π , η η .η, η η χ—ν 2 ν' . n η 2 b r i = 2 b η } >', < 2 b η e. /ssl 1=1 Hence, f{A(~\C) has measure zero. Since we can cover A with a countable number of such C we see that the measure of f(A) is also zero. • COROLLARY 2.1. Let U be an open subset of R " , and let f : U — R a C°° map. n
Σ
G
: LI X
R"-" I
F
I
R" ,
where p x is the projection onto thefirst component. Then g is evidently a C°° map. But U = UxOcUx R'~" has measure zero, and so f(U) = g(U χ 0) has measure zero in R'. • We next define a concept of measure zero for a subset of a c\infty manifold. DEFINITION 2.4. Let (M",9)) bean //-dimensional manifold, and lei A be a subset of M n . We say that A has measure zero if j> n ((/ri/l)cR" ha measure zero for an arbitrary chart (U l t ,
COROLLARY 2.2. Let V n and M"' be manifolds of respective sions η and m. η < m. Let f : V" — M u be a c\infty map. The has measure zero in M'" .
THEOREM 2.4. Let U be an open subset of R " . and let f : U — R a C°° map, 2n < p. Then for an arbitrary e > 0 there exists matrix A = (a ( j) such that
(i) |a• | < f , 1 = 1, ... . ρ, j = i.....
η . and
53. AN IMMERSION THEOREM
(ii) the map g : U -> R'' defined PROOF.
by g(x) = /( x) + Ax is an immers
Notice that
Dg( x) = Df(x) + Λ. Hence, we need to choose a suitable A so that the rank of Dg{x) at each point .v of U is η, i.e., A must be of the form Q - Df where Q is a (ρ, //)-matrix of rank η . Define maps F k : M{p , //; k) χ U — M(p , //) by '•\(Q, x) = Q- /)/(.v). Then Lemma 1.5 implies that M(//, η; k) χ U is a dilfcrcntiablc manifold of dimension k(p + η - k) + η and that the F k arc maps. Wc note further that as long as k < η, k(p + 11 — k) + 11 is monotone increasing with respect to A . Hencc, the dimension of the domain of does not exceed (// - 1 )(p + η - (η - 1)) + η = (2η - ρ) + ρη - 1 when Α < η . Now our assumption that ρ > 2// implies that this dimension is less than pn = dim M{p , tz). Hence, by Corollary' 2.2 the image of M(p. ιι : Α) χ U under F k has measure zero in M(p . n) . Thus, we canfind an clement I sufficiently close to the zero matrix in M(/;, η), which is not contained in any of the F k . A = 0, 1 η - I . This is the desired A . • §3. An immersion theorem We show in the present section that wc can immerse an η dimensional C\n itfy manifold in R 2 " . Set R^ = {.v 6 R|.v > 0 ) . DEFINITION 2.5. Let f : X — Υ , where X is a topological space and )' is a metric spacc. and let <5 : X —· R^ be a continuous function. We say that g is a δ -approximation of / if d( f(x) . g(x)) < r)'(.v) for all χ in V where d is the metric in Y .
c x THEOREM 2.5. Let M " be an η dimensional manifold, and le M" — R'1 be a C^ map. In < p. (liven a continuous function <5 : R + , there exists an immersion g : M" — R'' which is α ό - approxim of f. In addition, if the rank of f on a closed subset Ν of M" is η choose g such that (i) g\N = f\N . and (ii) g is homotopic to f relative to Ν .
Before proving the theorem wc need the following LEMMA 2.2. .·( locally compact topological is paracompact. PROOF.
space X with a countab
By assumption wc may take open sets L\. UT ...
U • compact. 1 = 1 . 2
II. l!MUCI)L)IN(ISOI' Γ "
30
MANIFOLDS
as a basis of X . We construct inductively a sequence of compact sets OO
X = (J A,, A( c int Aj+r 1=1 Put Λ, = U y . Assuming that the A- up to 1 = j have been constructed, we define Aj+ I . Let k be the smallest natural number such that Ai \subse U ] U · · · U U k , and put Al tA 2,...·
Ai+l=(U
t
U- -U(4)U(7,.
+ |.
Then the {Λ ( } obviously satisfy the above condition. Let W = {Wj} be an open covering of X . Since the set A/+l - int/f ( is compact, it can be covered byfinitely many W , and we canfind a finite number of open sets V satisfying: s r= 1
(i) Vj C W. for some j , (ii) ^ C i n t Λ / + 2 Setting Pi = ,... , V^ and 9° = P 0 U />, U • · · , we sec that L·? is a locallyfinite refinement of 'W . • From Lemma 2.2 it follows that all our topological or differential manifolds are paracompact. LEMMA 2.3. Let { T / J be an open covering of a c\infty manifold M" has an atlas {(Κ , hj)\j e J } with the following properties·. (i) J has cardinality K 0 . ('') { Vj I j € J } ' s a locally finite refinement of {(/,}· (iii) hj(Vj) = c ( 3 ) . (iv) M" = U ; 6 , Wj . where W. = h~ [(C"{ I)). Here C"(r) length 2 r.
= C " ( 0 , r) stands for
the η-cube centered
at 0 with
We select a sequence of compact sets as in the proof of Lemma 2.2, oo A,, A2, ...; M" = IJ Λ,, Aj \subsetint Aj+ I , 1=1 and construct a locallyfinite refinement of the {U l t} , keeping in mind that we must have (1) Ay(K;.) = C"(3) and (2) A/+l - ini/t,c U y /j~'(C"(l)). • PROOF.
2.4. There exists a c\infty
function ing: (i) φ(χ)= 1, ΛΓ€θΤ). (ii) 0 < φ{χ) < 1, .ν e C"{2) - c " ( l ) . (iii) (p{x) — 0, X 6 R " - C " ( 2 ) . LEMMA
φ : R" — R satisfying
53. AN IMMERSION THEOREM
51
I!' y =
/
-2-1
o
~
K 1
-
3
Λ
·
FIGURE 2.3 PROOF.
Define a function
λ
: R — R by (,-l/v
λ(χ)
=
and set ψ(χ) =
ν > 0. ν < C).
0.
;,(2 + . v ) / . ( 2 - . v )
x(2 + χ)λ[2 -
χ)
+ /(.ν — I) + Λ(—.ν — 1)
Now define φ by <»(-νι
-ν„> = Π ^ Λ ' ι >
(see Figure 2.3). • Wc say that ρ is a bell-shaped function. PROOF OF THEOREM 2.5. Assume the rank of / is η on A'. that is. the rank of / is η on some open neighborhood U of Ν. Then the family {(.', Μ - Ν) is an open cover of Μ . We select an atlas = (('',· hj)\J 6 ./ } according to Lemma 2.3. which is a locallyfinite refinement of ((,', Μ - Λ'} with Λ,(ΙΓ) = C'"(l) and //,(Γ) = ('"(3) (see Figure 2.4). Wc next set /;,((,',) = C"( 2) and rcindcx the ( (I , . //,)(/ e ./ ( so thai 1 < 0 if and only if 1 > 0 if and only if
Γ\subsetU . Γ C Μ - A'.
Noticing that t/( is compact, wc set < = minrJ(.v). Now we construct the desired g by induction. Put f a = I . then the rank of f 0 is //on U , and so it is η on U,
(.ν) + ?(.ν).·1.ν.
52
II. EMBEDDINGS 01" C' V
MANIFOLDS
the following conditions (i), (ii), and (iii) are met. (i) F a is of rank η on Κ = h k(N kl Π U k). By assumption the rank of f k_ toh k' is π on Κ. The Jacobian matrix of /·', gives DF A(X)
=
D(f
k_ i Ο Λ ; ' ( A - ) ) +
+ ψ(χ)Α.
ΫΦ(χ)Αχ
The map Φ ·. Κ χ Μ (ρ, η ; R) - Μ(ρ , η ; R) which assigns DF A(x) to (χ, /I) is continuous and the subset Μ {ρ , zz; //) consisting of matrices of rank // is open in Μ (ρ , tz; R). But since 0(A' χ 0)\subsetΜ (ρ, /z; n) we have Φ(ΛΤ χ A)\subsetΜ(//, /ι; //) for a small enough A (for an A close enough to 0 ). (ii) The desired A should also be sufficiently small that
\\Ax\\< e+, χ e c"(3). (iii) Finally, by Theorem 2.4 we may take A sosmallthat ,f k_ l°h k '(.v)+ Ax has rank η on C"{ 2). Having chosen the desired A as above, wc define for each k a map f k : M n - R" by _ j
+
k(x)
, A" 6 K, ,
Jk W~\r k. tw,
The map f
k
xeM-V
is well defined, i.e, for a point A* of V /»_,(*) +
k(x)
k
k.
- Uk ,
= /A_,(-v).
The rank of f k is by (i) η on N k_ t and by (iii) η on U k : therefore, the rank of f k is η on N k — Uxt+i^ 7 ;· Further, by (ii) the map f k is a J/2it-approximation of f k_ x . Define g : M" -» R" by g[x) = lim^^ f k(x) . This means the following.
Ii·) WHITNEY'S EMHEDDING THEOREM 1 .It" C R 1 " "
53
Recall thai /
N 0ch\GN
2C...,
= U
Wj>
j
M" = U N k. k Since the { F } is locallyfinite, for an arbitrary point .v of M" there exists a neighborhood U(x) of χ such that U(x) Π V / 0 for afinitely many j's, the largest among which wc call k ; then
*(*) = /*(*) = /;+,(*) = ·••. Clearly, the map £ is and of the rank // 011 M" . Moreover, £ is a ^-approximation of /. The construction of f k implies immediately that f k is homotopic to f k i while A' is kept fixed. Hencc, / is homotopic to g with Ν fixed. • §4 Whitney's embedding theorem I: ,\l" c R"" +l In this section we show that we can embed an /t-dimcnsional C x manifold M" in 2/i + 1-dimensional Euclidean space R"" + l . DEFINITION 2.6. A map f : Μ — Ν from a c r x manifold Μ to c x manifold Ν is a one-to-one immersion if / is an immersion and a one-to-one map. A one-to-one immersion is not necessarily an embedding. EXAMPLE. Reflect upon /: R — R 2 in Figure 2.5. We define a regular homotopy of immersions as used in Chapter 0 and which we shall discuss in detail in the next chaptcr. DEFINITION 2.7. Let M " and V p be C\n itfy manifolds of dimensions 11 and ρ . The immersions f , g : M" —> V p are regular/ ρ homotopic, f ~ g . Γ
if there exists a C\n itfy map F : Μ" χ / — V satisfies the following:
p
which with f {x) = /"(.v. 1)
(•) / 0 = / , /,=*• (ii) f t is an immersion for / e 1. We say that /' or the { i s a regular homotopy connecting f and g . Notice that the ~ is an equivalence relation.
FIGURE 2.5
54
. EMBEDDINGS
C
MANIFOLDS
LEMMA 2.5. Let f : M" —> R p be an immersion of a c , x l manifold Μ" 2n < ρ. Then for an arbitrary continuous function <5 : M" —> R + , there a one-to-one immersion g which is a δ -approximation of f . Further, is one-to-one in an open neighborhood U of a closed subset Ν of Μ" can choose g such that (i) g\N = f\N,and (ii) g is regularly homotopic to f relative
to Ν.
PROOF. On one hand, because / is an immersion, ihcrc exists an open covering {U
Then it follows immediately that φt is a c\infty function. For the sake of convenience we take VjCLJ if and only if 1 < 0. Now we define the desired g by induction. First put f 0 = f . Assuming next that we have an immersion f,_. : M" —> R'' which is one-to-one on Uy<* v j . we define f
k
: M" - R" by
where b k is a point of R P , which we select as follows: (i) b k is small enough to make f k an immersion. (ii) b k is so small that f k is a S/2 k-approximation of f (iii) Let N l n be a subset of Μ" χ Μ" ,
k
Ν 2" = {(Α-,.ν) e Μ" Χ M"w k(x) Φ
By assumption 2η < p, and so the measure of Φ(Ν 2") in R/; is zero. Hence, we want b k to be outside Φ(Ν 1η) ( if b k is small enough to satisfy the conditions (i) and (ii) it will work for the condition (iii)). The map f k satisfies may j k sauauta f a. (x) - tpAv) = 0, Λ Μ - -fΛ kΜ only = /*(*) (y)- =ο 0 if ifand and onlyif if { ^ The 'if part' is clear.
!H WHITNEY'S ΠΜΙΙΙ:Γ>Π1Ν(Ί TILL OKI Μ I ,LT" Ι Κ'
To show the converse notice that 0 = f k(x) - Λ(.ρ) = (f
k. t(x)
- f k_,0')) + (9 k{x) - v k(y))b k :
hence, if
( L(f)
ρ = lim f(x„)
\
= < ρ € R'' for some sequence (_v, , .ν,, ... ) V { which has no limit point in M" J
is the limit set of /.
LEMMA 2.6. Let M" be a c\infty manifold, and let f : M" - R'" be a continuous map. (i) f(M") is a closed subset of R" if and only if L(f) \subsetf(\f"). (ii) f is a topological embedding, i.e.. f is a topological map. if and o if f is one-to-one and L(f) Π f(M") = 0. PROOF, (i) Suppose f(M n) is a closed subset of R ' ' . Let Ρ € )' Then ρ = lim f(x n). But f(x n) e f(M") . and as f(M") is closed, r € f(M"). Conversely suppose L(f) C f(M"). Let ρ be a point in the closure of f(M"). For each natural number η there exists a point l'(.\ ) in C''(y. l/ti).
II. EMBEDDINGS OF
56
Consider a sequence x l,x 2,... Then
MANIFOLDS
in M" . Set A' = lim x n if lim λ" exists.
/(x) = /(limxJ = lim/(A-,) = .P,
and hence y e f(M"). If the sequence {A (i } has no limit point, wc have y 6 L(f ) \subset f{M") . (ii) Evidently a topological embedding / is one-to-one. If L(f)r\f(M") Φ 0 , there exists a point y of /„(/) Π f(M") with y = lim.γ , where the sequence {χ,, λ'2, ...} has no limit point. On the other liand, we can write y = f(x). Since / is a topological map, / ' is continuous, and hcncc the {jc, , x 2 , . . . } has limit point A'. This is a contradiction. Suppose now / were not a topological map, i.e. f ' : f(M") -> M" were not continuous. Then there would exist a point A' of M" such that f(x) € L(f) . But this contradicts f(M") η L(f) = 0 . • LEMMA 2.7. Let M " be an n-dimensional C°° manifold. Then exists a C°° map f : M" —> R" with L(f) = 0 . PROOF. Let {(F y , Λ .)} be a c\infty atlas of Lemma 2.3 with respect to the open covering {M"} of Μ" , and let φ be the C\n itfy function of Lemma 2.5. For each j > 0 let φ • : Μ" - » R be the C\n itfy function given in the proof of Lemma 2.5. Set /(A) =
£;>,(A-).
j> 0 The right-hand side of this equation makes sense because the { F } is locally finite. It follows that / is continuous. We want to show L(f) = 0 . Let {Α-, , .Υ, , ...} be a sequence in Μ" , which has no limit point. Then for any integer m > 0 there exists an integer 1 > 0 with Xj $ W ^ υ • · • U W m , and so .Y( g W •' for some j > Hence, f(Xj) > m\ hence, the sequence {/(A',), f(x^) . ...) has no limit point. Ο Now we are ready to prove THEOREM
dimensional subset.
2.6
(Whitney's embedding theorem). Let M" be an nC°° manifold. Then we can embed M" in R"" +l as
PROOF. Let / : Λ/" —» R1\subsetR 2 " + l be a C\n itfy map given in Lemma 2.7; L{f) = 0 . Let <5 : M" -> R + be the constant map δ(χ) ξ f > 0. Then by Theorem 2.5 there exists an immersion g : M" —· R 2 " f l which is a J-approximation of /. Further, by Lemma 2.5 there exists a one-to-one immersion h : M" —> R 2 r t + 1 which is a J-approximation of g . We have L{f) = 0 if the € > 0 is small enough. Hencc, Lemma 2.6 impies that It is 1 . Ο an embedding, and so h(M" ) is a closed subset of The embedding theorem of Whitney allows us to regard an //-dimensional C\n itfy manifold as a submanifold of R 2 " + l .
§5. THE THEOREM OF SARD
57
Whitney proved the above theorem in 1936. In the following sections we shall show that we can actually embed an //-dimensional manifold in 2n dimensional Euclidcan spacc R"" . §5. The theorem of Sard We prove Sard's theorem which we will need for the proof of Whitney's theorem on completely regular immersions in the next section. Sard's theorem is one of the most fundamental in differential topology. In general the set of critical values of a C\n itfy map is meager as we see in the following THEOREM 2.7 (Sard's theorem). Let U he an open subset of n-dimensional Euclidcan space, let f : U —> R'' be a C"x map, and set C = { A- € U\rank Then
f(U
of f at χ < ρ }.
) has measure zero in R''.
We follow the proof in Milnor's book [A3]. PROOF. Our proof is given by induction on η . When η = 0 the theorem is obvious. Let η > 1 and assume that the theorem is true up to η - 1 . For 1 = 1,2,... put j =1,2...,
ς = | A- € υ
» kf,
= 0,
Ox, Ox, • • • Ox,
p.
I < k < i.
1
Evidently we have CDC,DCrOCt3f
u |
D·.
.
We shall carry out our proof in three stages: Step 1. The measure of f(C - C,) is zero. Step 2. The measure of f(C l - C l+ ] ) is zero, 1 = 1 , 2 Step 3. For a sufficiently large k the measure of f(C k) is zero. Combining these steps we obtain Sard's theorem. Proof of Step 1. If ρ = 1 , we have C = C, . Hence. C - C, = 0 , and so the set f(C - C,) has measure zero. For ρ > 2 wc will need Fubini's theorem. FUBINI'S THEOREM ( 2 ) . Let A be a measurable
set in R ' = K 1 xR'+
For any point t of R1 if the measure of Α Π {1} χ the measure of A is also zero in R''.
It is enough to show that for an arbitrary .v in C - C, there exists a neighborhood F(x) of χ in U such that the measure of /(F(.v)nC) is zero. This is for the following reason. Since R" is locally compact and (~) Sternberg [A8]: also consult any introductory book on real analysis.
1
is zero in { / } x R
II. EMBEDDINGS OF C°° MANIFOLDS
58
R->
h(U)
R>
A Itlx//'"' -S(C')
FIGURE 2.6
paracompact it is Lindelof ( any open cover of R" has a countable subcovcr). Hence, oo c - c, C Μ V. , /(V. η C) has measure zero. (=1 Thus, c-c, =Ui,[^n( c - c ,)] and f(C - c,) = , /[Fn( c -C,)]. Therefore, f(C - C,) as a countable union of sets of measure zero has measure zero. We continue our proof. For χ ς. C - c , we may assume without loss Of, L R" by //(.v) = of generality that — Φ 0. Define a map h : U Ox, x n). Then h is regular at .v since
(dfJ0x Oh, Ox.
0
x\ x
OfJOx I
...
OfJOx,, 0
|f \
/ 0 0 Hence, h maps some neighborhood V of ,v diffeomorphically onto the corresponding neighborhood V' of h(x) . The restriction of goh~ 1 to V' isamapfrom V' to R p , and the set C' of critical points of g is h(l'nC) (see Figure 2.6). Hence,
g(C')
= g°h(VnC)
= (foh~
l)oh(VnC)
= f(VnC).
On the other.' and, for each point (I, x 2, ... , x n) of V' , g{i . x n) belongs to the hypeφIane {/} χ R" - '\subsetR" . So we have the restriction of g g •. {/} χ R"
1
Π V'
= .?!{'} * R
{/}xR /'-I II- I
55 THE THEOREM o r SARD
Now Og, Ox.
I *
0 Og'j/OXj
J '
By the induction hypothesis the set of critical values g'(C') of g' has measure zero in {1} χ R''~' ( C' denotes the set of critical points of g 1 in {/} χ R " - ' ). But wc have i'(C') η {/} χ R'"' = g'(C'). Hence, by Fubini's theorem g(C' ) has measure zero. Thus. /(I η Ο has measure zero. We have proved the first step. Proof of Slep 2. Pick .vn 6 (\ - (\ + , and assume o k+ ,f. 0x\ • · · Ox
#0.
Define w : U — R by w(x) = Then w is a C°°
Ox
- Ox
map and Ow/Ox r I / 0 . Here without loss of generality ι Λιι
wc may assume that 5, = I . Define a map // : L' — R" by Λ(.ν, Then h is a Oh, Ox.
v„) = (w(x).x
mapfO satisfying w/Ox f 0 V
ο
vj.
2
0w/0\\ ... I
Ow/i).x 0
H
\
ο
therefore, Ii is a homeomorphism of some neighborhood of .v(). Hence, we can show that / ( c t - c / + |) has measure zero following 'lie procedure of Step I. Proof of Slep 3. Wc show that if k > njp - I . the set /(C\ η I") has measure zero in R" . Here /" denotes an /(-cube of edge <5. Then as in the first step we can write 1=1 tofinish the proof. Let A' e C k and consider the Taylor expansion of f at ν : , k+1 /(A- + h) = f(x) = f{x)
+ _— D* +,/U + R{x , h).
+ <»') •
ο < 0
60
II. EMBEDDINGS OF C
MANIFOLDS
Then for χ e C k Π ί" , χ + A e /" , there exists a constant C such that
||Λ(*,Α)||<αΑ||*+Ι. Subdivide the cubc I" into r" //-cubes of edge <5//·, and let /, be the subcube which contains the point χ . Then each point of /, can be expressed as x + h,
Ρ || < \fn ( j j .
Hence, from the above equation we get ll/(·* + A) - /(x)|| = \\R(x, A)|| < c||A||* + l / rfS\\ k+ i
a
a = c(\fnS) Hence, we may cover f(C add up to V such that
k
η I") by a family of small cubes whose volumes
Recall however that k > n/p - 1 and so η - p(k + 1) < 0 . Hence, we have proved Step 3. • REMARK. We may use Sard's theorem to prove Theorem 2.4. Let M" and V 1' be C x manifolds of respective sions η and ρ , and let f : Μ" — V p be a C x map. Put COROLLARY 2.3.
C = { Λ € Μ" {rank Then
f(C)
dim
of f at χ < ρ }.
has measure zero in V 1' .
This follows routinely from Sard's theorem. §6. Whitney's theorem on completely regular immersions In this section we prove Whitney's theorem on completely regular immersions, which offers a stepping stone for the proof of Whitney's embedding theorem "an //-dimensional C x manifold can be embedded in R~" ". DEFINITION 2.9. Let M" be an η dimensional C™ manifold and let / : M" — R 2 " be an immeision. Wc say that / is completely regular satisfies the following: (i) / has no triple points. (ii) For p, and p 2 , ρ, / ρ,, /(/>,) = f(P 2) = 1 (df)
p{T p(M"))
Φ
(df)
p
(T
p
(M"))
= R (R2").
We say that / intersects transversely at q when / satisfies (ii). Here by " / has no triple points" includes " / has no quadraple points, quintaple points, etc."
if it
§6. WHITNEY'S THEOREM o n c o m p l e t e l y r e g u l a r immersions
61
THEOREM 2.8. Let M" be an n-dimensional C°° manifold, and let f : M" —» R"" be a c\infty map. For any continuous function δ : Μ" — R there exists a completely regular immersion g : M" — R 2 " which is a approximation of f . Further, if the restriction of f to some open neighborhood of a com set Ν is an completely regular immersion, we may choose g so that g| f\N. In addition if f is an immersion, then g may be chosen to be regul homotopic to f relative to Ν . PROOF. By Theorem 2.5 there exists an immersion / which is a δ/2approximation of / with f\N = f\N . f is completely regular on some open neighborhood of Ν . Select an atlas of R 2 " , .7'' = { ( C 2 " ( . y , , I). ψ ι)\ί = 1 , 2 , . . . } , where ψ < : C 2 " ( . V ( , 1) - C 2 " ( l ) . The family // = { i\{M" Ν) Π /~'(C2"(.y, , 1)) 11 = 1 , 2, ...} is an open covering of M" . We take an atlas { ( F , h,)\j e ./ } of M" given by Lemma 2.3 with respect to 7/ . such that for cach V the following conditions hold. ( •) /|F : V. — R"" is an embedding.
(**)
For some
morphism
such that /(F.) C C""( v; .1). there exists a dilfeo-
ψ) : c 2 " (
1) — R 2 " with ψ ι ° ψ ι °/C' ( ) C c 2 " ( I )nR" . (We think
of R" as R" = {(.Υ, x n. 0 0) 6 R 2 "} c R 2 " .) Rcindexing the { (I j, //() 11 € ./ } , as we did in the proof of Theorem 2.5. we assume / < o iff i;\subsetυ . ι > o iff , \ i " - \ . Wc write φ'^ = φ^ ο ψ. . We shall construct g inductively. Put g H = f . and assume that wc have defined : M" - R 2 " with = f\N . (a) Replacing the φ ( we may assume that g, satisfies the conditions («) and (•*) in place of /. (b) We assume that if N/ = Ν U (tj,<; " i) · '^en for any point ρ of the set (,?,|A,)"'(//) contains at most two points and in case ii contains two points g J\N j intersects transversely at p. Now we construct
: M" — R"" . Consider the map +1
°
0
' : C'"(3)
Define a projection π : R 2 " — R" by ττ(.ν, Then by the choice of we have f'ni
° 5/ °
•
c2
" ( I).
v, n ) = ( v ) M |
) ~ ' ( c " ( 3 ) )\subsetη '(Ο. 0
0).
y, u ).
62
II. EMBEDDINGS OF c°°
MANIFOLDS
and by Sard's theorem the set of critical values of the C\n itfy map no V'
j+ lo gj:[Uu
(U
—• C"(l)\subsetR"
W,)\ ng~\u x)
has measure zero. Hence, wc may select a point c e R" satisfying the following: (i) c q is sufficiently close to (0, 0,... , 0). (ii) c q is a regular value of π ο φ'. +, ° · (iii) If the points //, and p 2, /;, φρ 2, satisfy .?,(//, ) = .?,(//,)€ C2"(.v. , I), then
(χ)
=
map
2)
i I
^ 1,).
: M" — R~" by
χ e M" - h'l x(C" (1)),
<
I + a- e r , where φ is our bell-shaped function. Then the conditions (i), (ii), and (iii) for c q guarantee that g. + 1 is an immersion and that g . is a δ/ 2 q* approximation of g-. Further setting N' j+ l = Nj u " , and noting N', t , is compact, we see that for any point ρ of ), the set (^ + ,|Λ' / ' +1 ) -ι (/>) contains at most two points, and if it contains two points, then g. :+| intersects transversely at p. It is also routine that jfy +,|Al = / \N. In addition, by readjusting the φ ί we see that g J +, replacing / satisfies the conditions (») and (**). Finally, by examining the definition of g J+ i carefully we sec that g Ί and g j are regularly homotopic relative to Ν . Now define a map g : M" —· R'" by £(a-) = lim g (x) , l — oo which is what we wanted. • §7. Special self-intersections Our aim is to prove that a compact C\n itfy manifold M" of dimension η can be embedded in R 2 " ; to this end we construct a model of an immersion with a self-intersection (double point which intersects transversely). Consider η = 1 . Define a map η: R1 —> R2 by
y = χ - x/( 1 + χ2),
2 = 1/(1 +-Υ2).
This is a C\n itfy immersion as shown in Figure 2.7, with one self-intersection. From this we get an immersion β : R1 -» R2 with exactly one self-inicrscction such that for a sufficiently large r > 0, β is the identity map outside D'( 0, r) , the one-disk centered at 0 with radius r .
ii?, SI'IX'IAI. SFLF-INTtf KStiC I'lONS
63
More generally, for //, /i > 2, define a C x map <> : R" — R~" by "(•v,, .v,
v„) = (p,, .1-, 1
p,J.
1=2
1
»"
=
v„) e R".
It
1
.>', = V Γ
(,v, . .v,
//.
V,
ΰ·
= — ·
/ =
"•
2
where // = (1 + . y | " ) ( I + α ; ) · · · ( Ι + x~) . We show that the map η is an immersion. The Jacobian matrix (Dft)(x) of π is 2(1 //(Ι+.ΥΓ)
4.V..Y, //(I+.Y;)
0 0
ι
4·νι·ν„
"(I + Λ·;) 0 0
0 0
0 0
-2.Υ, m( 1 + a , )
A-2(l -A-f) M(I+a·-)
-2.y,
it( ι + χ ; ) a-,(i
-a-2:)
"( l + Λ-;)
«(I +A") -?ν·ϊ„· ·'·.('
A'„(l v
m(l+jrf)
)
-2A-|A-,A-„ t t ( i + a*2 )
H( 1 +.YM
J
Notice that not all entries of the first column arc zero, for if -2.YjV{/t(l + a*2)} is zero then .γ, = 0 so that 1 - 2(1 - Ap/{/t(l + .vp} = I - 2//t. The last equation is zero when u = 2, and hence we have some numbers different from zero among the χ,, ,Y, vn . Hence, at least one of the x ( (l - A:2)/{it(l + ,Yp}, 1 = 2,3 /;. is different from zero. Therefore. the rank of D(a)(x) is //, and so a is an immersion. Now wc look for double points of <>; wc want χ = (.v. . a . ,) and
64
II. EMBEDDINGS OF C°° MANIFOLDS
χ = (jr[, x 2 , ..., x' n) in R" such that a(jr) = a(x')
and χ / χ . Put
« = ( i + ( x ; ) 2 ) ( i + ( x ; ) 2 ) ·•·(!+(A·;,) 2 ), a(x)
• • · < yin) ·
= (J'I . y 2 '
A
( - V ' ) = (y'\
< y'i
y\ n )·
Since y\ = y,, 1 = 2, 3,...,//, we must have χ· = a, , 1 = 2,3 η. Further, y' n+ l = y n+ l implies u = u . and so (A:|)2 = χ* , that is, AJ ' = -A', . Hence, from y' n+ i = y n+ j , / = 2, 3,..., η, we get x t = 0, 1 = 2,3 //. Further, from = y,, we have -v, -
2a:
i =_
2A, -A·. + — 1 ,
u = 1
+ A-" =
2;
u
'
therefore, jc, = ±1 . Hence, the only intersecting poinl in the image of <• is u(l,0,...,0)=a(-l,0
The Jacobian matrix D(a)(x) (
0).
at χ = (± 1, 0
1 0
0) is
0 ^ 0
0
0
0
...
V 0
0
...
Tl/2 0 ... 0 ±1/2 0
0 ±1/2/
The 1th column represents the components of the vcctor (c/
•
1/2
-1/2
-1/2
1/2 V
1/2 :
-1/2/
which is evidently regular. Hence, the map <» intersects transversely at «(±1, 0, ... , 0).
§8. THE INTERSECTION NUMBER Of" A COMPLETELY REGULAR IMMERSION
65
From this we see that an immersion β : R" —< R 2 " with exactly one selfintersection exists and that β is the identity map outside D"( 0. r) for a sufficiently large r > 0 . §8. The intersection number of a completely regular immersion In this section we define the intersection number / f of a completely regular immersion / of an //-dimensional manifold M" in R " , and using the model constructed in the previous section wc show the existence of a completely regular immersion with arbitrary intersection number. DEFINITION 2.10. Let / : M" — > R~" be a completely regular immersion of an /i-dimensional compact C\n itfy manifold M". (i) The case where M" is orientable and η is even. Choose an orientation in M" . Suppose that /(/;) = /(<7). Let it, u n e T{.\l") and υ,,... ,v n e T q(M n) be ordered sets of lincaly independent vectors, which define the orientations of T l.U") and Τ (M") . Wc say that the selfintersection at /(/;) has positive type or negative type depending on whether the ordered set of vectors
(df )„(",)
(
«//),<«•„ι e ·/>,„,(«-")
defines the positive orientation or the negative orientation in R 2 " . and we define the intersection number of this sclf-inicrsection to be + 1 or -1 accordingly. The intersection number I f of I is the sum of intersection numbers of self-intersections of ./': /, 6 Z. (ii) The ease where M" is nonoricntable or η is odd. In this case wc define the intersection number /( g /,, in the same way as above. R E M A R K . For η = 1 compare the 11 in this section with the /, defined in Chapter 0.
2.9. Let M" bean η dimensional compact C x manifold (i) If Μ is orientable and η is even, then for an arbitrary integer in exists a completely regular immersion / : M" -· R " with 1, - m. (ii) If M" is nonoricntable or η is odd. then for any in € Z,. there ex a completely regular immersion f : M" — R 2 " with I = m THEOREM
PROOF. By Theorem 2.8 there exists a completely regular immersion f t) of M" in R " . Take a point v() of M" and sonic neighborhood Γ of x 0 to replace /„ by the map β with exactly one self-intersection defined in §7 or by the composite of β with the map r : R"" —· R"" defined by r(. ν, , x 2 , ... , .v,n) = (-.ν,. .γ, .v,n). The intersection number of the new completely regular immersion f is /. +1 or / - I . We repeal this 'II 'll proccss till wc get an immersion with intersection number m . •
II. EMBEDDINGS O
66
"
MANIFOLDS
§9. Whitney's embedding theorem II: M" c R"" In this section we prove Whitney's theorem "An //-dimensional compact manifold M" can be embedded in R 2 " ". THEOREM 2.10.
exists a regular
Let f : M" homotopy
be an immersion, η > 3. Then ther \ t e [0, 1 ]} of f satisfying the follo
—> R 2 "
{f
t
(0) f 0 = f, f x is a completely regular (1) The number of self-intersections
immersion. of f is two more than tha
//(i) M" is orientable, η is even, and the number of self interse f is greater than |/y|, or (ii) M" is nonoricntable or η is odd. and f at least one self-intersection, then there exists a regular homotop such that the number of self-intersections of f is two less than th (For more on regular homotopies see §1, Chapter III). PROOF OF THE FIRST HALF. Notice first that by Theorem 2.8 wc may assume / is a completely regular immersion. Take a sufficiently small /;disk D" in /(M"), D" \subsetf(M") \subsetR 2 ", and choose distinct points ρ and q in D" . (a) We try to grasp a whole picture with // = 1 . We take an embedding of £>' in R 2 as shown in Figure 2.8 and make two self-intersections by moving the disk around by a regular homotopy, say, q = 0 , ρ e I)' C R1 C R" . (b) Suppose D" is embedded in R"\subsetR~" . Let q = 0 and // be points of D" with ρ φ q. Assume ρ sits on the ,ν,-axis. Pull up ρ to the (jc, , A-n+|)-plane in the positive direction on the jr„+1-axis. Wc then position some neighborhood of χ parallel to the (χ,, .\'H+, .v,M)-planc. Next we move this neighborhood in the (χ,, , ... , .v,J-planc through the origin 0 (at this point the disk crosses itself) making sure not to crcatc any other intersections (see Figure 2.9). The only intersections arc on the ,ν,-axis; we have created two self-intersections.
FIGURE 2.8
(j1) WHITNEY'S EMBEDDING THEOREM II : Μ" c R'''
(i 7
FIGURE 2.9
A sketch (of a portion) of the proof for the latter half. We show how to eliminate two self-intersections in case (ii) where either M" is nonoricntable or η is odd, and how to eliminate two self-intersections of distinct types in case (i) where M" is orientable and η is even, in each case through a regular homotopy. Let /{ρ,) = /(p,) = q and /(//,) = f(p'-,) = q' , and assume that these selfintersections have opposite types. Lei C, . C\ be disjoint curvcs in Λ/" . C\ connecting p( and p. and noi passing through any other self-intersections of /, / = 1,2. Then B- = f(C {) is a curve connecting q and q . and Β = β, ϋ/J, is a simple closed curvc in f(M"). Take a two-cell a' with boundary Β such that F(M") η σ 2 = Β (Lemma 2.9). We next deform / through a' in a neighborhood of C, in M" so that the new image of C, will not meet B { . In this way we will remove the two self-intersections. The detail of this proof appears later. For now. using Theorem 2.9. we shall show the following THEOREM
manifold
2.1 1 (Whitney's embedding theorem). Lei M" he a closed C x of dimension n. Then we can embed M" in R : " .
PROOF. The theorem is routine for η = I as M 1 is a finite union of S1 . Case η = 2. We can embed the sphere S~ . the projective spacc Rl'~ and the Klein bottle K 2 in R4 . By the classification theorem for closed surfaces, Μ 2 is a connected sum of afinite number of copies of the above
68
II. EMBEDDINGS OF C'"" MANIFOLDS
surfaces. Hence, we can embed M 2 in R4 (cf. any elementary text covering the classification of surfaces.) ( 3 ) Let η > 3. By Theorem 2.9 there exists a completely regular immersions f 0 : M" —> R 2 " with If — 0. By Theorem 2.10 we can remove all self-intersections. Thus, we have obtained an embedding /, : M" — R"" . • In order to complete the proof of Theorem 2.10 we need the following three lemmas.
LEMMA 2.8. Let & = {B, p,\K\; R'" , Ο(///)} be an m-vector bundle over a polyhedron Let K' be a subcomplex of Κ . Assume that C,,...,C,_| are cross sections over \K\ such that for each ρ 6 |A'| C,_,(/>) form an orthonormal system, and (ii) is a cross section o such that for each point ρ e |A''|, ζ,(/>) (ρ). ζ,(ρ) form orthonormal system. Then if dim A" < m - 1. we can extend C, ove such a way that the C,(p) (p), (,(//) is an orthonormal sys PROOF. Fix a point ρ of |ΛΓ| and let R"' denote thefiber over ρ. Then the desired C,(p) may be a unit vector in the orthogonal complement of the (1- l)-subspace spanned by ζ,(ρ),... , (ρ), /f-, we want ζ ( (ρ) e C {{ζ,(ρ), ... , C,_,(P)}} X c R j , Where S™" is the unit sphere in the orthogonal complement of the space {{ζ,(ρ), ... , £,_,(/>)}} spanned by C,(p),... , C,_, (p). Hence our problem reduces to the problem of extending a cross section over \K'\ to |AT| in the (/// - i)-sphere bundle over . But the obstructions for this are in
Η\Κ,Κ'·ι
t,_,(£'"""))
(see for instance Steenrod [A7]), which is empty since we have dim A' < m — i, and so we can extend our cross section. • In what follows assume η > 3 . Let C(., 1 = 1, 2 , oc the curves given in (b) of the proof of Theorem 2.10 (Figure 2.10). Let M, and M, be suitable ( 3 ) H. Seifert and W. Threlfall, A textbook of topology. Academic Press. San Diego. CA. 1980; S. Lefschetz, Introduction to topology. Princeton University Press. Princeton. N.I. 1949.
59 WHITNEY'S EMBEDDING THEOREM II
Λ/" C R ? '
FIGURE 2.11
neighborhoods in M" of C, and c , , respectively. Put c
, = { Ρ,,ΙΟ < I <
1},
P,0 = //,. p„ = //.
We take the curves //. : [0, 1] — Μ" , p t(l) = //(ί, to be embeddings. Then the curvesqt : [0, 1] — R~" , q,{t) = q it , where /(/'„) = q„, arc embeddings as well.
L E M M A 2.9. Consider R 2 = {(,Y,P)|A\ Ρ 6 R } . Set .1, = { ( . Y . P ) 6 R |0 < Χ < I, Ρ = 0 } . Denote by A-, the arc of the circle of radius one centered at ( 1 / 2 , - \ / 3 / 2 ) , connecting the endpoints r and r of . I, in t halfplane y > 0. Let A = A{ U /(,. Let τ be a small neighborhood of in R 2 . and let τ be the union of τ and the finite region τ" bounded by (Figure 2.11). Then there exists an embedding ψ : τ — R " " satisfying following :
(i) V/(r) = (/, V(r') = q\ ψ(Α,) = Ι} ι, t = 1,2. (ii) ψ(τ) Π f(M") =Β . (iii) Τ ν.(ψ{ τ)) (Ζ Τ. (f (Μ")), for all q' 6 Β. PROOF. We first show that an embedding ψ as specified above exists with respect to the set x'. Put
Τ" = 7,(/(Λ/,)),
7-; = Γ ν(/(Μ:)).
Τ 2 = T
7",'= 7"2 η 7"",
r2' = 72nr2".
Changing / slightly we may assume that / maps some neighborhood of /), in onto the exponential image of some neighborhood of the origin of T" in /(Λ-/). We may assume further that / maps some neighborhood of C; in M ( onto the exponential image of some neighborhood of T* in T" (valid only very near q ), 1=1,2.
70
II. GMIII:l)DIN(>snl· ( " "
^"Ό
MANIKII DS
l' r x '
1
FIGURE 2.12
Now there exist a closed neighborhood F(r) of r in r and a linear map φ : V(r) —· T 2 mapping V(r) to a closed neighborhood of the origin in Τ 2 . It follows that the φ, when restricted to Ι'(/ )Π.·Ι(, maps a closed neighborhood of r in Λ( lo a closed neighborhood of the origin in Ί'' . Similarly, we define a linear map φ in some closcd neighborhood of ι '. Let r(-: [0, I ] —> A, C R 2 , 1=1,2 be a parametric representation of the curve A, such that Φ( γ ,(Ι)) = t/((/) wherever φ is defined. Forl€[0, 1] let u f(t) be a smooth cross section in the restriction Τ(τ)\Α, of the tangent bundle Γ(τ) of τ to A,, subject to the following: (0) exp(M,.(/))CT, 1 = 1 , 2 . w,(0) 6 T r(A 2), /t,(0) is pointed forward along Α-, . «,(1) 6 Τ γ·(Α 2), it, (I) is pointed backward along .-F. u 2(0) e T r(A t) , u 2(0) is pointed forward along .1,, » , ( 1 ) ε Γ . ( ^ , ) , «2(1) is pointed backward along . I r (ii) « , ( / ) e r M 0 ( τ ) , U i(t)t T r („(/!,.), i=i,2. (iii) «,(/) rotates counterclockwise as 1 goes from 0 to I ; it,(l) rotates clockwise as t moves from 0 to 1 (Figure 2.12). Denote by Λ ( (ί) the line segment in the direction of 11,(1) centered at /·,(!) of length p. Here we readjust I/(r) and V'(r') so that cach R^l) forms a part of their boundaries. We may also choose ρ small enough to have /?,(<,) η ft 2(<2) = 0 if /-,(/), r 2(t) i V(F)uVV). For r,(t) € [F(7)u V'(r')] U At , set v,(t) = (),„,(«,.(/)). Then v,(l) is a vector field in f(V(r) making it touch f(M")
Π V'(r')) , which wc want to extend to II, without at
Since φ is linear on V(r)
U F'(r'), we have
Φ(r i(t) + nu ί(t)) = q i(t)+<xv j(t),
r,(l) 6 F(7j U F V ) , |a| < p.
Using this identity we can extend φ to the closed neighborhood r' of A . Let ψ' be a continuous extension of φ over τ. By taking a smooth approximation, we may assume that ψ' is a smooth map. Since 2n > 5 , by Theorem 2.6, there exists an embedding ψ which is a ("'-approximation of ψ' . Hence, ψ is an embedding and for r' e A we have
will I NPY'S Ι ΜΙΙΙ·Ι>Ι>ΙΝΙΙ THPOKI M II
At" t. H''"
/I
Further, becausc 11 + 2 < 2/ι wc may assume ψ(τ) Of (Μ) = Β . Hence, a = ψ(τ) is the desired 2-cell. •
LEMMA 2.10. Suppose q and q are self-intersections of f of opposit types. Then there exist cross sections «», w2n in 7~(R~")|o , the restr tion of the tangent bundle 7"(R2") of R2" to π . satisfying the following: (0) For each q' in a, wt(q') '/'•>„(
" J ι(<7") = {di//) r-(e ,).
"',(
- (dij/) r.(<\).
(ii) For q' € B y . t V . t o ' ) 6 ?;.(/(A/,)).
wy(q') (iii) For q' e β,, ™„+l(<7*)
7;.(/(Λ/:)).
This lemma clarifies how the σ, the /(A/,), and the /(A/,) are positioned relative to each other in R 2 " . PROOF. Sincc ψ is an embedding, w y(q') and »'-,(
= {{',
i.tl»·
C
M2
'':„)l·
where { { · · · } } denotes the subspace of R 2 " spanned by { · · · } . For each point q' of /?, put F " - ' ((?*)
= {υ 6
(f(M
x))\v
1 T q.(B y) }.
Setting 98 x = U<,-e/j 'ι" (q' ), wc obtain an (η - 1 (-dimensional vector bundle 98 χ = {98 χ, π, . Β χ) over Β χ . As the base spacc B x is contractible. the bundle 9S X is trivial. Hence, there exist linearly independent vector fields, say, wx , t//j, ... , ι/'η+, , in 98 χ , which define an orientation of /(A/,) at each point q' of B x . Similarly for q' € B 2 setting
FT V ) = { υ e T v.(f(M
2))\v
Χ 7;.(β,)).
we obtain an (η - 1 (-dimensional vector bundle 98 2 = [98 2 . π,. B 2). where 9B 2 = υ,-gfl V 2~\q'). Again there exist linearly independent vcctor fields w2n ' n ^2' which define an orientation of ./(A/,) at each ' wn+ 2 point q' of B,. Here we assume xi> 2(q') lo be an clement of 7'.(/i,(. which points in the positive direction along B 2 . Let 98 = [σ χ R~" , ρ χ, σ) be an 2/z-dimensional trivial bundle over a. The restriction 98\B , of 98 over B 2 has η + I linearly independent cross sections ω , , ω , , w „ + , , . . . ,«/·,„. Further, if q' = q or q' = //'. the wx(q'),... ,iu 2n(q') are linearly independent. By Lemma 2.8 we can w'i
72
II. LMI1EDDIN0S OF (""' MANIFOLDS
extend the w3,... ,wn over B2 and for each q' e the 2// - 1 vectors «/,(?*), w2{q'), w3(q'),... ,wn(q"),w n+ 2(q') «>,„(?*) are linearly independent. Recall that the self-intersections q and q have opposite types. By suitable choices of wl,w i,... ,ttf„ +1 in 5, and w' 2 , wn+ 2,... ,w2l l in β, wc may assume that for q' = q or q' = q , the vectors «/,(
"'2»^*)
define the orientation opposite from the preassigned orientation in R " . Now we can deform v> 2(q) and w/,(t/') to w\(q) and w\(q) keeping them in Τ(σ) and out of Γ(ΰ,); thus, these vcctors remain linearly independent of the above vectors. Hence, the vectors «>,(?*),
w2(q'), ... , w2n(q')
define the positive orientation of R 2 " at q' = q or q' = q . Therefore, wc can extend the cross section w t over if, while keeping its linear independence with the other vectors. w„+\ o v e r n so Using Lemma 2.8 again we extend the sections w that w{,... ,uu n+ t are linearly independent at each point of π . Finally, we extend the cross sections wn+ 1,... , w2l l defined over B 2 to a while keeping their linear independence. This is possible sincc wc may think of β, as a smooth deformation retract of a. Hencc, wc have shown the lemma is valid. • The , as constructed above, have clarified the situation in a small neighborhood of a in R"n . PROOF OF THEOREM 2.10. Consider τ\subsetR"\subsetR " " . For each point r = (a,, a 2 , ... , a,J of R 2 " set r* = (a, , α,, 0 0) and In
ψ(Γ) = ψ{Γ'
+J2 ai ei) ι=3
2η
= Ψ^') + ι=3
For each point q' of σ, the vectors wt(q'),... ,w2n(q') arc linearly independent and they are C\n itfy in q' . Therefore, ψ when considered as a map from a neighborhood of a to R"" has the nonzero Jacobian matrix at cach point in its domain. Hence, we have the inverse . Setting Ν χ = ψ-\/(Μ
χ)),
N 2 = ψ-\/(Μ
2)),
we see that N { and N 2 are contained in a neighborhood U of τ in R~" . If we can deform /V, in U so IV, does not intersect N f (i.e. there exists an isotopy {/,:/€ [G, 1 ] } of the inclusion map 1: N 2 — U such that i 0 = i, f^iA^jJnA^, = C2>), the Camily {ψ ° i,) defines a deformation of / and the number of self-intersections of / decreases by two. Hence, in this case the proof will be complete. Set π(λ-,, a- 2 , ... , x 2n ) = (.ν,, 0, χ }
χ 2η).
59 WHITNEY'S EMHEDD1NG THEOREM II. A/" C R ; "
73
ι,)
=
r=0
FIGURE 2.13
-ι
υ
c
A
FIGURE 2.14
By the conditions (i), (ii), and (iii) of Lemma 2.10 and the definition of v„+|)" ti/(r) as given above for r' e .1, , 7"r-(ΛΓ,) is in the (_v,, v, plane. Hence T r.(n(N t)) is also in the (.ν,, .ν, ν , (-plane. Similarly T r.(n(N 2)) is in the (.ν,, x n+ 2 ,v,n)-planc. Hence. πΙΛ^ηπΙ.Υ,) is on the .ν,-axis. Let //(JC, ) be a C\n itfy function whose graph .ν, = //(.ν, I is the union of the set A-, and the Λ',-axia minus ihc set A { smoothed out at the points r and r (Figure 2.13). Take e > 0 such that ihe interior of Λ', consists of points whose distances from the (λ", , x2)-plane are each less than < . Consider a C^ function i' : R — R as follows (Figure 2.14): ΗΛ)| < 1 .
ι/(0) = I .
ι/(λ) = o if|;.|>f 2. Now define a map ()t : B,(r) = r - tu(x]
—> R"" by
+ ••• + .v,2,,)//(*, )e 2 ,
r = (.v,
v,,,) 6 R2".
By the definition of /', Ut is the identity map outside A',. Evidently the {Θ, : R* - R "} is a regular homotopy with (i Q = 1 . When ι = 1 . (1, maps the portion of N 2 on A2 to the halfplane .v, < 0 . But /i(W((A':)) = ;r(/V,( and ττ(/ν,)Ππ(/ν2) is on the x {-axis, and so A'|ni)|(/V,) must be on the .v,axis. However, 0 t (N 2 ) does not intersect the ,ν,-axis, and so tf,(.V,) must be empty. Further, since ν/(τ)η /(A/) = II no new self-intersection will arise if we take £ sufficiently small. This concludes the proof for the case that A/" is orientable and η is even. In other cases too we can remove pairs of self-intersections uy regular homotopies in just about the same way as above. To do this we only need
74
II. EMBEDDINGS OF C
MANIFOLDS
to show that the c , and C 2 can be chosen so that q and q have opposite types. The case Μ is not orientable. If we can take C, and C2 as above and q and q are of opposing types we follow the previous argument. If q and q is of the same type, we choose a curve C 2 from ρ Ί to so thai C2 U c 2 reverses the orientation in Μ . Then q and q' are of opposing type with respect to C, and C 2 . The case η is odd and Μ is orientable. Suppose q and q have the same type for C, and C 2 . Then choose a curve C[ from //, to p\ and a curve C, from p 2 to p\ as follows. The curvc c ( ' agrees with C( near the starting point and with C ; , i φ j , near the endpoint. The neighborhood M\ of C\ is chosen in such a way that M. agrees with Μ t near the point p t and with Mj near p\, j φ i. Orient M- t and M\ with the preassigned orientations near p t and p\. Then q and q have the opposing orientations with respect to (M\, M 2 ). Ώ
CHAPTER III
Immersions of C°° Manifolds In this chapter we discuss classifications of immersions by regular homotopies, a weaker version of classifications of embeddings by isotopies (with the Smale-Hirsch theorem as the central feature) which reduces the problem of immersions to the homotopy theory. §1. Immersions and regular homotopies Let M" and V 1' be C\n itfy manifolds of dimensions // and /; respectively. Denote by c\infty(M", V") the set of C°° maps from M" to I '' with the C°° topology. Write Imm(M", V n) for the set of immersions of M" in Γ'\ Sincc lmm(M", V) c c\infty(M", V'') , we give Imm(M". T r) the relative topology. DEFINITION 3.1. Let / and g be immersions of \t" in Γ ' ' . We say that / and g are regularly homotopic and write f — g if they belong to the same arcwise connected component of Imm(M" , Γ''). Evidently the ~ is an equivalence relation. I. Write Imm'(M", F'') for lmm(,W", I"'') with the relative topology of c\infty(M", V p). Then the identity map
REMARK c l
I : Imm(M, V) —· lmm'(,M, F) is continuous and induces the bijeclion 1, : /r0(Imm(M, F)) —- /r0(Imm'(M. ' )) (the surjeclivity is clear from the definition and the injectivity follows from the approximation theorem). Hence, wc may define /, g € Imm(M . F) to be homotopic if they can be connected by a curve in I mm1 (M . F): this was our definition in Chapter II. REMARK 2. The approximation theorem in Remark 1 goes as follows.
THEOREM 3.1. Let Μ αι,α V be C s manifolds. I < s < χ . and let f : Μ -* V be a C' map, 0 < r < s. Suppose that f is of class C" on A . A c Μ (/ is C s on some open neighborhood U of A). Then there exis a C s map g : Μ —» V which approximates f in the C' topology such g\w = f. Ac w c υ.
75
76
. IMMERSIONS OF
MANIFOLDS
FIGURE 3.1
We give our proof in the next section. REMARK 3. To show 1 ^ to be injective (Remark 1) wc apply the approximation theorem with A — Μ χ 01, / = [0, I], to the regular homotopy F : Μ χ I —> V in the c l topology . From the above remarks we obtain the following
PROPOSITION 3.1. Let f and g be immersions of M" in V'' . Then f and g are regularly homotopic if and only if there exists a homo with f 0 = f and f=g satisfying:
(i) For each t 6 [0, 1 ], f t: Μ" —> V 1' is an immersion. (ii) Set F ( = d(f t) : T(M) - T(M) and define a map F : T(M) {Ο, I] —• T(M) by F(v , t) = F ((v ). Then F is continuous. EXAMPLE. Look at the three immersions /, g, h : S' — > R in Figure 3.1. While / and h are regularly homotopic, / and g arc not. PROPOSITION 3.2. Let f and g be embeddings of M" in I , p . If g are isotopic, they are regularly homotopic.
f an
The proof is left to the reader. The basic problem in the immersion theory is the following: Given manifolds Μ and V , classify immersions of Μ in V by regular homotopies. A special case of this problem is the question of the immersibility of Μ in K; in Chapter II we stated the Smale-Hirsch theorem which gives a complete answer to this case in the language of the homotopy theory. In §3 we discuss the Smale-Hirsch theorem. §2. Spaces of maps: The approximation theorem
Here we put together facts about spaces of maps and reformulate the approximation theorem. Let Μ and Ν be C r manifolds, and let C'(M , Ν ) be the set of c r maps from Μ to N . In part D of §3 in Chapter I, we defined the C r topology in C r(M , Ν), which is also called the weak C r topology or the compact-open C' topology. We denote by C\ v (Μ, Ν) the space C r(M , N) with the weak topology.
52. SPACES OP MAPS: Π1Γ APPROXIMATION Till OREM
77
The space C' W(M . N) is a complete metric space and satisfies the second axiom of countability. If Μ is compact, c r . ( M , / V ) is locally contractible; in particular, C r„,(M , R m ) becomes a Banach space. When Μ is not compact, the weak topology does not regulate the behavior of maps "at a distance". For this reason a stronger topology is sometimes desired. We define a strong topology below which is also called the Whitney topology or the fine topology. Let Φ = {(ί',. V,)|le Λ) be a locallyfinite C' atlas of Μ ; that is, for each point .v in Μ there exists a neighborhood U(.\) such that onlyfinitely many L\ intersect L'(.x) . Let Κ = { A'.| 1 e Λ } be a family of compact sets A'( in Μ with A'; c C . Let Ψ = {( F , y/()| / e A } be a c r atlas of A', and let r. { / e Λ } be a set of positive numbers. For fe C r(M , N) with /(A,) c F. set
^V(f-
Φ , ψ, Α', ε) = { g € C'[ Μ . Λ')|(ί). (ii) }.
(i) # ' , ) c K , 1 e Λ, . (ii) ||θ Α '(ν,ο/ο^- | )(Α·)-//(ν/,ο ί ο ί ,;' ι )(.ν)|| 0. C' S(M . A') does not satisfy the second axiom of countability. These two topologies are induced on the space C V ( . U , A ) of smooth maps by the inclusion maps C\infty(A/. A') — C' U (M . A ) and c x ( . U . A') — C r s(M , A). The strong topology has the advantage that important subspaccs in differential topology are often open with respect lo this topology. THEOREM 3.2. The set \mm(Sl,N) open in C' S(M , N) . r > 1 .
of C' immersions
of Μ in
A'
PROOF. Since Imm'(M, A') = Imm'(M. A') η C'(M . A'), it suffices to show the case r = 1 . If / : Μ —> A' is a c l immersion, we choose a neighborhood A'(/; Φ, «.'', A', £) of / as follows. Let V'0 = {(l-'/(. V//()! β e « | be a c r atlas of Ν . Choose a C r atlas Φ = { ((',, φι e A ) of Μ saiisfying
(i) Ut is compact. (ii) For each 1 e Ν there exists β(ί) 6 Β such that / ( F ) C Ι / ( ( Ι | . Set F = Κ / ( ( ί ) , ψ j = ψ βίι ), ψ = { ( I , ψ, )| i e λ } . Consider a compact covering Κ = { A'J 1 6 A } of Μ . with A( c L\ . Then for each 1 in A , the set
,1, = {D^o/oiT'x.vlj.v e p,(A)}
is
78
IMMERSIONS OF
"
MANIFOLDS
is compact in the space of the one-to-one linear maps from R'" to R" . But the set of one-to-one linear maps are open in the space L ( R " ' , R " ) of linear maps from R ' " to R " . So there exists ε,. > 0 such that Τ e L ( R " ' , R " ) is one-to-one if ||Γ - 5|| < ε, for 5 € Λ,. Set ε = {ε ; | 1 e A } . Then each element of Ν (f; Φ , Ψ , Κ , ε) is a c l immersion. • A similar proof applies to the following 3.3. The set Embr(M,N) open in C' S(M , Ν), r > 1 . THEOREM
of C r embeddings
of Μ in Ν
Now we proceed to the following approximation THEOREM 3.4. Let U \subsetR m and V c R" be open. dense in C r s(U , F), 0 < r < oo.
Then
F) is
PROOF. C' s(U , V) is open in C r s(U , R " ) , so we need only to prove ihe theorem for V = R" . Let / e C\U , R") . For a neighborhood basis of / in C' S(U . R"). wc may choose the following sets
N(f,
Κ , ε) = { g : U
-
R"
,c
r
map|
- f\\
r K
< ε,, V/ € Λ } ,
where Κ = { K :\ i e λ } is a locallyfinite family of compact sets covering U , ε = {e.| / 6 Λ } is a family of numbers. We must show that with the fixed /. Κ and ε, C°°(U , R")N N(K,f, ε) Φ 0. Let {Λ·|1 6 A } be a partition of unity of class c\infty on U such that Supp(A;) is compact and K ( c Supp(/.(). Given a family {α (|1 e A } of positive numbers, there exist c x maps : Uj — R such that gj Define g : U —> R by ^(A:) = Σ, ^jWgjix) • Then g is of class c\infty . To evaluate ||Dkg(x) - ^ / ( x ) ! ! we notcice that if λ : U — R and φ : U — R are C k functions and if we define a map ψ by ψ(χ) = λ(χ)φ(χ) , then D k ψ(χ) is a linear function of if λ[χ) and D° φ(χ) , ρ , q = 0, 1 k , which does not depend on χ , λ, or φ . Hence for some constant Ak we have HO*(^)(*)|| < A k max ||//λ{*)|| · max ||0>(.v)||. 0
Write A = max{/l 0, ... , Ar) , and for each 1 in A set Λ, = { j €
φ0}.
3. CHARACTERISTIC CLASSES
Then A ( is afinite set, say A ( has ///, elements. Setting //,. = max{||A;|i, K \j e A,}. J®, = max{a ;| j 6 Λ,}, we obtain the following inequalities for χ € Κ ||D kg(x) - D kf(x)|| Ση= < Σ
0< k
||tfu/g,
-/υ||
Evidently we may choose α( so that ιιι,Αμ,β, < e r With such a choicc of
we have
IIS-/IU· < V for each / in Λ ( . • From the above theorem it is not difficult to derive the following approximation THEOREM 3.5. /cf Μ and Ν be C' manifolds. C\M , N) is dense in C' S(M . Ν). 0
1
< ν <
Χ
.
Then
§3. Characteristic classes This section is based on Wu [C23]. Wc discuss characteristic classes as a preparation for the further dcvelopcmcnt of the text (see Milnor and Slashed' [A5] for details). A. Cell divisions of Grassmann manifolds. Let κ""" be // + ///-dimensional Euclidean spacc. Let R(1 m denote the set of t/j-dimensional subspaccs of R'" + " and let R;l m denote the set of ///-dimensional oriented subspaccs of the same. A similar definition holds for C n M when we take C instead of R for scalars. DEFINITION 3.2. Wc call Rn , Rn , and C n m Grassmann manifolds. We topologize these sets as follows. We will do only R;; , but the resi are similar. Let y „ + m „, be the Stiefel manifold of orthonormal ///-frames in R""" . It is well known that r
n+m . m
We have a surjection π: V n+m. m
—» R n.m
. IMMERSIONS OF
80
MANIFOLDS
which sends an element {p , p m } of V n+m m to the subspace {{p n + m p m } } of R spanned by { ρ , , . . . , P„,} · We give R n w the quotient topology of V n+m m by π . Then it is easy to see that : 0(/l + W)/Q(/1) Χ Ο (///),
R_ where O'(tn)
is a subgroup of Ο(n + ///) of the form
( 1 Ο =
0
0
0 V
A 6 θ(///) € ο(// - 111) 0
DEFINITION 3.3. For natural numbers in and η , a map ω: {1 ,2,...,
m) —> {0, 1,2
/ι}..
which is monotone increasing in weak sense, 0 < ω(1) < ω(2) < · · · < ω(ιη), is called an (///, η ; w)-function or a Schubert function of type (///. //). The set of (/«, η; a>)-functions is denoted by Ω(/ιι, n) . DEFINITION 3.4. Define the dimension d(oj) of an element ο in Ω(/ιζ. //) by m d(w) = Σ,ω(ί). i=\ Special attention should be paid to the following notation for Schubert functions. We often write ω = (ω(1), ω(2) ω(/")) for an clement ω in Ω(//ζ, η). We also write ω" =(0,0
0, I
1),
0
0 ,k).
= ( 0 , . . . , 0, 2 7k . 2k
0 < k < m.
2),
2k = (0, ..., 0, 2k, 2k), ω 2k. 2k
0 < 2k < m, 0 <2k < η.
Henceforth, we put ω(0) = 0 , a>(rn + 1) = η for convenience. DEFINITION 3.5. Let ω e Ω(//ζ, η). We say that the number / is a jump point if ω(1) < ω(ί + 1).
§3. CHARACTERISTIC CLASSES
81
Given an element ω of Ω(/ζζ, η) we define the following new functions: ω ( ι ) = (ω(1),..., ω(1- 1), ω(1) - 1, ω(/'+ I), ... , ω(ιη)), (here 1 - 1 is a jump point of ω), ω(ί)
= (ω(Ι), ... , ω(1- 1J, ω(1) + 1, ω(ϊ + 1) ω(///)), (here /' is a jump point of '•)), ω* = (η - ω(///), /ι - ω(//ι - 1) η - ω(/// - / + 1)
η - ω( 1)),
that is, ω*(1) = η - ω(/// - 1 + 1). Now wc are ready to do ccll division of the Grassmann manifold Rn Let R* be the subspace of R" + "' = {(χ, =
.
m
,v„ + J| .v, e R } ,
" = -V,
= 0}.
DEFINITION 3.6. For an element ω of Ω(//, ///). set
= ί Λ ' e R„ „, | dim(A· η R' u , " + ') > , . , = 1.2 Wc say that U J
is the Schubert
m }.
variety of ω .
(u
The interior U l u of U w is a t/(w)-cell. For ω 6 Ω ( « . ,//) set {/(ω)} = {ω( 1) + I , ω(2) + 2
that is, 1(ω) = ω(1) + i. Put X ..,=
U eH..,)- cH„ l)
^M.JI·
Ί)
Here {{•··}} is the subspace of R""" spanned by { · · · ( . Further set {](&>)} = {1
ω( 1), ω( I) + 2
ω( 1) + ///. ω(2) + I . ω(2) + 3. ... , tu(//,) + ,;/ + 1 /, + /,,).
that is, (7(ω)} = {1,2
/// + //}- {/'(ω)}
(in this order).
Then Χ ω is given by the equation xjyw) = 0,
j = 1
it.
+
We denote by X w the space Χ ω oriented with the basis ordered as in (I) and by Χ ω the A'(U with the opposite orientation +
„
Let Ν
be the set of all elements of R w
η .in
whose normal projections to J
+
X w are nondegenerate and orientation preserving. The definition of N m is similar. Then we have t/(w)-dimensional open cells U = N ηZ7 , LJ = Λ' η V . to ω ω <ο (υ ιο Further, we have (U U V =V . v ω ω'
82
. IMMERSIONS OF
MANIFOLDS
ι Without orientation wc can characterize Ν ω or N ( u by m 1=1 7 = 7(ω),
7= 1(ω),
(2)
and we have i,, = 0,
7 > ω(1),
(3)
± for subsets of W contained in (7 . The above fact follows from (2) and (3). +
-
THEOREM 3.6 (Pontrjagin). The set { U l 0 , T/J ω e Ω(/ι, m) } gives a cell division of R n m . Ο
We denote this cell division by . Just as avove we obtain a cell division of R„ m by letting N w be the set of all elements of R n whose normal projection to X is nondegenerate and setting U=N, η £7 · tu tu ω * we see that ϋ ω is an open i/(a))-cell. THEOREM 3.6' (Pontrjagin). The set { (7 J ω 6 Ω,(η,ηι)) gives a cel division for R n m . We denote this cell division K { x ) . Henceforth, for simplicity we only discuss R,( m , the case for C„ „, being similar. The case R„ m is slightly more complicated with signs (for detail, see Wu [C23]).
PROPOSITION 3.3. The coboundary err
X—*
/,
where the sum runs through -1 depending on ω and i.
5U
t0
of the cell L' w is given by
/ • ,W(/) + l + /M+1 ,
.,
all i with ω(1) e Ω(η, in) and ηω
Now for ω with δΙΙ ω = 0 we denote by {ω} the cohomology class of Κ (χ ) represented by U w. If δ ϋ ω = 0(2) we denote the mod 2 cohomology class r. presented by U w
{ω}2. Β. Characteristic classes. Here we define characteristic classes of vector bundles and of C\n i fty manifolds. We shallfirst define characteristic classes of Grassmann manifolds R„ _ and R„ . π , ftt
η , ni
is
83
§3. CHARACTERISTIC CLASSES
DEFINITION 3.7. We call
^ ' = K"},6//*(R„ TF* = K " } 2 € / / * ( R
m
,Z2)
or
e//(R„ , „ , Z 2 ) ,
,.Z2)
or
e//(R„
Z2),
the k th Stiefel-Whitney class and the k th dual Sticfcl- Whitney tively. Wc call A"" = {w'"}e//'"(R ,Z) or /l"'(R ./.,) the Eitler-I'oincare 1
in '
> η.ΗΓ
'
x
ii ,m
clas
2'
class. The cohomology classcs I>U
= { < .2, } e //4A (R„.„,, Z)
or
e I f " (R„ „, , Z).
^ = K ; . . 2 , } € / A R „ , „ , Z ) or 6 /•/·"(R Z) are called the 4A'-th Pontrjagin class and 4A -th dual Pontrjagin class resp tively. Further C 2 ' = {ω"'} e //2* (C„ ,„, Z) is called the till Chern class. NOTE The above definitions agree with their usual definitions such as given in Milnor and Stasheff [A5], except Pontrjagin classes are off by two components. For instance the Stiefcl-Whitney classcs defined here satisfy the axioms for the Stiefel Whitney classes in the book of Milnor and Stashed" and so they must be the same by the uniqueness axiom. The same goes for our definition of Chern classcs, and so our Pontrjagin classes agree essentially with Milnor's Pontrjagin classes. Now we are ready to define characteristic classcs of vector b: ndlcs. Before doing so we need the following PROPOSITION 3.4. Let Κ be a k-dimensional locally finite complex, a let Ρ = \K\ be its polyhedron. Recall the natural map defined in C '„ R „.„, — Bo = Ιί™°(" + '">/°('"> * 0 ' ( " ) · which induces the map (/J, : [P. Rf| map ( i' n ). is a bijection.
mJ
— [/',
tf 0|Hlll·
H k < n. th
PROOF. This follows from the fact that the Sticfcl manifold θ(/ι + I )/θ(n) is homeomorphic to the //-sphere S" . •
COROLLARY 3.1. Let Κ be a locally finite k-complex. and let I' = |A' be its polyhedron. If k < η. there is a one-to-one correspondence equivalence classes of m-vector bundles ί over I' and [/', R ; ]. The above correspondence is given as follows. Recalling that R;l sists of /// dimensional subspaccs of R"'"'. we define En by £„.,„ = { ( * ·
m
con-
e R » . m - " e X } c R„.„, χ R " + w .
and define ρ : Ε — R by p(X , it) r η .in n .m J r vector bundle which we will denote by y n m [P . R(| m ] the equivalence class of f'y n m
= X . We now have an m' . We then assign to each { /'} e .
. IMMERSIONS OF
84
MANIFOLDS
Let Ρ = |/L| where Κ is a locally finite A-complcx, and let ξ be an ///-dimensional vector bundle over Ρ. DEFINITION 3.8. By Corollary 3.1 the bundle ζ is induced by a map / : Ρ -> R n m for a sufficiently large η , i.e., ξ ~ fy n m . Put \ν\ζ) = f'w x m(t.)=
fx
k
m
e H k(P, z 2),
W k (ξ) = f'W
e H m(P , Z),
k
6 H k(P, z,),
P* k (ξ) = f P ik e H U(P,
Z).
We call the kill Stiefel Whitney class, W k (ζ) the dual kill Whitney class, Χ"'(ξ) the Euler-Poincare class and P Ak (ζ) the A jagin class of ξ . They are characteristic classes of the bundle ζ . It is easy to see that these definitions do not depend on the choice of η . If we consider for example the natural embedding I . D Ο n,n+I η,m n+1.m' we have (/ η , Λ + 1 )*{ω™} 2 = {ω™} 2 .
We next define characteristic classes of C\n itfy manifolds. DEFINITION 3.9. Let Μ be an ///-dimensional C\n itfy manifold. By a characteristic class of Μ we mean a characteristic class of the tangent bun dle τ (Μ) of Μ : W k(M) = W k(r (M)),
W k(M)
= TF A'(t(M)).
X"\M)
P U(M)
= l> Ak(T(M)).
= Χ"'(τ
(Μ)),
These characteristic classes have the following properties. PROPOSITION 3.5. Let Κ be a locally finite Let ξ and η be vector bundles over Ρ. 1. If ξ ~ η , then
k-comple.x.
and let Ρ =
W k (ξ) Ρ*"(ξ) 2. If
= Ρ"(η),
ε is the trivial
Χ(ξ)
= Χ(η).
vector bundle, then
IF'(e) = 0. Ρ"{ε)
= 0,
.V(e) = 0,
i > 0, k >0, dime > 0.
3. For an ιπ -bundle ζ, we have PF'(£) = 0,
/>///,
/,4*(O 4. For an m-vector
by
Wtf)=l
= 0. 4k >m. bundle ξ. define the total Stiefel-Whitney
+ w'tf) + -- + W'"(4)6H'(P,
Z,).
§4. IMMERSIONS AND CHARACTERISTIC CLASSES
85
Then
Ψ(ξ@η)= W(f)HOf), where ζ Φ η is the Whitney sum of ξ and /;. This is called th duality theorem. 5. If a k-frame field exists for ξ . that is. ξ admits k linearly cross sections at every point, then Iν·"-^\ξ) = W m ~ M £ ) = ... = IF'" (ς) = 0. The proofs of 1-3 arc immediate from the definitions. Wc omit the proofs of 4 and 5. See, for example, Milnor and Slashed' [A5], Property 5 represents a geometric significance of the Stiefel-Whitney class. We may think of characteristic classcs as algcbraic expressions for the geometric properties of C\n itfy manifolds. §4. Immersions and characteristic classes In this section we give a necessary condition for the immersibility of an ///-dimensional C\n itfy manifold M'" in Euclidean (m + A)-space R'" +A in terms of characteristic classes. THEOREM 3.7. If we can immerse an m-dimensional in R " ' u . then we have TF'(M'") = 0,
i>k.
P A j (M"') = 0,
2J > k .
C\n itfy manifold
PROOF. Suppose that M'" is immersed in R"' + t . Let / : AI — BO(m) denote the classifying map for the tangent bundle r(M'") of M'" . Here BO(/ZZ) is the classifying spacc for the orthogonal group Ο(///): BO(/;z) = lim 0(m + n)/0(m) (cf. §5, Chapter I). Consider Rk sional subspaces in
R"' +a
m
, the Grassmann manifold of //z-dimen-
. We may think of R(
R k, nl = 0(m
χ O(n) .
+
m
as
k)/0(m)x0(k).
Let t k be the natural map from R;. to BO(///). Let g : M'" -» R ; m be the map which assigns to cach point Λ* of Λ/'" the tangent spacc Ί\.(ΑΙ'") to M"' at χ. Then we get the following homotopy commutative diagram (cf. Proposition 3.4): M"'
BO(///)
s \
/h R<.
and we have
.m
. IMMERSIONS OF
86
MANIFOLDS
(see the previous section concerning {ω"'} 2 , {co'y have
Hence, the theorem follows.
But for RA
in
we
i>k,
{<'} 2 = 0, K . 2 , } o = 0.
2j}0).
2 J>k-
•
COROLLARY 3.2. Lei RP" be the n-dimcnsional cannot immerse RI'" in R 2 " - 2 if η = 2'.
real projective
spa
PROOF. Suppose we immersed RP" in R 2 " - 2 . Then by Theorem 3.7 TF'(RF") = 0, i > η - 2. But we know that the Sticfcl Whitney classes of RP" arc as follows:
H'(RP"
, Z2) = Z 2[x] , degχ = I, .v"+l = 0, W(RP n) = (1 + ,v)" +l
(see for example Milnor and Stasheff [A5]). So
W{RP")
= (I +Λ-Γ""'.
Since η = 2 s we get W (RP ) φ 0 which is a contradiction. • The corollary is due to R. Thom and was incorporated in his thesis of 1952 (cf. [CI 8]). On the other hand, H. Whitney whose research in singularities of c x maps dates back to 1930, obtained the following THEOREM 3.8. Let η >2. Then manifold M" in R 2 " - 1 .
we can immerse an n-dimensional
The proof will be given in the next section. Combined with Corollary 3.2, Whitney's theorem offers the best result in a general setting. §5. The Smale-Hirsch theorem and its applications In this section we state the Smale-Hirsch theorem and its applications. This is the most fundamental result in the theory of immersions. The proof will be given in §9. DEFINITION 3.10. Let Χ, Y be topological spaces and let / : X — )' be a continuous map satisfying the following: (i) / induces a one-to-one correspondence between the arcwiscconnected components ot X and the arcwise-connected components of Y . (ii) For each x 0 of X , the induced homomorphism /.: π,(AT. x 0)~ is an isomorphism for all 1 > 1 .
n,(Y,
/(*„))
65 THE SMALE-HIRSCH THEOREM AND ITS APPLICATIONS
87
We then say that / is a weak homotopy equivalence —to be abbreviate by w.h.e. Let Μ and V be C\n itfy manifolds of dimension η and ρ respectively. Let Imm(A/, N) be the space of immersions of Μ in V with the C°° topology. Let Μοη(Γ(Μ), T{V)) be the space of monomorphisms of the tangent bundle T(M) of Μ into the langcni bundle 7"(F) of V with the compact-open topology. Here by a monomorphism of vector bundles mean a bundle map whose restriction to each liber Ι\(ΛΙ) is a vector space monomorphism (see §2, Chapter 1).
THEOREM 3.9. Let Μ and V be as above with η < ρ . Then the map d : Imm(M, V) —< Mon(/"(M), /(F)), / - df sending a C°°
map j to its differential
df
is a weak homotopy
This theorem which wc will refer as the Smale-Hirsch Theorem was fir proved by S. Smale in 1959 for the ease Μ = S" . F = R'', and later in 1960 by M. Hirsch for the general case. Starting with Smale's work, llirsch proved the theorem by induction on each simplex over each skeleton using C\n itfy triangulations (to be explained in the next scction) of c\infty manifolds. We save the proof for §9. For now we discuss some applications and corollaries of the theorem. DEFINITION 3.11. Let A.' and V be C\n itfy manifolds of dimensions η and ρ , η < ρ . Let / : Μ —> V be an immersion. By a normal r-frame field of or /(M) wc mean a cross section of the bundle associalcd with the normal bundle of /(M) in V , whosefiber is the Sticfcl manifold I7 , , . THEOREM 3.10. Let η < ρ .
(i) If wc can immerse M" in R'" r with a normal ι -frame held, can immerse M" in R''. (ii) Conversely, if we can immerse M" in R'' then we can imm in K' 11' with a normal r-frame field. PROOF, (ii) is clear,
(i) Suppose that / : M" -> R''" js an immersion of M" in R''" which admits a normal r-fraine bundle. Then we have /•r(R'+')~7W")®/V+'-\ r + where u '~" is the normal bundle of /(M") in R ' H f . Bui the existence of a normal r-frame field implies that p+r-n
rP-n
r
υ ~ς Φ ε/(Λί), where denotes the r-dimensional trivial vector bundle over f(M). the other hand, since η < ρ the natural map [At" , BO(//)] — [Μ" , BO(// + t-)] is a bijection. Hence, we have T{M n)®rr r ~ ζ,.
On
111. IMMERSIONS OF
88
"
MANIFOLDS
Thus, Μοη(Γ(Μ), T(R P)) is not empty. Hence, the Smale-Hirsch theorem implies that lmm(M" , R p ) is not empty. • From these theorems we obtain various solid examples. DEFINITION 3.12. We say that M n is parallelizable T(M") is trivial. COROLLARY 3.3. If
if its langcnl bundle
it can be immersed in R"
M" is parallelizable.
This is obvious from Theorem 3.10. COROLLARY 3.4. We can immerse a three dimensional Euclidean four space R 4 .
closed man
PROOF. Let M } be a closed three manifold. By Whitney's embedding theorem, we can imbed Μ in R 6 . It is enough to show that this embedding has a normal two-frame field. But the obstructions to the existence of a normal two-frame field lie in H ' ( M 3 , ,(K3 2 ) ) . Thefirst obstruction is the dual Stiefel-Whitney class W 2(M i), which is known to be zero (see Remark 1). The second obstruction is also zero since π2('/3 2) = 0 ( s e e Remark 2). Thus, there is a normal two-frame field. • •>
3
REMARK 1. The two dimensional dual Stiefel Whitney class W"(M ) of a three dimensional closed manifold is zero; we can show this as follows. Imbed M 3 in R 6 with the normal bundle u. Then τ ( Μ 1 ) φ // ~ t . By the Whitney duality theorem we have W\is)
= W'(
IF(T(M3))IF(//) = IF(E6) = I
. But
Λ/ 1 ).
Therefore, TF2(M3) =
But
W 2(M i) + (W
l
(M*))
2.
W 2 holds for any three-manifold. Hence, we get I F 2 ( M 3 ) = 0 . REMARK 2. Κ, 2 w50(3) Κ R F 3 implies /R2(F3 , ) = 0 . REMARK 3. A three-dimensional open manifold can be immersed in R4 ('). This fact follows easily from a theorem of Phillips which wc shall slate in §8. DEFINITION 3.13. A manifold Μ is open if each of its connccted component is noncompact. (IF1)2 =
COROLLARY 3.5. Let M" be an n-dimensional closed manifold. that η = 1(4). Then we can immerse M" in R . (')J. H. C. Whitehead, The immersion London Math Soc., 11(1960), 81-90.
of an open h manifotd
in Euclidean
As
3-space.
§6. C TRIANGULATIONS OP A C' MANIFOLD.
PROOF. Immerse M" in R2" and consider constructing a normal twoframe field. Thefirst obstruction to do so is W (ΛΓ), which turns out to be zero by Remark 1. We also have , (Kf 2 ) = 0 for // ξ 1(4) (sec Remark below). Hence, the second obstruction is also zero, and so we can construct a normal two-frame field. • REMARK. We see this by considering the homotopy exact sequence of the bundle ρ : 0(n)/0(n - 2) - θ(//)/θ(/ζ - 1 ) = S"' 1 . Now let us prove the theorem of H. Whitney which was stated in the previous section. PROOF OF THEOREM 3.8. (1) The case where M" is open. By the SmaleHirsch theorem it is enough to show that Μοη(7'(Λ/ ), r(R""~')) φ 0 . To do this we only need to show that the associated bundle of T(M"), whose fiber is I / 2n _ 1 n admits a cross section (see the remark below). Now the obstructions to do so sit in H'(M n , 7r|_|(F2n_, H )). But π ,(K , ) = 0, /'
Y„).
= iZ "· j' ι ADDENDUM. A. Phillips in his paper "Turning a sphere inside out" turns S 2 inside out in R 3 through smooth deformations allowing sclf-inlcrscctions (see A. Phillips [C14]). This fact was also made into a movie("). The article of Phillips together with the movie give a concrete demonstration for one particular ease of the statement "immersions of S 2 in R3 are all regularly homotopic". The quotation follows from the Smale-Hirsch theorem and the equalities zr2(F3 i 2 ) = jr2(SC>(3)) = 0. Y,
aj, xr
/ = 1
§6. C' triangulations of a C' manifold In this section we discuss C' triangulations of c r manifolds, a tool with which Hirsch generalized Smale's theorem on immersions of the sphere. ( 2 )N. L. Max. Turning
a sphere inside oiti. International Film Bureau Inc.. Chicago.
90
IMMERSIONS OF
MANIFOLDS
DEFINITION 3.14. Let Κ be a locallyfinite simplicial complex, and let |/F| be the polyhedron of Κ . Let X be a topological space. We say that (Κ , f ) is a triangu lation of X if f : |ΑΓ| —» X is a homeomorphism. Let (Κ , / ) , (AT,, y*,) be trianguiations of X . We say that (AT, , /j) is a subdivision of (Κ, f) if the map /"'o/.rlJg—
sends each simplex of linearly into a simplex of . We may also say that the triangulated space (Λ"; AT, , /j) is a subdivision of the triangu space (X; Κ, f ) . DEFINITION 3.15. If simplicial complexes A", and K 2 are isomorphic, wc say that the triangulated spaces (A",; K f , /,) and (X 2 ; K 2 , f 2) arc isomorphic, and write (A", ; * , , / , ) = (X 2 ; K 2 , f 2) (cf. Spanier(3)). We say (A,; AT,, _/j) and (X 2 ; Κ., f 2) are combinatorial/)' equivale when each has a subdivision isomorphic to a subdivision of the other. DEFINITION 3.16. A triangulated space (Χ \ Κ, f) is called a combinatorial η-cell if it is combinatorially equivalent to an //-simplex. DEFINITION 3.17. A triangulated space {X ; A, /) is an //-dimensional combinatorial manifold if the star at each vertex of Κ is a combinatori //-cell. We are now ready to define C r triangulations. DEFINITION 3.18. Let M n be an n-dimensional c r manifold and let (Κ, f) be a triangulation of M" . We say that ( A , / ) is a c r triangulation of M" if for each //-simplex a , f\o : a — M" is a c r map (this means that with Κ considered as a simplicial complex in Euclidian space R" of a high enough dimension we can extend / to a C' map in some neighborhood U{a) of a in R w ), and the degree of / at each point of Κ is η . If (Κ , / ) is a C' triangulation of Μ" = (M n , , the c r structure 3 is compatible with the tringulation (Κ , /). In 1940 J. H. C. Whitehead proved the following( 4)
THEOREM 3.11. Assume I < r < oo. Then: (i) A closed C' manifold admits a C' triangulation ( K,f ). (it) If(K,f) is a C' triangulation of a C' manifold M n , the tr lated space (Μ" \ Κ, f) is a combinatorial n-manifold. (iii) Let (Κ χ , f ) and [K 2, f 2) be triangulations of a C' manifold Then the triangulated manifolds (Μ" ; A', , f) and (M" ; A'2, f 2 binatorially equivalent. We omit the proof( 5).
( 3 )E. H. Spanier, Algebraic topology. MacGraw-Hill, New York, 1966. ( 4 )J. H. C. Whitehead, On C 1-complexes. Ann. of Math. 41 (1940). 809-824. ( s )See for instance Munkrcs, Elementary differentiaI topology. Ann. of Math. Stud.. P ton Univ. Press, revised ed., Princeton, NJ. 1966
57- GROMOV'S THEOREM
§7. Gromov's theorem The thesis of Gromov,(*) 1969, established a generalization encompassing the Smale-Hirsch theorem and the Phillips submersion theorem. The idea of the proof is essentially similar to the method Poenaru exploited in his proof of the Smale-Hirsch theorem(6), that is, the idea of handlebody decompositions. In this scction we first state Gromov's theorem. In §8, we shall citc some corollaries of Gromov's theorem: the Phillips theorem, the Gromov-Phillips theorem, and others. In §9 we prove Gromov's theorem. Let Μ be a C\n itfy manifold. Wc define a local diffeomorphism of Μ to be a diffeomorphism / : U — V , where U and F arc open subsets of Μ . We may composc two local diflcormorphisms / : U — IF and g : IF — V only when f(U) \subsetIF. The set 3>(M) of local diffcormophisms of Μ is a pseudogroup, which wc shall call the pscudogroup of local diffcomorphisms of Μ (see Chapter IV for the definition of pscudogroups). DEFINITION 3.19. Let (E, p, M) be a smoothfiber bundle. Wc say that a map Φ : 9ϋ(Μ) — 9(E) is an extension of &(M) to 9(E) if it satisfies: (I) For fe3l(M), f : U — V , the map 0(f) : n~\U) - // 1 (Γ) is a difTeomorphism and the following diagram commutes: — I ι ι II
<£(/)
Ρ (U)
· Ρ
(/
(I ) F
ι (2) Φ(1 ( ,)= l „ - , ( n . (3)
(p'v\n
Λ*) ! Ρ | v
U
AP~ lm
— P~\V)
I V
^U
Then Φ' is an extension. (*)Editor's note. For Gromov's thesis see M. L. Gromov, Smaduuskoc onnodrazhclukh seoekuu ν ukoioodrayukh. Uzv. Akad. Nauk serukh Mam. T. 33. Η-4. 1969. pp. 707-734. ( 6 )V. Poenaru. Regular homotopy and isotopy. mimeographed. Harvard Univ.. 1964.
92
. IMMERSIONS OF
MANIFOLDS
DEFINITION 3.20. Let (£, ρ, Μ) be a smoothfiber bundle. For an open set U in Μ let Diff\infty(i7) be the set of diffeomorphisms of U onto itself. We give DifT\infty(i/) the compact-open topology. An extension Φ of £?(M) to 3>[E) is continuous if for an arbitrary open set U of X the map Φ : Diff\infty(i/) —. Difr\infty(//"'([/)) is continuous with respect to the compact-open topology. In each of the above examples we have a continuous extension. In what follows, all our extensions will be continuous. Let (£,//, Μ) be a smooth fiber bundle, and denote by Γ°°(ρ) (or r\infty(£)) the set of C\n i fty cross sections in (£, ρ, M) . Let Ε' ω be a subbundle of the r-jct bundle (E r , ρ' , M) and write ρ Γ ω = p'\E r b). Denote by T^(p) (or Γ^(£)) the set of elements / of Γ°°(ρ) with the property that J rf(x) 6 Ε' ω for each χ 6 Μ . We give Γ°°(ρ) the C°° topology and Γ^(ρ) the relative topology. Then the map —
r°( P rJ
is continuous, where Γ°{ρ' ω) (also written Γ 0 (£ ( ^)) is the set of C° cross sections of (Ε' ω, ρ' ω, Μ) with the compact-open topolgy. THEOREM 3.12 (GROMOV'S THEOREM). Let (Ε, ρ, Μ)
be a smooth fiber
bundle and let (E r , p r , M) be the r-jet bundle of germs of of £. Assume (a) Μ is open, (b) Ε' ω is an open subset of E r , and (c) Ε' ω is invariant under the action of 9(Μ) that is, for fe9(M) we have Φ Γ (f)(E'j c E[ o. Then the map
c\infty
by the extension
is a weak homotopy equivalence. REMARK. This theorem does not work in general if Μ is a closed manifold. For example, consider Μ = S 1 and £ = T(S') = S' x R 1 . Then £' = /'(S 1 , R 1 ), and thefiber is j ' ( l , 1) = M(1, 1; R) = R. Let E^ be the associated bundle of £' , whosefiber is GL( I, R); £g is open in £ and is invariant under the action of 3>(S l) . Now r 7 ( £ ) = Imm(51 , R ' ) = 0 but Γ ο (Ε ι ο )φ0. §8. Submersions: The Phillips theorem This section features the Phillips theorem. Let Μ and V be c\infty manifolds of dimensions η and ρ respectively. Denote by Sub(M. F) the subspace of c\infty(M, V) consisting of all submersions of Μ in V . Denote by Epi(T(M),T(V)) the set of epimorphisms of the tangent bundle T(M)
§9. PROOF OF THE SMALE-HIRSCH THEOREM
93
of Μ in the tangent bundle T(V) with the compact open topology. Here φ : T(M) —> T(V) is an epimorphism if φ is a homomorphism of vector bundles, which is surjective on each fiber T X (M) (see §2, Chapter II). If /: Μ V is a submersion, its differential df : T[M) —> T(V) of / is an epimorphism. THEOREM 3.13 (PHILLIPS). Let Μ be an open manifold.
d : Sub(M, V) — Epi(T{M), f '
Then
the map
7(F)),
* df
is a weak homotopy equivalence. PROOF. Let η and ρ be the respective dimensions of Μ and V with η > p. Set Ε = Μ χ V , π = /?, . Then we get the onc-jcl bundle /;' = j\M , F) whose liber is ./' (/),//) χ V = Μ (//,//; R) χ I7 . Denote by Σ the subspace of all matrices in M(p, n \ R) each having rank less than or equal to ρ - 1. Then Σ is closed in M(p , n \ R). Put Ω = Μ (ρ, η ; R) - Σ. The subbundlc of corresponding to Ω χ V is invariant under the action of 3>{M) by the extension Φ 1 :3f(M) — 9(E [) defined by '(/) = df . Hence by Gromov's theorem, Ε1
j ' : r\infty(E) — Γ V ' , ) is a weak homotopy equivalence. A careful reflection on this fact yields natural isomorphisms φ, ψ making the following diagram commute: r\infty(/·)
Γ"(/φ χjΨ
φ
Sub(M, F) — E p i ( 7 ' ( M ) , T(V)). The theorem is obvious for //
Ο
§9. Proof of the Smale-Hirsch theorem The proof given here is due to Phillips [C14], Let Μ bean //-dimensional C\n itfy manifold and let Κ be a p-dimcnsional C\n itfy manifold. Assume // F a.^ equivalent if f 0 and f are regularly homotopic. Wc consider a fixed ρ - η dimensional vector bundle ν over Μ , and let ImmJA/, A') be the space of immersions whose normal bundles arc equivalent to //. Denote by Mon( / (Γ(Μ), T(N)) the set of monomorphisms φ of 7"(M) in T(V) such that φ'(Τ(ν)]φ(Αί)/φ(Τ{Μ))) is equivalent to ν . Here φ is induced by Φ-
94
111. IMMERSIONS OF
"" MANIFOLDS
Letting E(v) be the total space of ν we obtain the following commutative diagram Sub(£(i/), F) ——• Epi(7'(M) φ ν, T(V))
1
I
lmm v(M , V) —-—> Mon t(T(M),T(V)) where the vertical maps are naturally induccd. The first line is a weak homotopy equivalence by the Phillips theorem. Each vertical line is a homotopy equivalence. Thus, the second line is also a homotopy equivalence. Since the above statement is valid for every // , we have proved the SmaleHirsch theorem. • In Chapter IV we shall give another proof of the Smale-Hirsch theorem without using the Phillips theorem. §10. The Gromov-Phillips theorem Now we examine the Gromov-Phillips theorem which was independently proved by Gromov and Phillips at about the same time, and which later attracted much attention when used by Haefiiger in the proof of his classification theorem for foliations of open manifolds. Let Μ be an ///-dimensional and Ν an /i-dimensional C\n itfy manifolds. Consider a Ac-planefield on Ν, that is, a k-dimensional subbundle /; of the tangent bundle T(N) of N. Let u be the quotient bundle Τ(Ν)/η, and let π be the natural projection π : T{N)
— .
v.
Denote by Ερί(Γ(Λ/), v) the space of the cpimorphisms between T(M) and υ with the compact-open topology, and denote by Tr(M, //) the spacc (with C 1 topology) of all c l maps f \ Μ — Ν cach π ο df belonging to Ερί(Γ(Μ), //). Then we have the following THEOREM 3.14 (GROMOV AND PHILLIPS).
Lei Μ be an open manifold.
Then the map π ο df : Tr(M, η) —• Epi(7"(M), u), f^Kodf is a weak homotopy equivalence. PROOF. Set Ε = Μ χ Ν and π = ρ χ (the projection onto thefirst factor). Then we have Ε 1 = j \ M , N). Let Eg be the subbundle of one-jets which are transverse to the given Ac-plane field η . Then is an open subbundle invariant under the action of 3>(M) by Φ', where Φ' : 2)(Μ) — 3>(E X) was defined in Example 2 in §4. The Γ°(£,!) corresponds to Epi(7"(M), v).
§11. HANDLEBODY DECOMPOSITIONS OP
MANIFOLDS
95
This is shown in the diagram below C"(M.
d
N)
•> Horn (7'(M). T(N)) Ο Epi(7'(M), r)
\4, /;) «1 "Ι
«v
.-ι
1"
ψ II
R()°°(£) - I — » Γ ° ( £ ' )
•I 0 I (/'-") Hence, the theorem follows from Gromov's theorem.
•
§11. Handlebody decompositions of C\n itfy manifolds In preparation for proving Gromov's theorem, this section oilers a brief discussion of handlebody decompositions of C' x manifolds (for further details, see Smale [CI7a]). Let Μ be an /i-dimensional compact C\n itfy manifold, and let Q be a connected component of the boundary OM of Μ. Consider embeddings f :dD'xD"
J
— >Q,
1=1
k , 0 < s < //.
whose images are mutually disjoint. Introduce an //-dimensional compact c\infty manifold Γ = χ(Μ , Q\ /, , ... , f k\ s) as follows: the underlying manifold of V is
(a- , y) ~ /(Α-, .ν),
(a- , p) € OD] x D\f(x , V) e Q \subsetΜ" .
which has the usual smooth structure everywhere outside OD* x D" ' . By "smoothing out" the "corners" along dD* x D"~ s we will have the c v manifold V (Figure 3.2). We often write V = χ(Μ; f ( f k ; s) in ease OM" = Q. The embedded D' x D"~ s in V are called s-handles. DEFINITION 3.21. Lei V - χ(Μ, Q ;/,,···. f k ; s). Wc say that σ = {Μ, Q; /,, ... . f k ; s) is the presentation of V . A manifold with a presentation (D n ; /j f k ; s) is called a handlebody. We obtain the manifold I' = χ(Μ, Q;/,,.... f k ; s) by attaching s-handles D\ x D"~ s D' k x D k to Μ . More generally, we say that V = χ(Μ , Q; /j f k ; s) is a handlebody if Μ is a handlebody.
THEOREM 3.15. Let W be a compact manifold and let f : W —< R1 be C\n iy tf function such that the only singularities in /~'(-ε. ε) are nondeg of index λ lying in f~ (0). Further assume that Ν Π <) IF = 0 and
96
111. IMMERSIONS OF
MANIFOLDS
FIGURE 3.2
/ " ' ( - ε ) is connected. Then / " ' [ - ο ο , ε ] has a presentation ( Γ ' [ - ο ο , - ε ] , / " ' ( - ε ) ; / , , . . . , / , ; A).
of the fo
OUTLINE OF THE PROOF. Let /?,, ... , P k be the singularities of / in
f~ (0), and choose neighborhoods ^ of β,•, i = I, ... , k , which are mutually disjoint. Then in F , / can be written
/w = -Σ>,2 + Σχ1 ι=ι /=;,+1 with respect to the local coordinates χ = (χ jcf l), ||a|| < S , ό > 0. Denote by £, the (jc, , ... , x^-plane and by £ 2 the ( χ Λ + | , ... , .vn)-plane in V :. Then for a sufficiently small ε, > 0, £ , 0 / " ' [ - ε , , ε,] and D l are diffeomorphic (Figure 3.3). There exists a diffeomorphism φ: Τ 1 = Γ η / _ 1 [ - ε , , ε,] —> D x χ D"~ x ,
φ(ΤηΓ\-ε ])) = ϋΟ λ χ D"~ X, where Τ is some small enough tubular neighborhood of £, . Now comparing /"'[—oo, --ε,] and /~'[-oo. ε,] we see /~'[-oc, - ε , ] with a λ-handle attatched to it for each 1 agrees with / " ' [ - ο ο , ε , ] . Hence, we have proven the theorem. • DEFINITION 3.22. Let M" be an n-dimensional manifold and let / : M" —· R be a C°° function. We say that / is a nice function if ev?~y criti point β of / is nondegenerate and the index of β is equal to f(β) . THEOREM 3.16. Let M" bean n-dimensional admits a nice function. We omit the proof.
C\n iy tf
manifold.
Then
97
§12. PROOF OF GROMOV'S THEOREM
FIGURE 3.3
From Theorems 3.15 and 3.16 we see that an //-dimensional c x manifold without boundary is obtained by attaching handles to the //-dimensional disk D" one by one. §12. Proof of Gromov's Theorem In this section, following Haefiiger [B3], we prove Gromov's theorem('). The basic tool is handlebody decompositions of manifolds. Let Μ be an ///-dimensional C\n itfy manifold. If Μ is open, there exists a unique Morse function / : Μ -»[0, oo) such that the index of each critical point of / is less than /// (sec Chapter I). We arrange the critical points of / as follows (Figure 3.4): a t,a 2,
... ,
c,=/(a;).
1 = 1,2,...
c,, c 2 , ... are monotone increasing. (')Sec also V. Poenaru, llomolopy theory and differentiable Mathematics, vol. 197, Springer-Vcrlag, Berlin and New York. 1970.
singularities.
Lecture N
. IMMERSIONS OF
98
MANIFOLDS
There exist local coordinates (jc., ... , A*(h) about each a( (there is a local chart (U n,ip tt) with a ( 6 U l t ,
f = C i-X
2 i
2
2
x k + -V
t+ 1
2
+ •·•+.V,„,
where k is the index of (in other words, for each χ in f°
6
l / ( e , ) | * » = (.v,
xj , x 2 x + • • • + xl < | } .
Put Μ," , = M ( - W r In Figure 3.5 we identify V(a t) with „((/(
{(*./)!/<*,(*)}. for some C°° the union of
function g l:9M j_ l —(0, 1], Hcncc M( is difl'comorphic to and Ak , where Ak is diffeomorphic to D k χ D" l_l and
is diffeomorphic to a collariike neighborhood 13 of <)D k χ D'"~ (cf. Figure 3.5). We say that M-t is obtained by attaching a A'-handle to Λ/ _, (see Figures 3.5 and 3.6). Now Μ is the union of the following expanding sequence of compact
k
512. PROOF OF GROMOV'S THEOREM
99
FIGURE 3.5
manifolds with boundary: M,\subset· · ·\subsetM ( _,\subsetΜ,Ι,\subsetΜ,,\subsetΛ/"\subset· · · .
(0)
The three propositions below will yield Gromov's theorem. In what follows wc write (E(M),p, M) for (£,//, Μ), (£'(Μ),/Λ Μ) for (Ε' , ρ' , Μ), and a similar adaptation applies for (E' u(M) , //r , M). For AcM, (E(A),p,A) is the restriction of (E(M), ρ , Μ ) to A. Similar!; E r(A) is the restriction of E r(M) to A . PROPOSITION 3.6. Gromov's theorem is true if disk D'" ; that is , . r°° (£( D"' )) __ r°{E rJD"'))
Μ is the in-dimens
. IMMERSIONS OF
MANIFOLDS
is a weak homotopy equivalence. DEFINITION 3.23. Let Ε and Β be topological spaces, and let ρ : Ε —> Β be a continuous map of Ε onto Β . We say that (Ε , ρ, Β) is a fibration or afiber space if for any finite polyhedron Ρ and continuous maps F : Ρ χ [0, 1] -» Β and f : Ρ —> Ε with ρ ο f(x) = F(x), there exists a continuous map F : Ρ χ [0, 1] —> Ε satisfying (i) p°P(x,t) = F(x,t), (ii) F(x,0)=f(x). Ρ
Px'l
—
Ε
—ί—. Β
We also say that (Ε , ρ, Β) has the covering homotopy property Evidently afiber bundle is afibration (cf. Steenrod [A7]). PROPOSITION 3.7. Suppose that a C°° a C°° manifold Μ with a collarlike ary d Μ. Then the restriction maps ρ·.Χ Α[Ε
Γ ω{Μ^))-^Υ
are both homotopy equivalences
(CHP).
manifold \C is diffeomorphi neighborhood attached along
ΰ(Ε' ω(Μ))
and define
fibrations.
PROPOSITION 3.8. Let k < ///. Set A = D k χ D"'~ k . Β = D k p
χ D"'~
k
.
where
D k l n = {xeD k\\!2<\x\< I}. Then the restriction (i)
maps —
(ii) Ρ: r ° ( ^ M ) ) —are fibrations. Let ( Ε , ρ , Β ) and (Ε',ρ',Β 1) befibrations. A continuous map g:E — E' mapping eachfiber of Ε into somefiber of E' is called afiber map. In this case, the corresponding map g : Β — Β' between the base spaces is continuous and the following diagram commutes.
Β —ί— Β' LEMMA 3.1. Consider fibrations (Ε, g : Ε —> Ε', and the corresponding
Ρ
, Β) and ( Ε ' , Ρ ' . Β'), a fiber map map g : Β — Β' of base spaces.
101
§12. PROOF OF GROMOV'S THEOREM
that g is a weak homotopy equivalence. true. (a) If fiber
Then
g is a weak homotopy equivalence,
Ζχ
Km
Fx
the following
sta
then the restriction
o
'
is a weak homotopy equivavence. Here F x = p~ x (.ν), F' = p'~ 1 (g( (b) Conversely, if for each χ 6 Β the map g y : l-\ -—· F' {t ) i homotopy equivalence, then g too is a weak homotopy equivalence PROOF.
We consider the homotopy exact sequences of fibrations:
—,.(F) I — Tt,(F')
τt,{E)
π,(B) —
..[
,|
η,(Ε')
π,(β') —
/I,.,(/··) ^ I
//,_,(/:) — •·
.
π,_,(/·-')
//,_,(/:')
This diagram is commutative (Steenrod [A7]). Hence, the proposition follows from the "five-lemma". • Suppose two exact sequences of abetian groups commutative diagram :
T H E FIVE-LEMMA.
following
If
h , . hi . /z 4 , and lt i are isomorphism,
then so is h }.
We leave the proof to the reader. PROOF OF GROMOV'S THEOREM. The proof is by induction. Assume firsi that the theorem is true for a compact manifold Λ/ which consists of handles of indices less than k : that is, Proposition 3.6 is the starting point of our induction. Step 1. Let dim Μ = m, and construct a manifold M' by attaching a handle of index k to Μ, k < m. Assuming that the theorem holds for M, we shall prove that it is also true for A/'. As before wc let A/" be the manifold obtained from Μ by attaching a collarlike neighborhood along d Μ , that is,
A/' = A / "
UA,
ΑΓ DA = Β,
A as D k χ D"" d Β as Ο,
.
k
χ D""
k
C // χ θ " " * .
102
111. IMMERSIONS OF Γ"" MANIFOLDS
Consider the following commutative diagram:
"4
"Ι
ο
Γ^(£(β)) -!—. Γ°(E rJB)) By Proposition 3.8, ρ ω and ρ arcfibrations. By Propositions 3.6 and 3.7, the maps f : Γ\infty(£(/l)) - Γ*(E rJA)) and J' : Γ™(E(H)) - Γ°(/\ ' infty(/?) are weak homotopy equivalences, for we may think of II as the union of a 0-handle and a (k - l)-handle. Thus by Proposition 3.1, J' is a weak homotopy equivalence on each fiber. Look at the following commutative diagram.
"4 Γ?(Ε(ΛΓ))
"1 Γ°(Ε'
(2)
ω(λί"))
By restriction to A and Β we gel a morphism from diagram (2) to diagram (1). Γ„~(£(Μ')) /'„> !',„ Γ ω°°(Ε(Μ Γ„~(£(Α))— ^
> Γ»(£„;(Μ')) Jr x> Ί)) -j—» ΓalE wr(M~<))
(3)
*Γ 0 (£ ( /(Α)) / , ]/'
^Γ„~(£(β)) —
^
Γ 0(£„/(β)).
In (2) ρ ω and ρ arefibrations (cf. Proposition 3.7), so we have the homotopy exact sequences of fibrations. But by the induction hypothesis J r : Γ^(£(Μ^)) —> Γ°(Ε' ω(Μ")) is a weak homotopy equivalence. On the other hand the restriction of f : Γ^(£(Μ')) - Γ°(E rJM')) to eachfiber can be thought of via (3) as the restriction of the map in (1) to each fiber. The latter was a weak homotopy equivalence. Hence, it follows from Lemma 3.1 that J' : Γ^(£(Μ')) - Γ°(Ε'(Μ')) is a weak homotopy equivalence. Step 2. Since Λ/ is an open manifold, it has in general infinitely many handles. Decompose Μ into the sequence (0), M, C · ·· C A/(_, C M~l, C M ;\subsetM" C
(0)
§12 PROOF OF GROMOV'S THEOREM
103
which gives the following sequences of restriction maps <- ··· - C ( £ « - i ) ) - C ( £ ( " , ) ) - C ( £ « ) ) • Γ °(E rJM t)) - ··• - Γ°(£;(Μ,:,)) - Γ°(Ε' ω(Μ,)) - Γ °(E rJM~)) whose projective limits are l i i mC(£(M,)) = r\infty(E(A/)), limr°(£, >/,)) = Γ°(Ε'
ω(Μ)).
Hence, the theorem follows from Step 1 and the following lemma. LEMMA 3.2. We consider the following systems of topological spaces:
...
a i+>
. /I,
commutative
· At_ t
· ···
•
diagram for · .1,
where each vertical map _/ . : A: —. Β. is a fibration. homotopy equivalence, then the projective limit j = lim _/. : lim
If each
—· lim
is a/so a ivcaA: homotopy equivalence. We leave the proof to the reader (cf. Phillips [C 14]). It remains to prove the above three propositions. PROOF OF PROPOSITION 3.6. Thefiber bundle E(D'") is the product bundle D m χ F because D'" is contractible; therefore, wc identify cross scciions in this bundle with maps from D'" to /·" and the fiber of E'(D m) over 0 e D m with j' 0(D n\F)= \J f a xXD'" . F). yet
Evidently the restriction map ρ : Γ\E r JD
m ))
— Γ 0 (/ί ( >))
over the center 0 of D m is a weak homotopy equivalence. Hence, ii is enough to show that /zo/:r\infty(/· (//"))—· Γ ° ( Ε > ) ) is a weak homotopy equivalence. To show that (p o/). : n,{r^(E(D
m)))
— *,{Γ°(£>))>
is surjeclive, we think of the fiber F of Ε as a closed submanifold of Euclidean space R a (Whitney's embedding theorem). Let π : IF — /-" be a retraction, where IF is a tubular neighborhood of /·' in R . Let / : s' — r\infty(£ (>))\subsetJg(D"'
. F) \subsetJ r 0(D"' . R' V )
111. IMMERSIONS OF
104
MANIFOLDS
be a continuous map. We represent each jet of Jq(D"' of degree r and obtain a continuous map
, F) by a polynomial
Dm — R a ,
F-.S'x
where for each s e S' , the map F s : D"' •—» R A defined by F s(x) = F(s , χ) is c r differentiable, f(F s) is continuous in s , and j r (F,)(0) = /(s). Since F is continuous, there exists a neighborhood F(0) of 0 e D 1" such that F(S' χ V( 0)) C W . Set F' s = π ο FJF(O). Since E r w(D m) is an open subbundle, there exists a neighborhood (7(0) of 0 in K(0) such that Ε$'|(7(0) 6 Γ°(£'(t/(0))). Let A : D'" - {7(0) be an embedding which is the identity map in some neighborhood of 0. Define a map g : s' —> Γ°(E rJD "')) by g(s) = A - 1 ο F[ ο A . Then we have j'{g){0) = / . Here h is the induced map of A . Thus, we have shown that (Ρ ° f ) . ' s a surjection. The proof of injectivity is similar. • The proof of Theorem 3.7 is not very hard and is left to the reader. Hint: One can extend a cross section of Μ to a neighborhood U of Μ in M " . There exists an isotopy {/} of the identity map of M " satisfying the following: f
t
—» (7 is an embedding;
:
(ii) there exists a neighborhood V of Μ in M " such that ^ = x,
f,(x)
PROOF OF PROPOSITION 3.8.
χ € F , / e I.
In the following we simply write Γ°(Μ) for
Γ ω ( £ ( Μ ) ) and Γ(Μ)
for Γ°(E rJM)) . Set D\a.h\
=
RA'|a < |a-| < A } ,
= ( a'
= {
e R* |
A E RA|
|a
| < a },
|A| = a ).
Further, set A = χ , Β= χ D'" k . m The fiber bundle £ ( R ) is the product bundle R'" χ F (however, the action of the pseudogroup Z)(R"') of local diffeomorphisms of R'" is not necessarily trivial). Cross sections in this bundle are identified with maps from R"' to F . Let Ρ be a polyhedron, and let U be a subspace of R m χ Ρ . We say that a map / : { / — · £ is admissible if (a) / is of class C r on U η (R m χ {//}), V/; e Ρ , (b) j'(f) is continuous, and (c) /(/)(t/)C £'({/).
§12. PROOF OF GROMOV'S THEOREM
105
PROOF OF (ii). To show that Γ(Λ) — > Γ(β) is afibration, follow the argument that afiber bundle has the covering homotopy property, CHP (Definition 3.23). PROOF OF (i). The statement that Γ 0 ( Λ ) —» F 0 ( / I ) is afibration means the following. Given a polyhedron Ρ and continuous maps
—>Γ 0 (D* i 2 1 χ D"'~ k).
f:Pxl
So '· Ρ x (0) — Γ 0 (D* χ D"'- k). f(.\ , 0) = ρ ο g 0(. γ , 0), there exists a continuous map ^ :P χ /—. r 0(D
k
χ D'"~ k).
which extends g 0 with ρ ο g = f ; F χ 0 ———»
Γ
(D kxD""
k)
I" / Ρχί —L^
4 r 0(D* ,, X D"- k)
Equivalently wc can say that: given two admissible maps / : Of,
2|
χ D""
k
χ Ρ χ [0. 1] —* Γ.
g 0 : D k χ D'"~ k χ Fx {0} —« /•'. which agree on their common domains, there exists an admissible map g : D k χ D'"~ k χ /' χ [0, 1] —· /·" which extends both f and g 0 . We shall construct g in three stages. (a) We first extend / just a bit; that is, wc construct an admissible map f as follows. / : D kt
2|
χ D"'~
,, χ f\D\
k
χ
χ D""
k
χ Ρ x[ 0,1] — - P .
π< I ,
Fx [0, 1] = ./. χ Ρ χ {0} =
(b) We canfind an increasing sequence 0 = 10 < < · · • < ts = 1 such that for each η, 0 < // < s , there is an admissible map μ:ι defined in some neighborhood of D? ^ χ D"'~ k χ Ρ χ [t n , / f|+ | ] satisfying (i) μ η(χ,ν,ρ, U(D
kx 2])
t) = /(.ν, ρ,/;, 1), I = t a or ,v 6 U(D
is some neighborhood of
D kx
2) .
(ii) /t„(A-.p,/M) = //„(.Y.p,/;,i„), ,v €L'(S
k~').
k x 2 | ).
where
106
. IMMERSIONS OF
0
MANIFOLDS
I. /= [0.1)
1
FIGURE 3.7
k-1
-'"•
k
where U(S a ) is some neighborhood of S . If we do not require admissibility, such a map μ η can be uniformly constructed for each t. Since admissibility is an open condition, μ η is also admissible for a sufficiently small change in 1. Using the compactness of [0, 1], we may choose to construct the desired μ η . (See Figure 3.7.) Using the above μ 0, we can extend g to D 2 χ D"'~ k χ Ρ χ [0, /,] as follows. k p x e /J 0(x,y ,ρ,ι), |m2| , g{x, y, ρ, I) = g 0 (x,y,/>,o), χ i d|;, 21 . (c) Now we shall extend g 0 to g n inductively. Assume thai we have constructed the admissible map g„ : D\ χ D"" g n\D k xD""
2)
χ Ρ χ [0, I,,)—/· , xFx[0,g
k
SnWirft.H where
k
X D"" k
= g 0, X P
x
'.])
χ · · ·) is some neighborhood of D^, 2J χ D"'~
"
χ F χ [0, l n] .
Suppose μ η and f are defined on some neighborhood U C D k 2 χ D"'~ of D [ a χ {0}, and suppose U Π (D*, χ D m~ ) = 0 . Since k < m (this is becase Μ is an open manifold; see §4, Chapter I), m-k> 0, there is an isotopy At of D k χ D"'~ k , 0 < / < / „ , such that (1) zJ( is the identity outside U , in some neighborhoods of Sp χ 0 and χ 0 respectively, and for all t, t < lJ2 ; (2) (5,, x0) = S a x 0 for y satisfying β < γ < 1 . Wefirst construct g n+ i C = [{D\
in a sufficiently small neighborhood of the core χ 0)U (D k, 2] χ D'"- k)} x f x [ l , / J
k
FIGURE 3.8
as follows. In region I ( ||a || < β , 0 <'
In region III (y < Μ < 2 , l„ < 1 < /„ +| ), £„+,(*. J'.F. 0 =
(-ν, p).//, /).
where Λ and represent the actions of 0 ( J ( ) and Ψ(Λ( ) on thefiber /· ; Φ(Λ,) and Φ(Λ, ) are extensions of the diffcomorphisms and . In region IV (||JC|| < y, l„ < t < ),
In a small enough neighborhood V of the core C, the g /H , agree in the intersection of their domains. (Sec Figure 3.8.) Let //, be an isotopy of D k χ D m~ k such that (i) % = >· (ii) Λ, = 1 on [/(£>* 2 ] χ D m~ k) , (iii) h l(D kxD m~ k) \subsetK, t>t„/2, where l/(D* χ Ζ)"' - *) denotes some neighborhood of Dj, 2 j χ . Then it is enough to define the desired g fl +, by ,y.p,i)
= (Λ ( )"'^ 1+1 (Λ,(.ν, ρ).//./),
. IMMERSIONS OF
108
MANIFOLDS
where /z( is the action of the extension Φ(/ζ,) of the difTeomorphism h t on thefiber F . Q §13. Further applications of Gromov's theorem (A) Symplectic structures. Let Μ be an 2/z-dimensional C\n itfy manifold. We say that a closed differential two-form ω on Μ is a symplectic structure when ω" / 0. Here ω" = Μ Λ · · · Α oj . η A symplectic structure on Μ defines naturally an almost complex structure. In the following we shall explain this concept. Let Sp(2/z,R) bean 2/i-dimensional real symplectic group. Then we have 0(2η) η Sp(2n, R) = Sp(2//, R) η GL(/z, C) DEFINITION 3.24.
= GL(«,C)nO(2«) = U(/z).
DEFINITION 3.25. Let M 2" be an 2/i-dimensionaI C\n itfy manifold. An almost symplectic structure (or an almost Hamiltonian structur is a reduction of the structural group of the tangent bundle τ(Μ 2/1) to Sp(2n, R). DEFINITION 3.26. We say that a nondegenarate skew-symmetric bilinear two-form [ , ] : R 2 " χ R 2 " —» R on R 2 " is a linear symplectic structur that [ , ] is its skew scalar product. Here [ , ] is nondegenerate means that if [ξ, //] = 0, V// e R-", th ί = 0. EXAMPLE. Set R 2 " = { ( Ρ , , . . . ,p n, q s q l t)}, and define ω 2 by Ω 2 = Ρ , Λ <7, + • · · +
pn
Λ
q n.
ω2
Then is a nondegenerate skew symmetric two-form; so the scalar product [ , ] defined by [ί, η] = ω 2 (ζ , //) is skew. The two-form ω 2 is th c standar symplectic structure on R 2 " , which henceforth we will assume on R 2 " un otherwise stated. DEFINITION 3.27. A linear transformation S : R 2 " —» R 2 " is symplectic the skew scalar product is invariant under S; that is [Ξ(ζ),Ξ(η))
= [ζ,η],
νξ,ζ /eR 2 ".
The set of all symplectic transformations on R 2 " form a group, the 2n dimensional real symplectic group. If M 2 " has a symplectic structure ω, then for each point χ of Μ 2 " the tangent space Τ χ (Μ 2 ") enjoys a symplectic structure, and the structural group for the tangent bundle τ {Μ 2 ") of Μ 2 " reduces to Sp(2n , R); that is, AT" has an almost symplectic structure. On the other hand, since the above symplectic structure ω on M~" defines a symplectic structure on each tangent space Γ (Λ/"), the map I : T IM) Τ χ(Μ) defined on each Τ χ{Μ 2") by
513. FURTHER APPLICATIONS OF GROMOV'S THEOREM
109
K. »/] = (/£.»/). where ( , ) is the inner product of Τ χ(Μ) , satisfies = -1 . Hence, Τ χ(Μ) has a complex structure, which implies that we can reduce the structural group for the tangent bundle τ(Μ 2 ") of M 2 " tc GL(//,C). This also follows from the fact that U(//) and Sp(2/i, R) have the same homotopy type. EXAMPLE. Let V be an //-dimensional manifold. The cotangent bundle T'( V) = Ηοηι(Γ(Κ), R) of V is a 2/z-dimcnsional manifold which admits a symplectic structure. Taking local coordinates (.v, .v ) at each point λ* in V , we use the local coordinates (a-, λ π , dx dx n) at each point of T'(V) , and setting (_v, A'n, dx t dx n) = (//, p n, ... , q a) we define ω = dp Adq = dp } Adq t +
F dp n Λ dq n.
THEOREM 3.17. Let Μ bean In-dimensional open manifold. Then Μ has a symplectic structure if and only if Μ has an almost complex PROOF. Put Ε = Τ'[Μ) = Hom(7"(M), R). Let e] 0 be the subbundle of Ε 1 consisting of one-jets of one-forms α with (dtt)"/0. Then E'o is an open subbundle invariant under the natural action of ώ"(Μ). Now (/·.') is the space of one-forms with (da)" φ 0. Wc may regard as the space of two-forms β with β" Φ 0. On the other hand, we saw earlier that the set of two-forms β with β" φ 0 corresponds to an almost complex structure on Μ . Hence, the theorem follows from Gromov's theorem. • NOTE. This proposition is not valid if Μ is a closed manifold. One can define an almost complex structure on S b which has no symplectic structures.
(B) Contact structures. Let Μ be a (In + 1 )-dimensiona! C x manifold. DEFINITION 3.28. A one-form ω on Μ is a contact form if ω Λ (dw)" φ 0. A one-form a is a contact structure on Μ if
THEOREM 3.18. Let Μ be a (In + \)-dimcnsional open manifold A nec essary and sufficient condition for M to have a contact structure admit an almost contact structure. NOTE. Martinet showed that the theorem is true for three-dimensional closed manifolds( 8). ( )J. Martinet, Formes de contact sur varietes de dimension 209. Springer-Vcrlag, Berlin and New York. 1971, 142-163.
i. Lecture Nolcs in Math., vol.
110
. IMMERSIONS OF
MANIFOLDS
The theorem leads to a natural consideration: Is the same statement valid for almost complex structures and complex structures on open manifolds? We deal with this question in Chapter VI. We discuss applications of the Gromov-Phillips theorem in foliation theory in Chapter IV. ADDENDUM. In an historical perspective, Gromov's theorem and its applications are based on the covering homotopy method of Smale [ 17],
CHAPTER IV
The Gromov Convex Integration Theory This chapter is based on Gromov's work [C5]. Wc discuss a natural generalization of the Smale-Hirsch theory, which wc shall refer to as the Gromov convex integration theory. Wc feel that there arc various applications of this theory; in Chapter VI we will discuss one of them. §1. Fundamental theorem Let X and I) be η and q dimensional ( " x be a smooth fiber bundle. Let Ρ
manifolds. Let /) : .V — I
x' —' I
be thefiber bundle formed by the r-jets of germs of smooth cross sections in
(A', //, V) . Wc say that a subsel Ω of V is a dil/errniinl relation of order ι or a differential equation of order r. If the f-jet ./'(/') Γ — V' of
smooth cross section / : V —« X of (A', />. Γ) satisfies J'( ./')(!')\subsetΩ. we say that / is a solution of the differential relation Ω . If Ω is an open set wc arc considering a family of partial differential inequalities of order r, and if Ω is a closed set we will In· thinking of a family of partial differential equations of order / . We denote the set of cross sections in (A',//, Γ) by Sect(.V) ( ' ) . We give Sect(A) the relative topology of C\infty(I', X). We write Scct( \'r) for the set of cross sections of (A' r , ρ', I'), with the relative topology of f ' ( F . . V ' ) . Then the map ] ' \ Sect (A") —· Scct(.Vr). /-·/'(/) assigning t o each section its r-je1 becomes continuous. Set Sect(A\ Ω) = {σ e Scct(A'r)|rr(I') C Ω ). Sect(.V, Ω) = (/r'(Scct(.V'. Ω)). Then S e c t ( A ^ ) is the solution spacc of Ω. (') We wrote Γ(Α) in Chapter III; however, wc will use this for something else in Chapter V. Thus, this new notation. 111
112
IV. THE GROMOV CONVEX INTEGRATION THEORY
c«iv(y)
DEFINITION 4.1. We say that the w.h.e-principle holds for a differential relation Ω if the map J' : SectfA, Ω) —> SecttA^, Ω) is a weak homotopy equivalence, in which case (i) J r induces a bijection between arcwise connccted components, (ii) J r induces an isomorphism of homotopy groups
(J'), : 7r,(Sect(Af , Ω))
/ti(Sect(A,r. Ω)),
for cach 1,
1 > 1.
The fundamental theorem of Gromov gives a sufficient condition for the relation Ω to satisfy the w.h.e.-principle. For the moment we shall establish some notation for stating this fundamental theorem. DEFINITION 4.2. Let L be an affine space and Q \subsetL. We say that Q envelopes an element ,v of L if there exists a neighborhood U{.\) of ,v which is contained in the convex hull Conv(Q) of Q . Here the convex hull of Q is the smallest convcx set containing Q (Figure 4.1). If L is a Banach space and Q C Ω, the closure of the convcx hull of Ω is the smallest closed convex set containing Q . DEFINITION 4.3. Q is ample if each arcwise-connectcd component of Q envelopes every point of L . We agree that the empty set is ample. EXAMPLE,
(i)
Let
R",
L =
R"-1
=
{ (.v,
x n)
e R"|a'„ =
0}.
Then Q defined by Q = R" - R " - ' is not ample. ( i i ) Let
L =
R",
R"
- 2
=
{(a-
aJ
€ R"|a-„_, = * „
= ()).
Let
Q = R" - R"~ 2 ; then Q is ample. Let An be an affine transformation group of R" . A fiber bundle whose fiber is R" with the structural group An is called an affine bundle. DEFINITION 4.4. Let (Ζ , ρ, Κ) be an affine bundle. Suppose that the structual group An is reducible to Λ χ Aq . Further, we decompose the fiber Z k at each point k 6 Κ into z, = < ~ q χ Κ =
U
w*
R
Z·
Henceforth we shall write Rj* λ = {A} χ R' . A reduction of A to /I χ Aq together with a decomposition of eachfiber as above is called a direction of dimension q of (Ζ , ρ , Κ) .
SI. FUNDAMENTAL THEOREM
113
DEFINITION 4.5. Let (Ζ, ρ , Κ) be an affine bundle. By an affine embedding α : R' —» Ζ we mean a linear embedding of R' in the corresponding fiber. Further, suppose ( Ζ , ρ , Κ) has a direction of dimension q. We then say that an affine embedding a : R 7 —» Ζ is parallel to the given direction η embeds R ? in one of the R]J .. DEFINITION 4.6. Let ( Ζ , / / , K) be an affine bundle with an q dimensional direction and Q \subsetΖ , K Q C Κ . Wc say that Q is ample over K 0 if for any affine embedding η : R* - · Z t C Ζ , k € K 0 , parallel to the given direction, n " ' ( 2 ) c R ' is ample. Let {Χ, ρ, V) be a smoothfiber bundle whose base spacc andfiber are //-and (/-dimensional C\n itfy manifolds. Let (A'1 . ρ' , F) be thefiber bundle of one-jets of germs of sections of the bundle (X. ρ . F), and let (A' ,/; , X) be the natural fiber bundle associated with (A' 1 ,/; 1 , I'). The bundle (A'1 , p° , X) is affine. We have the following commutative diagram:
V 4.7. Let Κ be a codimension one submanifold of F . The submanifold I70 defines a (/-dimensional direction as follows. Consider the fiber (p°)' ](x) C A'1 over a point .v e A", //(.v) e l„\subsetF. Wc say ./'( /„) and j\f y) e (F°)~'(A') arc equivalent if /0|Fn and /,|IJ, define the same one-jet at p(x) . This equivalence relation decomposes the fiber (// )"'(.v) into equivalences classes, which define a (/-dimensional direction called a principal direction. We say that an affine embedding <• : R** — ,V parallel to a principal direction is principal. DEFINITION 4.8. Let (Χ , ρ , V) be a smoothfiber bundle as above. Lei (U , φ) be a chart of V , φ : U — R" , φ(χ) = (it, it,,). Set Γ = Cruo"1 {μ,. = 0). Then Κ is a codimension one submanifold which defines a q dimensional principal direction called a coordinate direction for the natural affine bundle over p~' U , 1=1,2 η. Now we are ready to state the fundamental DEFINITION
THEOREM 4.1 (Gromov's fundamental theorem). Let (X .p. F) be a fiber bundle, and let Ω c X 1 be an open set. Assume that for each χ exist a coordinate neighborhood Ν of χ and a chart (U . φ) of p such that (i) p(N) \subsetU, (ii) Ω is ample in every coordinate direction far (U . φ). Then the w.h.e.-principle holds for Ω.
114
IV. THE GROMOV CONVEX INTEGRATION THEORY
We postpone the proof of the theorem and present some simple consequences. Notice that we do not assume V to be open, whereas Gromov's theorem in Chapter 3 required that V be open; however, the condition on Ω was weaker in Chapter 3 than it is in the fundamental theorem in this section.
COROLLARY 4.1. Let (Χ, ρ, V) be a smooth fiber bundle, and let an open subset of X 1. If for an arbitrary principal affine emb Χ 1, α - ι ( Ω ) is ample in R ? , then the w.h.e.-principle holds for Ω
This corollary is immediate from Gromov's fundamental theorem.
COROLLARY 4.2. Let {Χ , ρ , V) be a smooth fiber bundle and let closed subset of A'1 . Suppose for an arbitrary principal affine R® - X' either α (Σ) c R® coincides with R® , or it is nowhere d its compliment is connected. Then the w.h.e.-principle holds fo
Corollary 4.2 follows immediately from Corollary 4.1. DEFINITION 4.9. Let ( A - 1 , //', V) be the bundle of one-jets of germs of smooth sections of (Α, ρ, V) . The fiber of this bundle is ./'(//, q) χ Y . Consider a closed subset Σ χ Y of J («, q) invariant under L 1 (//, q) . and let J z be the associated bundle of (A 1 , //', V) whose fiber is Σχ Y . Such a closed subset J z of A'1 is called a typical singularity.
COROLLARY 4.3. Let ( A ' , / / , V) be a smooth fiber bundle in which th space V and the fiber are C°° manifolds of dimensions η an Let Σ be a typical singularity of codimension greater than or eq A1 . Set Ω = Z c . Then the w.h.e.-principle holds for Ω .
This corollary follows from Corollary 4.2. §2. Proofs of the Smale-Hirsch theorem and Feit's thorem In this section we prove the Smale-Hirsch theorem and the Feit theorem. A. The Smale-Hirsch theorem.
THEOREM 4.2. (the Smale-Hirsch theorem). Let V and W be C\n itfy man ifolds of dimensions η and q respectively, η < q. Then the m to each immersion of V in W its differential
d : lmm(F, IF) —. Mon(F(F), 7'(H')), f^df is a weak homotopy equivalence.
PROOF. Set X = V χ fV and consider a map ρ : X — V which is the p jection of X onto its first component. The bundle (A',//, F) is smooth and there exist natural isomorphisms ψ and ψ making the following diagram
§2. PROOFS OF THE SMALE-HIRSCH THEOREM AND FEIT'S THOREM
115
commute: C\infty(F, W) - L - + Hom(T(V), T{W))
Sect(A') The bundle
(λ' 1 ,
Scct(A'')
//' , F) is equal to the following jet bundle. A'' = j \ v , IV)
./'(„, q)
Ρ I I71 .V = V χ II' ί I/'l V = V where j\n,q) = M(q,n\R) = {(q ,/i)-mairiccs over R}. Further, the structural group of (J 1 (I', IF), π, V χ Η-') is L*(n , q) = L [ (q) χ /.'(//) = GL(ij, R) χ GL(/i, R). Let Λ/() denote the set of (q. //(-matrices whose ranks are η . Then A/() is an open subset of M(q , η : R). which is invariant under the action of /_'(//,
F χ IF Now Ω is an open subset of A'1 . and by definition we obtain a subcommutative diagram of the above commutative diagram: Imm( V , IF) — M o n ( ' / ' ( F ) , ψj=
Sect(A', Ω)
/'(IF))
ν | =
ScclfA' 1, Ω).
Letting Σ = Ω'\subsetA 1 we see that Σ satisfies the hypothesis of Corollary 4.2, because if Σ 0 = M c a \subsetM(q , n ; R), then coding = 2. Hence the theorem follows from Corollary 4.3.
•
B. The Theorem of Feit. An analogous proof using Gromov's fundamental theorem works for the theorem of Feit. Let V and W be manifolds whose respective dimensions are η and q. Let / : V —> W be an c\infty map. For the natural number k and for each point χ of V , if the rank of / at ,v is greater than or equal to 0 . we
iv. ΤΗΙ·: GKOMOV CONVEX INTEGRATION THEORY
NO
say that / is a k-mersion. Let /c-meriF, IF) denote the topological space whose underlying set is the set of all A:-mersions of V in W , with the relative topology of C\infty(F, W) . Let k-mor(T(V), /"(IF)) be the spacc of homomorphisms of T(V) in 7"(IF), whose restrictions to each fiber are of rank greater than or equal to k , with the relative topology of Hom(7"( V) , T( IF)) (the compact-open topology). THEOREM 4.3
(the theorem of Feit). Let k < q. Then the map d \ k-mer( V , IF) —» k-mor(T(V),
Τ (IV)) ,
f^df is a weak homotopy equivalence. REMARK.
We cannot prove Phillips's theorem of Chapter 3 in this way. §3. Convex hulls in Banach spaces
In this section we discuss convex hulls in certain Banach spaccs as a preparatory step in proving Gromov's fundamental theorem. 1. Let Ρ = {ρ,,.».. , p n } be a set of nonnegative real numbers satisfying η Σ ρ , = >· (= I For Ρ and a number e , 0 < e < 1 , define a weakly monotone increasing piece wise-linear function θ = θ'' : [0, 1] — [0. 1] as follows. Let 0 < / , < / , < i 2
(l-e) '
P i,
I
I
<
η+ 1 Then 0 takes on the value i/(n + 1) in the interval [t :. t\\ and 0(0) = 0, 0(1) = 1 (see Figure 4.2). 2. Let β be a Banach space with the norm || ||. Let F be the set of continuous curves γ : [0, 1] —> Β , and define I : Γ —- Β by [ y(t)dt, 7 e Γ. Jo For results on the integration and differentiation of functions with values in a Banach space consult, for example, Riess & Sz.-Nagy ("). In the following, Conn(Q, b) denotes the arcwise connected component in a topological space Q , which contains b e Q. i(y)=
( 2 ) F. Riess & B. Sz.-Nagy, Lecons d'analvsc foiutionncllc,
Akadcmai Kiado. 1952.
117
5.1. CONVEX HULLS IN IIANACII SPACES
graph of ()'
ι ηs +ΝI
π+ 1
FIGURE 4.2
3. Let Β D Q, and let y g : [0, 1] —> (J be a continuous map. We consider somefixed Q for now. Define F(l c F as follows: Γ 0 = {y : [0, I] - Q| (i) y continuous, (ii) ;·(0) = y o(0). y( 1) = ;·„( I j. (iii) ; · - ; · „ , rcl{0. I}, in Q).
The set / ( Γ 0 )\SUBSETΒ has the following properties: /(Γ 0) is convex. /(Γ 0) is contained in the closure of the convex hull of Conn{ /(Γ 0) is dense in the closure of the convex hull of Conn((2. If Q is open in Β . so is /( Γ„).
LEMMA 4.1.
(a) (b) (c) (d)
PROOF OF (a). It is enough to show that for ρ + q = 1 , there exists a y ε Γ () such that
.
e Γ„. // > 0. ι/ > 0.
[ y(t)dt = // ί )\(t)dt + q I y 1(t)dt. Jo Jo Jo Setting u = pt we get
fo p^ t)d, =LM;)""· Setting υ = qt + I - q wc get [ qy,(t) dt = ί y, Jo ' Jr "
ν +q - I
d v.
Hence, we get + q- I On Ihe other hand, we have
dt.
GROMOV CONVEX INTEGRATION THEORY
118
This is because, setting 2s/p = 2 - 2t/p , wc have
(I)*· and setting w/p = 2t/p we get
CO-MjY^MJ)^· Hence, we have Ρ j
y,(t)dt + qj
y 2(i)dt
Therefore, if we set V(t)
[7,(21///), = | γ ι(2-2ΐ/ρ),
0
we get y ~ γ 2 , rel{0, 1} in Q (Figure 4.3), and because y 6 Γ 0 , the proof of (a) is complete. PROOF OF (b). Approximate the integral by a Riemann sum (cf. the proof of Proposition 4.2). PROOF OF (C). The point 0 = £ " = I F A of Conv(Conn(G,y0(0))), £ > , = 1, b ( e Conn(Q, y0(0)) can be estimated as follows. Choose y e Γ 0 such that
'GtttH·
and put yf = γ ° tff , Ρ = {ρ,, ... , ρ η } . Then we have fy ({t)dt—.b Jo The proof of (d) is evident.
•
(f-O).
. CONVEX HULLS IN ANAC
SPACES
119
4. We have the following well-known
LEMMA 4.2. Let Β be a Banach space, and Id Q t and Q 2 be conve sets in Β . If the closures of (2, and Q 1 agree, then (2, = (2,.
HINT. This follows from the separability of convex sets in a vector space. See Riesz & Sz.-Nagy ( 3 ) . The lemma also follows from For an open convex set Q in a Bana space Β . we have {Q)° = Q (*) • From Lemmas 4.1 and 4.2 we get the following PROPOSITION 4.1.
of Β. Then
Let Β be a Banach space and let Q be an open agrees with the convex hull of Conn((3. y „ ( 0 ) ) .
/(Γ0)
5. Now we want to present a basic lemma in Gromov's convex integration theoryLet Ω be an open subset of [0, 1] χ Β. For 1 6 [0. I] we denote by Ω, the set Ω η ({/} χ /}); Ω,\subsetΒ . LEMMA 4.3 (One-dimensional Lemma). Let φ η : [ 0 . I ] — Β be a continuous map whose graph is contained in Ω. Let /o:[0, 1)—> Β be a such that ,f ιr (i) -jf(0) =
(ii) for
l 0 e [ 0 . Ι], <Γοηη(Ω, ,
Then for an arbitrary E > 0 there exists a satisfying the following : (a) /(0) = / 0 (0).
0)· c l
map f : [0. 1
/(I) = / 0 (1).
(b) ||/-/0|| <e.
(if (c) The graph of ~ : [0, 1] — Β is contained in Ω . (d) There exists a homotopy { : [0, 1) — | r 6 [0, 1 ]} such th (<*) Ψ 0 = Ψ 0< Vi = dfl^t. (β) Ψ,(0) = φ ο(Ο), Ψ,(\) = Ψ 0(\ ).and (γ) the graph of ψ τ is contained in Ω . PROOF. ( I ) Case Ω = [0, 1] χ Q, Q is bounded open in Β . Here we divide [0, 1] uniformly into //+ 1 intervals, apply Proposition 4.1 to each interval [//(« + 1), (1 + 1)/(/; +1)], and construct a path (o, : [0, I] — Q such that ( ) Riess & Sz.-Nagy. Lecons if analyse fondionette. Akadcmai Kiado. 1952. ('') H. Eggleston. "Convexity", Cambridge Univ. Press. London and New York. 1958. Chapter I.
IV. THE GROMOV CONVEX INTEGRATION THEORY
120
(1) φ χ ~ φ 0 in Q,
(2) f \ (ϊΓΓτ)
( η Γ τ ) 1 = 0,
= Ψ°
(3)7ςνι(1)ί// = /
0
(ί±1)-/
0
(^
η, τ
) , 1 = 0,1
,.
Sublemma. Lei Β be a Banach space and let Q be a convex open of Β . Let f : I —» Q be a continuous map, where I = [0. 1 ]. Then f f die Q. Jo By assumption (ii) we may consider dt '
1+1 n+ 1'n+ 1
Co η ν (Conn ( < 2 , « > o ( - ^
Hence, by the sublemma we have ko
(«+ l)
/ o (//+l)
On the other hand, since (Proposition 4.1) Conv (I c o n n ( ( ? , , 0 ( ^ ) ) ) = / ( r 0 ) . there exists some
t + n+ 1'η+I
in placc of [0, 1) for defining
/W = /0(0)+
['φ, (Θ) d0. Jo Then / satisfies (a), (c), and (d). We see that / will satisfy (b) for a large enough η . (II) General case. We estimate Ω from inside; that is.
Ω' = υ m i=0
/+ 1 m
Q,C Β,
x f t c f i c [ 0 , 1 ]x B, (2, is as in (I).
Then by (I) the lemma is true for Ω', and so it is enough to consider Ω' — Ω . i.e. we take Ω' so that 0([0, 1]) C Ω' C Β. •
§3. CONVEX HULLS IN BANACH SPACES
121
6. PROPOSITION 4.2. Let Q be an open subset of R q . let Κ be a compact space, and let F be the space of continuous maps from Κ to Q. If arcwise connected, then the convex hull of F in C°(K . R+7) agrees space of continuous maps font Κ to the convcx hull Coxw(Q) of Q Conv(C°(K , Q)) = C°(K , Conv«2)).
(i) Wc show Conv( c "(A', <9)) C C°(K . Conv((>)). Suppose we have / e Conv(C°(A', Q)) . Then / = p.t\ + qf 2 ,// +
./; . ./; : Κ
~Q.
f : A — R" . but for .ν e Κ , f(x) = pf(x) + qf 2(x)eConv(Q) , so /eC°(A'. Conv(C?)). (ii) To show that <Λλ', Conv((?))\subsetc o n v ( c 0 ( A. Q)), let ψ e cn (AT, Conv(2)). For the map (p : A' — c onv((J) it suffices to construct a family of maps ψ ) : Κ χ [0. 1] —· Q such that as c — 0 [ i//,(k . t)dt —> (k). Jo The argument for this goes as follows. Since A is compact. Ii = C ( A'. R") is a Banach space and C°{K , Q) is open in H . For the ψ < as above, define
= t//,(k.i).
k e A . ι e [0. 1],
: A' —-> Β by A, (AT) = ί ψ, (k . t)dt, Jo
f > 0. k e A.
Then we get H, P„ : ^ —» [0,1] satisfying: η η P,> ο. £ > , = >. ^Ρ,(Α·)λ·, =9/(Α·). Ae A ι=1 ι=Ι
IV. THE GROMOV
122
INTEGRATION THEORY
0 FIGURE 4.4
(visualize the situation with a two-dimensional picture. Figure 4.4). We then choose a path γ : [0, 1] — Q such that γ(ϊ/(η + 1)) = ,γ(, 1=1,...,//, and put ψ, =y(O
f w0n.
where P{k) = {//,(£), ... , p n(k)} and θ is the function defined in §3.1. Now by following the proof of Lemma 4.1, (c) we can show that the {ψ ι } is the desired family . Ο 7. Consider a (/-dimensional vector bundle (Ζ , π . Κ). Assume Κ to be compact. We write π~ ι (k) = Z k . Let Ω\subsetΖ be open, and let φ 0 : Κ —• Ζ be a cross section of ( Ζ , π, Κ) such that
Ζ\π ο φ = 1, φ\Κ
0
= ψ 0\K Q , ip(K)c£l,
φ ~ ψ { ) rcl λ'
Then we have the following LEMMA 4.4.
The convex hull of
Γ0
in the space of sections of (
agrees with {φ :Κ—*Ω°\ποφ=:
φ\Κ
0
= φ 0\Κ 0).
PROOF. We show that our set is contained in the convex hull of Γ 0 . Let φ: Κ -* Ω° be a given section with ψ\KQ = φϋ\Κ0 . Just as in Proposition 4.2 in subsection 6 we start with sections χ/. Κ -> Ω homotopic to φ 0 relative to K Q and continuous maps p t Κ —> [0, I ], { p ( = 1 , Σ " = ι p r Υ, = φ Choose a path y(t) , I e [0, 1], in the space of cross sections Κ — Ω which
§4. PROOF OF THE FUNDAMENTAL THEOREM
123
agree with φ η on K g with the property y(/'/(" + ')) = x , • Now define ψί : Κ χ / — · Ω by ψ, (k , /) = y(k , 0 , W ( O ) · Then letting e - O w c get
f
Jo
V f(k,t)dt
The reverse inclusion is obvious.
— φ.
D
§4. Proof of the fundamental theorem In this section wc prove the fundamental theorem of Gromov's convex integration theory. 1. Lei 1 b e a compact C^ manifold and let F = I J, χ |0. I]. Lei }' = V χ R'' F be the product bundle. Cover I withfinitely many charts. We represent a chart by (//, un l . t), ι e [0. 1 ]. For a C map / : I·' — R" . set max ||/(«)||, ¥ Oil 00 ugl I <;
that one disregards Of/Ol for the former. 2. Main lemma. LEMMA 4.5.
Let Q be an open subset of Y and consider
Q = U Q„. Q„ = Q η (ν χ R V )\subsetR". I'd Suppose that maps f 0. φ„ : V — R" satisfy the following·. (i) The graph of
r.
Then for an arbitrary that (a) f\0V = f Q\0V . Of ovJJA 0u t Of Ot
ov.
- ) .
c > 0 there exists a
c l
map / : F —
11 - 1 .
du ,ov ,
(b) l l / - / 0 f < e .Ot (c) the graph of 0 f /01 is contained in Q. and
124
IV. THE GROMOV CONVEX INTEGRATION THEORY
(d)
There is no loss of generality in assuming that f Q(0 V) = 0 and = 0. ( 5 ) Let Β be the space of all c l maps g : I7, — R* such that g{dV
0)
= 0,
|£(ό)Ι/) = 0,
1= I , . . . , / / - I.
Then we may think of C' maps F —» R® as continuous maps [0, 1] —» Β . Let Ω = (J Ω, c [0, 1] χ B, t€ [0.1J Ω, = { g : V 0 — R® | g continuous, the graph of g C Q η (F0 χ {/} χ R " ) } . Now the lemma follows from Lemma 4.4 about convex hulls together with One-dimensional Lemma. • 3. Singularities in cubes. Let V be an //-dimensional c v manifold and let ρ : X —» V be a C 0 0 fiber bundle whosefiber is of dimension q . Let ρ 1 : X' —> V be thefiber bundle of one-jets of germs of sections in (Χ, ρ , V) . Let p° : X 1 -* X be thefiber bundle naturally defined by ρ 1 : X1 —> V. In this case X) is an affine bundle making the following diagram commute.
Now we assume that the base space V can be written as the product VQ χ Vx, where Fj is a one-dimensional manifold. Then thefiber bundle (X 1 , /zc, .Ϊ) has the direct sum decomposition X 1 = X 0 θ Χ , , where p 0 : XQ —> X and p, : Xt' —> X are affine bundles of dimensions q(ιι - 1) and q respectively. Let : X K —> X,J denote the natural projection. Then (Α'1 , π, , A'(]) is evidently an affine bundle whosefiber is of dimension q .
Let V be the η dimensional cube /" with coordinat u ). Let X — V χ R® V be the trivial bundle. Let Ω be ope
PROPOSITION 4.3.
(u t ,...,
( 5 ) Actually we should define H = {g : - R*| g(i) l' 0 ) = ,/0(<7l). however the proof will remain the same if wc assume this.
Vo(OVa)};
Ug/i>ii, (<>>„
54 PROOF OF THE FUNDAMENTAL THEOREM
125
X 1 and ample in each coordinate direction. Let f Q : V —> X and φ be C°° sections and assume ./' (/ 0) = φ ( ) on 01" . Then for any t > 0, there exists a c l section f : V — X which the following : (a) j\f) and
( y r |re[0. 1]) of sections in (V ,p
PROOF. We might construct the desired section f as follows. In the 1th step we consider the 1 th coordinate of Γ = I" and represent Γ as the product F = F o x[0, 1] with coordinates it, ",-ι·",,ι "„'• ", = 1, 1 = I /i. Then for the fiber bundle X = Γ χ R'' — I consider the decomposition of A'1 = A'J φ Λ',1 as discusscd above. The bundle A' — V is identified with the subbundle of the bundle A'1 — A'(J . whose fibers intersect 0(V) C A 1 . We then let Q \subsetV χ R'' correspond to Ω under this identification. By the assumption on Ω, the set Q satisfies the hypothesis of the main lemma . By the above method wefirst construct J\ — X .
IIfi - fi-tW < t r
' =1.2
//.
Further φ: is obtained from through a deformation leaving π, °φ, unchanged (recall π, : A'1 —> A^ ). So setting f = f 0 we have the desired section. Ο 4. Restatement of the assumptions of the fundamental theorem. We may assume i with the following. (i) There exists a chart (U , φ) of V such that l" is contained in U and φ(ΐ") is a cube in R® . (ii) U is contained in some chart of (A . ρ. F). By assumption Ω is ample with respect to each coordinate direction of I" in thefiber Yj.
126
IV. THE GROMOV CONVEX INTEGRATION THEORY
5. Proof of the fundamental theorem. Now the fundamental theorem follows from Proposition 4.3. We perform cubic subdivisions on the C\n i fty manifolds X and V as we do triangulation. We apply Proposition 4.3 to each cube in each skeleton to obtain the fundamental theorem. Sec § 6, Chapter 3, about C°° triangulations. REMARK. We can do without cubic subdivisions in step 5. This is done by using the theorem in Gromov-Eliashberg, Removal of singularities of smoot mappings , Izv. Akad. Nauk SSSR. 35 (1971), 600-627.
CHAPTER V
Foliations of Open Manifolds As an application of the Gromov-Phillips transversality theorem as discussed in Chater III we present in this chapter Haefliger's classification theorem for foliations of open manifolds, which brought the limelight back to the Smale-Hirsch theorem. We wilhfollow closely Haefliger's papers [B2, CIO]. A detailed discussion of foliations is found in The Topology of Foliations [A9]. Research in thefield of foliations continues to be active at present. §1. Topological groupoids In this section we define the classifying space B y of a groupoid F. DEFINITION 5.1. Suppose that a set Μ is divided into the coseis Μ . / . j = 1,2,... and satisfies the following: (i) If a e M tj and b € Μ j k , ab 6 M j k is defined. (ii) If a e M tj and b € M ik , a~'l> 6 M /k is defined and a(a '/>) = /'· (iii) If α e M tJ and b e M k j , a/;-1 e M j k is defined and (ab ' )l> = a . (iv) For a e Μ :·, b 6 Μ j k , and c € M k l , wc have ab(c) = a(/>c). We then say that Μ is a groupoid. DEFINITION 5.2. Suppose that a topological space Γ has a groupoid structure. We say that Γ is a topological groupoid if the groupoid operations (i). (ii), and (iii) are continuous. EXAMPLE. A topological group G is a groupoid. DEFINITION 5.3. Let X be a topological spacc. Let F = { ( U r , f . I·',·)} be a set of triples (U r,f, V,) of homeomorphisms f between open subsets V t and V f of X . We say that Γ = ((U f . f . F(.)} is a pscudogroup (of local homeomorphisms) of X if it satisfies the following: (1) (Χ, I ^ , A") € Γ. (2) If (U rf, V f) 6 Γ and U is open in U, , then (U. f\U. f(U))€ Γ.
(3) (U f, /, v f) 6 Γ, (U x.g. Κ ) e Γ. V f \subsetu e * (L'f . g ο /.
f(U
r)) 6 Γ. (4) [U f,f, v r) e Γ => (V f, /"' , Uf) 6 Γ. Similarly, we define a pseudogroup of local diffeomorphisms of a ifold X .
127
c x
man-
128
V. FOLIATIONS OF OPFN MANIFOLDS
Let β be a topological space, and let & = {(U f , /, F,)} be a pseudogroup of local homeomorphisms of Β . Denote by Yh the set of germs of / e 9 at b, and write Γ = (JA6« Γ/, · W e topologize Γ as follows (recall the sheaf topology): A basis for the open sets in Γ may consist of the sets {[f] x\x e U }, where each [f] x is the germ of / at χ e U , [U , f, V) eSs, and U runs through open subsets of B. Then Γ becomes a topological groupoid; we say that Γ is the topological groupoid (or simply groupoid) associated with S? . Denote by Γ the groupoid associated with the pscudogroup of local diffeomorphisms of R® and by Γ' the groupoid associated with the pscudogroup of analytic automorphisms of C . Assigning each element [f) x of Γ to the differential of / at x , we obtain a homomorphism of topological groupoids, u : Γ^ —< GL(t/, R). We also have the verbatim definition of the homomorphism u : F ( —» GL (
>Γ
such that for each χ e U l · η Uj η U k , y, k(x) = 7 u(x)y. k{x) • Two one-cocycles { U t , y tj \ i, j e J } and { U' k , y' k t\k , I e K) are equ alent if there is a family of continuous maps S j k : 1/ η U' k —- Γ satisfying: 5,k(x)v'ktM
= <*,/(*)'
-v e U, η u' k η
v j :(x) s,k (χ) = 6,k (χ) -ν 6 υ, η U j η u'k • This is obviously an equivalence relation. An equivalence class of one-cocycles is a Γ-structure on X . We denote the set of Γ-structures on X by Γ(Λ") or h'(X, Γ). When Γ is a topological group, each Γ-structure on X corresponds to an equivalence class of a principal G-bundle over X (cf. §2, Chapter I). Let / : Y —» X be a continuous map, and let a = {t/ , y tJ1 1, j e J } be a Γ-structure ο- X . Then / induces a Γ-structure on Y ; / V = {/-'(i;.),y ; 7 o/|,,
jeJ).
§3. Vector bundles associated with Γ -structures ·' ι Let Γ ? be the groupoid associated with the pseudogroup .9" of local diffeomorphisms of R i as defined in § 1.
§5. THE CONSTRUCTION OF THE CLASSIFYING SPACE FOR Γ-STRUCTURES
129
The following is another interpretation of a Γ^-structure on a topological space X . We consider the collection Ψ of systems ,9~ = {(U n , 6 .1 } of open sets and continuous maps satisfying: (i) {U n |α e A} is an open covering of X . (ii) Each f n : U n —» R® is continuous, <* e .4 . (iii) If U n Π Up # 0 , there exists a continuous map f such that for „v e U n η U p , fj.x) = f nf i • ,//((.v).
n/t
: U n η C'/( — Γ (
We say families ,9~ and !F' in Ψ arc equivalent and write !7 ~ '7~' if \J Sr belongs to &. Evidently ~ is an equivalence relation. An equivalence class under ~ is nothing but a Γ -structure on A'. In §1 we introduced the homomorphism r : Γ^ —· GL(t/, R) of groupoids by associating the germ of / at χ to the differential of / at .v. Hcncc. through the map ν we obtain a natural correspondence between restructures on X and (/-dimensional vector bundles over V (cf. §4. Chapter I). When the image /'(Γ^) is contained in a subgroup (τ-···οί GL()-85. (") L'ne construction dun universal pour classe assez large tie Γ-structures. Paris. 270 (1970), 640-642.
C.R. Acad.
V. FOLIATIONS OF OPEN MANIFOLDS
130
unit β{γ). We define Ε Γ to be the set of equivalence classes of infinite sequences (f 0 , x 0 , ί,, x , x„ ,...), /,€[ 0,1], 1=1,2 = 0 for all butfinitely many 1, ρ,€Γ, \P(Xi)
1 = 0,1 = P(Xj),
i,j=
1,2
as follows: ( < 0 , X „ , 1,, X , , . . . , /„,
X„, ...) ~(t'
ί , = ι;,
Q,
Xg, t\
,X,
'
)
1 = 0,,,...
[ 1, Φ 0 implies χ. = χ.. Henceforth we shall denote the equivalence class of (ί 0 , x 0 , /,, χ,, ...) ('ο*ο> 'ι*ι> · · · ) · We define maps l ( : Ε Γ —> [0,1] by (l 0 x 0 , Ι,Χ, , .. •) >— If, and maps χ,ιΐΓ'ίΟ, 1] —> Γ by (/„*„,/,*,,...)—·*,. These maps are well defined by the above definition of equivalence classes. We give ΕΓ the weakest topology making the t j and the x j continuous. Consider the natural action of Γ on £T, and denote by IS Γ the quotient space of Ε Γ by this action. More precisely wc have the following: by
ΒΓ = ΕΓ/ ~;
(i) /, = ?;, 1 = 0, 1,2 (it) /?(x,.) = β(χ\), 1 = 0, 1, 2 (iii) There exists y 6 Γ such that for all /' with I, φ 0 we have x,. = yx'
r
Sometimes we simply write (x, 1) ~ (x', l') ilf (χ, 1) = (γχ', I). V/e give Β Γ the quotient topology and denote the natural projection by ρ \ ΕΓ — ΒΓ. We call Β Γ the classifying space for Γ -structures The projection : £Γ — [0, 1] maps the equivalent elements under ~ to the same element; hence, it defines the projection u i : ΒΓ —> [0, 1 ], and we have u t ο ρ = . The classifying space Β Γ has a natural Γ-structure as follows: Set ^. = « " ' ( 0 , 1 ] ,
1 = 0,1,2
>0 : V , n Vj —' Γ ' •I (l 0x 0, 1| X )>— X,Xj
§6 NUMERABLE Γ-STRUCTURES
131
Then ω = { , i, j = 0, 1 , 2 , . . . } is a Γ-structure on ΒΓ. called the universal Γ -structure. § 6. Numerable Γ-structures DEFINITION 5.5. An open covering {11.\ j e J } of a topological space .V is numerable if there exists a locally finite partition of unity {/tj 1 6 / } with M~'(0,1 ] c 11. Any open covering of a paracompact space is numerable. DEFINITION 5.6. A Γ-structure [σ] on Α', σ = { U , y { )| 1, j 6 J }, is mi merable if its representative a can be chosen so that the {U ;} is numerable. We say that two numerable Γ-structures are numcrably homotopic if they arc connected by a numerable homotopy.
PROPOSITION 5.1. Let Β Γ be the classifying spacc and let oj be its Γstructure as defined in §5. Then (a) ω is numerable. (b) for a numerable Γ -structure σ on a topological spacc A . there a continuous map f : X —> ΒΓ such that Γ ω = σ . and (c) two continuous maps f Q and f of X to Β Γ are homotopic if a only if the Γ -structures f^ ω and /*ω arc numerably homotopic. PROOF, (a) Wecite Milnor's proof verbatim (J. Milnor('). Wcconstruct a locally finite partition of unity {?>,} on //Γ as follows. Define a map w : Β Γ —. [0, 1] by
ui,.(*>) = max i o , «,(//)- Σ u t (b) \ . then we have w. (0, 1] C V . For b € Β Γ. let m be the smallest 1 such that «,(//) / 0. We then have £ u,(b)= I, m
ujb) = wjb),
and Β Γ has an open covering ΒΓ = U" ; ,~'(0· ']• Since u ((b) = 0 for /
η < i, we have Σ
Uj(b')> 1/2 implies wi(b') = 0.
Therefore, N n(b) = { b
Ε "/(Λ')> '/2
( 3) Conslruciion of universal bundles II, Ann. of Math.. 63 (1956). 430-436. One will also find the proof on p. 54 in the book of Huscmollcr, Fiber bundles. McGraw Hill. New York. 1966.
V. FOLIATIONS OF OPEN MANIFOLDS
132
is a neighborhood of b, and N n(b)DuJ. '(0, I] = 0 if η < i. Hence, the open covering {ΐί.' ( -ι (0, 1]} of BY is locallyfinite, and so if we set v. =
Σ 1-WJ
'
the {i/ (} is the desired partition of unity. (b) We need ihe following
LEMMA 5.1. Let { I/.| C € J } be a numerable covering of X . Then exists a locally finite countable partition of unity { Ij η e Ν } such open set V n = ί ~1 (0, I] is a union of mutually disjoint open subset "r Again the verbatim proof is in the book Fiber bundles by D. Husemoller. PROOF. Let {?J.| 1 E 7 " } be a locallyfinite patrtion of unity on X with υ~'(0, 1] C {/,.. Then for each b e X , the set S(b) = { i e »',(/») > 0 } is finite. Further, for eachfinite subset S of Τ,
1F(S) = { b e X\V t{b) > Vj(b) , VI e s, Vj e Τ - S } is an open subset of X . The map it9 : Β — [0, I] defined by uJb) = max ( θ , min (»>,(//) - v (/>))! I ie.v./et-.v ' ' ) is continuous, and we have IF(S) = u s 1 (0. 1 ]. Let CardS denote the cardinality of S. Wc claim that if CardS = Card S' and S φ S' , then IF(S) η IF(S') = 0 . Let ieSS' . j 6 S' - S . Then tr(b) > Vj(b) for b e H 7 (S). and wy(//) > v ;(b) for b e IF(S'). But these two relations cannot occur at the same time, so the claim is true. Now put U
^i5'·
,1, m(^)=
Card S=m
Then u/-'(0, 1]= W
Σ
"sO»·
Card S =m
Set >Jb)
=
v>Jl>) Σ„>ο , "„( / ')'
Then 1~'(0, 1 ] = 1F(1; so the {1 (1} is the desired partition of unity. • To complete the proof of (b), let α be a numerable Γ-structurc on X . By Lemma 5.1 we may assume that a is defined by one-cocycles y n m on a countable open covering { ί/,| η = 0, 1 , ...} . Here U n = t~ 1 (0 . I ]. and the { l j is a partion of unity. The desired map f : X — BY may be defined by /(*) = [ ( ' o W W * ) · ( * ) y
mi
( * > .·•·)].
v e
L'n,
•
where [(/0(jc)x/h0(a'), .. -)] denotes the equivalence class of (l 0 (-v)y m0 (.v)—).
5 6. NUMERABLE Γ-STRUCTIJRES
133
Proof of (c). Denote by Β Γ ο ι ) a subset of Β Γ. which is the image under ρ : ΕΓ —> ΒΓ of the points (l0A'0, Ι,α-, , ...) in Ε Γ with l„ = 0 for η odd. Similarly wc define Β Γον (replacing 'odd' by 'even' in the above notation). We define /;od , //ev : ΒΓ - ΒΓ by A o d [(i 0 A- 0 ,/ | x | ,...)] = [(i 0 .v 0 ,0.l,.v | .0,. /tCV[(l0A-0
)].
, 1, A", ....)] = [(0 . !0.Y0 . 0 . ί,.Υ, . 0 . ... )].
The maps Λ0,1 and Λ ° are homotopic we have (/i H d)*to = (h^)'w = to .
LEMMA 5.2.
Further,
to the identity
PROOF. Define a linear function
- (1/2)") = 0.
-|i>. 11
r»n( I - ( 1 / 2 ) " " ) = 1.
Define a homotopy Λ"'1 : ΕΓ — FT as follows: ' hf(X,
I) = (/()A-„
/„V„,M„(.v)/,|f lA'„+, . (I - an(.V))l„,,.Y„t| . -"„(·ν))',„:·ν„.,....).
1 - (1/2)" < ν < I - ( 1 / 2 ) " " . . //^(Α·,1) = (Α-,1). Then wc have ι
{ (.'J i is an even natural number. Γ = /0 ' ( F ) } . Similarly, /*ω is defined by a cocyclc on the covering { l/.| 1 is an odd natural number, L \ = t\ ' ( F ) |. Let s( : V — Ε Γ be the continuous map defined by M'o-V 'r v i ' •] = ('o-v,~'-V V , " ' v i
m
134
V. FOLIATIONS OF O P N MANIFOLDS
then we can write J, ° f 0(x) = (t aMy
0i(x)
. ο, t 2(x)y
, 0, ...),
2i(x)
i i o / . W = (0,i1W)'liW,0,i3(x)r3i(x),...),
A- E U i , i even, Aret/,·, / odd.
Since /,'ω = /,'ω, if 1 ^ j'(mod 2) there exists a family of continuous maps y tj : U i Ρ U J , — > Γ such that the { y (J |l, j > 0} is a cocycle on the { f/l ' > 0 } . Now a homotopy connecting jj, and may be given by the following maps:
S {f s(x)
= [(1 - 5 ) R 0 ( X ) Y 0 , ( X ) ,
A-);'„(A-), ( I - S ) I 2 ( A - ) > - , , ( . V ) ,
513(ΑΓ)73,(Α-), ...], The proof of Proposition 1 is now complete.
A- €(/,..
•
§7. Γ-foliations Let Γ^ be the topological pseudogroup of germs of local C' difleomorphisms of R® jtwhere r is one of 0, 1, 2 , ... , oc , ω. Let Γ be an open pseudosubgroup of Γ^. DEFINITION 5.7. Let X be a c r manifold. By a Γ -foliation of A' we mean a Γ-structure on X represented by y = {((/,., J]) , y. | /, j 6 J }, such that each f. : t/, —> R® is a C' submersion. In other words, a Γ-foliation y is a system j ({/,., f t)\ 16 7 } of open sets and C' maps satisfying: (i) { Uj 11 6 J } is an open covering of X . (ii) f : (/, —< R® is a C r submersion. (iii) If {/,Π Uj Φ 0 , then for every point A' of U i η Uj , there exists a C r diffeomorphism g j j of some neighborhood of /(A ) onto the corresponding neighborhood of fXx) such that (a) the germ of at f(x) belongs to Γ, (b) fj = g.j ο f. in some neighborhood of A'.
Let X be a c r manifold and let , ·Ψχ be Γ-foliations of X . We say that ^ and ^ are integrably homotopic and write •9~0 ~ .!? DEFINITION 5.8.
if there exists a Γ-foliation y of A' χ [0, 1] such that the maps /', : X —Χ χ [0, 1) defined by i t(x) - (χ, t) satisfy the following: (i) r0
x
: T x(X)
is a surjection, where is the natural projection.
Τ αχ )(Χ
to y
if for any point χ
χ [0. 1]) -Ϊ* T f(x )(X χ [0, 1 ] ) / 7 > ( v ) ( L / ( t )
is the leaf of y
passing through f(x) , and π
5 8 THE GRAPHS OF F-STRUCTURFS
135
Evidently the relation ~ is an equivalence relation. i If X is a closed manifold, the Γ-foliations .?~0 and ^ of X are integrably homotopic if and only if there exists a c r diffeomorphism / of A' such that (i) (ii) f is isotopic to 1 v . We omit the proof (cf. Tamura [A9]). § 8. The graphs of Γ-structures Let Γ be a subset of Γ^ . DEFINITION 5.9. By a (/-dimensional Γ-foliated microbundlc mean an ordered triple (Ε , ζ , such that (i) ς consists of topological spaces and continuous maps, ζ : X — Ε —· Α',
over Λ wc
/; ο , = 1 .
(ii) % is a Γ-structure on·. Ε , W = j (U n. .(,)|(v e A } , such that for each a e A, the map (p\U n) χ f : U n —> A' χ R® is a homeomorphism of U it onto the corresponding open subset of Α χ R®. Fora C r manifold X , we may take the total spacc Ε to be a C' manifold and the maps 1 and // to be of class C r . Further,
Let X be locally compact and paracompact. Then f & on X , there exists a q dimensional Γ -foliated mic
PROPOSITION 5.2.
a Γ -structure dle [Ε,ξ,
ς : X -I— Ε — A such that & ~ 1*1Further the unique up to isomorphisms. In dles (E 0, ξ 0, and (£, , , i 0(A) and i,(X) together with a (i) the diagram
commutes, and (ii) h'&t ~ £? 0\U
0
germ of this microbundlc (Ε.ζ. ? other words, if we have two such there exist neighborhoods U u and homeomorphism h : U 0 — l\ such
.
We call the microbundle {Ε,ξ. W) as constructed above the graph of a Γ-structure . This idea was formulated by Haefiiger. PROOF OF PROPOSITION 5.2. Suppose we have a Γ-structure ,9~ represented by y = {(U i t , (0m), £„/;|t». β 6 A] . Consider the graph (H
)
m L th
136
V. FOLIATIONS OF O P N MANIFOLDS
U. FIGURE 5.1
U χ R® of
in U n χ R®
Ε = |J TJ ~ : u„ η u t Φ 0, (χ, .V) e r ,"(V , .') € Τβ. ft (x,y)-(*',/)
iff
( Λ = Λ ' , ι .p = g„ f(x)y • Define 1 : X —» Ε by i(x) = [(λ', (f (l(.v))], .v e U n . The construction of Ε implies the map 1 is well defined. Define // : Ε — X by //(.v. p) = .v . Notice that ρ ο /' = 1 . To define a Γ-structure on Ε, set 1Γ = π( '·Γ ), where π : U,( 7]( —· £ is a natural projection. Define ψη : IF — R® bv Φ„( χ' y) =y- Then the system & = {(H^ , ψΜ)|ί« € A } is a Γ-structure on Ε , and it is a routine to see i * f ~ . • §9. The Gromov-Phillips transversality theorem
We mentioned in Chapter III that the Gromov-Phillips transversality theorem is a major tool for the classification of Γ-foliations of open manifolds. Let A" be a C r manifold and let = {(U t, f), g l j \ 1. j e J } be a Γ-foliation of Μ. Denote by the normal bundle of 9"; that is, if f : Μ — ΒΓ is the classifying map of ν : Β Γ —• BGL(q , R) is the map as defined in § 1, and γ is the universal ^-vector bundle, then ~ (// ο f)'y q. Another interpretation goes as follows. v.Ψ |L/( =.f*(T(R")) and if U^L φ 0 , U l χ R® and Uj χ R® are pasted by dg t) along L\ η U t : (.v, r) ~ (•*' - y) iff jc' = JC, y -(dgij) x(y). Let X and Μ be C r manifolds, r > I . Let -9~ be a Γ-foliation of Μ , F={(U i,f i)\ieJ). DEFINITION 5.10. We say that a C r map / : X — Μ is transverse to the Γ-foliation if for each 1 e J the composite f of : /"'((/,) — R® is a submersion.
10. THE CLASSIFICATION THEOREM FOR Γ-FOLIATIONS OF OPEN MANIFOLDS
Denote by T r ( A ' , y ) the set of C r maps from .V to Μ which arc transverse to y . We give T r ( A ' , y ) the C 1 topology. Let τ(Χ) be the tangent bundle of A", and i^y the normal bundle of y . Denote by Epi (τ(Χ),υ.9~) be the set of bundle maps of τ(Χ) into i'7~ whose restrictions to each fiber are each surjective (such a map is called an epimorphism of vector bundles). We give Ερί(τ(Χ), i;.7) UK compact-open topology. Let Τ(Α/) be the tangent bundle of A/ and let π : τ(Μ) — j/y be the natural epimorphism. (Gromov and Phillips). Lei X be an open C' manifold, oc, ω. Let Μ be a C r manifold and let .7 be a Γ -foliation
THEOREM 5.1
r = 1,2
of A/ . Then the map Φ : Ύγ(Χ
, y ) —. Epi(r(A0 . u.7)
defined by Φ(.ί) = n(df) is a weak homotopy equivalence. • Theorem 5.1 follows routinely from Theorem 3.13 (the Gromov-Phillips theorem). As we remarked earlier the assumption that X is open is essential. §10. The classification theorem for Γ-foliations of open manifolds Let ω be the universal Γ-structure of the classifing spacc ΒΓ for Γstructures. Suppose Γ\subsetΓ^ , r > I . Let νω be the q dimensional vcctor bundle associated with ω.
THEOREM 5.2 (the classification theorem for foliations of open manifolds). Let X be an open C' manifold, r = 1.2 TC . ω. Then there exists a one-to-one correspondence between integrablc homotopy clas foliations of X and homotopy classes of the epi morpli isms of r(A') normal bundle vu> of ω . PROOF. The desired correspondence may be given as follows. Let .7 be a Γ-foliation of X . Let π : τ{Χ) —> v.9~ be a natural epimorphism. A priori, the foliation .9" is a Γ-structure, and so .7 is induced from the universal Γ-structurc ω: /'ω ~ y by some continuous map f : A' — ΒΓ. This defines a bundle map φ —-»//w,andthe map Ψ = φ ο π : r(A') — νω is a bundle epimorphism. We assign to the integrablc homotopy class of .7 the homotoppy class of Ψ . This correspondence is well defined. Wc want to show that this is a surjcction. Let ψ : τ(Χ) — ι/ω be an epimorphism of vector bundles, and let ψ = / : X — ΒΓ be the corresponding map of the base spaces. Let a = f'w: a is a Γ-structure on X. Let be the Γ-foliated microbundle over X corresponding to σ , ζ : Χ — Ε Χ . where Ε is a c r manifold. is a Γ-foliation of E, and ~ a. Then t*/>£ ~ f'vw. Here e
. FOLIATIONS OF O P N MANIFOLDS
138
epimorphism φ { : τ(Χ) φ : τ(Χ)
I Λ-
— » CvW and hence an epimorphism φ : τ(Χ) — / V f
I
— ι / f ,
I
——-
Χ
φ
=
Φ2
—» //g
ο φ,
"
— £ · .
Therefore, by the Gromov-Phillips transversality theorem there exists a c f map j : X —> Ε homotopic to i : X — Ε such that j is transverse to & and nodj : τ{Χ) — •i/f is homotopic to φ. Thus, = j ' f is the desired Γ-foliation, { y } {ψ}. Similarly we can show that the correspondence is injective. • We restate the above classification theorem in such a way that one can actually calculate. Recall Γ C Γ r . Alsorecallthe differential maps ν :Y
q
u :BV
—* GL(q , R) q
—. BGL{q
, R).
Now assume that //(Γ)\subsetG for some subset of GL{q , R). Then we have the induced map ν : BY —> BG and the following commutative diagram BY χ BGL(n-q,
σ
R) \ "χ ι BG χ // GL(H - q, R) |,,xl
BGL(q, /9
R) χ Β GL(/t - q, R)
Β GL(n, R)
where ® represents the Whitney sum and ρ is the induced map of G \subs GL(<7, R).
Let X be an open C r manifold of dimension n, r and let τ : X — BGL(n, R) be the classifying map for the tange dle τ(Χ) of X . Then there is a one-to-one correspondence betwe homotopy classes of Γ-foliations of X and homotopy classes of li BY χ BGL(n-q,R) : COROLLARY 5.1.
ΒΓ χ BGL(n
y
- q, R)
ι-
X -i-. BGL. Here by a lift of τ in BY χ BGL(n - q , R) we mean a coninuous map X -» BY χ BGL{n - q, R) making the above diagram commute. PROOF. There is a one-to-one correspondence between homotopy classes
§10. THE CLASSIFICATION THEOREM FOR Γ-FOLIATIONS OF OPEN MANIFOLDS
I 39
of lifts of τ in ΒΓ χ Β GL(/z - q, R) and the triples (/.»/, Φ) consisting of / : X -—> BY , a continuous map, //, an (// - q) dimensional vector bundle over A'. φ : f ι/ω® η —» τ(ΑΟ, an isomorphism of vector bundles. But there is a one-to-one correspondence between the set of triples as specified above and the set of homotopy classes of cpimorphisms of r(.V) onto t'to . Hcnce, the corollary follows from Theorem 5.2. • Because of Corollary 5.1, it becomes necessary to investigate topological properties of the classifying spacc βΓ in order to classify Γ-foliations of open manifolds. See Bott(4) concerning this subject. NOTE. Thurston showed that a similar classification holds for the foliations of closed manifolds ( 5 ) .
( 4 ) R. Bolt, Lcclures on characteristic classes and foliations. Lecture Noics in Math.. Vol. 279 (1972). Springer-Vcrlag, Berlin and New York. pp. 1-94. ( 5 ) W. Thurslon, The theory of foliations of codimension greater than one. Comment. Math. Helv. 49 (1974) 214-231; Existence of codimension one foliations. Ann. of Maih. 104 (1976). 249-268.
CHAPTER VI
Complex Structures on Open Manifolds In this chapter we discuss complex structures of open manifolds as applications of the Gromov-Phillips theorem of Chapter 111 and the Gromov convex integration theory of Chapter IV. §1. Almost complex structures and complex structures Let A" be a 2 0(2q) , A = [a,j)e U(n), a ,j
= bu +
P{A)
·
by , Cjj € R ,
(-c Σ)· * =
C
=
6.1. by an almost complex structure on a 2
flU *s /
(q) I Γ'
/
X — BO(2q) where BO(2q) and B\J(q) denote the classifying spaces of cr .tpact Lie groups 0(2q) and U( 0(2q). Then an almost complex structure on A" is a lift of τ to B\J(q) in the above diagram, that is a continuous map τ : X — » BU(q) such that ρ ο τ = τ. This definition 141
142
VI. COMPLEX STRUCTURES ON OPEN MANIFOLDS
is an easy consequence of the classification theorem for fiber bundles (cf. Chapter I). REMARK 2. Another interpretation. A vector bundle homomorphism J : τ(Χ ) τ(Α) with J 2 = - 1 of the tangent bundle τ(Χ) of a 2IJ-dimensional manifold X is an almost complex structure on X . The works by W. Wu [C23], A. Borel, F. Hirzebruch, T. Heaps, and so on include problems concerning the existence as well as the classification of almost complex structures on 2<7-manifoIds. DEFINITION 6.2. Let A" be a 2
:
Ψ^ί
η
V
— <Ρ,.( υ>. η
υ„)
is holomoφhic. Then the family W = { (U x , φ λ) \ λ 6 A } is called a complex structure of X . The pair (X , W) is a complex manifold. A complex structure ? on I defines naturally an almost complex structure on X , which is called the underlying almost complex structure of DEFINITION 6.3. We say that an almost complex structure a of a 2qdimensional manifold X is integrable if it is the underlying almost complex structure of some complex structure on X . The question 'Is the given almost complex structure integrablc?' was investigated in the case of open manifolds by Grauert, Brender, ctc. mainly through functional analytic methods; however, recently P. Landweber made much progress in this area using a geometric approach via the GromovPhillips theorem stated in Chapter III. M. Adachi also made some contributions towards the solution of this problem using the convex integration theory of Chapter IV. We shall discuss these in the following sections. §2. Complex structures on open manifolds
Let A be a ^-dimensional C\n i fty manifold. DEFINITION 6.4. Two almost complex structures Σ 0 , Σ, on A arc homotopic if they are homotopic in the sense of Remark I or Remark 2, and in this case we write a Q ~ σ, . Evidently = is an equivalence relation; wc call an equivalence class by ~ a homotopy class. DEFINITION 6.5. Suppose that A has two complex structures and which induce the integrable almost complex structures σ 0 and σ, , respectively. We say that and are integrably homotopic if a Q and rr, are homotopic, and if { σ | / € [0, 1]} is the homotopy connecting them, then for each 16[0, 1] σ ( is integrable. This is an equivalence relation.
§2. COMPLEX STRUCTURES ON OPEN MANIFOLDS THEOREM 6.1.
Let X be a 2q-dimensional = 0,
H\X,Z)
open manifold.
143
If
i>q + 1,
then every almost complex structure complex structure.
of X is homotopic to an integr
Let X be a 2q-dimensional open manifold. If Ii\X , Ζ) = 0, i>q+ I. then there is a natural one-to-one correspondence between integra classes of complex structures of X and homotopy classcs of alm structures of X . THEOREM 6.2.
Let Ρ be a polyhedron and let e", be the trivial //-vector bundle over Ρ . Let ξ 0, be vector bundles over P. We say that ς () and c, arc stably equivalent and write if there exist natural numbers r. s. such that ς 0 θ £r, ~ ί, θ (].. Clearly ~ is an equivalence relation. Wc let [i] denote the equivalence s
class containing ξ. We consider the inductive limits of the following classifying spaces: B q = lim B 0{n ] , B v = lim /*„„„. To a homotopy class [ij corresponds the homotopy class of some continuous map ,f:P — B iy On the other hand, wc can define a continuous map Ρ • βυ — * « , as the inductive limit of ρ : B U{lt ) —> /i ()(1||) . Thus, we call the lift f of i in the following diagram the complex structure of a stably equivalent ciass [c].
Λ /
r I" f
ρ —1 , n ι
n( )
Let η >2. Let M " bean n-dimensional C X manifold and assume that the stably equivalence class [τ(Λ/")] of the t dle τ (Λ/") admits a complex structure. Then Μ" χ R" " admits a structure. COROLLARY 6.1. Let M n bean n-dimensional orientable C v manifol 3 < η < 6. Assume that u mod 2 = W(M") for some integral cohomo class u S H-(M" , Ζ). where W 2(M") is the two-dimensional Sticfcl class of M" . Then Μ" χ R"~" has a complex structure. The corollary follows routinely from Theorem 6.3 and the following THEOREM 6.3.
144
VI. COMPLEX STRUCTURES ON OPEN MANIFOLDS
PROPOSITION 6.1. Let Κ be an n-dimensional polyhedron, 3 < η < The stable equivalence class [£] of a vector bundle ξ over Κ plex structure if and only if u mod 2 = Η /2 (ξ) for some integral c class ueH 2{K, Z). Here IV 2 (ξ) is the Stiefel-Whitney class in
See Milnor [A5] for the proof of this proposition. Now we shall give outlines of proofs of the theorems. To this end we first recall topological groupoids as discusscd in Chapter V. Let Γ^ be the groupoid of germs of local analytic isomorphisms of C , and let ΒΓ^ be the classifying space of Γ^-structures. Taking differentials yields a homomorphism of topological groupoids " : —» GL(<7, C). Further, we have the continuous map between classifying spaces, υ : BT^ —• BGL(q , C),
Φ
induced by thefirst ν . We are using the same symbol for both of the maps. We regard the second υ as afibration (moving it through homotopies), and denote by FT^ its homotopy fiber. The following construction in a more general setting may simplify our discussion. Given topological spaces X and Y and a continuous map / : X -> Υ, there exist afibration (Ε, ρ, Β) and homotopy equivalences φ : X -> Ε and ψ :Y —> Β such that the diagram
*·, -
V
Ί
ψ
-
Ε
1'
- Β is homotopy commutative. The homotopy type of the fiber F of the fibration (Ε, ρ, Β) is well defined. The construction is as follows. We may assume / to be the inclusion map X\subsetY by passing to the mapping cylinder C f of /. Then for (Ε , ρ, B) we take thefibration (Ω χ γ(Υ),ρ 2, Υ), where ,, is the path space Y
y(y)
-
= { « : [ 0 , 1]-^ Y | u is continuous, «(0) € X , «(1)€ Y)
with the compact-open topology and p 2[u) = u( 1) (cf. Spanier [A6]). Now taking this construction back to the realm of groupoids and classifying spaces, we obtain thefibration u : Β Γ^ — BGL(q , C) whosefiber is denoted by F T ' The groups GL(#, C) and GL(2q, R) have the following Iwasawa decompositions: GL(
GL(2q , R) « 0(2q) χ R " ( 2 " + " ,
. COMPLEX STRUCTURES ON OPEN MANIFOLDS
which commute with the standard homomorphism //. Ufa)
0(2T/)
GL(
/-Τ 0 1
/ /
/_ BGL(q,C) /
V
/
V τ A" — — . SGL(2
π ( (£Γ^)
= 0,
0
We assume Theorem 6.4 for the moment and prove Theorems 6.1. 6.2. and 6.3. PROOF OF THEOREM 6.1. Consider the above diagram. An almost complex structure on X is a lift τ of r in BGL(q , C). We want to lift the f to Β Γ^ . Notice that #GL(
,
1=1,2,...
(cf. Spanier [A6], Steenrod [A7]). Hence, Theorem 6.4 together with the hypothesis of Theorem 6.1 completes our proof. •
146
VI. COMPLEX STRUCTURES ON OPEN MANIFOLDS
PROOF OF THEOREM 6.2. The proof proceeds in the same way as the proof of Theorem 6.1; however, here the obstructions for f Q and f, to be homotopic as lifts lie in
H\X,n t(F φ),
1=1,2
So Theorem 6.2 follows. D PROOF OF THEOREM 6.3. Set X = Μ" χ R"~ 2 . Then because // > 2 , X is a 2(n-l)-dimensional open manifold, and H ' ( X , Z ) = 0, i > η . But by our assumption, the stable equivalence class of the tangent bundle t(M") of Μ" has a complex structure; hence, n — 2 > 0 implies that M"xR"~~ has an almost complex structure. To see this we turn to the following commutative diagram:
/t,(Z?U(/J — 1)) I". π.(ΒΟ(2η-2))
—n.(BV) [''·
1 < 11.
—π,(ΒΟ)
Hence, the theorem follows from Theorem 6.1. • We shall prove Theorem 6.4. in the following three sections. §3. Holomorphic foliations of complexifications of real analytic manifolds In this section we consider holomorphic foliations of complexifications of real analytic manifolds. Let Μ be a real analytic manifold. DEFINITION 6.6. By a complexification of the real analytic manifold Μ we mean a pair (1, CM) of a complex manifold CM and a real analytic embedding /: Μ —• CM , with a complex structure if = {( t/ 1 , ψ„) I <* € A } on CM satisfying φ, <(ί(Μ) η UJ = „(ί/„) η R". In other words, the pair (CM , M) is locally the pair (C" , R"). We state some facts about complexifications of a real analytic manifold in the follow-
Let Μ be a real analytic manifold. (1) Μ has a complexification (i,CM). (2) Let f : Μ W be a real analytic map from Μ to a complex m ifold W . Then we can extend f to a holomorphic map Cf : CM some complexification CM of Μ into W . (3) Let (;',CM) and ( l',C'M) be complexifications of M. Then exists an analytic isomophism h between an open neighborhood and the corresponding open neighborhood U' in i'(M) such that t THEOREM 6.5.
§3. HOLOMOPRHIC FOLIATIONS OF COMPLEXIFICATIONS
147
diagram commutes. . Μ
U \subsetCM
11 ^ " V C C'M
(4) For a complexification complex vector bundles
(i , CM) of Μ it'c have an isomorphism ~ T(M)
T(CM)\M
μ C.
We omit the proof. The above theorem implies that the germ of a complexification of a real analytic manifold is well defined. Henceforth, CM denotes either a complexification of Μ or its germ. DEFINITION 6.7. Let Μ be a c\infty manifold, let ΙΓ be a complex inanifoCand let f : Μ —> W be a C\n itfy map. We say that / is a C-suhmersion if the differential of /, df : T(M) — 7"(H"), induces an epimorphism of vector bundles C(df) : T(M) :•: C — - T( I F ) . REMARK 1. Let Ε be a real vector bundle and F a complex vector bundle. Let φ : Ε —> F be a homomorphism of real vector bundles. Then Iherc is a naturally induced complex vcctor bundle CW : Ε « C —· /·'. REMARK 2. Let Μ be a real analytic manifold. IF a complex manifold, and / : Μ —> W a real analytic map. If / is a C-submersion, then the extension of /, Cf : CM —» W , is a holomorphic submersion, that is. Cf is a holomorphic map of the maximal rank al each point of CM . DEFINITION 6.8. Let IF be a complex manifold. By a codimension q holomorphic foliation of W , wc mean a family .!?" = {( U t . f n ). f xj ! \ π . β A ) of pairs (('„,./,',), where {ί/ ( ΐ |«6.·Ι} is an open covering of" IF and the f n •.(7t —> C Q are holomorphic submersions, and the
lp • V
n
η U„ — Γ^
are continuous maps satisfying f a(x)=f
at°ffi(>x).
χe
u„ n
ν β·
The definition of the equivalence of codimension q holomorphic foliations and ^ is verbatim that of the equivalence of foliations in Chapter V. Further, let Μ be a real analytic manifold and CM a complexification of Μ . Let and be codimension q foliations of CM. Suppose that there exists a neighborhood U of Μ in CM such that and are
148
VI. COMPLEX STRUCTURES ON OPEN MANIFOLDS
equivalent. Then we say ^ and are germ-wise equivalent in Μ and write ~ ^ . Clearly, this is an equivalence relation. DEFINITION 6.9. Let Μ be a C\n ifty manifold. By a codimension q Cfoliation is meant a family & = {(V x , g j , g lf i \λ, μ e A } , where {ν λ\λ e Λ } is an open cover of Μ and the maps g x : V ; -» C* are C-submersions together with continuous maps
8Χμ
VX Π νμ
'
satisfying S l{x) = g Xt l°g,{x),
x e ϋ λ η υ μ.
In particular, if Μ is a real analytic manifold and for each A, is a c < u map, we call S? an analytic C-foliation. The 'equivalence' of C-foliations and the equivalence of analytic C-foliations are defined much the same way as the equivalence of foliations. Let Μ be a C" manifold, and let y be a codimension q holomorphic foliation of CM. Then Y\M is a codimension q analytic C-foliation of M.
LEMMA 6.1. Let Μ be a C w manifold. The map which assigns holomorphic foliation y of CM the restriction ,9~\M defines correspondence between germ-wise equivalence classes in Μ of co holomorphic foliations of CM and equivalence classes of codimen alytic C-foliations of Μ. PROOF, (i) Injectivity. Suppose we have two holomorphic codimension q foliations and of CM with ·9~ χ\Μ Μ. Then by the uniqueness of analytic continuations, and ^ are germ-wise equivalent in Μ . (ii) Surjectivity. Let S/ be a codimension q real analytic foliation of Μ ; & - {(Κι· g,J' g„p Ι α > P e A } . Wc want to extend .f to a holomorphic foliation of CM. Take a C 10 atlas {(Κ , φ η) |
φ χ g : V —. C" χ C". In the same way as we constructed the graph of a Γ-structure in Chapter V, we paste tubular neighborhoods of the (φ ttxg, t)(Y,) ' n C" x C together to get an (// + ?)-dimensional complex manifold W " + q , the holomorphic foliation y ' of W n+ q defined by the projection onto C q , and a C w embedding j : Μ —> IFn+® defined by χ g . Let Cj : CM —< I F n + i be a complexification of j . Since g n is a C-submersion we may assume Cj to be transverse to •9 r' . Then the pullback (Cj)'y' is the desired extention of & . • By virtue of Lemma 6.1 we may identify a codimension q holomorphic foliation of CM with its restriction to Μ .
3 HOLOMOPRHIC FOLIATIONS OF COMPLEXIFICATIONS DEFINITION 6.10. Let ^ and ^ be codimcnsion q holomorphic foliations of CM. We say that J^J and are integrably homotopic write ~ ^ if there exists a codimension q holomorphic foliation y of C(M X[0, 1]) satisfying the following:
(i) y|C(M χ 0)
and
y|C(Mx
(ii) For each t e [0, 1], y is transverse to C(M χ t) . A C'" manifold with boundary Μ χ [0, 1] has a complcxilication C(M χ [0, 1]) which contains C(Μ χ 1) as a complcx submanifold for each 1. Evidently, ~ is an equivalence relation; wc call an equivalence class of / this relation an integrable homotopy class. REMARK.
THEOREM 6.6 (classification theorem for foliations of CM ). Let M" be an n-dimensional open C w manifold. Consider the following
Bf- χ BGL(n Λ "
d
- q , C)
/
/
flGL(i,C)x80L(«-i,C)
® I
(6.D
4 ' / M" BGL(n . C) where r®C is the classifying map of the complexification τ(Μ) :• gent bundle τ(Μ). There is a one-to-one correspondence between classes of lifts of r®C in BY^ χ BGL(n - q . C) and integrable h of codimension q holomorphic foliations of CM . THEOREM 6.7. Let Μ be an n-dimensional compact C"' manifold, consider the following diagram BY* A " (6.2)
Μ
,
1·
/?GL(//.C)
where τ 5o C is the classifying map of the complexification i(M tangent bundle τ (Μ) of Μ . Then there exists a map from the se homotopy classes of holomorphic codimension η foliations of C set of homotopy classes of lifts of T®C to BY c n . We postpone the proofs of the two theorems above and prove Theorem 6.4. PROOF OF THEOREM 6.4. (i) First we show n t (FY^) = 0 . ι < q. Set Μ = S' χ .As 1 < q , Μ is a q dimensional open manifold. We have
150
VI. COMPLEX STRUCTURES ON OPEN MANIFOLDS
η = q; so the commutative diagram (6.2) becomes: SIBY^ ^
y
•
<1
7T
C
Ί
1
R , ~'
Μ = S' χ BGL(q , C) Since q — i > 0, Μ is parallelizable; hence, wc can take t « C as a constant map. Therefore each lift of t ® C in BT^ corresponds to some map of ί χ R'"' into F O n the other hand, CM have only one codimension q holomorphopic foliation, and it comes from the complcx struclure. Hence Theorem 6.6 implies that continuous functions from S' to /·' Γ^ has only one homotopy class, and so π ( (£Γ^) = 0, 1 < q . (ii) To show π (FT^) = 0, we utilise Theorem 6.7. Let Μ = S" . Then Μ is a compact C w manifold. We consider the following commutative diagram: ..·.{τ} € π η(ΒΟ{η)) n„(BV(n))
'•I
4-
κ η(ΒΟ) n n(BV), where 1. is the homomorphism induced by 1 : U(/t) —»'U and is the homomorphism induced by the natural map a : 0(/;) —> U(//). Here the map 1, on the right is an isomorphism. Let t be the classifying map of the tangent bundle r(M) of Μ . Since S" is parallelizable, 1,{τ} = 0 . Hence, from the commutativity of the above diagram we have {T®C}=a.{t)=0. Therefore, we may take r ® C to be a constant map. So there corresponds to a lift of r ® C in a continuous map S" —> FT^ . On the other hand, CM has only one holomorphic codimension // foliation, namely the complex structure, and so by Theorem 6.7, we get un (FI^) = 0, η > 1. • §4. The C-transversality theorem Let Μ, IF be C\n itfy manifolds and let y be a I^-foliation of IF. Then the normal bundle νy of y is a complex vector bundle. Let C r : r(IF) ® C — » b e the natural projection. Let Ε and F be vector bundles. Given a vector bundle homomorphism φ : Ε —> F , denote by C(<^>) : £®C —» F®C its complexification. DEFINITION 6.11. A C°° map / : Μ — > W is C-transversal to y if the composite r(M) ®C τ (IF) ® C —· u.9- of complex vector bundle homomorphisms is a surjection. We denote by CTr(M, y ) the space of C\n ifty maps f : Μ —> W which are C-transversal to y with c l topology.
54. THE C-TRANSVERSALITY THEOREM
151
Let CEpi(r(M), v!7) together with the compact-open topology be the space of vector bundle homomorphisms φ : τ(Μ) —> ν,Ψ" whose complexificaiions C(
τ(Η') -Ϊ-. u.7
sits in CEpi(r(M), u.7~) . Here π is the natural projection. THEOREM 6.8.
If
Μ is an open manifold,
the map
φ : CTr(M..7) — CEpi(r(A/). //.'/ ) sending
f to π ο df
is a weak liomoiopy equivalence.
This theorem follows from Theorem 3.14. Wc leave the details of the proof to the reader. See Theorem 6.9 tofill the gap between C 1 and C'" . Theorem 6.8 implies Theorem 6.6. The proof is verbatim the proof of the classification theorem for foliations of open manifolds in Chapter V. We proceed to the proof of Theorem 6.7, which is slightly more cumbersome. Here we apply Theorem 4.1 of Chapter IV. Let Μ be an ndimensional C'" manifold. Let (£,£,JT) be a Γ^-foliated microbundlc over Μ . Then we may think of Ε as a 3/i-dimensional C'" manifold and W as an analytic C-foliation of Ε . Let π : r(E) — νΐ be the natural projection. Denote by CTr" J (M,lf) the spacc, with the C' 1 topology, of C'" maps / : Μ — Ε such that the composite of complex vector bundle homomorphisms τ (Μ) ® C —» τ(Λ) co C —· v£ is surjective. If f € CTr"'(M , ίΓ), then ("
nod: CTr'"(M, induces the following
—. C Epi(r(M), v r£ ),
surjection
/r0(CTrω ( Μ , f ) ) — /rQ(C Epi(r(M), i'£~)). Let CTr'(M , W) be the spacc of c l maps / : Μ — Ε transverse to if . with the c l topology. Then CTr' u (M, f ) is a subspace of CTr (M . . By the approximation theorem (cf. §2, Chapter III) we have the following
152
VI. COMPLEX STRUCTURES ON OPEN MANIFOLDS
PROPOSITION 6.2. Let Μ be a compact C"' manifold. Then the inclus map i: CTr w (M, g) -» ΟΊτ\Μ, g) induces the surjeclion:
1. : π 0 ((ΤΓΓ ω (Μ, r ) ) — /r0(CTr'(M, g)). We omit the proof. See the approximation theorem in Chapter III. Now Theorem 6.9 results from Proposition 6.2 and the following 6.10 (C-transversality theorem). Let Μ be a compact Then the map
THEOREM
ifold.
C'" man-
nod: CTr' (Μ , g) —- C Epi(r(M), vg) , sending f to no df
induces a surjeclion
n 0(CTr\M
, g)) — /t0(C Epi(r(M), g)).
PROOF. Set X - Μ χ Ε, and let ρ : X — > Ε be the projection onto the first factor. Then (Χ, ρ, Μ) is a smoothfiber bundle. Denote by Sect(A") the space of C 1 sections in ( Χ , ρ , Μ ) with C 1 topology. Let (A"1, , M) be the one-jet bundle of germs of c l sections in (Χ,ρ,Μ). Let S(X l) be the space with the compact-open topology of continuous sections in (Λ"1 , p ] , M). Then taking one-jets we gel the continuous map •/' : Sect(A) — Sect(A'). Let C l(M , E) be the space of C 1 maps from Μ to Ε with the C topology, and let Hom(r(M), τ(Ε)) be the space of homomorphisms of the tangent bundle τ(Μ) of Μ into the tangent bundle τ(Ε) of Ε with ihe compact-open topology. Likewise we consider Ηοητ(τ(Μ), x(g)) - Then we have the commutative diagram
C'(M , E) — H o m ( r ( M ) , τ(Ε)) —2—. Horn (τ(Μ),υ£) U
Sect(A) . Sect(A') CEpi (τ(Μ),υ£) where d associates to a map / its differential df , η associates to φ the composite π ο φ, and φ and ψ are natural isomorphisms. Put 9 = π~'(0Ερί(τ(Μ), vg)) . Then we have the following commutative diagram C'(M, E)
——• Hom(T(M), τ(Ε)) — H o m ( r ( M ) , vg )
uj C T r ' ( M , f ) —-—> where
u}
uj
9
CEpi(r(M), vg)
is a surjection. Hence, the map — Ti0(CEpi(T(M),//g·))
§4. THE C-TRANSVERSALITY THEOREM
153
is onto. Thus, to prove the theorem it is enough to show that the map d, : w0(CTr'(A/, r j ) —. π 0 (Ο) is a surjection. Now choose an open subset Ω of A'1 which gives the correspondence by the isomorphism ψ as follows Hom(r(M), τ(Ε)) D
"I-
Sect(X')z>
Sect(A , Ω)
We then get the following commutative diagram: C'(M, E)
Horn (r(M). r(£)) ο
CTr'(Μ,/) —^ C
4 Sect (Χ, β )
—^Sect (Χ1. Ω) Sect (X 1 )
Seel (X)
where Sect(A', Ω) = (/') '(Sect(X' , Ώ)). Hence, it suffices to show that Gromov's theorem of Chapter IV is applicable to our Ω . Now A'1 is a jet bundle as shown below X = J (Μ, E) • J (//, 3//) = A/(3/i. //; R)
p'i
1
X =Μχ Ε
f| |p, A/ = A/ Here A/(3/i, n\ R) consists of the (3//, /i)-matrices over R. whose structural group is £'(//, 3n) = l'(3//) χ L\n) = GL(3/z. R) χ GL(t;. R). However, since Μ χ Ε is locally compact and paracompact. wc may assume its structural group to be 0(3η) χ 0(/i). Further, because {Ε , ζ , 8?) is a Γ^-foliated microbundle over the C'" manifold A/ , we can choose a c < " atlas in Ε as follows: set {(U
lt9 l)
|AeA},
φ, : υ, — R 3 " = R" φ R : " , φ λ0 ψ; ί\χ,ν)
= (.χ'. y)
if (Λ, Π υ μ φ Θ
154
VI. COMPLEX STRUCTURES ON OPEN MANIFOLDS
so that Ί where f = {(U
k,
=
X = P(X),
1 y' = s Xl l(x)y, ψ χ), g X/ 11 λ, μ e Λ } , and the maps 8ΊΜ'·
υληυμ
> Γ^;
8 λ μ · · Ρ ( ν λ ) η ρ ( υ μ ) - ^ GL(«,R) take on values in the Jacobian matrices of coordinate transformations of Μ . Let F be a subgroup of 0(3n) consisting of elements of the following form: /Α
ON1"
( A eO(n),
c)]2n
I Ce//(U(//))\subsetO(2//),
where // : U(//) —» θ(2/ζ) is the standard map. Then we can reduce the structural group of the above jet bundle to Fx θ(/ζ). Now wc define a closed subset Σ of M(3/i, n; R) as follows. Consider the natural correspondence: φ : Μ (In,
η;
R) —< M(n, //; C)
ί(ί) - ·'•»>•
Set Σ, = M(//, η ; C) — GL(n , C). Then Σ, is a closed subset of real codimension two. Write Σ 2 = ^"'(Σ,)· The set Σ, is also a closed subset of M(2n, η ; R) of real codimension two. Put M(3n , η; R) D Σ = M(n, η ; R) φ Σ,. Then Σ is a closed set of codimension two Further Σ is invariant under the action of F χ 0(n) . Hence, we may consider a sub-jet bundle J Σ of the jet bundle j ' ( M , Ε) , whosefiber is Σ. Then J z is also a closed codimension two subset of J X(M, E). Put Q = y ' ( M , E) - J L. The definition of Ω implies that Sect(Arl , Ω) corresponds to ζ? under ψ . Therefore, Sect(A"^) corresponds to CTr'(M,l?) under ψ. Now Theorem 4.1 applied for Ω confirms that y' : Sect(X, Ω) —* SectX1, Ω) is a weak homotopy equivalence. Hence, we have proved the theorem.
•
55. NOTES
155
§5. Notes
1. Almost complex structures on closed manifolds are not in general integrable; Frohlicher showed the existence of an nonintegrable almost complex structure on S b . See A. Frohlicher, Zur Differentia/geometric' der komplexe Structuren, Math. Ann. 129 (1955), 50-95. Later Van de Ven showed that there exists an almost complex structure which is not integrable on a four-dimensional manifold. See A. Van dc Ven. On the Chern numbers of certain complex and almost complex man Proc. Nat. Acad. Sci. USA. 55 (1966), 1624-1627. Recently S. T. Yau showed the existence of a four-dimensional closed parallelizable manifold (hence, an almost complex manifold) without complex structure. Also N. Brothcrton constructed an example similar but different from the Yau's. See S. T. Yau, Parellelizable manifolds w ithout complex stru ture, Topology 15 (1976), 51-53, and N. Brothcrton, Some parallelizable four manifolds not admitting almost complex structure. Bull. London Math. S 10 (1978), 303-304. The above works by Van de Ven, Yau, and Brothcrton arc all based on Kodaira's work ('). 2. Almost complex structures of open manifolds of dimensions less than or equal to six, on the other hand, can be made integrablc through homotopies by Theorem 6.1 (Adachi [Cl, C2]). So far no examples of open manifolds with almost complcx structures not admitting complex structures arc known.
( 1 ) K. Kodaira. On the structures ofcompsct complex analytic surfaces 1. Amcr. .1. Math. 86 (1964), 751-793: II. Amer. J. Math. 88(1966). 687-721; HI. Amcr. .1. Malh. 90 ( 1968). 55-83; IV. Amcr. J. Math. 90 (1968). 1048-1066.
CHAPTER VII
Embeddings of C^ Manifolds (continued) In this chapter we discuss embeddings centered around Haefliger's theorem which is the most fundamental in the theory of embeddings. Haefliger's theorem contains the classical theorem of Whitney given in Chapter II as a special case. Haefiiger gave two proofs for his theorem. Here we take up the second proof and introduce briefly a generalization of Whitney's method of eliminating double points of completely regular immei'sions given in Chapter II and the works on embeddings of complexes by van Kampen ('). Wu [C24], Shapiro [C16], etc. §1. Embeddings in Euclidean spaces Let V be an η dimensional C\n itfy manifold and let Ar be the diagonal set of V χ V , that is, V χ V ο Δν DEFINITION 7.1.
= { ( x , A-)|.v G V }.
Wc say that a continuous map F : V χ V - Ay — S"'~ '
is equivariant
or Z 2-equivariant
if F satisfies the relation
F(x , y) = - F(y , χ),
χ, ν e V , χ Φ y.
A homotopy {F,} , F t : V χ V - Δ,. —> S"'~ is equivariant if for each 1 € [0, 1], F t is an equivariant map. Two equivariant maps F and G are equivariantly homotopic if there is an equivariant homotopy connecting F and G \ in this case we write F ~ G or F ~ G . Evidently ~ is an c z, <· equivalence relation. Let / : V —» R'" be an embedding, and define the associated map f as follows: / : F x V-Ay — .S"·"', n
( ' ) E. van Kampen, Komptexe
) )
\f(x)
in euktidisclien
157
-/(.') I ' Raumcn. Hamburg Abh. 9 ( 1933), 72-78.
158
VII. EMBEDDINGS OF C°° MANIFOLDS (CONTINUEDI
Then / is clearly an equivariant map. Further, if two maps /, g : V — R'" are isotopic, their associated equivariant maps / and g are equivariantly homotopic.
c\infty THEOREM 7.1. Let V bean n-dimensional manifold. The as of each embbedding f : V —» R'" to its associated equivariant ma V χ V - Δν —> S'"~' gives rise to a map Φ of equivalence classes :
{/: V - R"\ embedding) / = {F : Fx V - Av —» S'" ' , equivariant
map )/ ~ .
The map Φ enjoys the following properties: (a) 7/ 3(/i + 1) < 2m , Φ is a surjeclion. (b) If 3(/i + 1) < 2m , Φ is a bijection. Henceforth the range for (//, ///) with 3(// + I) < 2m will be referred to as the stable range. We postpone the proof of this theorem till in a later section; here we state some facts which follow routinely from the theorem;'^ DEFINITION 7.2. We define an equivalence relation ~ in V χ V — Α.. by (χ, y) ~ (y, χ), "χ, y e F , x / y , and denote by F* the quotient space of V χ V - Ay under ~ . Wc say that F* is the reduced symmetric square of V . Consider the following relation in (F χ F - A, ) χ S"'~' : (x, y; s) ~ (y, χ; -s) ,
x , y e F , χ φ y, s e S'" '.
Let Ε be the quotient space of (F χ F -A) χ S'"~'
. Define ρ . Ε — F* by
p([x, y; s]) = [χ, y]; then (F,//, F*) is afiber bundle whosefiber is with the structural group Z 2 , which is an associated bundle of the double covering space F χ V - Av —> V' . The following lemma is trivial.
LEMMA 7.1. There is a one-to-one correspondence between equivar motopy classes of equivariant maps of V χ F - F ( into S'"~' and ho classes of sections in the fiber bundle (Ε, ρ, I" ).
This lemma together with Theorem 7.1 reduccs the question of the existence of embeddings of an //-dimensional c\infty manifold V in R'" and their classification by isotopies within the stable range, to the question of the existence of cross sections of thefiber bundle (Ε, ρ , F") and their classification by homotopies.
COROLLARY 7.1. Within the stable range the classification of the dings of an n-dimensional C\n iy tf manifold V in Euclidean space R depend on the C°° structure of V .
This is evident from the above comment.
51. EMBEDDINGS IN EUCLIDEAN SPACES
15")
Within the stable range any two embeddings sphere S" in Euclidean space R'" are isotopic.
COROLLARY 7.2.
dimensional
o
The reduced symmetric space ( 5")* of the n-dimen S" has the homotopy type of the real projective space R P
LEMMA 7.2.
sphere PROOF.
Consider the following commutative diagram. A- 6
S" ——
II [AT]
e RP"
S" χ S" - A
.
(S ")'
•
3 (.Ν, - A )
I T β
[{Λ-, - Λ - ) ]
7.2. We show that if F = S" any pair of cross sections s 0 and s, of (Ε , ρ, V') arc homotopic in the stable range. The obstructions for this homotopy arc in the cohomology groups PROOF OF COROLLARY
/7 | .(^'"" 1 )),
H'(V\
1=1,2
with local coefficients associated with the covering V χ Γ - J,. — I '* (Steenrod [A7]). We have // 0. Now Corollary follows from Theorem 7.1 and Lemma 7.1. • PROOF OF THEOREM 2.3.
LEMMA 7.3. 7/ V is k-connected. the cohomology groups with loca ficients associated with the double covering Γ χ Γ - Δν — F* sa following : H\V\ Π^'"' 1)) = 0. ) ι > 2η - k.
Note that the lemma is independent of the choice of m . Let Z,' bexmcoFX and Z ; rthe twisted integer system associated with the double cover V χ V - Δν — F*, and let Z" be the other. Since 7c(Sm~ ) is afinitely generated Abclian group it is a direct sum of cyclic groups. Hence, it suffices to show //'(I'* , Ζ w Z ( ) ) = 0, 1 > 2η - k . for every prime number ρ. Looking at the cohomology exact sequcncc whose coefficients are the exact sequence REMARK.
PROOF.
0 — z' — z' — z' » z n — 0. we see that it is sufficient to show T7'(F*. Ζ') = 0. 1 > 2// - k . Consider the Thom-Gysin exact sequence for I' χ Γ -A, -— Γ° regarded as δ sphere bundle (cf. Spanier [A6], Steenrod [A7]) ··• —> H'{V
x C - j ) —> H'(V'
— H i+\V χ V
-Δ)—.···.
, V') —» //' +I (F* · Z")
160
VII. EMBEDDINGS OF
°° MANIFOLDS (CONTINUED
The proof will be complete when we show 7/'(F χ F - A) — 0 , /' > 2/7 - k , since in this case we have by the exactness H'(V' , Z') = H ,+\v' , z i>2n - k , and these groups are 0 for 1 > 2//, where dim F* = 2/i. However, by the Lefschetz duality theorem (Spanier [A6]) we have χ F — Δν) = H 2n_ j(V χ V , Av).
H'(V
Using the Kunneth formula and the fact that F is Ar-conncctcd, wc see that the right-hand side is zero for 2n - k < i < In . • PROOF OF (a). We assume that F is a closed //-manifold which is kconnected. If m = 2n — k then by the assumption 2k + 3 < η, the pair (η, m) satisfies the inequality 3(n + 1) < 2m. The obstructions to the existence of cross sections of (Ε, ρ, F") lie in the cohomology groups 1=1,2,...
H\V , / 7 ( _ | ( y " - 1 ) ) ,
with local coefficients associated with the covering V χ V - Av —> F* (cf. Steenrod[A7]). But by Lemma 7.3 we have
H\V' , n^^S"'"'
1))
= 0,
j > 2/z - A·.
Hence, the above cohomology groups are 0 for all 1 > 0, and so we have proved part (a) of the theorem. PROOF OF (b). The obstructions for cross sections s0 and s, of (E, //, V') to be homotopic lie in the cohomology groups 7/'(F", n ^ y " ' ' 1 ) ) ,
1=1,2...
with local cofficients associated with the covering spacc F χ F - A,. —> V* . But by Lemma 7.3 these groups are zero for all i > 0. Hence, we have proved part (b) of the theorem. • §2. Embeddings in manifolds
In the remaining sections of this chapter, we follow Haefliger's paper [C9]. DEFINITION 7.3. Let X and Y be topological spaces, and let F : Χ χ X —• Υ χ Y be a continuous map. We say that F is equivariant or Z 2-equivaria if for each point (χ,, x 2) of Χ χ X , we have
F(x,,x2) = ( y | ,y 2 )e Fx F, F(x
2,x i)
We say that F is an isovariant
= (y 2,y [). map if F is equivariant and satisfies
F-'(A
y)
= Ax,
where Αχ and Ay are the diagonal sets of X and Y respectively. A homotopy F t : Χ χ X -* Y x. Y is equivariant if for each / e [0,1], F is an equivariant map. Two equivariant maps F and G are equivariantly homotopic, F ~ G , if there is an equivariant homotopy connecting them.
52. EMBEDDINGS IN MANIFOLDS
161
A homotopy F : Α χ X —> Υ χ Y is isovariant , if F is isovariant for each 1 e [0, 1]. Two isovariant maps F and G are isovarianily homotopic F ~ G , if there is an isovariant homotopy between them. Evidently the ~ i ' c and the ~ are equivalence relations. i DEFINITION 7.4. Let /, g : X — Y be continuous maps, and suppose there arc two homotopies {/zr} and {h[) between them. By a homotopy connecting these tyvo homotopies wc mean a continuous map // : Λ' χ / χ / —> Υ , / = [0. 1], such that if h t (x) = H(x,
τ, I), wc have the following
(') Α,.Ο-Α,. ''r.l = Λ τ · (ii) Λ0 ( = Λ0 = //; = /, /), ,=/), =h\ = g.
When such a homotopy exists we say that the homotopies {/; r} and {h' T} arc fiamotopic and write {/i t} ~ {h[} . Clearly this is an equivalence relation THEOREM 7.2. Let F he an n-dimensional Μ be an m-dimensional manifold. -
closed
manifold,
an
(a) Suppose 3(// + 1) < 2m . Then a continous map f : V — Μ topic to an embedding g if and only if there exists an equivarian I f ·. V χ V —· Μ χ Μ , satisfying
the following
:
(i) T70 = / χ / . (ii) //, is an isovariant map. Further, we can choose g so that g χ g is isovarianily homotop (b) Suppose 3(// + 1) < 2m. Let f. g : V —> Μ be embeddings. a homotopy {/ r} between f and g is homotopic to an isotopy if a there exists an equivariant homotopy
Η χ <: Γ χ V —. Μ χ Μ .
r, 1 € [0. 1]
such that 0) "r.0 = . W r · (ii) ΙΙ χ , is an isovariant
homotopy.
The proof of Theorem 7.2 shall be given in the next section; here wc state one of its easy consequences in the following
COROLLARY 7.3. The classification of the embeddings of F in Μ stable range does not depend or. the C\INFTY structures of I' and Μ
The corollary is obvious by Theorem 7.2.
162
VII. EMBEDDINGS OF
°° MANIFOLDS (CONTINUED
A(x)
1
FIGURE 7.1 PROOF OF THEOREM 7.1.
We look at the following diagram; Μ
{/: V - . R m , embedding}/
= R'" :
{// : V χ V - R'" χ R'" , isovariant}/ ~ /
'
/
Φ\
!
λ
: V χ V - Δ,,
, equivariant}/ ~
Here we defined Φ in Theorem 7.1, and we define "F via Theorem 7.2:
χ
/ΐ,(Λ-, , Χ , ) - //,(.<•-, , .V,)
,χ 2) = |/ι,(χ,, χ 2 ) - Α,(.ν,
where Η(χ χ, χ 2) = (Λ, (Α", , Χ 2 ), h 2(x i , χ 2)). Then Η is an equivariant map. Further, the above diagram commutes. Hence, if A is a bijection, Theorem 7.1 follows from Theorem 7.2. We show that A is surjective. Consider an equivariant map φ: V χ V - Δ,
S"
and define //, :V χ V
\Γ χ Κ"
as follows. Choose a sufficiently small tubular neighborhood U of the diagonal set Δν in V χ V , Δν C U C V χ V . Let λ : R1 — R1 be a C\n i function such that (i) A is monotone increasing, (ii) Λ(λ) = 0, x < 0 , λ(χ) = 1, jc > 1 , (iii) 0 < A(x) < 1, xeR 1 (Figure 7.1). Define a C\n ifty function μ on U , μ : U —> R . by p(x,y) = X(p((x,y),A
v)),
where ρ is a Riemannian metric in U . Nov set 1(0,0).
,,=,2.
163
§3. THE PROOF OF THEOREM '..2
Then Ηφ is an equivariant map which satisfies Η φ '(z) R «) = Δν . Set Λ(Η φ) = Η φ ; we then have O tv-
\ '
=
\φ(
' χ ·>) + Φ(χι • Λ"·>) , χ]) + ^(.ν,ΤΖ)!
Χί
χ =
*(Λ|'
In a similar manner we can show that A is injcctive.
:)·
•
§3. The proof of Theorem 7.2 As in Chapter 2, for the proof of Theorem 7.2 wefirst consider immersions and then go on to embeddings. Let V and Μ be manifolds of respective dimensions η and m , and let /0 , f t : V —< Μ be embeddings. It is evident that if J' 0 and /, are isotopic, then they are regularly homotopic. We have the following routine LEMMA 7.4. Let f : V —> Μ be an immersion. Μ χ Μ , the set Δν is open in (/ χ . THEOREM 7.3. Let V bean n-dimensional compact let Μ be an m-dimensional C°° manifold.
For c x
f χ / : Fx manifold,
(a) Assume 3(/; + 1) < 2m. An immersion f : Γ —» Μ is regularl homotopic to an embedding f t if and only if there exists an equ motopy II, : V χ V — > Μ χ Μ swc/i t/;al
(i) //„ = / χ /, //, 1s isovariant. and (ii) for all t in [0, 1], Δν is an open subset of 1I~\a k i) . Further, we can take f χ f to be isovarianily homotopic to II (b) Assume 3(/J + 1) < 2m. Let f a , f : V — A/ be embeddings. regular homotopy [f] connecting f Q and f is regularly homoto isotopy {f'} between f 0 and f if and only if there exists an e homotopy li : V χ V — Μ χ Μ satisfying the following : (i) 77; Q = / χ f , If | is a// isovariant homotopy . (it) "o., = /o*/o. , = j, x/,. (iii) For alt t and τ in [0, 1], is an open subset of 1Γ\(Δ u). Here by a regular homotopy connecting { f ) and topy {F r ( } satisfying (ii) For each τ and I in [0, 1], Fr
(
wc mean a homo-
is an immersion.
We shall prove Theorem 7.2 assuming Theorem 7.3. Butfirst we give a preparatory discussion on the Haefliger-Hirsch theorem. Denote by Μοη(Γ(Κ), Γ(Μ)) the set of all bundle monomorphisms of T(V) in T(M) with the compact-open topology. We say that a continuous map φ : T(V ) —» 'F(M) sending afiber to the correspondingfiber is isovariant
VII. EMBEDDINGS OF
MANIFOLDS (CONTINUED
if for each χ e V the restriction of φ to T X(V)
satisfies
φ(-ξ) =-<ρ(ξ), φ~'(0) = 0. Denote the set of all isovariant maps from T(V) to T(M) by l(T(v),T(M)). This set is also given the compact-open topology. Let f 0 and f be equivariant maps from neighborhoods U a and i\ of Δν in V χ V to Μ χ Μ : f
0
: U Q —> Μ χ Μ,
/, : (/, —• Μ χ Μ.
We say that f Q and /, are equivalent if there exists a neighborhood U of Αν , Ay C U C U 0 Π ί/,, such that /0|t/ = ./,11/ . Let l(A,.,A u) denote the set of equivalence classcs of isovariant maps from neighborhoods of J, to Μ χ M. We give /(Ay, Αλ/) the compact-open topology. Let lmm(F,M) be the set of all immersions of V Γη Μ endowed with the cl-topology. Denote the set of topological immersions of V in Μ by TOP-Imm(K, M); this set too is given the compact-opcn topology. We consider the following diagram: Mon(7(F). 7'(M)). Imm(F, M)
/(/•(F). 7'(M)) HA...
A..)-
TOP-lmm(V, M) Here d is a continuous map which assigns to an immersion f its differential df; δ assigns to / the germ of / χ / : V χ V — > Μ χ Μ in J r ; i and j are inclusion maps. We define θ as follows. Give V and Μ each a complete Riemannian metric. ( 2 ) Then wc have the map exp : 'F(I') — V sending X e T X(V) to the end point of the geodesic starting at .v of length ||X||. Define e v : T(V) - V χ V by e r ( A ) = (ex pA\exp(-*)). The restriction of the map e v to some neighborhood U of the zero section V of T(V) defines a difTeomorphism between U and some neighborhood of Av . We have a parallel situation for e k f : 7"(M) —» Μ χ Μ . Now define Θ on I(T(V) , T(M)) by θ(ψ) = ί Μο ψοε-;\
ψ e I(T(V)
, T(M )).
and let θ be the the restriction of θ to Mon(7"(F), T( Μ )). ( 2 ) We can always do this. See Nomizu and Ozcki. The existence metrics, Proc. Amer. Math. Soc. 12 (1961). 889-891.
of complete
Riemannian
§3. THE PROOF OF THEOREM .2 THEOREM
7.4. (i)
The diagram above induces the following
commu
diagram : 7T((lmm( V. Μ))
—-—. π,(Μοη(Γ(Κ), 7"(Λ/)))
j'.
j".
7[,(TOP-Imm(l'. Μ)) — <
π,(/(Γ(Γ). 7'(.t/)l)
τι,(/(4ρ., J,,)) t ^ " " 5 '
(ii) <5. ο l.v a bijcction for 3/; + I < 2/» . δ. ο is a surjeclion for 3/1 + I < 2m . PROOF, (i) The commtativity in the triangle is obvious by definition. The commutativity in the rectangle on the left follows from the fact that the tangent bundle of V is equivalent to the normal bundle of Δ, in F χ I
T(V)
— U \subsetV χ F
where J(x) = (λ". Λ"). (<5 assigns to a map f : Μ — Γ its topological differential df.( }) (ii) This follows from the three facts. (a) θ is a bijcction. (b) j* is a surjection when 3/J + I < 2m . and it is a bijcction when 3// + 1 < 2m. (Haefiiger and Hirsch [CTII]; this is profoundly related to the homotopy groups of Stiefel manifolds). (c) d< is surjectivc when //
I. Topology 3 Suppl. (1964). 53-80.
166
VII. EMBEDDINGS OF C°° MANIFOLDS (CONTINUED
V
V FIGURE 7.3
less than or equal to δ(χ). If we take δ small enough, these geodesies starting at different points of Av do not intersect. Let δ be sufficiently small so that we may assume our isovariant homotopy satisfies the following: a) Hf
= H t\U
b) Hf
= //, on Av .
s
near 1 = 0, 1= 1 .
Let λ : [0, 1] -» R be a C\n ifty function (Figure 7.2) such that (i) λ(0)= I, A ( l ) = 1, (ii) 0 < X(t) < 1, 0 < t < 1 . We now define a new equivariant homotopy H t by R, =
'
ί W
Ο" iVt)e, on
u i s.
On U s-U l s we define H\ as follows. Suppose that a point ζ of U 0-U )g is on a geodesic starting at (χ, χ) in Δν. and orthogonal to A v . Suppose further that this geodesic intersects dU l s and 0U S at r, and z 2 , respectively. Then H' t(z ) is defined to be the point on the geodesic (unique if <5
§ . PROOF OF THEOREM .
167
is small enough) connecting //,''(?,) and //,(r-,) in M x M such that ζ, ζ :zz 2 = //f i(z1)//;( = ) ://,'(z)//,(r:) (see Figure 7.3). The map //(' is an equivariant homotopy which is equal to / χ / for 1 = 0 and to 77, for 1 = 1 . Further, J,. is open in 7/(' '(J,,) • Hence we have shown part (a) of Theorem 7.2. The proof of (b) is similar. Π §4. Proof of Theorem 7.3 Now wc shall outline the proof of Theorem 7.3; the method wc use here is a generalization of the way we eliminated double points of a completely regular immersion. A. Construction of a standard model for a deformation eliminating double points. We construct a model having the following properties: L, /.' are C°° mar.ifolds, Φ : L —> /-' is an immersion and Φ( : /. — L' is a regular homotopy yvith Φ0 = Φ such that (1) Φ, is an embedding. (2) 0 t\K c = Φ„|ΚΓ , A' a compact subset of L . We start with the following three objects (Figure 7.4): (i) A compact manifold D with afixed-point free involution .7 : I) — D, ,/2 = I ./(.ν) ^ A" for all A' 6 D . (ii) A function A : Ο — [-1 , I) such that (a) A ο J = A ; (b) A~'(-l )=0D, (c) άλφ 0 on A - '(0). (iii) A vector bundle ξ = (L,//, D) . Consider 1 = [-1 , +1] and let θ' = 1) χ // ~ , where (d , t) ~(J(d),-i),
d e D, ι e I.
Then 7)' is afiber bundle with the base spacc D/J and fiber / . We construct a vector bundle L' over D' as follows. Let η : Dxl — DxD be a map defined by <*(d , 1) = (d , J{d)) , and let ξ = (ί,ρ,ΰχ1),
ξ^α'(ξχξ):
that is, I = {(/,, l m , 1)1 l d e L d , /,,„, e F , U ) , ι e I , d e D j. where /.,, denotes thefiber over d . Define L' by L! = L/ ~, Va > ' w w - ' t f
"')·
We denote the elements of £>' and Z.' containing ( £>',
, /, (ι/| . 1).
168
VII. EMBEDDINGS OF C°° MANIFOLDS (CONTINUED
*
DP
"
FIGURE 7.4
is the desired vector bundle. Define an immersion φ : D — 1)' by ψ (d) = [ά,λ(ά)]. Identifying D and D' with ihe zero-sections of L and L' . we extend ψ to an immersion Φ ·. L —. L' , *(/,) = [(,. O.A( R satisfying 0 < μ(ά) < i(d) + I , p(d) + p(J(d))>2k(d), μ(ά) = Oif k(d) < -1/2. A function μ satisfying the above specifications exists. For instance consider a C°° function /?:£>—· R such that 0
+ 1/2)β(ά) .
= [ά,λ{ά)-ίμ{ά)\.
1€[0, 1],
§ . PROOF OF THEOREM .
169
Then the constitute a regular homotopy such that φ 0 = φ and is an embedding. If [d , k(d) - p(d)\ = [d' , X(d') - fi(d')] , wc p(d)) ~ (d' , X(d') - p(d')) . Hence, by definition d — d' or
d' = J(d) , X(d') - fi(d') = ~U(d) - /i(d)). Since X(d') = k(d) , wc must have 2/.(d) = p(d) + p(J(d)), d = d' . We assume the structural group of the vector bundle L is an orthogonal group, i.e., we think of L as a Euclidean vector bundle so that eachfiber of L has a Euclidean metric. We denote by ||//(|| the length of an element of the fiber L (/ over d. Let t» : R — R be a bcll-shapcd function (Figure 7.5): (i) a(R) = [0. IJ. (ii) «(0) = 1 .
0 l(l l l) = [l d.O,X(d)-n(\l
tl\)l,i(d)].
Then the {Φ,} is a regular homotopy such that Φ 0 - Φ and Φ, is an embedding. Further, Φ( does not move outside some compact set.
B. Selection of a model. In the hypothesis of Theorem 7.3 (a) yvc may assume that / : V — Μ is in general position, that is, the map f χ / : V χ V —> Μ χ Μ is transverse to Δλι at each point of V y. V - A,.. This fact follows from the Thom transversality theorem (cf. §5, Chapter 1). Wc may also assume that no triple points exist since in < 2m . This can be shown in a similar way as in the proof of Theorem 2.8. We may assume that an equivariant homotopy 7/( is defined for ι e [-1, 1] = I and satisfies the following: (1) The C\n itfy map Η : V χ V χ 7 —· Μ χ Μ , //(υ,, v 2,t) = Η,(ν,, ·//,) is transverse to J u outside Δν χ / in which case Δ = H~\d sl) - J, χ / is a closed submanifold of V χ V χ / . (2) Let p. , ρ, : V χ V χ 7 be the projections to thefirst and second components. Then ρ, - ρ χ\Δ : Δ —» V is an embedding.
170
Ν», EMBEDDINGS OF C°° MANIFOLDS (CONTINUED]
(3) Let /: V χ V χ I -» V be the projection onto the third component. Then 0 is not a critical value of ί\Δ : Δ —» I . The conditions (1) and (3) are satisfied by Thorn's transversality theorem (§5, Chapter I). Here we do not require any restrictions on dimensions. As for condition (2) notice that 3(/z + 1) < 2m implies 2 d i m J < d i m I / , and hence, we may use Whitney's embedding theorem (§4, Chapter II) (wc can redefine H t so that ρ, \Δ is an embedding). Now to construct the model Φ ( : L —» L' suggested in A we take for (i), (ii), and (iii) the following: (i) D = ρ ι(Δ) = ρ 2(Δ), J =// 2 op"'. (ii) A : D —> /, A = /op~'. (iii) The normal bundle of Ο in Κ for the ξ = (L , ρ, D) . We have ρ χ(Δ) = ρ 2(Δ) and J°J= \ since the H l are equivariant. C. Application of the model. We shall construct diffeomorphisms —. V,
ψ' ·. L' r —> Μ, - - .j.
where L ( ( respectively, L' ( ) is an e-neighborhood (respectively, c -neighborhood) of D (respectively, D' ) in L (respectively, L' ) such that (a) /ο ψ = ψ'οΦ, (b) /-'(!F'(L;)) =
/,(«) =
Av),
ν i f(L
t ).
Then the {/,} is a regular homotopy such that f Q = f and /, is an embedding. This shows the "sufficiency" part of Theorem 7.3 (a). The "necessity" part is obvious. Constructing these Ψ and Ψ' constitutes the hardest part of the proof. We shall accomplish this in three steps. Step 1. We construct embeddings ψ :D
— >
V ,
ψ': D' —> Μ
such that
(e0) / ° V = (/°
T d(D)
7; (D #)
PROOF OF THEOREM and / ( F ) η ψ\ΐ)')
=
f(
171
V(D)).
As before we let ρ, , p., : V χ V χ / —> V be the projections onto the first and second components. Similarly, we define p\ and ρ, : Μ χ Μ — Μ. The map y/ shall be the inclusion of Din Κ . As for the ψ' the restriction ψ' = ψ'\φ(Ω) is defined by the condition (a„ ). Set A = {[rf, 1] € D'|0 < / < X(d) } . and - l] = P\ ° //(
[t/ . i) 6 ,1.
Then ψ'α is a continuous extension of over A. Since I is a deformation retract of D' . wc get an extension ψ : D' -* Μ of ψ' α . On the other hand, we have dim A) < m - //. and so, we can extend ψ' 0 :
= f(D)
(this follows from Lemma 5.2 in Shapiro [C 16]). Finally, applying the Whitney embedding theorem with 2dim // < dim Μ wc obtain an embedding ψ' : D' — Μ satisfying («) ψ'~ψ'. (β) ψ'\υ (D) = ip',\U(D) , where U(D) is some neighborhood of I), (γ) W- ¥(D)) Π /(//) = //. The condition (y) follows from dim D + dim Γ < dim Μ . The condition (β) implies the condition (c0 ). and the condition (;·) implies the condition (b0 ). Step 2. We construct vector bundle monomorphisms ψ ·. L—• T(V),
ψ : L' —* T(M)
such that (i) ψ\D = ψ, ψ\ϋ' = ψ'. (ii) (df) ο ψ = ψ ο Φ , (iii) ip(L) (respectively, v'(L') ) is the normal bundle of T(D) (respectively, Τ(ψ'(ϋ'))) in T(V)\D (respectively, Τ(Μ)\ψ'(θ')). Recalling that L is the normal bundle of D in V we simply take the inclusion map for ψ . To construct ψ wefirst define a C\n itfy map ξ : D —> Μ by i(d) = i//'[d , 0]. The map satisfies ζ ο J = ζ . LEMMA 7.5. Denote by N(V , D) the normal bundle of D in there exists a vector bundle monomorphism
ζ : N(V , D)®j N(V,
D) — T(M)
F.
Th
172
VII. EMEDDINGSOF C ° MANIFOLDS (CONTINUED
such that (0) ζ induces ζ,
(1) (2) iv„ij
W)-mv<)-ww J W).
deD 0=x~\o).
Here N(V , D)® d N(V , D) means (1 χ J)'(N(V , D) χ N(V , £>)). PROOF. Let π, and π 2 be the restrictions of //, and /;, to Δ . Since π 2 is an embbedding of A in V , the normal bundle N(V χ V χ /, A) is the Whitney sum of T(V)\D{D = //,(/})), N(V , £>)(£> =/>,(zf)) , and the trivial bundle e : N(V
χ V χ I , A) ~ x\N(V , D)®/r,'A'(l / , D) φ n\T(D)
θ e.
Hence, we get a vector bundle isomorphism N(V χ V χ I , A) • A ( F , /))©, N(V , £>) φ 'F(D) φ t
1
'1
ί - 1 Β On the other hand, Η is tranverse to ΑΛ/ on A ; therefore, N(V
χ V χ /, A) ~ (//|J)*A'(M χ M,A m).
But we have N(M χ Μ, Ait) ~ (p\)'T(M). From these bundle isomorphisms we get a bundle monomorphism E:N(V, D)®j N(V , D)®T(D)&e . T(M)
D Define a map σ: N(V , D)®j N(V aO dJ J{d rt d,e)
. Μ , D) Φ T(D) φ e — by =
(l
Jid rl d,t Jul re),
where l d (respectively, ) is a vector normal to D at d (respectively, J(d))\ l d e T d(D) and = (dJ) d{t d ). Then a is a vector-bundle involution and satisfies Ξ ο σ = -Ξ because H t is equivariant. Finally, Ξ , restricted to the portion over D 0 in N( V , £>)® y N(V , £>)®0 reduces to (df) ®y (-df) φ 0. The construction of ψ' in thefirst step implies the existence of a homotopy connecting p\ ο Η ο and ζ : D —> Μ, which is invariant under J and which is left fixed in D 0 . Therefore, Ξ is homotopic to a monomorphism Ξ' satisfying: (i α) Ξ' induces ξ . (ii/?) Ξ'οσ = -Ξ'. ('ii y) Z'\D 0 = SID 0.
84. PKCXIF OP Till OKI Μ 7 t
1/3
Now we want to construct the desired ξ . Before doing so we need the following
SUBLEMMA. Let Β be a complex, and let A be a subcomplex of Β. Let Ε = £, φ E 2 and Ε' = E\ φ E' 2 be Whitney sums of vector bundles ove Suppose that σ : Ε — Ε' is a bundle isomorphism whose restriction is the direct sum ο® φ σ 2 of bundle isomorphisms
σ° : E {\A —• E\\A ,
: E 2\A — E' 2\A.
We assume that σ® can be extended to an isomorphism that dim Β < d\m(thefiber
σ { : Ε, —
of E-, ).
Then there exists a bundle homomorphism (i) σ 2 is an extension of σ 2. and (ii) there exists a homotopy connecting on A.
a, : F, — E 2 such that a and ίτ,φσ,. which is left
We leave the proof to the reader. Now back to the proof of Lemma 7.5. In the sublemma wc take E t = T(D)®c/{l„,e)~{-tj
(rf ).
E 2 = N[V , D) ®j N(V , D)/
-<'). (l,J
Md ))
~ (~l
Jul r
-'„)•
Then and £, arefiber bundles over D/J . Denote by ζ/. I : D/J — .ip'(D') \subsetΜ the induced map of ζ : D — Μ . Set
Ε\ = (ζμ)'Τ(ψ'(ϋ')). E 2 = (i/J)'N(M.
ifi'(D')).
Then as dim£) < 2(// - dim/)) wc may apply the sublemma to obtain Lemma 7.5. Ο We next construct ψ : L' — T(M ). Recall that L' is the quotient of the bundle (L®j L) χ [0, I] - D χ [0. 1] by the equivalence relation (l d , l J(d ), t) ~ ( ~ l J { d ) , ~l d . -I) • Hence, it suffices to construct a bundle monomorphism Z-)x[0, I ] —> N(M
, ψ\θ'))
satisfying the following: (1) χ induces χ : D χ [0, 1] ^ D' , χ (d , t) = v'[d . 1], (2) x\(L ®jL)xO = £. (3) (a) x(l d,0,X(d)) = df(l d), s.(d)> 0. (b) χ(0. -I d,-;.(d)) = df(l d), k(d)< 0.
VII. EMBEDDINGS OF c°° MANL'OLDS (CONTINUED)
174
Wefirst construct the restriction χ, of χ to ( L ® ; 0) χ [0, I]. By the requirements (2) and (3a) has already been defined on Dx{0) and on the set (d , X(d)) , X(d) > 0. The obstructions to extending over D χ [0, 1] are in the homotopy groups ^m—dim D' , η-dim £>) '
1 < dim D ,
where V is the Stiefel manifold of a-framcs in R^ . But wc knoyv P.I i < ρ - q. In our present case these homotopy groups all vanish if dim D < in - dim D' - (// - dim D) , 0 < m - η - dim D' , that is, if 3« + 1 < 2m since dim D' — dim D + 1 = 2n - m + 1 . But 3// + 2 < 2m by our hypothesis. Thus, the above homotopy groups arc zero for 1 < dimD = 2n — m . Hence, we can extend χ, over D x [ 0 , I ]. We next extend χ χ to χ so that the requirements (2) and (3)(b) will be satisfied. This is not difficult because the set D χ [0, I] is a deformation retract of the set D χ {0} U {(t/, X(d))\X(d) < 0 }. Now define ...
.
(1
j X(ljJj
{j r0.
I > 0,
Constructions of Ψ and Ψ'. First, wc construct Ψ in l. t for a suffic small e > 0 such that (M) xlT x(L f))
= MT xlL t)).
xeD
(consider the exponential map exp ). On the other hand, for φ(.\) = ,v' e if there exists a map ψ' χ from some neighborhood U[. of x' in L' to Μ such that f ο ψ = ψ'
χ
οΦ
(we can do this by choosing a suitable chart to apply the implicit function theorem). From this we construct a C\n ifty map ψ' of some neighborhood U' of D' in L! inlo Μ such that (dr')
X,(T x.(u'))
= φ\τ χ.(υ')),
χ'
e D\
f οψ = ψ' ο Φ . We can choose ί so small that Ψ'\ L' f is a diffeomorphism of Z.' in U' . Thus, we have shown the sufficiency of (a) in Theorem 7.3. We leave the rest of the proof for the reader to continue in the original paper of Haefiiger [C9],
54 PROOF OP THEOREM 7.3
I7S
THE PROOF OF THEOREM 7.3 (b). The method of proof is identical to that of the above. Replace V by V χ [0, 1], Μ by Μ χ [0, 1], and / by F : V χ [0, 1] - Μ χ [0, 1] defined by F(v. I) = /,(/;). Then the condition 3(/J + 2) < 2(/;i + 1) becomcs 3(/i + I) < 2m . •
Afterword Chapter 0 featured Whitney [C20] as a means for giving the reader an intuitive preview of the book. In Chapter I we dealt with the fundamentals of differential topology—C' manifolds, C' maps,fiber bundles, and other related concepts: we limited our discussion both in selection and scope to the parts essential for this book. For more in-depth information on the subject, we rccommcnd Di/icrcniicibk· manifolds by Y. Malsushima.( ' ) In Chapter II we discusscd cmbeddings of manifolds, our focal point being Whitney's embedding theorems [C19], [C2I]. Here we used, among other references, the book of Tamura.(") Our topics in this chapter included a method for eliminating double points in completely regular immersions. We saved Haefliger's generalization of this method for Chapter VII. The theme of Chapter III was immersions of C v manifolds. Here our discussion centered around the Smale-Hirsch theory—a natural generalization of (the topic in) Chapter 0—and included the theorems of Phillips and Gromov. The theorem of Gromov encompasses the submersion theorem of Phillips and is a generalization of the Smale-Hirsch theorem. Consequently, we presented the proofs of these theorems as corollaries of Gromov's theorem. For the proof of Gromov's theorem we followed Haefliger's presentation of the subject [B3], We also mentioned that Gromov founded his theorem on Smale's "homotopy covering technique" [CI 7], "taking an idea out of an old wise man's paper".(*) In Chapter IV we introduced yet another generalization of the SmaleHirsch theory. Gromov's integration theory [C5], unlike his theorem in Chapter III, does not require openness for the base spaces of jet bundles. This constitutes an essential difference between these two works. Here we used a report paper of Shigeo Kawai. In our opinion this chapter points to a promising future direction for the subject of this book. We presented in Chapter V an application—Haefliger's classification the( 1 ) Y. Matsushima. Differcnliablc manifolds. Marcel Dckkcr. New York. 1472. ( 2 ) I.Tamura. Differential topology. Iwanami Shotcn. Tokyo. 1978 (Japanese! ( ' ) Editor's note. The phrase in quotes was added in translation by Ihc auihor
178
AFTERWORD
orem for foliations of open manifolds —of the theorem of Gromov in Chapter III. We recommend Tamura [A9] for the fundamentals of foliation theory. We devoted Chapter VI to complex structures on open manifolds as an application oi'the theorems of Gromov in Chapters III and IV. The integrability of almost complex structures on open manifolds of arbitrary dimensions remains unsolved to date. We gave an outline of a proof of Haefliger's embedding theorem in Chapter VII. As we mentioned above, we had wished to shed some light on the fact that Haefliger's proof is a natural extension of Whitney's method as used in eliminating double points of immersions; we are somewhat dubious as to what extent we succeeded in do;ng so. We feel thatfinding a sufficient condition, independent of connectivity, for the existence of embeddings of manifolds is a major open problem in this field. We did not mention Haefliger's alternative proof for his embedding theorem [C8]. This is based on the so-called Whitney-Thorn theory concerning singular sets of diflerentiable maps. We also omitted any solid application of the Smale-Hirsch theory to manifolds; for this we refer the reader to Smalc [B9] and James [B4], From the historical perspective we marvel at the evolution of the simple problem of Chapter 0 as developed throughout our book to its present stage, and we expect further progress in the future. Addendum. A videotaped version of the main topic of Chapter 0 is available.( 3)
( 3)
Regular hoinolopies
in the plane. International Film Bureau Inc.. Chicago. Illinois.
References
A . BOOKS
[ΑΙ] [A2] (A3] (A4| [A5] [A6] [Λ7] [A8] [A9]
Μ. W. Hirsch. Differential topology. Grad. Texts in Math., vol. 33. Springer-Verlag Berlin and New York. 1976. J. Milnor. Morse theory, Ann. of Maih. Stud., vol. 5 I. Princeton Univ. Press. Princeton. N.J.. 1963. , Topology from the differentiable viewpoint . The Univ. Press of Virgin lottesville. VA„ 1965. . Lectures on the li-cobordism theorem. Princeton Math Notes. Princeton U Press. Princeton. N.J.. 1965. J. Milnor and J. Stashed". Characteristic classes. Ann. of Math. Stud., vol. 76. Princeto Univ. Press. Princeton. N.J.. 1974. Ε. H. Spanier. Algebraic topology. McGraw-Hill. New York. 1966. N. Stcenrod. The topology of fibre bundles. Princeton Math. Scr.. vol. 14. Prince Univ. Press. Princeton. Ν J.. 1951. S. Sternberg. Lectures on differential geometry. Prentice-Hall. Englcwood Clills. 1964. I. Tamura. Topology of foliations: An introduction. Trans!. Math. Monographs, vol Amcr. Math. Soc.. Providence. R. I.. 1992.
B. L P C T U R F NOTES, SURVF.VS.
[Bl] [B2| [B3J [B4| (B5) (B6) [B7] [B8] [B9| [BIO]
etc.
A. Haefliger, /'lodgements de vurietc dans le domaine stable. Scminairc Bourh (1962-631. Expose 245. llomnlopy and imcgrabiiity. Lecture Notes in Math., vol. 197. Springer-Verlag. Berlin and New York, 1967, pp. ; 33-163. Lectures on the theorem of ( Iromov . Lecture Notes in Math., vol. 209. Sprin Vcrlag, Berlin and New York. 1972, pp. 128-141. 1. M. James. Τ wo problems studied by llopf. Lecture Notes in Math., vol. 279. Springc Vcrlag. Berlin and New York, 1972. M. G. Gromov, A topological technique for the construction of solutions of d equations and inequalities. Actcs Congrcs internat. Math. 2 (1970). 221-225. J. Milnor. DifferentiaI topology. Mimeographed Notes. Princeton Univ.. Princeton. N J.. 1958. A. Phillips. Turning a sphere inside out. Sci. Amcr. 214(1966). 112-120 V. Poenaru, llonwtopy theory and differentiable singularities. Lecture Notes i vol. 197. Springer-Verlag. Berlin and New York, 1971, pp. 106-132. S. Smalc. A survey of some recent developements in differential topology. Math. Soc. 69 (1963), 131-145. R. Thom. Le classification des immersions d'apres Smalc. Seminaire Bourhaki. ber. 1957. Expose 157. 179
180 [B11)
REFERENCES
R.Thom and H. Levine, Singularities of differentiable mappings. Lecture Notes in Ma vol. 192, Springer-Verlag, Berlin and New York, 1971, pp. 1-89. C . PAPERS.
[CI]
M. Adachi, A note on complex structures on open manifolds, J. Malh. Kyoto Univ. (1977), 35-46. [C2] Construction of complex structures on open manifolds, Proc. Japan Acad. S Malh. Soc. 55 (1979), 222-224. [C3| S. D. Feit, k-mersions of manifolds. Acta Math. 122 (1969), 173-195. (C4) M. Gromov, Transversal mappings of foliations into manifolds. Izv. Akad. Nauk 33 (1969). 707-734; English transl. in Math. USSR-Izv. 3 (1969). (C5] , Convex integration of differential relations. Izv. Akad. Nauk SSSR .37 (19 329-343. [C6] M. Gromov and Y. Eliaschbcrg, Removal of singularities of smooth mappings. Izv. Aka Nauk SSSR 35 (1971), 600-625; English transl. in Math.'USSR-Izv. 5 (1971). [C7J A. Hacfliger, Differentiable imheddings. Bull. Amcr. Malh. Soc. 67 (1961), 109-112. [C8J I'longements diI]erenttables de varit'ies dans varices. Comment. Math. I l (1961), 47-82. (C9) , I'longements differentiables dans le domaine stable, Commcni. Malh. Hc (1962), 155-176. [CIO] Feullitages sur les van en's ouvertes. Topology 9 (1979), 183-194. [CI 1) A. Haelliger and M. Hirsch, Immersions in the stable range. Ann. of Math. 75 (19621. 231-241. [CI 2] M. W. Hirsch, Immersions of manifolds. Trans. Amcr. Math. Soc. 93 (1959). 242-276. [C13] P. Landwcbcr. Complex structures on open manifolds. Topology 13 (1974). 69-76. [C14] A. Phillips. Submersions of open manifolds. Topology 6 (1967). I 71-206 [C15] Smooth maps transverse to a foliation. Bull. Amcr. Math. Soc. 76 (1970). 792 797. [CI6J A. Shapiro, Obstructions to the imbedding of a complex in a Euclidean space. I. Th obstruction, Ann. of Math. 66 (1957). 256-269. [C17] S. Smalc, Classification of immersions of a sphere into Euclidean space. Ann. of 69 (1959). 327-344. [CI 7a] Generalized I'oincare's conjecture in dimension greater than four. Ann. 74 (1961), 391-406. (CI 8] R. Thom, Espaces fibres en spheres et Carres de Steenrod. Ann. Sci. Ecole Norm. S (1952), 109-181. [C19] H. Whitney, Differentiable manifolds. Ann. of Malh. 45 (1936). 645-680. [C20J On regular closed curves on the plane. Compositio Math. 4 (1937). 276-284. (C2!) The sell-intersections of a smooth n-manifold in In-space. Ann. of Math (1944), 220-246. [C22] , The singularities of a smooth n-manifold in (2« - 1 )-space. Ann. of Math (1944), 247-293. [C23] W. -T. Wu. Sur les classes caracteristiques des structures Jibrees spheriques. Sci. Indust.. vol. 1183 (1952), Hermann, Paris, pp. 1-89. [C241 On the imbedding of manifolds in a Euclidean space. Sci. Sinica 13 (196 682-683.
Subject Index
Admissible map. 104
Connecting homotopy. 161 Contact form. 109 structure. 109 Convex hull. 112 Coordinate functions. IS neighborhood. 8. 18 system. 8 C oordinate bundle. IS (/-image of. 26 restriction of. 20 smooth. 19 trivial. 22 Coordinate direction. I 13 Coordinate transformations. I system of. 23 Covering homotopy properly. Covering space. 17 Critical point nondegenerate. 38 Critical value. 13 ( ' map. 7. 12 Cross section. 20 C topology. 37 C triangulation. 90 Curve. 13
Aflinc
bundle. 112 embedding. 111 Almost complex manifold. 141 structure. 141 Almost complex structure. 27 Almost contact structure. 109 Almost Hamiltonian structure. 108 Almost symplectic structure. 108 Ample, I 12 Alias. 8 oriented. 8 Base space. 18 Bell-shaped function. 51 Bundle associated. 28 induced. 27 principal. 28 universal, 31 Bundle map. 20 Bundle space. 18 C-foliation analytic. 148 C-submcrsion. 147 C-transversal. 150 C-transversality theorem. I 51 C-foliation. 148 Characteristic classes. 84 Chart. 8 Chcrn class. 83 C * map. 7 Classification theorem for foliations. 149 Classifying space for 0(//). 30 Combinatorial //-cell. 90 manifold. 90 Compact-open C topology. 76 Complex manifold. 142 Complex structure. 142. 143 Complexification. 146
Degree of a map //: .V1 - V . 4 ^-approximation of a map. 49 Dilfeomorphism. 12 local. 91 Differentiable map. 12 map of class C . 7 Diirerenliable structure. 8 Differential equation. 111 of a map. 15 relation. I I I Dimension. 80 Direction of dimension. I 12
181
182
SUBJECT INDEX
Effective, 18 Embedding, 12 Epimorphism, 93 Epimorphism of vector bundles. 137 Equivalent atlases, 8 bundles, 19, 22 Equivalent one-cocycles, 128 Equivariant, 160 homotopy, 157, 160 map. 157 Equivariantly homotopic (equivariant maps), 160 Eulcr-Poincare class, 83 Extension continuous, 92 of local diffcomorphisms, 91 Face of a simplex, 31 Fiber, 18 Fiber bundle, 19 smooth, 19 Fiber map, 100 Fiber space, 100 Fibration, 100 Γ-foliated microbundlc, 135 Γ-foliation, 134 Γ-structurcs, 128 classifying space for, 130 homotopic, 129 numerable, 131 universal, 131 General position, 31 Grassmann manifold. 29, 79 Gromov's theorem. 92 . 113 Groupoid, 127 topological, 127 Handlcbody, 95 Hessian, 38 Holomorpohic foliation, 147 nomotopic almost complex structures. 142 Immersion, 12 completely regular, 60 Index of nondcjcncrate critical point. 38 Intcgrable almost complex structure, 142 Intcgrably homotopic, 134 almost complex structures. 142 holomorphic foliations, 149 Intersection number of a completely regular immersion, 65 Isotopic embeddings, 45 Isovariant bundle map. 163 homotopy, 161 Isovariant map. 160
Isovarianlly homotopic (isovariant maps).
161
Jet bundle. 36 Jump point, 80 A--connected space, 46 Klein bottle, 16 A'-mcrsion, 116 Knot theory, 46 Lie group. 19 Limit set of a map. 55 Linearly indepcndenl points, 31 Mobius strip, 10, 16 Manifold almost complex. 27 c r, 8 differentiable, 8 orientation of. 9 parallclizablc. 26 presentation of. 95 smooth, 8 Mapping transformations. 21 Measure zero. 47, 48 Monomorphism of vector bundles. 87 Morse function. 40 Nice function. 96 Noiulegencrale critical point. 38 Normal r-frr.melield. 87 /i-simplcx. 31 Numerable open covering. 1.31 Numcrahly homotopic. 131 ()nc-cocycle. I 28 One-to-one immersion. 53 Open manifold, 88 Open submanifold. 9 Orlhonoima! k-frame. 29 Parallclizablc manifold. 88 Polyhedron. 32 Pontryagin class, 83 Principal alfinc embedding. I I 3 Principal direction, 113 Problem der gcslalt, 46 Problem der lagc, 45 Product bundle, 22 Product manifold, 10 Proper map. 41 Pscudogroup, 127 I'scudogroup of local diffcomorphisms. 91 Pullback. 27 Rank of a differentiable map, 12 Reduction, 26
SUBJECT INDEX Regular closed curve lift of, 1 Regular closed curves deformation of, 3 equivalence of, 1 regularly homotopic, 3 Regular homotopy, 3. 53 Regular homotopy (between homotopies), 163 Regular value, 13 Regularly homotopic immersions. 53, 75 r-equivalent jets, 35 maps, 32 r-cxtcnsion of a map. 36 Ricmannian metric. 26 r-jct, 33 o f / at Λ, 35 regular. 33 Rotation number, 4 Sard's theorem, 57 Schubcrt function, 80 variety, 81 Self-Intersection, 62 of negative type. 65 of positive type. 65 .v-handlcs, 95 Simplicial complex. 32 Skew scalar product. 108 Smale-Hirsch theorem. 87, 114 Stable range, 158 Stably equivalent. 143 Stiefel manifold. 29 Stiefel-Whitncy class. 83 Structural gioup, I X
183
Tangent space, 13 The transversality theorem of Gromov and Phillips. 137 Theorem of Feit, 116 Thom transversality theorem. 42 Topological groupoid associated with Χ. 128 Topological manifold. 7 Topological transformation group. 17 Topology fine. 77 strong. 77 Whitney. 77 Torus. 10 Total space. 18 Total Sticfcl-Whiiney class. 84 Transition functions. 18 Transverse intersection. 60 Transverse map. 42 Transverse to ·'/". 134 Transverse to the Γ-foliation ./ . 136 /-regular map. 42 Triangulated spaces combinatorial!) equivalent. 90 isomorphic. 90 Triangulation. 90 Twisted torus. I 7 Typical singularity. 114 Underlying almost complex structure. 142 Universal vector bundle. 31 Vector bundle. 30 homomorphism of. 32 Vector field. 26
enlargement of, 26 Subdivision of a triangulation. 90 Submanifold. 10 Submersion, 12 Symplectic transformation. 108 Symplcctic structure. 108 linear, 108 standard, 108 System of local coordinates. 8
Weak ( ' lopoloj'A 7Λ Weak homolop) equivalence. Κ7 w.h.e-principlc. I I 2 Whitney dualiiy theorem. 85 topology. 77 Whitney's embedding theorem. 46. 56. 67 Whitney's theorem on completely regular immersions. 60 Whiiney-Graustein theorem. 5
Tangent bundle, 25
/^-equivariant. 157. 160
