From Calculus to Cohomology
.
de Rham cohomology and characteristic classes
Ib Madsen and
J~rgen
Tornehave
University q{Aarhus
,,~~I CAMBRIDGE
::;11
UNIVERSITY PRESS
v
Contents .:....
',)
.
' ~)<,
fJ
-.
,~
f"J J
.
Preface Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter II
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Appendix A Appendix B Appendix C Appendix D References Index ...
Introduction . . . . . . . . The Alternating Algebra. de Rham Cohomology. . Chain Complexes and their Cohomology The Mayer-Vietoris Sequence . . . . . Homotopy . . . . . . . . . . . . . . . . . Applications of de Rham Cohomology Smooth Manifolds . . . . . . . . . . . . Differential Forms on Smoth Manifolds . Integration on Manifolds Degree, Linking Numbers and Index of Vector Fields. The Poincare-Hopf Theorem . . . . . Poincare Duality . . . . . . . . . . . . The Complex Projective Space ern . Fiber Bundles and Vector Bundles. . Operations on Vector Bundles and their Sections Connections and Curvature . . . . . . . . . . . . . Characteristic Classes of Complex Vector Bundles The Euler Class. . . . . . . . . . . . . . . . . . . . . Cohomology of Projective and Grassmannian Bundles Thorn Isomorphism and the General Gauss-Bonnet Formula Smooth Partition of Unity. . . . .
Invariant Polynomials . . . . . . .
Proof of Lemmas 12.12 and 12.13
Exercises
. . . . . . .
. vii
. 1
. 7
15
25
33
39
47
57
65
83
97 II 3
127 139 147
157
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243
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283
'j
I
,I j.
j}
~
VB
PREFACE
This text offers a self-contained exposition of the cohomology of differential forms, de Rham cohomology, and of its application to characteristic classes defined in terms of the curvature tensor. The only formal prerequisites are knowledge of standard calculus and linear algebra, but for the later part of the book some prior knowledge of the geometry of surfaces, Gaussian curvature, will not hurt the reader. The first seven chapters present the cohomology of open sets in Euclidean spaces • and give the standard applications usually covered in a first course in algebraic topology, such as Brouwer's fixed point theorem, the topological invariance of domains and the Jordan-Brouwer separation theorem. The next four chapters extend the definition of cohomology to smooth manifolds, present Stokes' the orem and give a treatment of degree and index of vector fields, from both the cohomological and geometric point of view. The last ten chapters give the more advanced part of cohomology: the Poincare-Hopf theorem, Poincare duality, Chern classes, the Euler class, and finally the general Gauss-Bonnet formula. As a novel point we prove the so called splitting principles for both complex and real oriented vector bundles. The text grew out of numerous versions of lecture notes for the beginning course in topology at Aarhus University. The inspiration to use de Rham cohomology as a first introduction to topology comes in part from a course given by G. Segal at Oxford many years ago, and the first few chapters owe a lot to his presentation of the subject. It is our hope that the text can also serve as an introduction to the modern theory of smooth four-manifolds and gauge theory. The text has been used for third and fourth year students with no prior exposure to the concepts of homology or algebraic topology. We have striven to present all arguments and constructions in detail. Finally we sincerely thank the many students who have been subjected to earlier versions of this book. Their comments have substantially changed the presentation in many places. Aarhus, January 1996
1.
INTRODUCTION
It is well-known that a continuous real function, that is defined on an open set of R has a primitive function. How about multivariable functions? For the sake of simplicity we restrict ourselves to smooth (or Coo -) functions, i.e. functions that have continuous partial derivatives of all orders. We begin with functions of two variables. Let f: U ---; R2 be a smooth function .defined on an open set of R2 . Question 1.1
Is there a smooth function F: U ---; R, such that
(1)
-;- = hand -;- =
of
of
VXI
VX2
Since
h,
where
0 2F
02 F
aX2aXI
aXl aX 2
f
= (h, h)?
we must have
ah
(2)
aX2
ah
=
aXI'
The correct question is therefore whether F exists, assuming (2). Is condition (2) also sufficient? Example 1.2 Consider the function f: R2
f(xI,x2)=
---;
-X2
(
f
= (h, h) satisfies
R2 given by Xl)
xi+xrxi+x~ .
It is easy to show that (2) is satisfied. However, there is no function F: R2 -{0} ---; R that satisfies (1). Assume there were; then
(2Jr d
Jo
dBF(cos B, sin B)dB = F(l, 0) - F(l, 0) =
o.
On the other hand the chain rule gives
d . dF . dBF(cosB, smB) = dx . (-smB)
= - h(cosB,
of
+ ay . cosB sinB) . sinB + h(cosB,
sin B)· cosB
This contradiction can only be explained by the non-existence of F.
= 1.
1.
Note that rot
0
grad
= O.
3
INTRODUCTION
Hence the kernel of rot contains the image of grad, Ker(rot)
= Kernel
of rot
Im(grad) = Image of grad
Since both rot and grad are linear operators, Im(grad) is a subspace of Ker(rot). Therefore we can consider the quotient vector space, i.e. the vector space of cosets 0: + Im(grad) where 0: E Ker(rot):
HI(U)
(3)
= Ker(rot)jlm(grad).
Both Ker(rot) and Im(grad) are infinite-dimensional vector spaces. It is remark
able that the quotient space HI (U) is usually finite-dimensional.
We can now reformulate Theorem 1.4 as
(4)
HI(U)
=0
whenever U ~ 1R 2 is star-shaped.
On the other hand, Example 1.2 tells us that HI (R2 - {O}) -=I O. Later on we shall see that HI (1R 2 - {O}) is I-dimensional, and that HI (1R2 - U7=1 {Xi}) ~ IRk. The dimension of HI (U) is the number of "holes" in U. In analogy with (3) we introduce
HO(U) = Ker(grad).
(5)
This definition works for open sets U of I~k with k 2: 1, when we define grad(J)
=
(:f
1""
Xl
Theorem 1.5 An open set U ~ Rk is connected
of )
Ox n
'
if and only if HO(U)
= R
Proof. Assume that grad(J) = O. Then f is locally constant: each Xo E U has a neighborhood V(xo) with f(x) = f(xo) when X E V(xo). If U is connected, then every locally constant function is constant. Indeed, for Xo E U the set
{X
E
Ulf(x) = f(xo)} = f-I(J(xo)).
is closed because f is continuous, and open since f is locally constant. Hence it is equal to U, and HO(U) = R. Conversely, if U is not connected, then there exists a smooth, surjective function f: U --+ {O, I}. Such a function is locally 0 constant, so grad(J) = O. It follows that dimHO(U) > 1. The reader may easily extend the proof of Theorem 1.5 to show that dimHO(U) is precisely the number of connected components of U.
5
INTRODUCTION
1.
Example 1.7 Let S = {(Xl, X2, X3) E 1R31xi + X§ = 1, X3 = in the (Xl, X2 )-plane. Consider the function
o}
be the unit circle
f(:1:I' X2, X3)
_ (-2XIX3 -
2
, .2
2
x 3 +(x l +x 2 -1)
on the open set U
=
xi + x§ -
-2X2X3 2'
2
2
2
X3 +(X I +x 2 -1)
2' . 2
2
1
2
)
x 3 +(x l +x 2 -1)
2
R3 - S.
One finds that rot(f) = 0. Hence f defines an element [1] E HI(U). By
integration along a curve, in U, which is linked to S (as two links in a chain),
we shall show that [1] i- 0. The curve in question is
,(t)
= ( VI + cos t,O, sin t ),
-Jr:S
t
:s Jr.
Assume grad(F) = f as a function on U. We can determine the integral of F(r(t)) in two ways. On the one hand we have
fit
j
7r-€
-7r+€
d -d F(r(t))dt = F(r(Jr - f)) - F(r( -Jr
t
+ f))
-+
°
for
f
-+
°
and on the other hand the chain rule gives
:tF(r(t)) = !t(,(t)) . ,~(t)
°
+ J2(r(t))· ,~(t) + h(,(t)). ,~(t)
= sin 2 t + + cos 2 t = 1. Therefore the integral also converges to 2Jr, which is a contradiction.
Example 1.8 Let U be an open set in IRk and X: U -+ IRk a smooth function (a smooth vector field). Recall that the energy A')'(X), of X along a smooth curve ,: [a, b] -+ U is defined by the integral
A')'(X) =
l
b
(X o,(t) , ,'(t))dt
where ( ,) denotes the standard inner product. If X = ~,( b), then the energy is zero, since
(X 0 ,(t) , ,'(t)) by the chain rule; compare Example 1.2.
=
d
dt ~(r(t))
grad (~) and ~,( a)
=
7
2.
THE ALTERNATING ALGEBRA
I~.
Let V be a vector space over
A map
f:VxVx···xV-d~
....
,
J
k times is called k-linear (or multilinear), if f is linear in each factor. Definition 2.1 A k-linear map w: V k ---+ IR is said to be alternating if w(6, ... ,~k) = 0 whenever ~i = ~j for some pair i I- j. The vector space of alternating, k-linear maps is denoted by Alt k (V).
We immediately note that Altk(V) = 0 if k > dim V. Indeed, let el, ... , en be a basis of V, and let W E Altk(V). Using multilinearity,
w(6"",~k) =W(LAi,lei""'LAi,kei) with AJ = Ail,l the elements eh,
= LAJw(eh,···,ejk)
Ajk,k' Since k > n, there must be at least one repetition among , ejk' Hence w( cil' ... , ejk) = O.
The symmetric group of permutations of the set {I, ... ,k} is denoted by S(k). We remind the reader that any permutation can be written as a composition of transpositions. The transposition that interchanges i and j will be denoted by (i, j). Furthemiore, and this fact will be used below, any permutation can be written as a composition of transpositions of the type (i, i+1), (i, i+1)o(i+1, i+2)o(i, i+1) = (i, i + 2) and so forth. The sign of a permutation: sign: S(k)
(1)
---+
{±1},
is a homomorphism, sign(o- 0 T) = sign(o-) 0 sign(T), which maps every transpo sition to -1. Thus the sign of 0- E S(k) is -1 precisely if 0- decomposes into a product consisting of an odd number of transpositions. Lemma 2.2
If W
E Altk(V) and
0-
E
S(k), then
W(~a(l)"" ,~a(k)) = sign(o-)w(6,··· ,~k)' Proof. It is sufficient to prove the formula when
0-
= (i, j). Let
Wi,j (~, ~') = w(6, .. · ,~, ... , (, ... , ~k),
e
with ~ and occurring at positions i and j respectively. The remaining ~v E V are arbitrary but fixed vectors. From the definition it follows that Wi,j E Alt 2 (V). Hence Wi,j (~i + ~j, ~i + ~j) = O. Bilinearity yields that Wi,j (~i, ~j) + Wi,j (~j, ~i) =
O.
0
2.
unction w(6,· .. ,~k) = terminants.
Lemma 2.7 A k-linear map W is alternating if w(6,··., ek) = 0 for all k-tuples with e,; = ~i+I for some 1 :::; 'i :::; k - 1. Proof. S(k) is generated by the transpositions ('i, 'i of Lemma 2.2,
l
w(6,···, ei. ei+I, ... , ek)
) - w2(6)WI(6)·
+ 1),
W
+ wD
+ W~ 1\ w2 (AWl) 1\ W2 = A(WI 1\ W2) = WI 1\ AW2 WI 1\ (W2 + W~) = WI 1\ W2 + WI 1\ W~ (WI
I. Since a (p, q)-shuffle cardinality of S(p, q) is
W2 E Altq(V), we define
1\
W2
= WI
1\
W2
for WI,W~ E AltP(V) and W2,W~ E Altq(V).
Lemma 2.8 If WI E AltP(V) and W2 E Altq(V) then WI
,~a(p+q»)·
Proof. Let
7
E S(p
+ q)
7(1)
noreover:
L 1\
7(q We have sign( 7)
W2 E Altp+q(V).
o when 6
6·
We let
1\
W2
= (-1)Pqw2
1\ WI.
be the element with
= P + 1,7(2) = p + 2, ... , 7(q) = P + q
+ 1) =
1, 7(q
= (-1 )pq.
+ 2) =
2, ... , 7(p
+ q) =
p.
Composition with 7 defines a bijection
S (p, q) =
0
is alternating.
It is clear from the definition that
~(}(p+q).
1),···
and by the argument
= -w(6,···, ei+I, ei, ... , ek).
Hence Lemma 2.2 holds for all () E S(k), and
.... , p + q} satisfying
9
THE ALTERNATING ALGEBRA
--=.
S ( q, p);
()
f--+ ()
0 7.
Note that
W2 (~aT(I)' ... ,eaT(q»)
= W2 (ea(p+I), ... ,ea(p+q») WI (eaT(q+I),"" eaT(p+q») = WI (ea(I)"'" ~a(p»).
.. , ~a(p+q») is zero, since
n with the transposition
Hence
W2
e
1\ WI (~I, ... , p+ q )
L
sign((})w2(~a(1),···, ea(q»)WI (ea(q+I)"'" ea(p+q»)
aES(q,p)
L
), ... , ~a(p+q»)
0"
sign( (}7 )W2 (eaT(l) , ... , ~aT(q) )wI(eaT(q+I), ... , eaT(p+q»)
aES(p,q)
= (-1 )pq
(p+l) , ... eTa(p+q»)'
L aES(p,q)
Pq
cr(p + 1) = 1, we see that e tenus in the two sums be alternating according
o
sign(() )WI (ea(I), ... , ea(p»)W2 (ea(p+I), ... ,~a(p+q»)
e
= (-l) WI 1\ w2(6, .. ·, p+q ).
0
2.
II
THE ALTERNATING ALGEBRA
dtr(V) then
Note that in this formula {0"(1), ... , O"(p)} and {CT(p + 1), ... , cr(p + q)} are not ordered. There are exactly S (p) x S (q) ways to come from an ordered set to the arbitrary sequence above; this causes the factor ;;b, p.q. so WIAw2 = WI /\ W2.
lutations
An R-algebra A consists of a vector space over R and a bilinear map f..l= A x A -+ A which is associative, {l(a,/-l(b,c)) = /-l(/-l(a,b),c) for every a,b,c E A. The algebra is called unitary if there exists a unit element for /-l, /-l(1, a) = /-l( a, 1) = a for all a E A.
0"
with
).
Definition 2.11 (i) A graded R-algebra A* is a sequence of vector spaces A k , k = 0,1 ..., and bilinear maps /-l: Ak x Az -+ Ak+Z which are associative. (ii) The algebra A* is called connected if there exists a unit element 1 E Ao and if E: R -+ Ao, given by E(r) = r· 1, is an isomorphism. (iii) The algebra A* is called (graded) commutative (or anti-commutative), if /-l(a, b) = (-l)kl/-l(b,a) for a E Ak and bE Az.
, S(p, q, r) given by
lIld
0"
E S(p, q,
r)
... ,p+q+r}
The elements in Ak are said to have degree k. The set Alt k (V) is a vector space over R in the usual manner:
'-+O"OT -+ 0" 0 T.
(WI
+ w2)(6,···, ~k) = wI(6,···, ~k) + w2(6,···, ~k) ('\w)(6'''',~k) ='\w(6,"·,~k),
,\ E R
q The product from Definition 2.5 is a bilinear map from Altp(V) x Alt (V) to Altp+q(V). We set Alto(V) = R and expand the product to Alto(V) x AltP(V) by using the vector space structure. The basic formal properties of the alternating forms can now be summarized in
p+I)'" ., ~a(p+q+r))
rep))
aT(p+q+r))]
Theorem 2.12 Alt*(V) is an anti-commutative and connected graded algebra.D
, ~u(p+q))
Alt* (V) is called the exterior or alternating algebra associated to V.
I
(2). Quite analogously
Lemma 2.13 For I-forms WI, ...
,W p
E AltI(V),
ying the second equation
IS
o one can often see the
tI),···
,~a(p+q))·
(WI /\ ... /\ wp)(6, . .. , ~p) = det
Proof. The case p to Definition 2.5,
=
(
WI(6)
wI(6)
W2;~I)
W2;6)
wp (6)
wp (6)
WI(~p) ) W2(~p)
...
Wp(~p)
2 is obvious. We proceed by induction on p. According
2.
,
THE ALTERNATING ALGEBRA
Note from Theorem 2.15 that Altn(V) ~ R if n = dim V and, as mentioned earlier, that Altp(V) = 0 if p > n. A basis of Altn(V) is given by Ell\ .. . 1\ En. In particular every alternating n-fonn on IR n is proportional to the form in Example 2.3.
~j, .. . ~p) 1
lere ~j has been omitted. ie first row. 0
A linear map f: V
(Wi(~j)) = 1. Conversely, me of them, say w p , as a then
---+
W induces the linear map AltP(J): AltP(W)
(4) E Altl(V) are linearly ill choose elements ~i E
'p
= 0,
Wi
I- 0 if and only if they
1 ••• ,
---+
Altn(V)
is a linear endomorphism of a I-dimensional vector space and thus multiplication by a number d. From Theorem 2.18 below it follows that d = det(J). We shall also be using the other maps
. We have proved Wp
AltP(V)
---+
by setting AltP(J)(w)(6, ... , ~p) = w(J(6), ... , f(~p)). For the composition of maps we have Altp(g 0 J) = AltP(f) 0 AltP(g), and AltP(id) = id. These two properties are summarized by saying that AltP(-) is a contravariant functor. If dim V = nand f: V ---+ V is a linear map then Altn(J): Altn(V)
p-l 1\
13
Altp(J): AltP(V)
o
En the dual basis of
---+
Altp(V).
Let tr(g) denote the trace of a linear endomorphism g.
Theorem 2.16 The characteristic polynomial ofa linear endomorphism f: V is given by
n-p)
---+
V
n
det(J - t)
= I: (-1)i tr (Altn-i(J)) t i , i=O
where n
,emma 2.13 gives
... ,ip } ... ,ip }
I-
{jl,
,jp}
= {jl,
,jp}
=
dimV.
Proof. Choose a basis el, ... , en of V Assume first that el, ... , en are eigen vectors of f,
f(ei) = Aiei, i = 1, ... , n .
.2 and (3) we get
Let E1, ... , En be the dual basis of Altl(V). Then \ ... 1\ Ea(p)
AltP(J) (Ea(l) 1\ ... 1\
:enerates the vector space a relation
Ea(p)) = Aa(l)'"
and tr AltP(J)
=
I:
Aa(p)Ea(l) 1\ ... 1\ Ea(p)
Aa(l) ... Aa(p)·
aES(p,n-p)
Aa E IR
On the other hand det(J - t)
o
= II(Ai -
t)
This proves the formula when
= I: (-1t- P (I: Aa(l) ... Aa(p)) tn-p. f
is diagonal.
.....
~ fJ
=
...
,~
~
<::I
~
:t:
~
~
~
,~
~
I~
...=:l
~
..e
:c~
,-.,.
""""
,-.,. C"l
I
Co)
~ ~,
-:
"'-~ ~
15
V, then both sides of the obvious for the left-hand
DE RHAM COHOMOLOGY
In this chapter U will denote an open set in IR n , {el,"" en} the standard basis and {El, ... ,E n } the dual basis of Alt l (lR n ).
) Altp(g)
= tr
3.
AltP(J). Consider the
(V) }.
ex linear, then D is dense lot give a formal proof of ltries can be approximated e characteristic polynomial lt eigenvalues are linearly ; for such a matrix, which
Definition 3.1 A differential p-fonn on U is a smooth map w: U The vector space of all such maps is denoted by OP(U).
Dxw: IR n
f (i.e. the
~nt in f). Since both sides :ince the equation holds for D
R is dense in the set of es cannot be approximated 11 matrix. Therefore in the lear maps, even if we are
AltP (lR n ).
If p = 0 then AltO (IR n ) = IR and OO(U) is just the vector space of all smooth real-valued functions on U, OO(U) = COO(U, IR). The usual derivative of a smooth map w: U ~ Altp(lR n ) is denoted Dw and its value at x by Dxw. It is the linear map
with E D with dn ~
~
~
Altp (lR n ),
d
Ow
(Dxw)(ei) = -dw(x + tei)t=O = -;-(x). t UXi In Altp (IRn) we have the basis EJ = Ei Ei p ' where I runs over all sequences with 1 ::; il < i2 < ... < i p ::; n. Hence every w E OP(U) can be written in the form w(x) = "L,W[(X)E[, with w[(x) smooth real-valued functions of x E U. The differential Dxw is the linear map j
fer
(1)
Dxw(ej) =
/\ ••• /\
L ~~: (x)EJ , j = 1, ... , n. [
The function x 1--+ Dxw is a smooth map from U to the vector space of linear maps from Rn to AltP (lR n ). Definition 3.2 The exterior differential d: OP(U) ~ OP+l(U) is the linear op erator
p+I d x w(6,···,(p+I)
=
~
L..,,(-1)
1-1
A
D xw((z)(6,···,(I,.··,(p+I)
1=1
with (6"",~I, ... ,(p+l)
= (6,···,(I-l,(I+I,···,(p+d·
p It follows from Lemma 2.7 that dxw E Alt + I (IR n ). Indeed, if (i = (i+I, then p+l ~ 1-1 L.." (-1) DxW((I)(6,···, (1,"" (pH) A
1=1
. 1
A
=(-lr- D xW((i)(6,···,(i, + (_l)i D x W((i+I)(6, =0
,(p+I) , ~i+l, ... , (pH)
-=
0
rn
'~ '-l
= ~
~
E-t
"::l '-l ,;;..,
c:l
Z
~
.,= .::: u
0
==
~
-..
-.. ,.....;
C'l
~.
I
"
~" -! )
........
-,
3.
17
DE RHAM COHOMOLOGY
The exterior product in Alt*(Rn) induces an exterior product on n*(U) upon defining en dXi E n1 (U) is the ~neral, for f E nO(U),
(Wl/\ W2)(X) = Wl(X) /\ W2(X),
The exterior product of a differential p-form and a differential q-fonn is a differential (p + q)-form, so we get a bilinear map
/\: np(U) x nq(U)
(n For a smooth function
'" EJfdxi. f..JBX;
--+
np+q(U).
f E COO(U, R), we have that
(fwd /\ W2
= f(Wl
/\ W2) =
WI /\
f W2.
This just expresses the bilinearity of the product in Alt* (~n). Also note that f /\ W = fw when f E nO(U) and W E np(U).
l)
f[
= dxf( ()f[
Lemma 3.6 For
WI E
np(U) and W2 E
nq (U),
d(Wl /\ W2) == dWl/\ W2
...
+ (-I)PWl/\ dW2·
,~k," '~P+l)
o
Proof. It is sufficient to show the formula when then WI /\ W2 = fg f [ /\ fJ, and
WI =
ft.[ and W2
d(Wl/\ W2) = dUg) /\ f[ /\ fJ = ((dJ)g + fdg) /\ f[ = dfg /\ f[ /\ fJ + fdg /\ f[ /\ fJ = df /\ f[ /\ gfJ + (-ll ff[ /\ dg /\ fJ = dWl /\ W2 + (-l)PWI /\ dW2.
= gfJ.
But
/\ fJ
0
= (il, ... ,ir,k, ... ,ip). ~P+1(U) --+ n p+2(U) is
Summing up, we have introduced an anti-commutative algebra n*(U) with a differential,
d: n*(U)
--+
n*+1(U),
dod
=a
and d is a derivation (satisfies Lemma 3.6): (n*(U), d) is a commutative DGA (differential graded algebra). It is called the de Rham complex of U.
-fn /\ f[.
:hat
=j
Theorem 3.7 There is precisely one linear operator d: np(U) 0,1, ... , such that (i) f E nO(U), df = 1\
t.[
= O.
o
(ii) dod = 0 (iii) d(Wl/\ W2)
-3£
fl
--+
np +1 (U),
+ ... + #tfn
= dWl /\ W2 + (-l)PWI /\ dW2 if WI
E
np(U).
p=
...
="'I'J.
=
'C'$ C.I
-t
= ~
C'$ ~=
~
'
0
I~
C.I
....
~
~
=
,...., ---
.:=
C'l ---
u
~
I
~
~
....
.
~
-' ,)
3.
erties. Conversely assume We will show that d' is
cular d' Xi = dXi for the i hat d' Xi = ci, the constant (iii) gives d'E[ = O. Now ~ (iii),
f
/\E1
d = d' on all of DP(U).D
las
:+ dh/\ E3 =
Oh)
C3 /\ EI.
OXI
U -+ Alt l (1R 3 ) is the :ly, we have already noted i 0 d(Xi) = 0 by Theorem l1utativity, Ei/\Ej = -Ej /\Ei Ei:
093)
-72 + -0 ;2
CI /\ E2 /\ c3·
X3
is the quotient vector space
(U))
components of U. A connected component of U is a maximal non-empty subset W of U that cannot be written as the disjoint union of two non-empty open subsets of W (in the topology induced by Rn). An open set U ~ IRn has at most countably many connected components (in each of them one can choose a point with rational coordinates.)
Lemma 3.9 HO(U) is the vector space ojmaps U -+ IR that are constant on each
Proof. A locally constant function f: U -+ IR gives a partition of U into the mutually disjoint open sets f-l(c), c E IR. Consequently f: U -+ IR is locally 0 constant precisely when f is constant on each connected component of U. It follows that dimRHO(U) (considered as a non-negative integer or 00) is precisely the number of connected components of U. The elements in DP(U) with dw = 0 are called the closed p-forms. The elements of the image d(DP-I(U)) C DP(U) are the exact p-forms. The p-th cohomology group thus measures whether every closed p-form is exact. This condition is satisfied precisely when HP(U) = O. A closed p-form w E DP(U) gives a cohomology class, denoted by
[w] = W + dDP-I(U) E HP(U), and [w] = [w'] if and only if W - w' is exact. In general the vector space of closed p-forms and the vector space of exact p-forms are infinite-dimensional. In contrast HP(U) usually has finite dimension. We can define a bilinear, associative and anti-commutative product
kernel of
HP(U) x Hq(U)
(3)
by setting (WI
:U)) . ~
19
connected component oj U.
= dw.
aft _ ( OX3
DE RHAM COHOMOLOGY
[WI]
[W2] =
[WI /\
-+
Hp+q(U)
W2]. It is well-defined because
+ dTJt) /\ (W2 + dTJ2) = WI /\ W2 + dTJI /\ W2 + WI /\ dTJ2 + dTJI /\ dT12 = WI /\ W2 + d (TJI /\ w2 + (-l)PWI /\ TJ2 + TJI /\ dTJ2)'
We want to make U -+ HP(U) into a contravariant junctor. Thus to a smooth map ¢: UI -+ U2 between open sets UI C IR n and U2 C IR m, we shall define a linear map
HP(¢): HP(U2)
,R) with vanishing deriva
such that:
maps. such that ql rv q2 if there = ql and a(b) = Q2. The sets, namely the connected
(4)
-+
HP(U I ),
HP(¢2 0 ¢t) = HP(¢t) HP(id) = id.
0
HP(¢2)
We first make D* (-) into a contravariant functor.
......
0
=
'1'l
,~
0004
<;,)
:e
~
~
~;::l
<;,)
~
,;;.,
0
,"= .:::
z
~
::
"""' C'l
"""' ..-I
u
~
I
~
.....
--,
.
~
-' "-- ~.
3.
sets and ¢: U1 ---t U2 a 1P (Ud is defined by (w)x
= wc/>(x)'
This shows (i) when p > 0 and q > O. If p = 0 or q = 0 the proof is quite analogous, but easier. Property (ii) is contained in the definition of ¢* for degree O. So we are left with (iii). We shall first show that d¢*(f) = ¢*(dJ) when f E 00(U2). We have that m af m af
df = L ax Ek = L ax /\ Ek,
k=l k k=l k
it the analogue of (4) is Dx¢(~p)),
D x (¢),
for ¢: U1
---t
when Ek is considered as the element in 01(U2) with constant value Ek. From (i) and (ii) we obtain
:: idl!p(u),
U2 is an inclusion, since
¢*(dJ) =
f
k
= LmLn (a - f k=l l=l
=
t
l=l
ERn. Then
aXk
a(f 0 ¢) El aXl
In the more general case w because dEl = O. Hence
hi
k=l 0
¢ ) (a¢ _k ) aXI
= d(f 0 = fEI
¢)
El
0 ¢) /\
k
= Ln
(t ~~k EL) l=l
(m L
l=l
k=l
(a - f aXk
0
I
a¢ ) ¢ ) _k aXl
El
= d(¢*(f)).
f /\ EI, Lemma 3.6 gives dw = df /\ EJ,
=
¢*(dw) = ¢*(dJ) /\ ¢*(EI) = d(¢* J) /\ ¢*(EI) = d(¢*(f) /\ ¢*(EI)) = d(¢*w).
o
: d¢i( ().
f (::
¢* ( : : ) /\ ¢*(Ek) =
k=l
e have that
l¢k (I) ek )
21
DE RHAM COHOMOLOGY
The second last equality uses Lemma 3.6 and the fact that d¢*( EI) = 0:
:ons
d¢*(EI) = d(¢*(EiJ /\ ... /\ ¢*(Eip ) ) =
~tor
that satisfies the three
since
d¢*(Eik)
L
(-l)k-l¢*(Ei l ) /\ ... /\ d¢*(Eik) /\ ... /\ ¢*(Eip ) = 0
= 0 by Example 3.11
and Lemma 3.5.
D
In the following it will be convenient to use the notation of Example 3.3 and write Rn . Then
dx I
instead of the (constant) p-forrn EI be written as
1)) )
¢(~a(p+q))) ] r(p+l) , ...
= dXil
,~a(p+q))
= Eil
/\ ... /\ dXip /\ ... /\ Eip ' An arbitrary p-forrn can then
w(X) = LWI(X)dxI and Example 3.11 becomes ¢*(dYi) = d¢i when Yi: U2 ---t R is the i-th coordinate function and ¢i = Yi o¢ the i-th coordinate of ¢; cf. Theorem 3.12.(ii),(iii).
....
~
e
XJ
,~
-4
i;"l
:Q
~
~;::l
~
i;"l
~
....
z
'
:r:
..c::
~ I~
0
----. n
----. C'l
u
~
I
~-
~,
-'
3.
.. , /'n), and let
23
DE RHAM COHOMOLOGY
Theorem 3.15 (Poincare's lemma) IfU is a star-shaped open set then HP(U) = 0 for p > 0, and HO(U) = IR. Proof. We may assume U to be star-shaped with respect to the origin 0 E and wish to construct a linear operator
Sp: SlF(U)
lx n ) ~))
~ b (t) ), /,' (t)) dt.
--+
lI~n,
S1P-1(U)
such that dSp + Sp+ld = id when p > 0 and Sld = id - e, where e(w) = w(O) for w E S10(U). Such an operator immediately implies our theorem, since dSp(w) = w for a closed p-form, p > 0, and hence [w] = O. If p = 0 we have w - w(O) = Sldw = 0, and w must be constant. First we construct
e 1.8.
Sp: S1P(U x R)
ts in Rn . Then
w where I
=
=L
fr(x, t)d:cj +
L gJ(x, t)dt /\ dXJ
(i1,oo.,i p) and J = (j1, ... ,jp-1). We define
¢*(1:1) /\ ... /\ d¢*(x n ) /\ ... /\
S1P-1(U).
Every w E S1P(U x R) can be written in the form
.,. /\ dXn'
ld x 1
--+
L Ci
Sp(w) =
dXn.
1
gJ(x, t)dt) dXJ.
Then we have that
"(1
~ ~ dSp(w) + Sp+ld(w) = LJ I
= ?jJ(t)x,
where ?jJ(t) is a
c·
"'1..
1
ogJ(X, ox- t) dt ) dXi /\ dXJ
° t Ofr(x,t)) + ,,( ~ Jo ot dt dXj J ,I.
=L
a linear map =
1
(1
1
f:: Jot
"(
Ofr~:, t) dt) dXj
L fr(x, l)dXj - L fr(x, O)dXj.
We apply this result to ¢* (w ), where definition is independent of I+¢*(dv) = ¢*(w)+d¢*(v).
¢: U x R --+ U, ¢(x, t) = ?jJ(t)x and ?jJ(t) is a smooth function for which
{q (¢) [W2])
lism of graded algebras.
?jJ(t) = 0 ?jJ(t) = 1 { o ~ ?jJ(t) ~ 1
if t ~ 0 if t ~ 1 otherwise.
ogJ ) OXi dt dXi /\ dXJ
::>
'I'J.
-t
:t: ~
... =
'C':l Q
C':l
~;:l
Q
;.:l
,~
::>
l~
z:
:= ~
~
,-..... ,....;
,-.....
r",-'-"
G'l
..t:
I
U
"'-j
25
1(U)
as above. Assume
~d7/J(t)Xip + 1/J(t) dX ip)
4.
CHAIN COMPLEXES AND THEIR HOMOLOGY
In this chapter we present some general algebraic definitions and viewpoints, which should illuminate some of the constructions of Chapter 3. The algebraic results will be applied later to de Rham cohomology in Chapters 5 and 6.
Pdx/. A sequence of vector spaces and linear maps
A.LB~C
(1)
p>O p=O.
o
is said to be exact when 1m f = Ker g, where as above Ker 9 = {b E Blg(b)
= O} (the kernel of g)
Imf = {j(a)la E A}
(the image of j).
Note that A .L B ---+ 0 is exact precisely when f is surjective and that 0 ---+ B ~ C is exact precisely when 9 is injective. A sequence A* = {Ai,d i }, (2)
... ---+ Ai-1 d~l Ai
!
Ai+l
1
d:
Ai+2 ---+ ...
of vector spaces and linear maps is called a chain complex provided di +1 0 di = 0 for all i. It is exact if Ker di = 1m di - 1 for all i. An exact sequence of the form f o ---+ A ---+ B
(3)
9
---+ C ---+ 0
is called short exact. This is equivalent to requiring that
f is injective,
9 is surjective
and
1m f
= Ker g.
The cokernel of a linear map f: A ---+ B is
Cok(j) = B /Imf. For a short exact sequence, 9 induces an isomorphism
"" C. g: Cok(j) ~ Every (long) exact sequence, as in (2), induces short exact sequences (which can be used to calculate Ai) 0---+ 1m d i - 1 ---+ Ai ---+ 1m d i ---+ O.
~
L'iJ. -..
~ ·.~··.·-~t. ff~i. i 4.
:;y
J*
~ace
!). ~n
{(ai, 0), (O,bj)} is a
exact sequence of vector ~, and B ~ A EB C.
lce 9 is surjective there basis of B: For b E B g. Since Ker 9 = 1m f,
'6
27
CHAIN COMPLEXES AND THEIR HOMOLOGY
Lemma 4.3 A chain map
i-I
...
=
f: A*
---+
B* induces a linear map
H*(f): HP(A*)
---+
HP(B*),
for all p.
Proof. Let a E AP be a cycle (dPa = 0) and [a] = a + 1m dP- l its corresponding cohomology class in HP(A*). We define f*([ a]) = [fP(a)]. Two remarks are needed. First, we have d~ JP(a) = fP+l d~(a) = fP+l(O) = O. Hence fP(a) is a cycle. Second, [fP(a)] is independent of which cycle a we choose in the class raj. If [all = [a2] then al - az E 1m d~-t, and JP(al - a2) = JP d~-l(x) = d~-lfP-l(x). Hence fP(al) - fP(a2) E Imd~-l, and JP(aI),f P(a2) define the same cohomology class. 0 A category C consists of "objects" and "morphisms" between them, such that "composition" is defined. If f: Cl ---+ C 2 and g: C 2 ---+ C 3 are morphisms, then there exists a morphism go f: Cl ---+ C 3 . Furthermore it is to be assumed that ide: C ---+ C is a morphism for every object C of C. The concept is best illustrated by examples: The category of open sets in Euclidean spaces, where the morphisms are the smooth maps. The category of vector spaces, where the morphisms are the linear maps.
f(ai)'
n of {bj} and {f(ai)}' It f independent. 0 dP -
1
. ---+
AP
dP
---+
AP+l
---+ ••• }
The category of abelian groups, where the morphisms are homomor phisms. The category of chain complexes, where the morphisms are the chain maps. A category with just one object is the same as a semigroup, namely the semigroup of morphisms of the object.
~d to be closed} and the } be exact). The elements
sists of a family fP: AP
---+
~ chain map is illustrated
~
Every partially ordered set is a category with one morphism from c to d, when c :::; d. A contravariant functor F: C ---+ V between two categories maps every object C E obC to an object F(C) E ob V, and every morphism f: Cl ---+ C2 in C to a morphism F(f): F(Cz ) ---+ F(CI) in V, such that
F(g
...
0
f) = F(f)
0
F(g),
F(idc) = idF(c).
A covariantfunctor F:C ---+ V is an assignment in which F(f): F(Cl) and F(g 0 f) = F(g) 0 F(f), F(idc) = idF(c).
---+
F(Cz ),
.
....
\
-~~,~~
4.
;y
~tween categories. The
:ovariant ones preserve
b1 E BP-l with gP-l(b 1) = c. It follows that gp(b - diJ-l(bl)) = O. Hence there exist a E AP with fP(a) = b - diJ-1(b 1). We will show that a is a p-cycle. Since JP+l is injective, it is sufficient to note that fp+l (d~ (a)) = O. But
jP+l (d~ (a))
~
29
CHAIN COMPLEXES AND THEIR HOMOLOGY
= diJ (JP(a)) = diJ (b -
d~-l(bl))
=0
linear maps from ---+ Hom(Cl, A) is nt functor from the
since b is a p-cycle and dP 0 dP- 1 = O. We have thus found a cohomology class [a] E HP(A), and f*[ a] = [b - d~-l(bl)J = [b]. 0
a covariant functor
One might expect that the sequence of Lemma 4.4 could be extended to a short exact sequence, but this is not so. The problem is that, even though gP: BP ---+ CP is surjective, the pre-image (gP)-l(c) of a p-cycle with c E CP need not contain a cycle. We shall measure when this is the case by introducing
~s and smooth maps, space of differential
Definition 4.5 For a short exact sequence of chain complexes 0 C* ---+ 0 we define &*: HP(C*) ---+ HP+l(A*) to be the linear map given by
Dvariant functor. xactly the de Rham lant.
---+
_
0 is exact
o lexes the sequence
1 (
A*
.L B* .!!...
d~ ((gP) -1 (c)) ) ] .
There are several things to be noted. The definition expresses that for every b E (gP)-I(c) we have d~(b) E Im(Jp+l), and that the uniquely detennined a E AP+l with fp+l(a) = diJ(b) is a (p + I)-cycle. Finally it is postulated that [a] E HP+l(A*) is independent of the choice of bE (gP)-I(c). In order to prove these assertions it is convenient to write the given short exact sequence in a diagram:
o BP .!!... CP
r
&* ([ c]) = [(JP+ 1
e de Rham complex
---+
o _
1
1
1
v-I AP-l _/V-I BP-l ..fl..-.. Cp-l _
Id~-l
Id~-l
v
V
Id~-l
AP --.l-- BP -1L- CP -
Id~~ Id~
AP+1
r+ BP+1 l
,
gV+r,
Cp+l _
o o o
111 The slanted arrow indicates the definition of 8*. We shall now prove the necessary assertions which, when combined, make 8* well-defined. Namely:
a))]=O assume for lb] E HP(B) is surjective, there exists
I,
(i) If gP(b) = c and d~(c) = 0 then diJ(b) E Imfp+l. (ii) If fP+l(a) = diJ(b) then d~+l(a) = O. (iii) If gP(bI) = gP(b2) = c and JP+l(ad = diJ(bi) then [all HP+l(A*).
= [a2]
E
·'>
c
11:3 """. ~ .~ ~ \:lI):~
.......
~
-.
"~ . . ~.~ ~,~
'"
~
..c,~
~R
.,.
..J
~
...
..........~•. ;
~~"'."
..;,!;>,..~ -r~-
4.
Y
= 0, and Ker gp+l
(a)
= d~+l fP+l(a)
Definition 4.10 Two chain maps f, g: A* ----t B* are said to be chain-homotopic, if there exist linear maps s: AP ----t BP-l satisfying
= =
at d~(bd - d~(b2)
: Up+l
r
31
CHAIN COMPLEXES AND THEIR HOMOLOGY
= 1 ( d~ (b2)) +
dBS + SdA = f - g: AP
----t
BP
for every p. In the form of a diagram, a chain homotopy is given by the slanted arrows.
lexes (the dots indicate
_
AP-l _
_
BP-l _
AP _
1/-~
AP+l _
1/-
AP+2 _
1/-
9 /
1
9 /
BP _
BP+l _
BP+2 _
The name chain homotopy will be explained in Chapter 6.
Lemma 4.11 For two chain-homotopic chain maps f, g: A*
J* = g*: HP(A*)
----t
----t
B* we have that
HP(B*).
Proof. If [a] E HP(A*) then
t.
(/* - g*)[a] = [JP(a) - gP(a)] = [d~-ls(a)
(A*) is exact.
B(b))]
Conversely nd a E AP, such that
[d~-ls(a)]
= O. D
linear maps
sp: OP(U)
= D
id*
= c.
Conversely assume gP+l fP+l(a) = 0, and D
----t
Op-l(U)
with dP-1S p + Sp+ldP = id for p > O. This is a chain homotopy between id and o (for p > 0), such that
'+l(B*) is exact.
=
=
Remark 4.12 In the proof of the Poincare lemma in Chapter 3 we constructed
= O.
. Hence g*[b - fP(a)]
+ sd~(a)]
= 0*: HP(U)
----t
HP(U),
P
> O.
However id* = id and 0* = O. Hence id = 0 on HP(U), and HP(U) = 0 when > O.
p
Lemma 4.13 If A * and B* are chain complexes then HP(A* EB B*)
= HP(A*) EB HP(B*).
tant
~
l
A * L B* sequence
----t
~ C*
r. HP+l (B*)
----t
°
Proof. It is obvious that
D
and the lemma follows.
----t • • •
Ker (d~tBB) = Ker d~ EB Ker d~ P 1 P 1 I m (d P-l) AtBB = I m dA- EB I m dB-
D
c
ii
.N
-§o -e
'6
~
£"'t-l -.
"".
33
5.
THE MAYER·VIETORIS SEQUENCE
This chapter introduces a fundamental calculational technique for de Rham cohomology, namely the so-called Mayer-Vietoris sequence, which calculates H*(UI U U2) as a "function" of H*(UI), H*(U2) and H*(UI n U2). Here UI and U2 are open sets in IR n. By iteration we get a calculation of H*(UI U' .. U Un) as a "function" of H*(Ucr ), where 0: runs over the subsets of {I, ... , n} and Ucr = Uil n ... n Uir when 0: = {il,' .. , i r }. Combined with the Poincare lemma, this yields a principal calculation of H* (U) for quite general open sets in IR n . If, for instance, U can be covered by a finite number of convex open sets Ui, then every Ucr will also be convex and H*(Ucr ) thus known from the Poincare lemma. Theorem 5.1 Let UI and U2 be open sets of IR n with union U = UI U U2. For v = 1,2, let i y:Uy ---> U and jy: UI n U2 ---> Uy be the corresponding inclusions. Then the sequence
o ---> DP(U) is exact, where IP(w)
jP
--->
DP(UI) EEl DP(U2)
= (ii(w), i2(w)),
Proof. For a smooth map ¢: V
--->
JP
--->
DP(UI n U2)
JP(WI, W2)
= j{(wI) -
Wand a p-form w
DP(¢)(w) = ¢*(w) = 'L.(fI
0
--->
j2(W2).
= 'L.fIdx/
¢)d¢i I A ...
A
0
E DP(W),
d¢i p '
In particular, if ¢ is an inclusion of open sets in Rn, i.e. ¢i (x)
= Xi, then
d¢iI A ... A d¢i p = dXiI A ... A dXip' Hence
¢*(w) = 'L.(II
(1)
0
¢)dx/.
This will be used for ¢ = iy,jy, v = 1,2. It follows from (1) that IP is injective. If namely JP(w) = 0 then ii(w) = 0 = i 2(w), and i~(w)
= 'L.(fI 0 iy)dxj = 0
if and only if fI 0 i y = 0 for all I. However II 0 il = 0 and fI 0 i2 = 0 imply that fI = 0 on all of U, since UI and U2 cover U. Similarly we show that Ker JP = 1m IP. First,
JP
0
IP(w)
= j2'i 2(w) - )iii(w) = j*(w) -
j*(w)
= 0,
_. "
5.
~
35
THE MAYER-VIETORIS SEQUENCE
Theorem 5.2 (Mayer-Vietoris) Let Ul and U2 be open sets in Rn and U = Ul UU2. There exists an exact sequence of cohomology vector spaces
Ker JP. To show the
Uv ),
••• ---+
HP(U)
r HP(UI) EEl HP(U ) ---+ J. o· HP(UI n U2) ---+ HP+1(U) 2
---+
Here 1*([ w]) = (ii[ w], i 2[w]) and notation of Theorem 5.1.
which by (1) translates U2. We define a smooth
J*([ wd, [W2])
=
---+ •••
Ji[ wd - j2'[ W2] in the 0
Corollary 5.3 If Ul and U2 are disjoint open sets in Rn then
1*: HP(UI U U2) ---+ HP(U l ) EEl HP(U2)
is surjective. To this end U2}, i.e. smooth functions
is an isomorphism. Proof. It follows from Theorem 5.1 that
l(X)
=
1 for x E U (cf.
P2} to extend
f
to Ul and smooth function by
F: OP(UI U U2) ---+ OP(Ul ) EEl OP(U2) is an isomorphism, and Lemma 4.13 gives that the corresponding map on coho 0 mology is also an isomorphism.
Example 5.4 We use Theorem 5.2 to calculate the de Rharn cohomology vector spaces of the punctured plane R2 - {O}. Let
T2
,UPPu(Pl)'
Ul = R2 U2 = R2
-
I Xl 2: 0, X2 = O} I Xl ::; 0, X2 = O}. such that HP(UI) = HP(U2) = 0 for
{(Xl,X2) {(Xl, X2)
2
lPPU(P2)' ~cause Pl(X)
+ P2(X) =
1. r I, we can apply the above :he functions fr,v: Uv ---+ IR, OP(Uv )' With this choice
o
1 U2)
These are star-shaped open sets, HO(Ul ) = HO(U2) = R. Their intersection
Ul n U2
> 0 and
R = R~ u R:
is the disjoint union of the open half-planes X2 > 0 and X2 < O. Hence
HP(UI n U2) =
(2)
{ ~ EEl R
if p > 0 if p = 0
by the Poincare lemma and Corollary 5.3. From the Mayer-Vietoris sequence we have J. o· ---+ HP(Ul ) EEl HP(U2) ---+ HP(UI n U2) ---+ o
t exact sequence of chain a long exact sequence of : us that
)HP(O*(U2))'
= R2 -
p
••
HP+l(R 2 -{0})
!.: HP+l(Ul ) EEl Hp+1(U2) ---+ •••
For p > 0,
o ---+ HP(UI
n U2) ~ HP+l (R 2 - {On
---+
0
;
.,.J
~'
,
'"
.~.' ': ' '
£"'tol
--~
'~-~i' ... '... . ..
'
..
..••...
'
.....•'.
.;~)"~
5.
: 0 for q ~ 2 according
..
'.
THE MAYER-VIETORIS SEQUENCE
let V = Ul U ... U Ur-l, such that U the exact sequence
=V
37
U Ur. From Theorem 5.2 we have
HP-l(V n Ur) ~ HP(U) ~ HP(V) EB HP(Ur ) fO(U2)
~
which by Lemma 4.1 yields
l) EB H 1 (U2)'
HP(U)
r H- 1 (U v ) = 0, 1° is sequence (3) reduces to
1)£H 1 (R 2 - {O})-O. ~) ~ R, and since 1° is s Ker JO ~ R, so that JO ,xactness
-{O}).
l.
.ted 1 U2)
itant functions. If
Ii
is a
-f21ulnu2
'red by convex open sets
If r = 1 the assertion on is proved for r - 1 and
[So
~
Im8* EB Imr.
Now both V and V n Ur = (Ul n Ur) U··· U (Ur -l n Ur) are unions of (r -1) convex open sets. Therefore Theorem 5.5 holds for H*(V n Ur ), H*(V) and H*(Ur ), and hence also for H*(U). 0
,
""
39
6.
HOMOTOPY
In this chapter we show that de Rham cohomology is functorial on the cate gory of continuous maps between open sets in Euclidean spaces and calculate
H*(R n
-
{a}).
Definition 6.1 Two continuous maps fv: X ---* Y, v = 0,1 between topological spaces are said to be homotopic, if there exists a continuous map
F:X x [0,1] such that F(x, v) = fv(x) for v
=
---*
Y
0,1 and all x E X.
This is denoted by fa ~ iI, and F is called a homotopy from fa to iI. It is convenient to think of F as a family of continuous maps it: X ---* Y (0 ::; t ::; 1), given by ft(x) = F(x, t), which defonn fa to iI.
Lemma 6.2 Homotopy is an equivalence relation.
JJ
Proof. If F is a homotopy from fa to iI, a homotopy from iI to fa is defined by G(x, t) = F(x, 1 - t). If fa ~ iI via F and iI ~ fz via G, then fa ~ fz via
.
_ {F(X, 2t)
H(x, t) Finally we have that f
~
G(x, 2t - 1)
0::; t::; 1
1::; t ::; 1.
Lemma 6.3 Let X, Y and Z be topological spaces and let fv: X ---* Z be continuous maps for v = 0,1. If fa ~ it and 90
9v: Y
90
0
fa
~ 91
0
o
f via F(x, t) = f(x). ---*
Y and then
~ 91
iI.
Proof. Given homotopies F from fa to iI and G from 90 to 91, the homotopy H from 90 0 fa to 91 0 iI can be defined by H(x, t) = G(F(x, t), t). 0 Definition 6.4 A continuous map f: X ---* Y is called a homotopy equivalence, if there exists a continuous map 9: Y ---* X, such that 9 0 f ~ idx and f 0 9 ~ idy. Such a map 9 is said to be a homotopy inverse to f. Two topological spaces X and Yare called homotopy equivalent if there exists a homotopy equivalence between them. We say that X is contractible, when X is homotopy equivalent to a single-point space. This is the same as saying that idx is homotopic to a constant map. The equivalence classes of topological spaces defined by the relation homotopy equivalence are called homotopy types.
6.
)y Rm . If, for the con 1m from fo(x) to h(x) ~py F: X x [0, 1] -+ Y
41
HOMOTOPY
°
Indeed, cP* (dt 1\ dx J) = since the last component (the t-component) of cP is constant; see Example 3.11. Analogously, for
cPi(w) =
L fI(x, l)dx/.
In the proof of Theorem 3.15 we constructed
Sp: np(U x R)
ontractible.
np - 1 (U)
such that
(dSp + Sp+ld)(w)
(1)
hen
, a smooth map. wtopic, then there exists ,(x) for v = 0,1 and all to hJ.
oth map f: U -+ V. We n h(x) to f(x) for every s function 1P: R lstruct
-+
-+
[0,1]
= cPi(w) -
Consider the composition U '!::. U x IR .!... V, where F is a smooth homotopy between f and g. Then we have that F 0 cPo = f and F 0 cPl = g. We define
Sp: np(V) to be Sp
= Sp
0
-+
np-1(U)
F*, and assert that dSp + Sp+ld
= g* - j*.
This follows from (1) applied to F*(w), because
dSp(F*(w))
+ Sp+ldF*(w)
= cPiF*(w) - cPoF*(w) = (F
Furthermore Sp+ldF*(w)
0
cPl)*(W) - (F
0
cPo)*(w)
=
g*(w) - j*(w).
= Sp+lF*d(w) = Sp+ld(w), since F* is a chain map.
I(t)). for t ~ ~, H is smooth : to approximate H by a : the same restriction on I) = H(x, v) = fv(x). 0
D In the situation of Theorem 6.7, Lemma 4.11 states that f* = g*: HP(V) -+ HP(U). For a continuous map cP: U -+ V we can find a smooth map f: U -+ V with cP c::= f by (i) of Lemma 6.6, and by Lemma 6.2 and the result above we see that f*: HP(V) -+ HP(U) is independent of the choice of f. Hence we can define
cP* = HP(cP): HP(V)
9 then the induced chain
by setting cP* = f*, where f: U
-+
-+
HP(U)
V is a smooth map homotopic to cPo
Theorem 6.8 For pEl and open sets U, V, W in Euclidean spaces we have (i) If cPo,
cPl: U
-+
V are homotopic continuous maps, then
very p-form w on U x R
cPo
= cPi: HP(V)
-+
HP(U).
(ii) If cP: U
1\ dXJ.
-+ V and 1P: V -+ Ware continuous, then (1P 0 cP)* = cP* 1P*: HP(W) -+ HP(U). (iii) If the continuous map cP: U -+ V is a homotopy equivalence, then
: (x,O), then O)dx/.
cP*: HP(V) is an isomorphism.
-+
HP(U)
0
l~ ~- 6 ......H :"'-...... -t; Q
..c. ,~
;t; ~
,,-J
~
~
.£"ot-l
.~ .... :~.
_~
Q..
~o
~ ~>,~ .. ~ .~_l .
....
(I) ..•.
1....,~;,
6.
Proposition 6.11 For an arbitrary closed subset A of R with A isomorphisms HP+l (Rn+l - A) ~ HP(R n - A) for p ~ 1 n HI (Rn+l - A) ~ HO(R - A)/R . 1 HO(R n+1 - A) ~ R.
Lemma 6.2 gives that 100th ¢ and 1jJ, follows
I g: V
-+
W with ¢ ~
43
HOMOTOPY
1= Rn
we have
f
we get
Proof. Define open subsets of Rn +1 = Rn x R,
) 1jJ* .
UI = Rn x (0, (0) U (R n - A) x (-1, (0) U2 = Rn x ( -00,0) U (R n - A) x (-00,1). Then UI U U2 = Rn+l - A and U1 n U2 = (R n - A) x (-1,1). Let ¢: U1 -+ U1 be given by adding 1 to the (n + l)-st coordinate. For x E UI, U1 contains the line segments from x to ¢(x) and from ¢(x) to a fixed point in Rn x (0, (0). As
r,
verse to ¢*.
o
lomotopy type of U.
In
Jhism h: U \*:
HP(V)
-+ --t
in Example 6.5 we get homotopies from id u1 to ¢ and from ¢ to a constant map. It follows that U1 is contractible. Analogously U2 is contractible, and HP(Uv ) is described in Corollary 6.10. Let pr be the projection of U1 n U2 = (Rn - A) x (-1,1) on Rn - A. Define i : Rn - A --t U1 n U2 by i(y) = (y,O). We have pr 0 i = idRn_A and i 0 pr ~ idulnu2' From Theorem 6.8.(iii) we conclude that
V between HP(U) for
pr*: HP(R n - A)
--t
HP(UI
n U2)
is an isomorphism for every p. Theorem 5.2 gives isomorphisms l), as h- l : V
--t
&* : HP(U1
U is a
o
t, then HP(U) = 0 when
n U2)
--t
-
A)
for p ~ 1. By composition with pr* one obtains the first part of Proposition 6.11. Consider the exact sequence
o
--t
HO(Rn+l - A) ~ HO(UI) tB HO(U2)
!:. HO(U1 n U2) ~ I
HP+l (R n+1
HI (R n + l
-
A)
--t
O.
= idu to a constant map
a continuous curve in U, 'O(U) = R by Lemma 3.9. map. Hence by Theorem
An element of HO(UI) tB HO(U2) is given by a pair of constant functions on UI and U2 with values al and a2. Their image under J* is by Theorem 5.2 the constant function on U1 n U2 with the value al - a2. This shows that Ker &*
o
= 1m J* = R . 1 ,
and we obtain the isomorphisms
HI (Rn+l - A) ~ HO(U1 Lbspace Rn x {O} of Rn+l 19 of constant functions.
We also have that dim(Im (1*))
n U2)/R.I
~ HO(R n - A)/R . 1.
= dim(Ker (J*)) = 1, so HO(R n+1 -
A) ~ R.O
c
i3
.go
'G
'"
.-J
.:-,J
6.
Theorem 6.13 For n
rave a diffeomorphism
~
45
HOMOTOPY
2 we have the isomorphisms
HP (R n
_
{O}) ~
{R
o
if p = 0, n - 1 otherwise.
le induced linear map
- A)
Proof. The case n = 2 was shown in Example 5.4. The general case follows 0 from induction on n, via Proposition 6.11.
mmutative diagrams, in
An invertible real n x n matrix A defines a linear isomorphism IR n a diffeomorphism
R:
fA: Rn
L..R
n +1 -
A
{O}
-+
Rn
Rn, and
{O}.
-
Lemma 6.14 For each n ~ 2, the induced map fA: Hn-I(R n - {O}) Hn-I(Rn - {O}) operates by multiplication by det A/ldet AI E {±1}.
ti1
4
-
-+
-+
VI Proof. Let B be obtained from A by replacing the r-th row by the sum of the r-th row and e times the s-th row, where r i= sand c E R,
t)1 ~ VI nV2
B
= (I + cEr,s)A,
where I is the identity matrix and Er,s is the matrix with entry 1 in its r-th row and s-th column and zeros elsewhere. A homotopy between fA and f B is defined by the matrices
A) ) 8* ([w1) = -8*([wD for ",ith w = jr(WI) - j2'(W2). '*([w]) = [7] where 7 E 1,2. Furthermore we get
W2) - j2(R;WI)
(I
+ teEr,s) A,
0 ~ t ~ 1.
From Theorem 6.8 it follows that fA = f B. Furthermore detA = detB. By a sequence of elementary operations of this kind, A can be changed to diag (1, ... , 1, d), where d = det A. Hence it suffices to prove the assertion for diagonal matrices. The matrices
. ( 1, ... ,1, 1dI Id1td) ' dlag
O~t~l
»]) = [R*7]. Hence I.
>I 0
Ro = pr and therefore
yield a homotopy, which reduces the problem to the two cases A = diag(l, ... , 1, ±1), so fA is either the identity or the map R from Adden 0 dum 6.12. This proves the assertion. From topological invariance (see Corollary 6.9) and the calculation in Theorem 6.13, supplemented with
UI n V2)
HP(RI _ {O})
ls forced to be the identity 8* [w]. This completes the
o
we get
~ { ~ Efl R
if p if p
=0 i= 0
...
'It
"
47 ~omorphic.
7.
APPLICATIONS OF DE RHAM COHOMOLOGY
°
ssumed to map to 0, .od IRm - {O}. Hence
o
Let us introduce the standard notation 71 71 D = {x E 1R Illxll S
5
1
71 -
= {x
E 1R
A fixed point for a map f: X
Illxll =
71
----7
1}
(the n-ball)
I}
(the (n - 1)-sphere)
X is a point x E X, such that f(x)
= x.
roof of Addendum 6.12.
Theorem 7.1 (Brouwer's fixed point theorem, 1912) Every continuous map f: D 71 ----7 D 71 has a fixed point.
o o act rows. It is not hard
Proof. Assume that f(x) #- x for all x E D71. For every x E D 71 we can define the point 9 (x) E 5 71 - 1 as the point of intersection between 5 71 - 1 and the half-line from f (x) through x. sn-1
Dn f(x)
lsider the diagram
x
:U1 n Uz)---O
g(x)
l-m (Ul nUz)---O 1
(2) of the proof of the
We have that g(x)
= x + tu, whereu = II~-~~~(II' and t
= -x·
u + V/ 1 - Ilxll z + (x . u) z.
Here x . u denotes the usual inner product. The expression for g(.7:) is obtained by solving the equation (x + tu) . (x + tu) = 1. There are two solutions since the line determined by f (x) and x intersects 5 71 - 1 in two points. We are interested in the solution with t 2: O. Since 9 is continuous with g\Sn-l = idsn-l, the theorem follows from the lemma below. 0
Lemma 7.2 There is no continuous map g: D 71
----7
5 71 -
1 ,
with glsn-l = idsn-l.
Proof. We may assume that n 2: 2. For the map r: R71 - {o} ----7 1R 71 - {O},r(x) = x/llxll, we get that idRn_{o} ~ r, because 1R 71 - {O} always contains the line segment between x and r(x) (see Example 6.5). If 9 is of the indicated type, then g(t . r( x)), 0 S t S 1 defines a homotopy from a constant map to r. This shows
7.
;y
Hn-1(R n - {O}) = 0,
For p E /Rn - A we have an open neighborhood Up
o Up = {x
: {x}.l, the orthogonal lector field on 5 n is a . every x E 5 n . ~ with v(x)
i- o for
-
f(x)
all
{o}
n
R I d(x,p) <
E
Rn - A of p given by
~
~ d (p, A)}.
These sets cover IR n - A and we can use Theorem A.I to find a subordinate partition of unity cPp. We define g by
g(x) =
>r field w on Rn
49
APPLICATIONS OF DE RHAM COHOMOLOGY
where for p E IR n
-
{
if x E A
L
cPp(x)j(a(p))
if x ERn - A
pERn-A
A, a(p)
E
A is chosen such that
d(p, a(p)) < 2d(p, A). Since the sum is locally finite on Rn - A, g is smooth on IR n - A. The only remaining problem is the continuity of g at a point Xo on the boundary of A. If x E Up then
on
c)
ial map iI, iI(x) = -x. Rn + 1 - {O}), which by rna 6.14 evaluates f{ to ld v with
-X2m, X2m-l).
1
+ d(x, p) < d(xo, x) + "2 d(p, A)
:::; d(xo, x)
1
+ "2 d(p, xo).
Hence d(xo,p) < 2d(xo, x) for x E Up. Since d(p, a(p)) < 2d(p, A) :::; 2d(xo,p) we get for x E Upthatd(xo,a(p)):::; d(xo,p)+d(p,a(p)) < 3d(xo,p) < 6d(xo, x). For x E Rn - A we have
L
g(x) - g(xo) =
o
cPp(x)(f(a(p)) - f(xo))
pERn-A
lblem": find the maximal one may have on 5 n . , independent if for every l1dent.)
24a +b , where 0 < b< 3. n :tor fields on 5 is equal
'f: A
4
d(xo, p) :::; d(xo, x)
---+
Rm continuous,
and (1)
Ilg(x) - g(xo)1I :::;
f(xo)ll,
p
where we sum over the points p with x E Up. For an arbitrary E > 0 choose 8 > 0 such that Ilf(y) - f(xo)11 < E for every yEA with d(xo,y) < 68. If x E Rn - A and d(x,xo) < 8 then, for p with x E Up, we have that d(xo, a(p)) < 68 and Ilf(a(p)) - f(xo)11 < E. Then (1) yields
Ilg(x) - g(xo)11 :::;
L cPp(x) .
E
= Eo
P
= f.
md for x E Rn we define
L cPp(x)llf(a(p)) -
Continuity of g at Xo follows.
o
Remark 7.5 The proof above still holds, with marginal changes, when Rn is replaced by a metric space and Rm by a locally convex topological vector space.
.
.- .~ -5 I
...
{.
..-l
...
let ¢: A - t B be a tself, such that
Lap h: IRn
-t
IRm . A
''is
'"
~.
7.
r
\
51
APPLICATIONS OF DE RHAM COHOMOLOGY
Proof. By induction on m Proposition 6.11 yields isomorphisms HP+m(R n+m - A) ~ HP(lR n - A) Hm(R n+m _ A) ~ HO(lR n - A)jR·l
(for p > 0)
for all m 2 1. Analogously for B. From Corollary 7.7 we know that R 2n - A and 1R 2n - B are homeomorphic. Topological invariance (Corollary 6.9) shows that they have isomorphic de Rham cohomologies. We thus have the isomorphisms HP(R n - A) ~ HP+n(R 2n - A) ~ HP+n(R 2n - B) ~ HP(R n - B)
for p > 0 and
Analogously we can le a homeomorphism
H°(l~n - A)jR·1 ~ H n (R 2n - A) ~ H n (R 2n -
B)
~ HO(R n - B)jR· 1. 0
For a closed set A ~ Rn the open complement U = Rn - A will always be a disjoint union of at most countably many connected components, which all are open. If there are infinitely many, then HO(U) will have infinite dimension. Otherwise the number of connected components is equal to dim HO(U).
A that
), ¢(x)) = (On, ¢(x)).
Corollary 7.9 If A and B are two homeomorphic closed subsets ofRn, then Rn-A and Rn - B have the same number of connected components.
)f vectors of the form
Proof. If A -I Rn and B -I Rn the assertion follows from Theorem 7.8 and the remarks above. If A = Rn and B-1 Rn then Rn+l - A has precisely 2 connected components (the open half-spaces), while Rn+l - B is connected. Hence this case cannot occur. 0
o
;losed subsets A and B ~ - t 1R 2n .
. from Lemma 7.6 with hes the two factors. 0 l
R2n - A and R 2n - B. Rn - B. A well-known
meomorphic to S2, but ous other examples are
n
~ed subsets of R . If A
Theorem 7.10 (Jordan-Brouwer separation theorem) If L: homeomorphic to sn-l then
~
Rn (n 2 2) is
(i) Rn - L: has precisely 2 connected components U1 and U2, where Ul is bounded and U2 is unbounded. (ii) L: is the set of boundary points for both Ul and U2.
We say U1 is the domain inside L: and U2 the domain outside L:.
Proof. Since L: is compact, L: is closed in R n . To show (i), it suffices, by Corollary 7.9, to verify it for sn-l ~ IR n . The two connected components of Rn - sn-l are
Illxll < I} and max Ilxll, the connected
b n = {x By choosing
T
=
ER
n
xE~
TW = {x E R
n
W
= {x
set
'11xll
> T}
n
ER
Illxll
> I}
.'~ <;- _.... <:; t:v:l .' ...... . .tj Q ;tj ...c ~§o.
....
.
,
....J
~
.
...eQ. 'ta ~<
.'.~ ...e ...... ,n
....
-.
.ss:;;ti.:' ~ .•;: . "'U'
7.
lGY
53
APPLICATIONS OF DE RHAM COHOMOLOGY
Corollary 7.13 (Invariance of domain) If V ~ Rn has the topology induced by Rn and is homeomorphic to an open subset of Rn then V is open in Rn .
md the other component
1 V of p in Rn. The set ~sponding proper closed :ed, so by Corollary 7.9 '2 E U2 , we can find a d ,(b) = P2. By (i) the osed set ,-I(E) ~ [a,b] belong to (a, b). Hence . for ,([a, Cl)) ~ Ul and [a, Cl) and t2 E (C2' b], :>ws that p is a boundary
Proof. Assume that m < n. From Corollary 7.13 applied to V, considered as a subset of Rn via the inclusion Rm ~ Rn, it follows that V is open in R n . This 0 contradicts that V is contained in a proper subspace.
th k ::; n, then IR n
Example 7.15 A knot in R3 is a subset E ~ R3 that is homeomorphic to 51. The corresponding knot-complement is the open set U = R3 - E. We show:
Corollary 7.14 (Dimension invariance) Let U ~ Rn and V open sets. If U and V are homeomorphic then n = m.
o
-
o
Proof. This follows immediately from Theorem 7.12.
A
~
Rm be non-empty
HP(U) ~ { R
if 0 ::; ~ ::; 2 otherwise. According to Theorem 7.8, it is sufficient to show this for the "trival knot" 51 ~ R2 C R3. First we calculate
o
.9 it is sufficient to prove ler. 0 n
open set and f: U ~ R n in RH , and f maps U
HP(R 2 - 51) = HP (b 2) EEl HP(R 2 - D 2).
(2)
Here b 2 is star-shaped, while R 2 - D 2 is diffeomorphic to R 2 - {O}. Using Theorem 3.15 and Example 5.4 it follows that (2) has dimension 2 for p = 0, dimension 1 for p = 1, and dimension 0 for p 2: 2. Apply Proposition 6.11.
: same will then hold for s proves continuity of the phere.
An analogous calculation of H*(Rn - E) can be done for a higher-dimensional knot E ~ Rn , where E is homeomorphic to Sk (1 ::; k ::; n - 2). See Exercise 7.2.
S. It is sufficient to show
Proposition 7.16 Let E ~ Rn (n 2: 2) be homeomorphic to 5 n U2 be the interior and exterior domains of E. Then
ementary theorems about 2: 2. Jet Ul and U2 be the two O. They are open; Ul is Rn - f(D) is connected. in U1 or U2. As f(D) is , - f(D) ~ U2. It follows
HP(UI) ~
{Ro
ifp = ~ otherwIse
o
{R0
Proof. The case p = 0 follows from Theorem 7.10. Set p > 0 there are isomorphisms
HP(UI) EEl HP(U2) ~ HP(R n
-
E) ~ HP (R n
and let
ifp =~, n otherwIse. ~V
-
Ul
1
= Rn - Dn.
5n-
1
and
For
)
~ HP(b n ) EEl HP(~V) ~ HP(~V).
The inclusion map i : W ~ Rn inverse defined by 'en though it is not known U U2, we must have that
and HP(U2) ~
l
-
{O} is a homotopy equivalence with homotopy
g(x)
= Ilxll....+ 1 x.
7.
:>GY
55
APPLICATIONS OF DE RHAM COHOMOLOGY
Example 7.18 One can also calculate the cohomology of "lR n with m holes", Le. the cohomology of
From Theorem 6.8.(iii) om Theorem 6.13 yields
m
n
V =R
(U Kj).
-
j=1
p rt {O, n - I}. On the are 0 or 1, so it suffices
and that the bounded set iagram of inclusion maps
The "holes" Kj in IR n are disjoint compact sets with boundary L:j, homeomorphic to sn-l. Hence the interiors Kj = K j - L:j are exactly the interior domains of L:j. One has (3)
HP(V) '"
Gm
if p = 0 if p = n - 1 otherwise.
We use induction on m. The case n = 1 follows from Proposition 7.16. Assume the assertion is true for m-l
Vl = IR n
-
(U
Kj
).
j=1
Let V2 = IR n - K m . Then VI U V2 = IR n and VI n V2 the exact Mayer-Vietoris sequence
R
= V. For p 2: 0 we have
HP(lR n ) !:.. HP(Vl) ttl HP(V2) ~ HP(V) l(U2) #
o
o.
gest that VI is contractible the case. In Topological n = 3, where Vl is not SI - t VI, which is not ible either. Corresponding states that there exists a
llu]
and
hib
2
will map
Vl
on in Schoenflies' theorem n ~d that E is flat in R , that ) ¢: sn-l X (-0,0) - t Rn
-t
O.
0 then HO(lR n ) ~ IR and 1* is injective. We get HO(VI) ~ R by induction and HO (V2 ) ~ IR from Proposition 7.16. The exact sequence yields HO(V) ~ R. If p > 0 then HP(Rn) = 0 and the exact sequence gives the isomorphism HP(Vl ) EEl HP(V2) ~ HP(V).
If p
=
Now (3) follows by induction.
~
'"
57
8.
SMOOTH MANIFOLDS
A topological space X has a countable topological base, when there exists a countable system of open sets V = {Ui liE N}, such that every open set can be written in the form UiEI Ui, where I ~ N. For instance IR n has a countable basis for the topology given by
V
= {bn(X;E) I x = (XI, ... ,X n), Xi
E Q;
E
E Q,
E>
O}
where b (x; E) is the open ball with center at x and radius E.
A topological space X is a Hausdorff space when, for arbitrary distinct x, y EX,
there exist open neighborhoods Ux and Uy with Ux n Uy = 0.
Definition 8.1 A topological manifold M is a topological Hausdorff space that
has a countable basis for its topology and that is locally homeomorphic to IR n .
The number n is called the dimension of M.
Remark 8.2 Every open ball bn(O, E) in Rn is diffeomorphic to Rn via the map
q, given by
q,(y) = {~an(7rIIYII/2E)
. y/llyll
if y if y
# 0, Ilyll < E =
O.
(Smoothness of q, and q,-l at 0 can be shown by means of the Taylor series at 0 for tan and Arctan.) Thus in Definition 8.1 it does not matter whether we require that M n is locally homeomorphic to IR n or to an open set in Rn.
Definition 8.3 (i) A chart (U, h) on an n-dimensional manifold is a homeomorphism h: U --+ U', where U is an open set in M and U' is an open set in Rn . (ii) A system A = {hi: Ui --+ UI liE J} of charts is called an atlas, provided {Ui liE J} covers M. (iii) An atlas is smooth when all of the maps
hji = hj
0
hi l : hi(Ui n Uj)
--+
hj(Ui n Uj)
are smooth. They are called chart transformations (or transition functions) for the given atlas. Note in Definition 8.3.(iii) that hi(Ui n Uj) is open in Rn .
Two smooth atlases AI, A2 are smoothly equivalent if Al U A2 is a smooth atlas.
This defines an equivalence relation on the set of atlases on M. A smooth structure
on M is an equivalence class A of smooth atlases on M.
'b
."..J.
'"
.c-,J
8.
lsisting of a topological
M instead of (M, A).
1+1
Illxll = I}
is an n
n+ 1) charts (U±i, h±i)
,n I
Xi
< O}
, ... ,Xi, ... ,Xn+I). The rse map is
i,.··
,Un)
tions are smooth.
59
SMOOTH MANIFOLDS
Since chart transformations (by Definition 8.3.(iii)) are smooth, we have that Definition 8.7 is independent of the choice of charts in the given atlases for M 1 and M 2 . A composition of two smooth maps is smooth. A diffeomorphism f: Ml -+ M2 between smooth manifolds is a smooth map that has a smooth inverse. In particular a diffeomorphism is a homeomorphism. As soon as we have chosen an atlas A on a manifold M, we know which functions on M are smooth. In particular we know when a homeomorphism f: V -+ V' between an open set V c M and an open set V' ~ IRn is a diffeomorphism. We can therefore define a new maximal atlas, A max , associated with the given smooth structure:
A max = {f: V
-+
V' I V ~ M n open, V' ~
IR
n
open, f diffeomorphism}.
The inverse diffeomorphisms f-1: V' -+ V will be called local parametrizations. From Remark 8.2 it follows that every point in a smooth manifold M n has an open neighborhood V ~ M n that is diffeomorphic to IR n .
e define an equivalence From now on chart will mean a chart in the maximal atlas.
Alternatively one can the canonical projection
)n.
Definition 8.8 A subset N c M n of a smooth manifold is said to be a smooth submanifold (of dimension k), if the following condition is satisfied: for every x E N there exists a chart h: U -+ U ' on M such that x E U
(1)
where
pen.
U+i)'
We define Ui =
with U+i n U-i = 0. ;entative in U+i and one lOrphism. We define -i
and a continuous map M1 if there exist charts X E Ul and f(x) E U2,
(2
r2 ' at all points of !vI1 then
IRk C IRn
and
h(U n N) = U' n IRk,
is the standard subspace.
It is easy to see that a smooth submanifold N of a smooth manifold M is a smooth manifold again. A smooth atlas on N is given by all h: Un N -+ U'n IRk, where (U, h) are charts on M satisfying (l).
Example 8.9 The n-sphere sn is a smooth submanifold of IR n+1. In fact the charts (U±i, h±i) from Example 8.5 can easily be extended to diffeomorphisms satisfying (l). Definition 8.10 An embedding is a smooth map f: N -+ M such that f(N) C M is a smooth submanifold and f: N -+ f (N) is a diffeomorphism. Theorem 8.11 Let M n be a smooth manifold of dimension n. There exists an embedding of M n into a Euclidean space IRn+k. This result will be proved below for a compact M, but let us first note that N n = f(M n ) satisfies the following condition:
.~·.1
.,.j
.r,J.
~
""
...";t
8.
~ Rn + k , an open set
'"
We define a smooth map f: M
SMOOTII MANIFOLDS
---+
61
Rnd +d by setting
f(q) = (h(q),··., fd(q), 1>l(q), ... , 1>d(q))·
:0 V) and such that
. surface when n = 2). )rphic to an embedded ition, then the implicit nse of Definition 8.8. in Theorem 8.11 can her hand k cannot be
Assuming f(ql) = f(Q2), we can by (i) choose j such that ql E Uj. Then 1>j(q2) = 1>j(Ql) =I- 0, Q2 E Uj, and by (ii), ql = Q2. Hence f is injective. Since M is compact, f is a homeomorphism from M to f(M). Let 7fl:
U{
---+
IR n(d-l)+d ,.
Define a diffeomorphism hI from dijJeomorphically onto
rna A. 7 we can find a ,) ~ V', such that 'l/J is f h(p).The smooth map
) coincides with hand (;0 E coo(Rn,R) with dIet
o choose 1>p and f p as in nite number of the sets I functions i ~ d)
IR nd+d
7f2:
IRn (d-l)+d
---+
be the projections on the first n coordinates and the last n( d - 1) + d coordinates, respectively. By (ii) 7fl 0 f = h is a diffeomorphism from Ul to U{. In particular 7fl maps f(Ud bijectively onto U{. Hence f(U l ) is the graph of the smooth map gl:
'Jld. For p E M there
Rnd+d ---+ Rn ,
7f
= 7f2 0 f
gl
l 1 (UU
hl(X,y) = (X,y-gl(X)),
0
(hIUJ-
l .
to itself by the formula
x E U{, Y E Rn(d-l)+n.
We see that hI maps f(U l ) bijectively onto U{ x {O}. Since f(U l ) is open in f(M), f(Ud = f(M) n WI for an open set WI ~ IR nd +d, which can be chosen to be contained in 7f 1l (UU. The restriction hll wl is a diffeomorphism from WI onto an open set W{, and it maps f(M) n WI bijectively onto W{ n Rn , as required by Definition 8.8. The remaining f(Uj ) are treated analogously. Hence f(M) is a smooth submanifold of Rnd +d. Note also that flul: Ul ---+ f(Ul ) is a diffeomorphism, namely the composite of fllul: Ul ---+ U{ and the inverse to the diffeomorphism f(Ul) ---+ U{ induced by 7fl. The remaining Uj are treated 0 analogously. Hence f: M ---+ f(M) is a diffeomorphism. Remark 8.13 The general case of Theorem 8.11 is shown in standard text books on differential topology. To get k = n + 1 (Whitney's embedding theorem) one uses Theorem 11.6 below. In the proofs above one can change "smooth manifold" to "topological manifold", "smooth map" to "continuous map" and "diffeomorphism" to "homeomorphism". This will lead to the theorem below, where the concept (locally flat) topological submanifold is defined in analogy to Definition 8.8, but with a homeomorphism instead of the diffeomorphism h. Theorem 8.14 Every compact topological n-dimensional manifold is homeomor 0 phic to a (locally flat) topological submanifold of a Euclidean space Rn+k.
On a topological manifold M n we have the IR-algebra CO(M, IR) of continuous functions M ---+ R. A smooth structure A on Mn gives a subalgebra t
Uj ~ Rn .
COO((M, A), R) ~ CO(M, R)
.,.
8.
ructure A on M (and I the notation, and the d by COO(M, R). This ructure on M. This is the identity maps idM
~n
smooth manifolds N
nooth maps is smooth. ,2 applied to p = g(q) :>od V of pin M, such ~n subset of Rn. For the
=g*(Ji)
E
COO(N, IR),
:>n M, 9 is seen to be
o ~ttempts
to classify n omorphism and home opological manifold is 2 there is a complete -e two infinite families
-------~...------------~--~~"'\
~'"
<:=~>. "
..............
\
\
)
_ ~ /
[Hirsch], [Massey]. care conjecture, which omotopy equivalent to ogical 3-manifold M 3 looth 3-manifolds also
''ti
SMOOTH MANIFOLDS
63
are diffeomorphic. In the mid 1950s J. Milnor discovered smooth 7-manifolds that are homeomorphic to S7, but not diffeomorphic to S7. In collaboration with M. Kervaire he classified such exotic n-spheres. For example they showed that there are exactly 28 oriented diffeomorphism classes of exotic 7-spheres. In 1960 Kervaire described a topological 10-manifold that has no smooth structure. During the 1960s the so-called "surgery" technique was developed, which in principle classifies all manifolds of a specified homotopy type, but only for n ~ 5. In the early 1980s M. Freedman completely classified the simply-connected com pact topological 4-manifolds; see [Freedman-Quinn]. At the same time S. Don aldson proved some very surprising results about smooth compact 4-manifolds, which showed that there is a tremendous difference between smooth and topolog ical 4-manifolds. Donaldson used methods originating in mathematical physics (Yang-Mills theory). This has led to a wealth of new results on smooth 4 manifolds; see [Donaldson-Kronheimer]. One of the most bizarre conclusions of the work of Donaldson and Freedman is that there exists a smooth structure on R4 such that the resulting smooth 4-manifold "1R 4 " is not diffeomorphic to the usual R4 . It was proved earlier by S. Smale that every smooth structure on IR n for n -=I 4 is diffeomorphic to the standard Rn .
'-:>
'"
65
9. DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
In this chapter we define the de Rham complex D* (M) of a smooth manifold M m and generalize the material of earlier chapters to the manifold case. For a given point p E Mm we shall construct an m-dimensional real vector space TpM called the tangent space at p. Moreover we want a smooth map j: M ---t N to induce a linear map Dpj: TpM ---t Tf(p)M known as the tangent map of j at p.
Remarks 9.1 (i) In the case p E U <;;;; Rm, where U is open, one usually identifies the tangent space to U at p with Rm. Better suited for generalization is the
following description: Consider the set of smooth parametrized curves ,: I ---t U with ,(0) = p, defined on open intervals around O. An equivalence relation on this set is given by the condition ,~ (0) = ,&(0). There is a 1-1 correspondence between equivalence classes and Rm , which to the class [,] of, associates the velocity vector ,'(0) E IR m . (ii) Consider a further open set V <;;;; Rn and a smooth map F: U ---t V. The Jacobi matrix of F evaluated at p E U defines a linear map DpF: Rm ---t Rn . For,: I ---t U, ,(0) = p as in (i) the chain rule implies that DpF(,'(O)) = (F 0 ,)'(0). Interpreting tangent spaces as given by equivalence classes of curves we have DpF(bD = [F
(1)
In particular the equivalence class of F
0 ,].
0 ,
depends only on [,].
Let (U, h) be a smooth chart around p E Mm. On the set of smooth curves Ct: I ---t M with Ct(O) = P defined on open intervals around 0 we have an equivalence relation
Ctl
(2)
tv
Ct2 {:} (hoCtI)'(O) = (hOCt2)'(0).
This equivalence relation is independent of the choice of (U, h). In fact, if (fJ, h) is another smooth chart around p, one finds that
(h
0
CtI)'(O)
= (h 0
Ct2)'(0) {:}
(h 0
CtI)'(O)
= (h 0
Ct2)'(0)
by applying the last statement of Remark 9.1.(ii) to the transition diffeomorphism F = h 0 h- l and its inverse.
o
"5
"
.,...;
~
"
)LDS
9.
luivalence classes with
r space defined by the
h p E U, then
,(0) ,
DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
TpM is well-defined. m, where F = h 0 7;,-1,
1
67
Suppose MID ~ !R I is a smooth submanifold with inclusion map i: M ~ !RI. Definition 8.8 implies that Dpi: TpM m ~ TpR I ~ RI is injective. In this case we usually identify TpM with the image in Rl, consisting of all vectors a'(O) where a: I ~ M ~ Rl is a smooth parametrized curve with a(O) = p. For a composite rpo f of smooth maps f: M ~ N, rp: N ~ P and a point p E M we have the chain rule, immediately from Lemma 9.3(i),
Dp(rp 0 J) = Df(p)rp 0 Dprp.
(3)
ce class of a.
\
Remark 9.4 Given a smooth chart (U, h) around the point p E M m we obtain a basis for TpM
(&:1\"'" (&:m)p' where C9~i)P is the image under D h (p)h- 1: Rm ~ TpM of the i-th standard basis vector ei = (0, ... 1, ... ,0) E RID. A tangent vector X p E TpM can be written uniquely as
(4)
;:::M.
Xp =
fai(&~Jp i=l
in tenus of representing
where a = (0.1,"" am) E RID. If X p = [a]' where a: I ~ U with a(O) = pis a representing smooth curve, we have a= (hoa)'(O).
:hart around f (p) in N,
n:
Given f E COO(M, R) we have the tangent map
Dpf: TpM
(5)
~
Tf(p)R
~
R.
The directional derivative Xpf E R is defined to be the image in R of X p under (5), i.e. Xpf = (J 0 a)'(O). In terms of f 0 h- 1 we have by the chain rule efined on the open set ram is linear and given re linear isomorphisms, utative. The formula in
o be identified with Dph :;'rom now on we write larly D h (p)h- 1 : Rm ~
ID &fh- 1
d
Xpf
=dt (fh- 1
0
ha(t))/t=o
=:L
ax' (h(p))ai.
i=l
t
In particular
(6)
( ~) &Xj
1 f = &fh- (h(p)). p
8xj
Under the assumptions of Lemma 9.3.(ii) there is a similar basis (&/&yJ f(p) , 1 ::; j ::; n, for Tf(p)N and the matrix of Dpf with respect to our bases for TpM
9.
)L!lS
Specializing this to the
69
DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
we can write 9 = gi 0 gil 0 9 = gi 0 h, where h = gil 0 g: g-l(gi(Wd) is a smooth map between open sets in Euclidean space. Thus
g*(W) = (gi
0
h)*(w)
=
-+
Wi
h*(gi(w))
in a neighborhood of z, and the right-hand side is a smooth k-form by assumption.
o
with transition function The exterior differential
d: nk(M)
tes.
gent vector X p E TpM. with certain coefficient hood of p E U, X is ndent of the choice of IE M, X is a a smooth
-fonns on TpM, where as function of p. Let a smooth chart, where
-+
n k+ 1(M)
can be defined via local parametrizations g: W is a smooth k-form on M then (8)
dpw = Alt k+1((D xg)-1)
0
-+
M as follows. If w = {w p hEM
dx(g*w),
p =
g(x).
It is not immediately obvious that dpw is independent of the choice of local parametrization, but this is indeed the case: Given a local parametrization g, then any other locally has the form 9 0 ¢, with ¢: U -+ W a diffeomorphism. Let 6, ... , ~k+! E TpM. We choose VI,· .. , Vk+l E Rn so that Dx(g 0 ¢)(Vi) = ~i. We must show that
dyg*(W)(WI"'" wk+d = dx(g
0
¢)*(W)(VI,"" 'Uk+!)
where ¢(x) = y and Dx¢(Vi) = Wi. This follows from the equations
(g 0 ¢)*(w) = ¢*(g*(w)) d¢*(r) = ¢*d(r), where r = g*(w); see Theorem 3.12. It is obvious that dod = O. Hence we have defined a chain complex
).
. .. -+
e value at x is for k =
0).
etrizations with N
E W. We show that gi(Wi). Close to z
-+ . " .
¢*(r)p = Altk(D p¢)(rq,(p)),
r
E
nk(N);
¢*(r)p = rq,(p)
for k = O.
One defines a bilinear product w 1\ r by (w 1\ r)p = wp 1\ rp, (10)
1\:
nk(M) x nl(M)
-+
nk+I(M).
One shows by choosing local parametrizations that ¢*w and w 1\ r are smooth. It is equally easy to see that (11)
E
.:!:. nk(M) .:!:. nk+I(M)
We have nk(M) = 0 if k > dimM, since Altk(TpM) = 0 when k > dim TpM. A smooth map ¢: M -+ N induces a chain map ¢*: n*(N) -+ n*(M) , (9)
lfonns on TpM is said I parametrization. The particular, nO(M) =
n k- 1(M)
d(w 1\ r) = dw 1\ r + (-1) kw t, dr wl\r= ( -1) kl rl\w
for w E nk(M), r E nl(M).
o
·13
....-J
j
.~
,.,.
~
'"
.";f
9.
OLDS
manifold M, denoted
f)
~
Hp+q (M) which : 0 for p > n = dim M
DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
71
Lemma 9.10 Let V = (Vi)iEI be an open cover oj the smooth submanifold Mn. Suppose that all Vi have orientations and that the restrictions oj the orientations Jrom Vi and Vj to Vi n Vi coincide Jor all i =J j. Then M has a uniquely determined orientation with the given restriction to Vi Jor all i E I. The proof is a typical application of a smooth partition of unity in the following form:
induces linear maps
Theorem 9.11 Let V = (Vi)iEI be an open cover oj the smooth manifold M n ~ IRI. Then there exist smooth Junctions rPi: M --+ [0,1] (i E 1) that satisfy
lnctor from the category ~raded anti-commutative
(i) SUPPM(rPi) ~ Vi Jorall i E I. (ii) Every p E M has an open neighborhood where only finitely many oj the Junctions rPi (i E 1) do not vanish. (iii) For every p E M we have LiEI rPi(P) = 1.
)rientable, if there exists Such an W is called an
Proof. Since M has the topology induced by IRI , we can choose an open set Vi ~ IR I with Vi n M = Vi for each i E I. By applying Theorem A.I to U = UiEI Ui we get smooth functions 7/Ji: U --+ [0, 1] with
if T = j . w, for some orientation of M is an
dXl 1\ ... 1\ dx n , which
, ... , bn of TpM is said depending on whether
The sign depends only ientation forms on M n , I(M) with j(p) =J 0 for tation at p, if j(p) > O. d bases of TpM. If M
e:
i there are precisely 2
o
an orientation of V is f. Conversely we have:
(i) SuPPu(1/Jd ~ Ui. (ii) Local finiteness. (iii) For every x E V, L iEI 7/Ji(X) = 1.
Let rPi: M --+ [0,1] be the restriction of 'lj.'i; conditions (i), (ii) and (iii) of the theorem follow immediately. o
Proof of Lemma 9.10. Let the orientation of Vi be given by the orientation form Wi E on (Vi), and choose smooth functions rP( M --+ [0, 1] as in Theorem 9.1 1. We can define W E on (M) by
W=
L rPi Wi, iEI
where rPiWi is extended to an n-form on all of M by letting it vanish on M - Su pp AJ ( rPi)' This is an orientation form, because if p E Vi ~ M and b1 , ... ,bn is a basis of TpM, with wi,p(b 1 , . .. ,bn ) > 0, then wi',p(b 1 , ... , bn ) > 0 for every other if with p E Vi, and in the formula
wp(b 1 , . .. , bn ) =
L rPi(p) wi,p(h, ... , bn ), t
all terms are positive (or zero). Thus W is an orientation form on M, and bl' ... , bn are positively oriented with respect to w. The orientation of M determined by W has the desired property.
..
';s
~'---~-
9.
LOS
mtation of the required In). Since both T p and ) and T determine the
DIFFERENTIAIJ FORMS ON SMOOTH MANIFOLDS
73
where el, .. " en is the standard basis of IR n . These functions are called the coefficients of the first fundamental form. For 1: E W the n x n matrix (gij (x)) is symmetric and positive definite.
o
etween manifolds that )* (wz) is an orientation orientation-reversing), I (resp. -WI).
between open subsets of Rn . It follows from only if det( Dx
0 and only if all Jacobi
we can find a chart amorphism when U is We call h an oriented I oriented charts of M as of M consisting of lctions will be positive.
t
A smooth manifold equipped with a Riemannian structure is called a Riemannian manifold. A smooth submanifold M n <,;;; IR I has a Riemannian structure defined by letting ( , ) p be the restriction to the subspace TpM <;;; IR l of the usual inner product on RI .
Proposition 9.16 If Mil is an oriented Riemannian manifold then M n has a uniquely determined orientation form VOIM with VOIM(bl, ... ,bn ) = 1
for every positively oriented orthonormal basis of a tangent space TpM. We call VOIM the volume form on M. Proof. Let the orientation be given by the orientation form W E 0,n(Mn). Consider two positively oriented orthonormal bases b1 , ... , bn and b~, ... , b~ in the same tangent space TpM. There exists an orthogonal n x n matrix C = (Cij) such that n
c" b· b'i -- "", ~ ~J J'
j=1 and w p E AltnTpM satisfies wp(b~, ... ,b~) = (detC)wp(bl, ... ,bn ).
(12)
Mil, then Mil has harts.
is on
-preserving diffeomor :fined by the restriction rna 9.10. 0 metric) on a smooth r, for all p E M, that ltion f: W ----> M and
I
Positivity ensures that det C > 0; but then det C = 1. Hence there exists a function p: M ----> (0, (0) such that p(p) = wp(b 1 , ... , bn ) for every positively oriented orthonormal basis b1 , ... , bn of TpM. We must show that p is smooth; then volu = p-lw will be the volume form. Consider an orientation-preserving local parametrization f: l¥ ----> Mn and set
Xj(q) =
(a~j\ =
Dqf(ej) E Tf(q)M
for 1"5: j "5: nand q E W
These form a positively oriented basis of Tf(q)M. An application of the Gram-Schmidt orthonormalization process gives an upper triangular matrix A(q) = (Uij(q)) of smooth functions on W with Uii(q) > 0, such that n
(13)
bi(q) =
L Uij(q)Xj(q),
i = 1, ... , n
j=1 is a positively oriented orthonormal basis of Tf(q)M. Then Jr the functions j "5: n,
(14)
po f(q) = Wf(q)(b1 (q), ... , bn(q)) = (detA(q)) Wf(q) (Xl (q), ... , Xn(q)) * = (detA(q))(j w)q(el,.'" en)'
This shows that p is smooth.
0
,
'"'
<
"to
'L
·.1'i ,~.
"'-~.'-
t~
.>,.f
9.
OLDS
f
,~~~~",~?(:~}?~~
DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
75
Now we have
in local coordinates:
0
Dxr( v) = { Ilxll
dXn,
-1
if v E Rx if v E (Rxl
so that D x r(1J) = Ilxll-lw, where W is the orthogonal projection of v on (Rx).1. Letting Wi be the orthogonal projection of Vi on (Rx).1 we have
ametrization and
n
so that p(p) = 1 for all
Wx(Vl,"" vn-d = IIxll- det(x, WI,···, 'Wn-l) = Ilxll-ndet(x, VI,···, 1Jn-l) n = Ilxll- wo x ( 1J l,' .. , vn-d· Hence the closed form
la for
W
is given by n
dXn.
h (q)
yields (with
(19)
W
6
1 "'" = Ilxll n (-1) i-I Xi dXl/\ ... /\ dXi /\ ... /\ dXn'
Example 9.19 For the antipodal map
(q) akl(q)·
A: sn-l
~
Sn-l;
Ax = -x
we have :re G(q) = (gjl(q)). In [i aii(q) > 0, we obtain
o by
A*(volsn-l) = (-1tvolsn-l , and A is orientation-preserving if and only if n is even. In this case we get an orientation form T on RlP n - l such that 7f* (T) = volsn-l, where 7f is the canonical map 7f: sn-l ~ Rpn-l. For x E sn-l,
TxS n - l D~A TAxS n - l
. Altn-l(R n ), we have
is a linear isometry. Hence there exists a Riemannian structure on Rpn-l characterized by the requirement that the isomorphism
T x sn-l Dx'!r T'!rex) Rpn-l ~
./\ dXn.
then x, WI, ... , Wn-l Hence WOISn-l = i*(wo) , sn-l given by Wo, the If and only if the basis
-1
is an isometry for every x E sn-l. If n is even and RlP n - l is oriented as before, one gets 7f*(VOlRpn-l) = VOlsn-l. Conversely suppose that Rpn-l is orientable, n 2: 2. Choose an orientation and let VOlRpn-l be the resulting volume form. Since D x 7f is an isometry, 7f*(VOlRpn-l) must coincide with ±volsn-l in all points, and by continuity the sign is locally constant. Since sn-l is connected the sign is constant on all of sn-l. We thus have that
. Then (17) implies that vith Wlsn-l = vols n- 1 by the map r(x) = x/llx\\.
vn-I))
, Dxr(vn-d)·
7f*(VOIRpn-l) = 8volsn-l, where 6
= ±l. We can apply A*
and use the equation 7f
0
A
(-1t8volsn-l = 8A*(volsn-l) = A*7f*(vol Rpn-l) = (7f 0 A)*(volRpn-l) = 7f*(VOlRpn-l)
= 7f to get = 8volsn-1.
This requires that n is even and implies that Rpn-l is orientable if and only if n is even.
...
~
.
~
'b
~;::-~,~: ~
"\
9. DIFFERENTIAL FORMS ON SMOOTH
FOLDS
the Cartesian product pair of charts h: V - t V' x k: U x V - t V' X V' th atlas on M x N. For
'1
77
MANIFOI~DS
Proposition 9.22 Let M n ~ IR n + 1 be a smooth submanifold of codimension 1. (i) There is a 1-1 correspondence between smooth normal vector fields Y on
M and n-forms in nn(M). It associates to Y the n-form W =
Wy
Wp(W1 , ... , n n ) = det(Y(p), WI, ... , W n ) T
for p E M, Wi E TpM. (ii) This induces a I-I correspondence between Gauss maps Y: M
.h) and (V, k). The tran : k) satisfy the condition ~ation of M x N. If the M) and (J E nn(N), the w)l\pr'N((J), where prM
nifold M n ~ Rn + k of mal vector space TpM 1. n open set W ~ M is a very pEW. In the case lve length 1. Such a map
ts an open neighborhood ; j :; k) on W such that Ir every pEW. ist smooth tangent vector of TpM, cf. Remark 9.4. I x (n + k) determinant
J
E
-t
sn and
orientations of M . Proof. If p E M then Y(p) = 0 if and only if wp = O. Since Wy depends linearly on Y, the map Y - t Wy must be injective. If Y is a Gauss map, then wy is an orientation form and it can be seen that wy is exactly the volume form associated to the orientation determined by wyand the Riemannian structure on lv1 induced from Rn +1 . If M has a Gauss map Y then (i) follows, since every element in nn(M) has the form j . Wy = wfY for some j E COO(M, R). Now M can be covered by open sets, for which there exist Gauss maps. For each of these (i) holds, but then the global case of (i) automatically follows. An orientation of M determines a volume form VOIM' and from (i) one gets a Y 0 with wy = VOIM. This Y is a Gauss map.
Theorem 9.23 (Tubular neighborhoods) Let Mn ~ Rn+k be a smooth submani fold. There exists an open set V ~ lF~n+k with M ~ V and an extension of idM to a smooth map r: V - t M, such that
Ilx -
(i) For x E V and y E M, if Y = r(x).
r(x)11 :;
Ilx - YII,
with equality if and only
(ii) For every p E M the fiber r-1(p) is an open ball in the affine subspace p + TpM 1. with center at p and radius p(p), where p is a positive smooth function on M. If A1 is compact then p can be taken to be constant. (iii) If E: M - t R is smooth and 0 < E(p) < p(p) for all P E M then
SE
n neighborhood W of Po e basis
given by
= {x E V Illx - r(x)11 = E(T(X))}
is a smooth submanifold of codimension 1 in Rn+k.
We call V (= Vp ) the open tubular neighborhood of M of radius p
W) Proof. We first give a local construction around a point Po E M. Choose normal vector fields Yl, ... , Yk as in Lemma 9.21, defined on an open neighborhood W of Po in M for which we have a diffeomorphism j: Rn - t W with j(O) = Po. Let us define cI>: Rn+k - t Rn+k by
)) , Gram-Schmidt orthonor that Y1 , ... , Yk have the
o
k
cI>(x, t) = j(x)
+ I>j Yj(j(x)) j=l
(x
ERn,
t E Rk ).
...
'a
-',
[fOLDS
(po).
f TpoMJ... By the inverse There exists a (possibly) > 0, such that
9.
79
DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
Now all of M can be covered with open sets of the type Vo with the associated smooth maps TO which satisfy Condition (20). If (VI, rl) is a different pair then TO and 7'1 will coincide on Va n VI. On the union V' of all such open sets (of the type above) we can now define a smooth map r: V' -+ M, which extends idM, such that part (i) of the theorem is satisfied. We have rWo = ro for every local (VO, ro). If in the above we always choose EO ~ 1, then the fiber r-I(p) over apE M will be an open ball in p + TpM J.. with radius p(p) for which 0 < p(p) ~ 1. Thus we have satisfied (i) and (ii), except that p might be discontinuous.
The distance function from M,
dM(X) = inf
yEM
.et Va <;;; ~n+k. The map which extends idwo' so th center at p and radius own we can arrange that
11
~
Ilx-yll
e as follows. of Po in Rn + k such that .VO <;;; TV where M n Va f(z of Wo so that (in the where intKi denotes the
Ilx - yll.
is continuous on all of Rn+k. For x E V', (i) shows that
dM(X) =
Ilx -
r(x)1I
(x
E
V').
If p E M and x E p + TpM J.. has distance p(p) from p, we can conclude that dM(X) = P(p). In this case (i) excludes that x E V', so x lies on the boundary of V'. Hence the distance function
d: M
R;
-+
d(p) = inf
z.g'V'
lip - zll
satisfies 0 < d(p) ~ p(p) and is continuous. By Lemma A.9 the function ~d 0 r: V' -+ R can be approximated by a smooth function '!/J: V' -+ R such that
11'!/J(x)-~dor(x)11 ~ idor(x) for all x E V'. In particular
i d(x) ~ '!/J(x) ~ ~ d(x)
Kz) an E E (0, EO] such that Ie open set
Hence the restriction p = '!/JIM: M -+ R is a positive smooth function with p(p) < p(p) for all p E M. When M is compact, the same can be achieved for the constant function which takes the value p = ~ min d (p). If we define pEM
(x)11 < E},
I> Eo lefined on M, attains a Consider such a Yo E Kz
< E~ EO. se 9.1), but then x E Va Ie satisfied by replacing
when x E M.
V
= {x
E
V'
Illx -
r(x)1I < p(r(x))}
both (i) and (ii) hold for the restriction of r to V. It remains to prove (iii). It is sufficient to show that s~ n Va is empty or a smooth submanifold of Va of codimension 1. The image under the diffeomorphism
S
= {(p, t)
E
Wo
X
Rk II/til
= E(p) < EO}.
The projection of S on W o is the open set
U = {p
E Wo
I E(p) < EO}
<;;; M.
The diffeomorphism ¢: U x Rk -+ U x Rk given by ¢(p, t) U x Sk-I to S. This yields (iii).
= (p, E(p)t)
maps 0
~.
"
~.~
, ~"!"IIO..
~
..•.. ':..•.
1·_~ .~ ..
~
LDS
9.
ition in the case where lber E, 0 < E < p, we Mby
:nce of tubular neigh with i: M -+ V the iUch that r ° i = idM.
Hd(M)
-+
Hd(V) is
i Mn all cohomology
of Rn+k by Theorem s compact we can find nion U = UIU .. . UUr \1 -+ U and a smooth ~ shows that
5.5.
[]
81
Proof. Choose tubular neighborhoods (Vv , iv, r v ) of M v , v = 1,2. Lemma 6.3 d implies that i2 ° fo ° rl ~ i2 ° h ° rl· Hence H (i2 ° fo ° TI) = H d(i2 ° h ° rl), so that
Hd(rI)
- with centers at p and Joundary points of N E ~al-valued function on x E SE with r(x) = p.
DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
° Hd(Jo) ° H d(i2) = Hd(rl) ° Hd(JI) ° H d(i2).
Since Hd(rl) is injective and H d(i2) is surjective, we conclude that Hd(Jo) = Hd(h). If 1;: MI -+ M2 is continuous, we can use Lemma 6.6.(i) to find a smooth map g: VI -+ V2 with 9 ~ i2 0 1;orl' For the smooth map f = r2ogoil: MI -+ M2, Lemma 6.3 shows that f ~ r2 ° (i 2 ° 1; ° rl) ° il = 1;. 0
Remark 9.27 As in the discussion preceeding Theorem 6.8, the de Rham coho mology can now be made functorial on the category of smooth submanifolds of Euclidean space and continuous maps. Theorem 6.8 and Corollary 6.9 are valid (with the same proofs) with open sets in Euclidean space replaced by smooth sub manifolds. By Theorem 8.11 the same can be done for differentiable manifolds in general. Corollary 9.28 If Mn ~ IR n +k is a smooth submanifold and (V, i, r) an open tubular neighborhood, then Hd(i): Hd(V) -+ Hd(M) is an isomorphism with Hd(r) as its inverse. Proof. We have r ° i = idM and i ° r ~ idv, as V contains the line segment between x and r(x) for all x E V. By Proposition 9.26.(i) we can conclude that Hd(r) and Hd(i) are inverses. 0 Example 9.29 For n 2: 1 we have
Hd(sn)
~ {~
if d = 0, n otherwise.
Let i: sn -+ Rn+l - {O} be the inclusion and define r: Rn+l - {O} -+ sn by r(x) = Tfxrr Then r ° i = idsn, i ° r ~ id Rn+l_{O} and Hd(i) is an isomorphism. The result follows from Theorem 6.13.
of Euclidean spaces. IS,
then
rI).
Remark 9.30 Let UI and U2 be open subsets of a smooth submanifold M n ~ RI . Using Theorem 9.11, the proof of Theorem 5.1 can be carried through without any significant changes. As in Chapter 5 this gives rise to the Mayer-Vietoris sequence
smooth map. -+
HP(UI U U2)
I"
-+
HP(U I ) EEl HP(U2)
J"
-+
HP(UI n U2)
0"
-+
HP+l(UI U U2)
-+
.,..J, r-,).
?
,
1'~
f~~
~
~'
~ ~
_
-;::
'&
.'
--'l,
_
~~ ...
~-~.
~
~
IIIFOLDS
83
ology of Rpn-l (n :2: 2).
10. (Tx S n -
1
INTEGRATION ON MANIFOLDS
)
Let M n be an oriented n-dimensional smooth manifold. We define an integral
1M: n~(Mn)
L)
1[") is equal to the set of p
I:op(sn-l)
--+
np(sn-l)
~s
~-1 )
R
on the vector space of differential n-forms with compact support. Next we shall consider integration on subsets of Mn, and Stokes' theorem will be proved. Finally we calculate H n (M n ) for an arbitrary orientable compact connected smooth manifold M n . In the special case where M n = IR n (with the standard orientation) we can write w E n~(lRn) uniquely in the form
)).
1
--+
w = j(X)dXl /\ ... /\ dx n ,
into a direct sum of two
where j E C=(R n , R) has compact support. We then define
r j(X) dX l/\ ... /\dx n = JRnr j(x)d/-l n , JRn
'-1).
lorphisms
where d/-l n is the usual Lebesgue measure on Rn . The same definition can be used when w E n~(V) for an open set V ~ Rn, since wand j are smoothly extendable to the whole of Rn by setting them equal to 0 on IR n - 8UPPV(W).
~ (sn-l))
(sn-l). Combining with
isms (See Example 9.29)
Lemma 10.1 Let ¢J: V --+ W be a diffeomorphism between open subsets V and W of IR n , and assume that the Jacobi determinant det (Dx¢J) is of constant sign 8 = ±1 for x E V. For w E n~(W) we have that
i
¢J*(w) = 8 .
lw
w.
Proof. If w is written in the form
{O}) w = j(X)dXl/\ .. , /\ dX n -1 )
... -x of Rn into itself. m by (_1)n. Using (24) with n even
with j E
C~(W,
R), it follows from Example 3.13.(ii) that
¢J*(w) = j(¢J(x)) det (Dx¢J) dXl /\ = 8j(¢J(x))ldet (Dx¢J) \ dXl /\
/\ dX n /\ dx n·
The assertion follows from the transformation theorem for integrals which states that o j(x)d/-l n = j(¢J(x)) Idet (Dx¢J) I d/-l n.
r
Jw
r
Jv
_1 ".;
<:)
"t
....,i.
.~...
.
-
~
.,
11 _~ ..
,~
10.
onal smooth manifold M n
INTEGRATION ON MANIFOLDS
85
Lemma 10.3 (i) (ii)
I M W changes sign when the orientation of M n If w E
n
n~ (M )
is reversed. has support contained in an open set W c M n , then
fM W= fw W'
ort contained in U, where
when W is given the orientation induced by M. (iii) If ¢: N n ---+ M n is an orientation-preserving diffeomorphism, then we have that
t, i.e. such that su PP M (w ) be chosen as above and la..!. the right-hand side is " h) is another positively
,W
= h(U n U)
mt.
Since
given by
~-l)*w) c;;; W
at v.
ined by (1) is independent
fM for
W
=
L
¢*(w)
E n~ (M).
Proof. By a partition of unity, we can restrict ourselves to the case where su PP M (w) is contained in a coordinate patch. All three properties are now easy consequences of Lemma 10.1 and (1). 0
Remark 10.4 In the above we could have considered integrals of continuous n-forms with compact support on M n . If the orientation of M is given by the orientation form IJ E nn(M), a continuous n-form can be written uniquely as jIJ, where j E CO(M, R). The support of jIJ is equal to the support of j. The integral of (1) extended to continuous n-forms gives rise to a linear operator
subordinate to an oriented fa:
, and where only finitely
W
C~(M, R)
R;
---+
fa(J) =
fM jIJ.
This linear operator is positive, i.e. fa (J) 2:: 0 for j 2:: O. By a partition of unity it is sufficient to show this when supp(J) C;;; U, where (U, h) is a positive oriented Coo -chart. Then we have that fa(J) =
r
j
0
h-1(x)¢(x)d/Ln,
Jh(U)
), applied to a Ua with erator on D.~ (M). If, in y oriented Coo -chart, the U, h). This yields
where ¢ is determined by (h-1)*(IJ) = ¢(X)dXl /\ ... /\ dx n . Since ¢ is positive
we get fa(J) 2:: O.
According to Riesz's representation theorem (see for instance chapter 2 of [Rudin])
fa determines a positive measure /La on M which satisfies
fM j(x)d/La = fM jIJ,
j E
C~(M, R).
d properties. Uniqueness
o
The entire Lebesgue integration machinery now becomes available, but we shall use only very little of it in the following.
l~'~
..
-~
'0
_N ....,; -.. .'-.",tj .~. ;tj ,;t) ,~; ....
.
"..J
r,.) .
~
~
"
~
.
...:;.. .j
jj
~ ,§-~
~ ....... ,n"
.~ -~,
.....
.-." ~.
~'\
.,""""'_.0...""£.'"
X:·--.J
10.
1>8
le form VOIM will determine ~asure on Rn. For a compact
subset N ~ M n is called a submanifold with boundary, round p, such that
87
INTEGRATION ON MANIFOLDS
Lemma 10.6 Let N ~ Mn be a domain with smooth boundary. Then oN is an (n - I)-dimensional smooth submanifold of M n . Suppose M n (n ;::: 2) is oriented. There is an induced orientation of oN with the following property: if pEoN and VI E TpM is an outward directed tangent vector then a basis V2, ... ,vnfor TpoN is positively oriented if and only if the basis VI, V2, ... ,Vn for TpM is positively oriented. Proof. Every smooth chart (U, h) in M that satisfies h(U be restricted to a chart (U n oN, hi) on oN: hi: U
n oN ~ h(U) n ({O} x Rn -
I
n N) = h(U) n R~
can
).
These charts have mutual smooth overlap according to the above. This yields a smooth atlas on oN. interior or an exterior point ,(U) is contained in an open , If P is a boundary point of (U,h) and (V,k) be smooth g transition diffeomorphism
Suppose M n is oriented. Then, possibly changing the sign of X2, we can choose a positively oriented chart (U, h) of the considered type around any pEoN. The resulting smooth charts (UnoN, hi) on oN have positively oriented transformation diffeomorphisms, and they determine an orientation of oN that satisfies the stated property. 0
Remarks 10.7
n V)
(i) We want to integrate n-forms W E n~(M) over domains N with smooth boundary. In view of Remark lOA we can set
fN
Rn- ,
W
=
fM INW
IN
nme. = (
r
) Since
where is the function with value I on N and zero outside N. Alternatively, one can prove an extension of Lemma 10.1 which uses the following version of the transformation theorem: Let ¢: V ~ W be a diffeomorphism of open sets in Rn that maps R~ n V to R~ n W, and let j be a smooth function on W with compact support. Then
JR':..nw
R~
into
R~,
oN is said to be outward th h(U n N) = h(U) n Fe .ate. This will then also be
j(x)df-Ln
=
r
JR'.:_nv
j¢(x)ldet Dx¢ldf-Ln.
(One could approximate both integrals by integrals over Wand V upon multiplying j by a sequence of smooth functions 'l/Ji with values in [0, I] and converging to IR':...) (ii) In the case n = 1, Lemma 10.6 holds in the following modified form. An orientation of oN consists of a choice of sign, + or -, for every point pEoN. Let VI E TpM be outward directed. Then p is assigned the sign + if VI is a positively oriented basis of TpM, otherwise the sign is - .
L-,J
~-..
_
T
s-
. . . . " :
~
;: --- '" ~
- #':'-~ ~~- ~
_
lS
10.
en f has compact support
-
~
....
~
~
-,.
r
~-
89
INTEGRATION ON MANIFOLDS
W
=
and
r JN
dw
=
r
Jh(U)naR~
(h- 1 )*(w) =
r (hJh(U)nR~
1)*(dw)
=
r
JaR~
r
JR~
K
dK.
Hence the proof reduces to the special case where M = Rn , N = w E O~-1 (R n ). This case is treated by direct calculation. We define +
~
Let K E n~-I(Rn) be the (n - I)-form that is (h- 1 )*(w) on h(U) and 0 on the rest of WI. By diffeomorphism invariance we then have that
JaN
itokes' theorem below.
"
1·•.• .·~ ~
?
'til
' " ~
...J
W.: and
n
Pl
W
L Ji(x) dXI/\ ... /\ d;;i /\ ... /\ dX n
=
i==l
and choose b > 0 such that SUPPRnJi ~ [-b, bt, 1 :S i :S n. Using Theorem 3.12, WlaR~ =
h(O, X2,· .. , x n ) dX2 /\ ... /\ dx n .
Hence
ain with smooth boundary ed orientation. For every
r
(3)
JaR~
w =
Jh
(0, X2, ... , x n )df1n-l.
By Theorem 3.7 we have
~
dw=L.J
(
-1
)i-l aJi d
ax.
i==l
XI/\ dx 2/\.·./\dxn·
t
Hence er to make the necessary e ! E n~ (M) with value cides with w on N, both we may assume that w
r
(4)
dw =
t
JRn. t=l
(_l)i-l
r
JRn
oJi df1n. aXi
For 2 :S i :S n we get
1
00
type of Definition 10.5 he formulas
-00
a!i
~(XI, . .. , Xi-I,
t, Xi+l,· .. , xn)dt
uX t
=!i(XI, ... ,Xi-l, b,Xi+I,""X n ) - Ji(XI, .. ·,Xi-l, -b,Xi+I,."'Xn )
=0, and then by Fubini's theorem
PPM(W) ~ U and (U, h) ore the chart (U, h) is
(5)
k
O!i ~ dlln = R~ UXi
° (2 < -
i -< n) .
•
...
:~J~~]~.f~i;~!,,,<~,-
10.
)8
INTEGRATION ON MANIFOLDS
91
Example 10.12 The volume of sn-l can be calculated by applying Stokes' theorem to D n with the standard orientation of IR n and the (n - 1)-fonn on Rn given by
-!I(-b,X2,""X n )
n
Wo
=
" '" ~
. 1 XidxI/\ ... /\ ---(_1)1dXi /\ ... /\ dx n .
i=l
) dJ-Ln-l.
Hows.
D
Since WOlsn-l
= volsn-l and dwo = ndxI /\ ... /\ dX n we have that Vol(sn-l)
ld and w E n~-I(M) then
=
r
}sn-l
Wo
=
r dwo = nVol(D
JDn
n ).
By induction on m and Fubini's theorem, it can be shown that Vol(D 2m +I)
Vol(D 2m ) = 1fm m!'
Ie way of showing that the )w that
2m
1 m + m!1f (2m + 1)!
=
2
Vol(s2m)
=
This yields a d-dimensional compact :: nd-I(M), then Corollary
1
Vol (s2rn-1 ) =
21f
rn
(m -1)!'
2m
2
1
+ m!1f
rn
(2m)!
J. ~
and 1.7. It can be shown that [w] = 0 if and only if
We conclude this chapter with a proof of the following:
Theorem 10.13 If M n is a connected oriented smooth manifold, then the sequence e closed (n - 1)-fonn on (8) i /\ ... /\
dx n .
;; the volume fonn volsn-l, Ie from Remark 10.10 that rem 6.13, [w] is a basis of
2)
n~-I(M) ~ n~(M) .& IR
->
0
is exact.
Corollary 10.14 For a connected compact smooth manifold Mn, integration over M induces an isomorphism
1M: Hn(M n) -=. R.
D
under this isomorphism is
In (8) it is obvious that the integral is non-zero and hence surjective. It follows from Corollary 10.9 that the image of d is contained in the kernel of the integral. We show the converse inclusion.
.""
10.
Let h E n
W
dXI . f\
= O. f\
We must find ... f\ dx n , and let
dx n .
C~(lRn)
93
INTEGRATION ON MANIFOLDS
be the function n-l
(12)
h
=f -
L
aJi aXj'
j=l
A function fn E C~(Rn) with afn/axn
=
j
Xn
(13)
fn(X1, ... ,Xn-I,X n ) =
h is given by h(XI,. ",xn-l,t)dt.
-00
It is obvious that fn is smooth, but we must show that it has compact support. To this end it is sufficient to show that the integral of (13) vanishes when the upper limit X n is replaced by 00. Now (10), (11) and (12) yield that n-l
O. There exist functions
h(X1, ... , Xn-l, t) = f(XI, .. . , Xn-l, t) -
L
ago ax} (Xl,"" Xn-1)P(t)
J j=l = f(X1, . .. , Xn-1, t) - g(X1,' .. 1 xn-dp(t).
Finally from (9) it follows that
f: f:
len a smooth function
l by setting =
;tion with
J f(x)d/Ln =
rle
n)dx n .
h(X11"" Xn-1, t)dt f(X1, . .. , Xn-1, t)dt - g(XI, . .. , xn-d
o.
= 1,
and define fj E
SjSn-l.
p(t)dt
o
Lemma 10.16 Let (Ua)aEA be an open cover of the connected manifold M, and let p, q E M. There exist indices ai, ... , D:k such that (i) P E Ua1
and
(ii) Uai n UQi + 1 tively). The function ral sign. Furthermore = J fd/Ln = O. Using ith
f:
i-
q E Uak 0 when 1
sis
k - 1.
Proof. For a fixed p we define V to be the set of q EM, for which there exists a finite sequence of indices ai, ... , Ci.k from A, such that (i) and (ii) are satisfied. It is obvious that V is both open and closed in M and that V contains p. Since M is connected, we must have V = M. 0 Lemma 10.17 Let U ~ M be an open set diffeomorphic to Rn and let W ~ U be non-empty and open. For every w E n~(M) with suppw ~ U, there exists a '" E n~-I(M) such that supp '" ~ U and supp(w - d",) ~ W. Proof. It suffices to prove the lemma when M = U, and by diffeomorphism invariance it is enough to consider the case where M = U = Rn.
,. 110,
".J
"
C"tJ
.,
"
10.
INTEGRATION ON MANIFOLDS
95
Lemma 10.15 implies that Theorem 10.13 holds for W, Le. there exists a TO E rl~-l (W) that satisfies
· Then
w.
(W - dK)IW
= dTo·
Let T E rl~-l(M) be the extension of TO which vanishes outside of SUPPW(TO). Then W - dK = dT, so T + K maps to wunder d. 0
o ~
M be non-empty and ) with supp(w - d/'i,) ~
~ M diffeomorphic to , diffeomorphic to Rn , 'e use Lemma ]0.17 to hat
:Sk-l).
diffeomorphic to Rn. )p(Wj - dFcj) ~ W For
~t
o
ith JM W = O. Choose 10.18 we can find a rollary 10.9,
L
(; =0.
'b
.,
"...;.
.c-.-> . ~
97
11.
DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS
Let f: N n --t M n be a smooth map between compact connected oriented mani folds of the same dimension n. We have the commutative diagram
Hn(M) HnU) Hn(N)
~l
(1)
R
deg(f)
~l
R
where the vertical isomorphisms are given by integration over M and N respec tively; cf. Corollary 10.14. The lower horizontal arrow is multiplication by the real number deg(J) that makes the diagram commutative. Thus for w E on(M),
j~j*(w) =
(2)
deg(J)
1M w.
This formulation can be generalized to the case where N is not connected:
Proposition 11.1 Let f: N n --t M n be a smooth map between compact n dimensional oriented manifolds with M connected. There exists a unique deg(J) E R such that (2) holds for all w E on(M). We call deg(J) the degree of f. Proof. We write N as a disjoint union of its connected components N1, ... , N k and denote the restriction of f to Nj by Ji. We have already defined deg(Ji); we set k
(3)
deg(J) =
L deg(Ji). j=1
Thus for w E on(M),
1 N
L 1. fJ(w) = Lj=1 deg(Jj) 1w = deg(J) 1w. j=1 k
j*(w) =
k
N]
M
Corollary 11.2 deg(J) depends only on the homotopy class of f: N
0
M
--t
M.
Proof. By (3) we can restrict ourselves to the case where N is connected. The assertion then follows from diagram (1), since Hn(J) depends only on the homotopy class of f. 0
o
~
~~'
ti
.,..J .-::-..
<
....ce
FIELDS
. '~." , .< ,
11. DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS
rnooth maps between n d P are connected. Then
Then
f (5)
99
is a Lebesgue null-set in Rm.
Note that x E U belongs to 5 if and only if every m X m submatrix of the Jacobi matrix of f, evaluated at x, has determinant zero. Therefore S is closed in U and we can write S as a union of at most countably many compact subsets K S; S. Theorem J 1.6 thus follows if f(K) is a Lebesgue null-set for every compact subset K of S. We shall only use and prove these theorems in the case m = n, where they follow from
J)) deg(g)
""
.
~
~CTOR
"'.
1.. .'....'J~.' .'
~
l
o
w.
nected compact orientable ~ an orientation of M and >rientation leaves deg(f)
Proposition 11.7 Let f: U --+ Rn be a C1-map defined on an open set U S; Rn, and let K S; U be a compact set such that det (Dxf) = 0 for all x E K. Then f (K) is a Lebesgue null-set in Rn. Proof. Choose a compact set L S; U which contains K in the interior, K S; Let C > 0 be a constant such that sup IIgrad~ Ji II ~
(4)
~EL
This follows from an the concept of regular :lue for the smooth map
~s
c
L.
(1 ~ j ~ n).
Here fj is the j-th coordinate function of f, and Let
II II
denotes the Euclidean norm.
n
T
=
II [ti' ti + aJ i=l
le complement of f (N
r:
N n ----t J\;j1n
n
)
the set of
n open subset W S; M m is reduces Theorem 11.5 the Lebesgue sense) are coordinate patches and gue null-sets is again a g result:
be a cube such that K S; T, and let E > O. Since the functions aJi/aXi are uniformly continuous on L, there exists a 8 > 0 such that (5)
Ilx - yll
~ 8
af ' (x) - ar (y) I ~ '* aXi aXi _J
l
_J
E,
(1 ~ i,j ~ nandx,y E L).
We subdivide T into a union of Nn closed small cubes T 1 with side length and choose N so that
(6)
diam(T1) =
a'j; :s 8,
T 1 n K =1=
0 '* Tl
~,
S; L.
For a small cube Tl with T z n K =1= 0 we pick x E Tz n K. If yETI the mean value theorem yields points ~j on the line segment between x and y for which
L n
(7)
fj(y) - Ji(x) =
i=l
map defined on an open Since
~j E
afj ax' (~j)(Yi - Xi)' t
T z S; L, the Cauchy-Schwarz inequality and (4) give
Ifj(Y) - Ji(x)1
:s Clly - xII
'Cl
"
11. DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS
CTORFIELDS
101
Proof. For each q E f-1(p), Dqf: TqN ---t TqM is an isomorphism. From the inverse function theorem we know that f is a local diffeomorphism around q. In particular q is an isolated point in f-1(p). Compactness of N implies that f-1(p) consists of finitely many points q1,.' . ,qk. We can choose mutually disjoint open neighborhoods Wi of qi in N, such that f maps Wi diffeomorphically onto an open neighborhood f(Wi ) of p in M. Let
me N
k
U=
k
(n f(Wd)
U Wi).
- f(N -
i=1
i=1
Since N - U7=1 Wi is. closed in N and therefore compact, f(N - U7=1 Wi) is also compact. Hence U is an open neighborhood of p in M. We then set Vi = Wi n f-I(U). 0
Xi).
:nce
'e may choose an affine
Consider a smooth map f: N n ---t Mn between compact n-dimensional oriented manifolds, with M connected. For a regular value p E M and q E f-1(p), define the local index (11) Ind(J; q) = { 1 if Dqf.: TqN ---t TpM preserves orientation -1 otherwIse. Theorem 11.9 In the situation above, and for every regular value p,
deg(J) =
L
Ind(J; q).
qEj-l (p)
than fanlf. Then (8) all points q E IR n whose I in H with radius ar;F sgue measure JIn on IR n
=f-
In particular deg(J) is an integer. Proof. Let qi, Vi, and U be as in Lemma 11.8. We may assume that U and hence Vi connected. The diffeomorphism flv;: Vi ---t U is positively or negatively oriented, depending on whether Ind(J; qi) is 1 or -1. Let W E nn(M) be an n-form with
c
SUPPM(W)
Nn'
~ U,
1M W = 1.
Then SUPPN(J*(w)) ~ f-I(U) = VI U ... U Vb and we can write .be T[ with T[ n K # 0 in such small cubes T[, e assertion. 0 smooth map f: Nn ---t many points q1, ... , qk. gi in Nn, and an open
k
J*(w) = LWi' i=l
where Wi E nn(N) and SUpp(Wi) ~ Vi. Here wilV; = (J1V;)*(wlU)' The formula is a consequence of the following calculation: deg(J)
= deg(f)
1 =1 W
M
k
= ?=Ind(J;qd k.
z=l.
k
J*(w)
=L
i=1
N
L
1
k
Wi = L
N
i=1
1
(J1V;)*(WIU)
V;
k
WIU = LInd(J;qd. z=l
0
<
"...J
l' j.~.'. ~.
c-,.J
~
OCTOR FIELDS
...
~
'
<
~
103
11. DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS
Proposition 11.13 s that deg(f) = 0 (in the
(i) Ik(KI,J d) = (_l)(d+l)(I+l)lk(J d,KI ) (ii) If J and K can be separated by a hyperplane He IR n + 1 then Ik(J, K)
=
O. D
between oriented smooth ?e a compact domain with int union of submanifolds
I
(iii) Let gt and h t be homotopies of the inclusions go: J -+ IR n +l and ho: K -+ IR n +l to smooth embeddings gl and hI, such that gt( J) n ht (K) = 0 for all t E [0,1]. Then lk( J, K) = Ik(gl (J), hi (I{)). (iv) Let
L Ik(J; Ki)
= O.
i=1
Proof. We look at the commutative diagram IJIJ,K.
sn
K x J IJIK,~
sn
J x K
lT
'*(dw)
lA
where T interchanges factors and A is the antipodal map Av follows from Corollary 11.3 upon using that
=0 D
ler linking numbers, and
pact oriented connected I ~ 1 satisfy d+l = n.
deg(T)
= (_l)di,
deg(A)
=
-v. Then (i)
= (-It+ 1 = (_l)d+i+l.
In the situation of (ii) the image of 'lJ will not contain vectors parallel to H, and the assertion follows from Corollary 11.10.
Assertion (iii) is a consequence of the homotopy property, Corollary 11.2. Indeed,
a homotopy J x K x [0,1] -+ sn is given by
(ht(y) - gt(x)) / Ilht(Y) - gt(x)ll· Finally (iv) follows from Proposition 11.11 applied to the map F: J x P
y-x y - xii' Remark 9.20) and sn is :ation of Rn+l. We note J or K is reversed.
F(x, y) = (
-+
sn with
xii,
and to the domain X = J x R with boundary components J x Qi. Indeed, Ii = FIJxQi has degree deg(!i) = Ik(J, Ki). D Here is a picture to illustrate (iv):
l)j~-
0
'fO .
~~."" .'~ . ~'~ ~ . .~6 . ..c-§.y i. ~ .. ~.~ I:)., . . '~: roe
......
'"
.",.
~...l;.\
....
'
CI)"
·~7~,
II.
:TOR "'IELDS
>~_-~f~j~~~Jt~~
DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS
105
Proof. We apply formula (2) to the map \.II = \.II J,K and the volume form w = vols2 (with integral 47T) to get lk(J, K)
(12)
= deg(\.II) = ~ [ 47T
We write \.II = r
0
f
r:
E
1R3
-
\.II * (vols2 ).
with
f: .J x K
For x
.JJxK
1R 3
---t
-
{O}, r*(vols2)x
1R3 - {O}; f (ql , q2) = q2 - ql {O} ---t Sz; r(x) = x/llxll. E
Alt Z (1R 3 ) is given by
r*(vols2)x(v, w) = det (x, v, w)/llxI13 (cf. Example 9.18). The tangent space T(qlm)(J x K) has a basis {'V(ql) , W(q2)} and Df(ql,q2)(V(qd) = -v(qd, Df(Ql,Q2) (w(qz)) = w(qz). ply that J and K cannot
R3 where J and K are
Therefore (13)
'1'* (vols2) (Ql ,q2) ev(ql)W (qz)) = r* (vols2 )q2-ql (-v(ql), w(qz))
= IIq1 - qzll-3 det (ql - qz, 'V(ql), w(qz)) .
. Let us choose smooth
Igle traversing of J and consider the set
-2,
,\ > O}. mgent vectors to J and
The integral of (12) can be calculated by integrating (0: X p)*\.II* (vols2) over the period rectangle [0, a] x [0, bJ. This yields Gauss's integral. For p E SZ, I (p) is exactly the pre-image under \.II. Thus p is a regular value of \.II if and only if the determinant in (13) is non-zero for all (ql' q2) E I(p), and the sign 8(ql, q2) is determined by whether D(Ql,Q2) \.II preserves or reverses orientation. Assertions (ii) and (iii) now follow from Theorems 11.5 and 11.9. D
Remark 11.15 In Theorem 11.14.(ii), after a rotation of R3 , the regular value p can be assumed to be the north pole (0,0,1). The projections of J and K on the Xl, x2-plane may be drawn indicating over- and undercrossings and orientations, e.g. Figure 2
1,,8'(v)) du dv. 1t J
2)
E
I(p).
q2), where 6(Q1, q2) is
K
c
~
.. 1:)
~
.
'lo
'"
.-I
11 ~ ...",
.~.h----\
107
11. DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS
CTORFIELDS
sses over (and not under) ltation of the curves and picture
Proof. Let i: sn-l ----t IR n - {O} be the inclusion map and r: IR n - 1 - {O} ----t sn-l the retraction r(x) = x/llxll. We have ~(F; 0) = degFJ, where Fl = r 0 F 0 i. The lemma follows from the commutative diagram below, where Hn-1(i) and H n - 1 (r) are inverse isomorphisms:
H n- 1(lR n _ {O})
Hn-1(f)
lHn-1(r) K
Hn-1(R - {O}) Hn-1(r)11Hn-1(i)
Hn-l(sn-l)
Hn-1(sn-l)
Hn-1(fl)
D Given a diffeomorphism ¢: U ----t V to an open set V ~ Rn and a vector field on U, we can define the direct image ¢*F E Coo(V, Rn) by ¢*F(q) = Dp¢(F(p)),
p = ¢-l(q).
J-Jemma 11.18 If F E COO (U, IR n ) has 0 as an isolated singularity and ¢: U
lk(J,K1 ) =-1.
is a diffeomorphism to an open set V
~
----t
V
IR n with ¢(O) = 0, then
ance with (iv) of Lemma ~(¢*F;
les of vector fields. U ~ Rn , n ~ 2, and let ro for F is also called a I with
~
----t
of p, and by Corollary
~(F,
0).
Proof. By shrinking U and V we can restrict ourselves to considering the case where 0 is the only zero for F in U, and where there exists a diffeomorphism 'IjJ: V ----t IR n . The assertion about ¢ will follow from the corresponding assertions about 'IjJ and 'IjJ 0 ¢, since 'IjJ*(¢*F)
smooth map F p : sn-l
0) =
= ('IjJ 0
¢)*F.
Thus it suffices to treat the case where ¢: U ----t Rn is a diffeomorphism and where Y = ¢*F E COO (IR n , IR n ) has the origin as its only singularity. Let Uo ~ U be open and star-shaped around O. We define a homotopy : Uo x
[0,1]
----t
IR
n
(DO¢)X
;
t(x) = (x, t) = { ¢(tx)/t
if t if t
=0 i= O.
I.
For x E Uo, of F at 0, and is denoted
1
¢(x) as its only zero. Then
td r (n 8¢ ) n Jo dt¢(tx)dt= Jo ~Xi8xi(tx) dt=~Xi¢i(X),
=
where ¢i E COO (Uo, Rn) is given by
¢i(X)
R.
=
18¢
1°
-
8Xi
(tx) dt.
<
.-I .. ~
'"
i' ~ ..
ICTOR FIELDS
~
2
,
11. DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS
109
Figure 4 t
I
= -1
t
= +2
set W with Uo x [0, 1] ~
o an open subset of Rn .
4>t(Uo): ~4>t)-1 Y(4)t(x)).
Jo
and X o = (A- 1 ) .. y,
x [0, 1]
Let X be a smooth tangent vector field on Mn and let Po E M n be a zero. Let
F = h*(XIU) E COO(h(U), Rn )
~ sn-l given by
for a chart (U, h) with h(po) = O. If DoF: Rn ~ Rn is an isomorphism, then Po is said to be a non-degenerate singularity or zero. Note that by the inverse function theorem F is a local diffeomorphism around 0, such that 0 is an isolated zero for F. Hence Po E M n is also an isolated zero for X.
)"Y; 0).
)Y
0
A: Rn
Lemma 11.20 If Po is a non-degenerate singularity, then ~
Rn. This
~(X,po) =
sign(detDoF) E {±1}.
Proof. By shrinking U we may assume that h maps U diffeomorphically onto an open set Uo ~ Rn, which is star-shaped around 0, and that F is a diffeomorphism from Uo to an open set. As in the proof of Lemma 11.18 we can define a homotopy n
)oth Yand (A-l ).. Y to
G: Uo x [0, 1] ~ R
D d on the manifold Mn, l; ~(X;Po) E Z of X is
;
DOF G(x, t) = { F(tx)jt
if t = 0 if t 0/= 0,
where G can be extended smoothly to an open set W in Uo x R that contains Uo x [0,1]. Choose p > 0 so that pD n ~ Uo. We get a homotopy G: sn-l X [0, 1] ~ sn-l
,
G(x, t) = G(px, t) j IIG(px, 011,
between the map Fp in Definition 11.16 and the analogous map A p with A = DoF. It follows from Corollary 11.2 that
h(po) = O. loes not depend on the Jlane by drawing their
~(X;
po)
=
~(F;
0)
= deg(Fp ) = degAp = ~(A; 0).
The map fA:R n - {O} ~ Rn - {O} induced by A operates on H n - 1 (R n - {O}) by multiplication by ~(X;po); cf. Lemma 11.17. The result now follows from Lemma 6.14. D
<
.,.J
~,
. - ~::.--...
""
'"
.J
,L"(J.
'''
~
III
11. DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS TOR FIELDS
~n,
with only isolated total index of X over
Lemma 11.25 Suppose F E coo(Rn, IR n ) has the origin as its only zero. Then there exists an F E COO(lR n , IR n ), with only non-degenerate zeros, that coincides with F outside a compact set. Proof. We choose a function ¢ E Coo (IR n , [0, 1]) with
s pER for X. If M is
if
I
¢(x) = { 0 if n an open set U ~ Rn, I with smooth boundary
inf
1::;llx119
and choose w with Ilwll < c. For 1::; zeros of P belong to the open unit ball
p-l(O) =
'(x)ll· ndary ax is the disjoint ~re aDj, considered as J the one induced from
Ilxll >
We want to define F(x) = F(x) - ¢(x)w for a suitable w E IR n . For we have F(x) = F(x). Set
c=
lse disjoint closed balls
Ilxll ::; 1 Ilxll 2: 2. 2
IIF(x)11 > 0
Ilxll ::; 2, IIP(x)11
bn .
Since
c-Ilwll > O.
2:
P coincides with F -
Thus all
w on
bn,
b n n F- 1(w).
We can pick w as a regular value of F with Ilwll < c by Sard's theorem. Then DpP = DpF will be invertible for all p E P-l(O), and P has the desired properties. D Note, by Corollary 11.23, that
i(F;O) =
(14)
~ i(P,P) PEP-l(p)
and Corollary 11.3. D ~Fj
F
Here is a picture of F and
in a simple case:
R) depends only on D
efor every pEaR that e Gauss map which to r to oR. Then
Figure 5
/ ;,
~::::<\
,\\;,,'\
\ \\" /,' ~~"oL,!11 ,-_/ j \ / ( ~--
~~_, '-,; !K.--; . . ) I
o
\\.'
\
" j /i
I"
\"
\ \\\ \
\1,,\ l "'o, "
_--,,'
__
~
)/~\[\ r- ;h r ' \~\ \ ,\, \
/11/1/
, are homotopic. Since the desired homotopy
'\\" '. \ ' \
/ /;!:
\"\\' /;/// \ \'~\", ~,";///;I
~
\ \
=-
--'-_'
_,J \ \
/'
/ I II I /1
1
1
//1/
~/I / /,J I /! ( '....,'-. ,,J-l
,...'~ " \ 1/ \\ 'I / /
)1/ 1
\
I ; /
\f .- (--- ""---
\ \
\\, '
WI/ \\\\;,
1;/ / /""" '\\\ \
F
F
\
\. \
/
"" \ \ '..
//
The zero for F of index - 2 has been replaced by two non-degenerate zeros for ft, both of index -1.
~
"
:TOR FIELDS
113
, compac~ manifold M n tor field X on M having
re diffeomorphic to IR n 11.25 on the interior of m (14).
12.
In the following, Mn ~ ~n+k will denote a fixed SIDooth-submanifold. If the cohomology of M n is finite-dimensional (e.g. when Mn is compact), the i-th Betti number is given by (1)
bmanifold and let N E be te by g: 8Nt ---t sn+k-l ?oth vector field on M n
THE POINCARE--HOPF THEOREM
bi(M) = dimR Hi(M n ).
The Euler characteristic of Mn is defined to be (2)
X(M n ) =
n
L (_l)i bi(M). i=O
This chapter's main result is:
non-degenerate zeros. lve a smooth projection N€ C N C IR n + k , and
IS
Theorem 12.1 (Poincare-Hopf) Let X be a smooth vector field on a compact manifold M. If X has only isolated zeros then Index(X) = X(M).
nd only if q E M and 8NE pointing outwards. Corollary 11.24
By the final result of Chapter 11 it is sufficient to show the formula for just one such vector field X on M. We shall do so by making use of a Morse function
Mn . Given f E COO(M, R), a point p E M is a critical point for f if dpf = O. oil
ary zero of X. In local Rn, X can be written
Proposition 12.2 Suppose that p E M is a critical point for f E COO(M, R). (i) There exists a quadratic fonn d~f on TpM characterized by the equation
d~f(o/(O))
10f
= (f 0
a)"(O)
where a: (-8,8) ---t M is any smooth curve with a(O) = p. (ii) Let h: U ---t Rn be a chart around p and let q = h(p). Then the composition
Rn D~-l TpM :ity on TpM..l, and by 'ix (8fd8xj(O)) (with jegenerate zero for F
o
r!lf R
is the quadratic fonn associated to the symmetric matrix
8 2 f 0 h- 1 )
( 8Xi8xj (q) .
~
.".J
"
e,J. -~--,,~
12.
,= f
0
h -1. A direct
,,~(t).
iating once again and
THE POINCARE-HOPF THEOREM
Proof. Let g: IR n --t M be a local parametrization and Yj: Rn --t IRn+k, j = 1, ... , k smooth maps, such that Y1(X), ... , Yk(X) is a basis of Tg(x)Ml.. for all x E IR D • By Lemma 9.21 we know that M can be covered by at most countably many coordinate patches g(U) of this type. Therefore it suffices to prove the assertion for g(U) instead of M. We define <1>: Rn+k --t IRn+k by k
(4)
<1> (x,
t)
= g(x) + L
tj Yj(x)
(x
E
IR n , t E IRk).
j=l
;(0). adratic form from (ii). D
i) and let F = it 0 h- 1 Ie proof above can be
By Sard's theorem it suffices to prove that if Po is a regular value of <1>, then f becomes a Morse function on g(U). Set k = fog: Rn --t IR; we can show instead that k becomes a Morse function on IR n . We have k(x)
(5)
1
= 2"
(g(x) - Po, g(x) - po),
where ( , ) denotes the usual inner product on IRn+k. Since (ogjoXi(X), Yy(x)) = 0, it follows by differentiation with respect to Xj that
rj (0),
_ / ~ Oyy) \OXi'OXj'
2
/ 0 9 y; ) _ \OXiOXj' y -
(6)
lcobi matrix associated rs 1'(0) and ,,'(0). By lentity
I
115
From (5) we have ok OXi
(7)
\g(X) Po,
::i)'
and therefore
R) is said to be non
>le. We call f a Morse The index of a non :ubspace V ~ TpM for
(8)
/ og
02k
---- = \OXj
09 ) , OXi
/
+ \g(x) -
Furthermore, by (4), 0<1> og =+L ox' ox' k
(9)
J
functions by:
;f ---.
R defined by
02 g ) Po, OXiOXj .
J
ty
y=l
oYy ox·' J
--
0<1> ot y
= Yy.
Assume that Po is a regular value of <1> and let x be a critical point of k. It follows from (7) that g(x) - Po E Tg(x)Ml... Hence there exists a unique t E IRk with k
(10)
g(x) - Po = -
L t y Yy(x), y=l
~
.-
'"
~
~---~~~
12.
:arly independent at the i (9) give
THE POINCARE-HOPF THEOREM
117
Theorem 12.6 Let p E M n be a non-degenerate critical pointfor f E Coo(M, IR). There exists a Coo-chart h: U --+ h(U) ~ Rn with p E U and h(p) = 0 such that n
f
0
r,
h- 1(x) = f(p) + L 8i X
x E h(U),
i=1
=
\:~ ,
where 8i = ±l (1 :S i :S n) (By an additional permutation of coordinates we can put f into the standard form given in Example 12.5.)
::i)'
ith the vectors from (9)
Proof. After replacing f with f - f(p) we may assume that f(p) = O. Since the problem is local and diffeomorphism invariant, we may also assume that f E Coo(W, R), where W is an open convex neighborhood of 0 in Rn and that 0 is the considered non-degenerate critical point with f(O) = o. We write f in the form n
f(x)
= LXi gi(X);
gi(X) =
1 r af(tx) Jo dt.
i=1
Since gi(O) = *f(0) = 0, we may repeat to get n ~
matrix
gi(X)
gij(X) =
= L Xj gij(X); j=1
t
Jo
agi(SX) ax'J ds.
On W we now have that n
al point.
n
f(x) = L LXi Xj gij(X), i=1 j=1
o
where gij E Coo(W, R). If we introduce hij = ~(gij + gji) then (hij) becomes a symmetric n x n matrix of smooth functions on W, and ... x 2n ,
n
(11)
f(x) = L i=1
n
LXi Xj hij(X). j=1
By differentiating (II) twice and substituting 0, we get Xn
),
a
2 f ~-!)-
UXiUXj
.,2) igin is non-degenerate origin as its only zero
(0) = 2hij(0).
In particular the matrix (hij (0)) is invertible. Let us return to the original f E Coo(M, R). By induction on k, we attempt to show that the Coo -chart h from the theorem can be choosen such that f 0 h- 1 is given by (II) with
(hij) =
(~ ~) ,
o
.e
'<;')
,@'
.;oJ
.~
..
~
..ce' ",
ItM
12.
~±1,
... , ±1), and E some
loth functions. So suppose
Oi
119
THE POINCARE--HOPF THEOREM
Definition 12.7 Let j E C=(M, IR) be a Morse function. A smooth tangent vector field X on M is said to be gradient-like for j, if the following conditions are satisfied: (i) For every non-critical point P E M, dpj (X(p)) > O. (ii) If p E Mn is a critical point of j then there exists a C=-chart h: U h(U) ~ IR n with p E U and h(p) = 0 such that
= ±1
at the minor E is invertible variables in Xk, ... ,Xn , so , continuity we may assume W. Set
h-1(x) = j(p) - xi - ... and h*X1u = grad (J 0 h- 1 ).
j
0
A smooth parametrized curve 0:: [
- xl + xl+1 + ... + x;,
Hence one gets (f 0 o:)'(t) critical points, then j 0 0:: [
:)
x E h(U),
M is an integral curve for X, if
--->
o:'(t) = X(o:(t))
1)
---+
for t E [.
= dn(t)j(X(o:(t))). --->
If 0:(1) does not contain any R is a monotone increasing function by condition
(i).
r seen to be OYk/OXk(O) = diffeomorphism III around
Lemma 12.8 Every Morse function on M admits a gradient-like vector field.
c):
Proof. We can find a C=-atlas (Un, hn)nEA for M which satisfies the following two conditions: n
+
(i) Every critical point of j belongs to just one of the coordinate patches Un' (ii) For any 0: E A either j has no critical point in Un or j has precisely one critical point p in Un, hn(p) = 0, and j 0 h;;l has the fonn listed
n
L L
xixjhij(x)
i=k+1 j=k+1 2
in Example 12.5.
)
Let X n be a tangent vector field on Un detennined by X n = (h;;l)* (grad (f 0 h;;1)). Choose a smooth partition of unity (Pn)nEA subordinate to (Un)nEA' and define a smooth tangent vector field on M by
n
L xixjhij(x) =k+l
X =
L
PnXn,
nEA where PnXn is taken to be 0 outside Un. If P E M is not a critical point for j then, for every 0: E A with p E Un and q = hn(p), we have ;tep.
o
dpj(Xn(p)) = dq(f
0
h~l)(gradq(f
Indeed, there is at least one 0: with Pn (p) 2.6 P is the only critical i10rse function then j has lill always be at least one ximum (index ,\ = n).
dpj(X(p)) =
0
h~l))
> 0 and
L Pn(P) dpj(Xn(p)). n
We see that X satisfies condition (i) in Definition 12.7.
> O.
<
-.-~
~
,.;
"..;
.~
~
~1
~....~
·~".~(~~.frw
12.
E
:l:
A with P E Ua' It
hood of P, and condition ,atisfied. 0 the local index of vector
TIlE POINCARE-HOPF TIlEOREM
121
Proof. It is a consequence of Theorem 11.27 that any two tangent vector fields with isolated singularities have the same index. Thus we may assume that X is gradient-like for j. The zeros for X are exactly the critical points of j, and the claimed formula follows from Lemma 12.9. 0 It is a consequence of the above theorem that the sum
a smooth tangent vector ot a critical point for f. ) = 0, then
Definition 12.7.(ii) and
n
L
(13)
(-1)'\~
),=0
is independent of the choice of Morse function j E COO(M, IR). Given Theorem 12.11, the Poincare-Hopf theorem is the statement that the sum (13) is equal to the Euler characteristic; cf. (2). We will give a proof of this based on the two lemmas below, whose proofs in turn involve methods from dynamical systems and ordinary differential equations, and will be postponed to Appendix C.
lic to Rn and chosen so lities 0, ~long
to the same open
Let us fix a compact manifold M n and a Morse function j on M. For a E IR we set
M (a)
= {p
E M
I j (p) < a}.
Recall that a number a E IR is a critical value if j-l (a) contains at least one critical point.
Lemma 12.12 If there are no critical values in the interval [aI, a2], then M(al) and M(a2) are diffeomorphic.
)
laps from U - {po} to
o E COO(M, R), one can
I.
nifold and X a smooth ~t j E COO(M, R) be a ~dex A for f. Then we
Lemma 12.13 Suppose that a is a critical value and that PI, ... , Pr are the critical points in j-l(a). Let Pi have index Ai. There exists an E > 0, and disjoint open neighbourhoods Ui of Pi, such that (i) PI,···, Pr are the only critical points in j-l ([a - E, a + ED. (ii) Ui is diffeomorphic to an open contractible subset of IRn. (iii) Ui n M (a - E) is diffeomorphic to S),; -1 X Vi, where Vi is an open contractible subset of IR n -),; +1 (inparticularUinM(a - E) = 0 if Ai = 0). (iv) M(a + E) is diffeomorphic to Ul u ... U Ur U M(a - E). 0
Proposition 12.14 In the situation of Lemma 12.13 suppose that M (a - E) has finite-dimensional cohomology. Then the same will be true for M (a + E). and r
x(M(a
+ E)) = x(M(a -
E))
+L i=1
(-1),;.
12.
Ifollary 6.10 imply that
123
THE POINCARE-HOPF THEOREM
Here the sum runs over the critical points p E f-l(aj), and A(p) denotes the index of p. We can start from M(bo) = 0. An induction argument shows that dimHd(M(b j )) < 00 for all j and d. The sum of the formulas of (14) for 1 :s: j :s: k gives
ows that Ui n M(a - f) es
X(M) = X(M(bk)) =
2: (_l)A(p) p
o
where p runs over the critical points.
'i
n M(a -
f), it has a T
Ii
= X(U)
-2: (_1)
A
The Poincare-Hopf theorem 12. I follows by combining Theorems 12.1 I and 12.16.
i.
Corollary 12.17 If Af n is compact and of odd dimension n then X(M n ) =
i=l
o.
I) and the lemma below,
o
h manifold. If U, V and me is truefor UuV, and
Proof. Let f be a Morse function on M. Then - f is also a Morse function, and - f has the same critical points as f. If a critical point p has index A with respect to f, then p has index n - A with respect to - f. Theorem 12.16 applied to both f and - f gives
V).
X(M)
=
n
n
A=O
A=O
2: (_l)A CA = 2: (-It-Ac A.
The two sums differ by the factor (-1) n, and the assertion follows.
)
~
HP(U n V)
e alternating sum of the lual to zero; cf. Exercise
o
Example 12.18 (Gauss-Bonnet in 1R 3 ). We consider a compact regular surface S ~ R3, oriented by means of the Gauss map N: S -+ S2. The Gauss curvature of S at the point p is K(p)
manifold M n , then
uJex A.
al values. Choose real bk > ak. Lemma 12.12 of the choice of bj from de Rham cohomology, I 12.14, and l)A(p)
[]
-+ ...
= det (dpN);
TpS
=
{p}.l
= TpS2.
Sard's theorem implies that we can find a pair of antipodal points in S2 that are both regular values of N. After a suitable rotation of the entire situation we can assume that p± = (0, 0, ± 1) are regular values of N. Let f E COO(S, IR) be the projection on the third coordinate axis of R3 . The critical points pES of f are exactly the points for which TpS is parallel with the XI, x2-plane, i.e. N(p) = P±. At such a point p, the differential of the Gauss map dpN is an isomorphism. Hence K(p) i= o. A neighborhood of pin S can be parametrized by (u, v, f (u, v)), and in these local coordinates the Gauss curvature has the following expression: 2
K =
2
-1
82f
(1 + (:~) + (:~)) det ( a;~ 8u8v
2
8 f 8u8v
82 f 8v 2
)
.~- 0 ,~ ~ ......... u .~ 't3 .~ .. .... ,.() ''tS. "t3 .'~ ~
,...;
"
.~."
"""'---
~ '100:... ~ ~.,t3
~
~. ..~.~
.5q.,
:,
'
):f'
12.
minant in the expression )int for f. If K(p) > 0 (p) < 0 the determinant , to get
N(p)
=
P±, K(p) <
-1 (P+)
mE POINCARE-HOPF mEoREM
I K (p) < O}
125
Example 12.20 (Morse function on IIlpn) Real functions on IIlpn are equivalent to even functions f: sn -+ Ill, i.e. f (x) = f (- x) for all x E sn. Let us try n
f(x) =
O}.
11.9
~
I: aiX; i=O
for x = (XO, Xl, ... , Xn ) E sn s;;; lI~n+l, and real numbers ai. The differential of f at x is given by n
dxf(vo, ... , vn ) = 2
I: aiXiVi, i=O
where v - (VO, VI,
... , Vn )
E T.TS n so that n
Alt 2 (Tp S)
I:XiVi = O. i=O
K(p)vols. Hence
is a critical point for f if and only if the vectors x and (aoxo, a1 Xl, ... , anx n ) are linearly independent. If the coefficients ai are distinct, this occurs precisely for x = ±ei = (0, ... , ± 1, ... ,0), and f has ex actly 2n + 2 critical points. The induced smooth map i: IIlpn -+ III then has n + 1 critical points lei]' In a neighborhood of ± eo E sn we have the charts h with Thus x
2
= 41fdegN.
la
h±l(Uj, ... ,Un) =
function f: T -+ R is a Gauss curvature of Tis
7
18). Hence f is a Morse respectively. Theorem = dim H 2 (T) = 1, we
(±JI- L~=l u;'
Uj, . .
,un).
and in a neighborhood of 0 E Rn ,
f
0
h±l(Ul,""
un)
= ao ( 1 -
n
I: u; i=l
)
n
n
+ I: ai u ; = ao + I: (ai i=l
- ao)ur
i=l
The matrix of the second-order partial derivatives for f 0 h±l (at 0) is the diagonal matrix diag(2(al - aO),2(a2 - ao), ... ,2(an - ao)). Hence ±eo are non-degenerate critical points for f; the index for each is equal to the number of indices i with 1 ~ i ~ nand ai < ao. An analogous result holds for the other critical points ±ej. For simplicity, suppose that ao < al < a2 < ... < an' Then the two critical points ±ej for f: sn -+ R have index j. The induced function Rpn -+ R is a Morse function with critical points [ej] of index j. We apply Theorem 12.16 to Since c>. = 1 for 0 ~ A ~ n, we get
i:
f.
x(lIlpn) This agrees with Example 9.31.
= {~
if n is even if n is odd.
-":>- 0 1':::i.""-1.~
.~
.~
'.
.~,~ .~<:
. .e-,.>.
.... '..c....e '.l:S.. .'
'-1.:/_,.
~~
.....ce .Et .....I:e l:l..~ ,.n ,-f
:
~ .... f,I,l
.f--:~..1iIi'~
..,.
~j.
,~
",.J
co.,)
''\,
1~.. •.· 2 •~'_i.-.~.
-..
"
J~~
127
13.
POINCARE DUALITY
Given a compact oriented smooth manifold M n of dimension n, Poincare duality is the statement that (1)
HP(M)
~
Hn-P(M)*,
p E
7L
where Hn-P(M)* denotes the dual vector space of linear forms on Hn-P(M). The proof we give below is based upon induction over an open cover of M. Thus we need a generalization of (I) to oriented manifolds that are not necessarily compact. The general statement we will prove is that (2)
HP(M)
~ H~-P(M)*
where the subscript c refers to de Rham cohomology with compact support. For a smooth manifold M we let n~(M) be the subcomplex of the de Rham complex that in degree p consists of the vector space n~(M) of p-forms with compact support. The cohomology groups of (n~(M), d) are denoted H;(M), i.e.
H~(M) = Ker(d: n~(M)
-t
Im(d: n~ l(M)
n~+l(M))
-t
n~(M))
Note that when M is compact n~(M) = n*(M), so that H;(M) = H*(M) in this case. The vector spaces H;(M) are not in general a (contravariant) functor on the category of all smooth maps. However, if !.p: M - t N is proper, I.e. if !.p -1 (K) is compact whenever K is, then the induced form !.p* (w) will have compact support when w has. Indeed SliPPM!.p*(W) C
!.p-1(sUPPNW)
and !.p* becomes a chain map from is an induced map
n~(N)
H~(!.p):
Hr(N)
to
-t
n~(M).
Hence by Lemma 4.3 there
H~(M)
and H~ (- ) becomes a contravariant functor on the category of smooth manifolds and smooth proper maps. A diffeomorphism is proper, so Hg (M) ~ Hg (N) when M and N are diffeomorphic.
~
"
,/,,"','
.!"'t-'
13.
129
POINCARE DUALITY
defined by setting
:ally constant functions st be identically zero on [n particular Hg (M) = 0 \.1) = IR for such a man-
I, then we have the iso
II-compact.
i*(w)lv = w,
i*(w)\u-supp(w) = O.
for w E n~(V). We get a linear map i*: H~(V)
(3)
--+
H~(U)
which is called the direct image homomorphism. Given a second inclusion j: W --+ V, (i 0 j)*(w) = i* 0 j*(w), so that (H~( -), i*) becomes a covariant functor on the category of open subsets and inclusions of a given manifold. There is also a Mayer-Vietoris theorem for this functor. Indeed, if UI and Uz are open subsets of M with union U, and i v:Uv --+ U, jv: UI n Uz --+ Uv are the inclusions, then the sequence
o --+ n~(UI n Uz) ~ n%(uI) EB nHUz) ~ n~(U) --+ 0
(4)
is exact, where
= n, so we may assume Hereographic projection,
of n* (sn) consisting PO· , = 0, by Example 9.29, iT = w. We must show Suppose W is an open = O.
~x
W, say Tlw
= a.
Iq(WI' wz) = ih(WI)
+ iz*(wz)
and
Jq(w) = (jh(W), -j2*(W)).
We leave the verification of the exactness to the reader, with the remark that surjectivity of I q uses a smooth partition of unity on U subordinate to the covering {Uv }; cf. Theorem 5.1.
Theorem 13.3 (Mayer-Vietoris) With the above notation there is an exact se quence ..• --+
H%(UI n Uz) ~ H%(UI) EB H2(U2) ~ HZ(U)
0. Hg+I(UI n Uz)
Proof. This follows from Theorem 4.9 applied to (4).
--+ ....
o
But
= 0,
and that Tlw is a Now choose a smooth lue 1 in a smaller open n be extended to all of extended form and let
In comparing Theorem 13.3 with Theorem 5.2 the reader will notice that the directions of all arrows have been reversed. For later use, let us explicate Definition 4.5. o*[w] E HZ+l(UI n Uz) is defined as follows: write w = WI +wz with Wv E ng(U) and sUPPu(w v ) C Uv. Then dWI and -dW2 agree on UI n U2 and the common value T = dWII UlnU2 = -dwzl UlnU2 is a closed form in ng+l(Ul n U2) that represents o*[w].
[]
Id let i: V
--+
U be the
Proposition 13.4 Suppose {Ua I 0: E A} is a family of pairwise disjoint open subsets of the smooth manifold M with union U. Then there are isomorphisms (i) Hq(U) --+ IlaEA Hq(Ua ); (ii) EBaEA Hg(Ua ) --+ Hg(U);
[W]
~ [i~(w)]
([wa])aEA ~
LaEA i a * ([w a])
13.
131
POINCARE DUALITY
Theorem 13.5 (Poincare duality) For an oriented smooth n-dimensional manifold, D~ is an isomorphism for all p.
The proof is based upon a series of lemmas.
)aEA
Lemma 13.6 Suppose V
:I>a*(Wa) we give
~
U
~
M n are open subsets. Then the diagram
HP(U)
IT n*(Ua )
lD~
the
HP(i~
HP(V)
lD~
.,
H-;:-P(U)* ~ H-;:-P(V)*
o
wmplex.
to its dual vector space
:xact sequence of vector
7/;* = O. The other lp defined on 1m1P can )m's lemma when C is p*
0
n 13.3 to get the exact
J'
-+ H~(UI
n U2 )
-+ ...
commutes. Proof. Let W E np(U), r E n~-p(V) ,be closed forms representing cohomology classes [w] and [r]. Then
DV
HP(i)([wJ)([rJ)
0
i'
0
)aEA.
:t defines a bilinear map
a bilinear map
DV ([i*(w)J) [r]
=
Dfr ([wJ) ([rJ) = Dfr ([wJ) [i* (r)] =
HP(UI n U2)
~
(M)*.
1
w /\ i*(r).
HP+1 (U)
lDP+l u
DP~n~
l
H-;:-P(U1 n U2)* (-l)p+~a! H-;:-p-l(U)* is commutative. Here f)* is the boundary in the Mayer-Vietoris sequence of Theorem 5.2, and f)! is from (5). Proof. Let w E np(Ul n U2) and r E n~-p-l(U) be closed forms. We write w = ii(Wl) - j2(W2) with Wv E np(Uv ) and jv: U1 n U2 --+ Uv the inclusions. Let K E n p+1(U) be the (p + I)-form with i~(K) = dw v , where i v : Uv --+ U are the inclusions. Then K represents f)* ([wJ) so that
Dfr+1f)* ([wJ)([rJ) = Dfr+1([KJ)[r] =
Wl/\ W2
i*(w) /\ r
Lemma 13.7 For open subsets U1 and U2 of Mn with union U the diagram
i)) to obtain a bilinear
\1
fv
Since sUPPu(w /\ i*(r)) c suppu(i*(r)) = suppv(r) we may as well in the second integral just integrate over V. But the n-forms i*(w) /\ r and w /\ i*(r) agree on V. D
:al) - j~(a2)
or space dual of ih, etc. on 13.4.(ii) implies the
=
1
K/\ r.
It was pointed out after the proof of Theorem 13.3 that a representative for f)*[r] E H-;:-P(U1 n U2) can be obtained by the following procedure: write r = rl + r2 with
rv
E n~-p-l(U)
and
suppu(rv ) c Uv
c ~
..
~
't'.
~
......J
~
'to
".
j'
. . 'c-"
~
~
h:---J
13.
- p )-form that represents
nU2
w 1\ a.
r
Dt
Dt
Dt
Dt.
The proof of (ii) uses the commutative diagram
HP(U)
-l)Pwv 1\ dTv ,
I1aEA HP(Ua )
ID~
dW21\ T2
JU2
133
This is a commutative diagram according to the two lemmas above. Our assump tions are that ffi and nu2 are isomorphisms for all p, and it follows l 2 1 by the 5-lemma (cf. Exercise 4.1) that so is
sign (-1 )P+ 1 . We have
+
POINCARE DUALITY
l
rrDP u"
H;:-P(U)* - - I1aEA H;:-P(Ua )* and by
The horizontal maps are isomorphisms by Proposition 13.4 and the right-hand vertical map is an isomorphism by assumption. To prove (iii), we use that HP(U) ~ HP(lR n ) and H;:-P(U) ~ H;:-P(lR n ) together with Theorem 3.15 and Lemma 13.2. We only need to check that D& : HO(U) -+ H;:(U)* is an isomorphism. The constant function 1 on U is mapped to the basis element
L:H~(U)
21\ dT2.
-+
R
), and we have
r
of H;:(U)* ~ IR.
D
W2 1\ ]2* (a)
U2
r
H(W2) 1\
Ju1 nU2
Theorem 13.9 (Induction on open sets). Let M n be a smooth n-dimensional manifold equipped with an open cover V = (VjJ) jJEB' Suppose U is a collection of open subsets of M that satisfies the following four conditions
a
D
Ipose that U1 , U2 and = U1 UU2 • mbsets of Mn. If each ,n U = UaUa . ) Rn satisfies Poincare
(i)
0
E U.
(ii) Any open subset U ~ VjJ diffeomorphic with Rn belongs to U. (iii) If U1 , U2, U1 n U2 belong to U then U1 U U2 E U. (iv) If U1 , U2, ... is a sequence of pairwise disjoint open subsets with Ui E U then their union Ui Ui E U. Then M n belongs to U. The proof is based upon the following lemma, where the term relatively compact means that the closure is compact.
Lemma 13.10 In the situation of Theorem 13.9, suppose U1 , U2 , ... is a sequence of open, relatively compact subsets of M with ~
w+\U)_
ID~+J
+18' ~ . H;-P-I(U)_
(i)
n Uj E U for any finite subset J,
jEJ
(ii) (Uj) jEN is locally finite.
'b
"...;
'"
£"t->
. .__·.~,":~?'f
13.
But then Theorem 13.9.(iii) implies that the union I Ujm
U for every set i) and condition (iii) of true for sets of m - 1 E
135
POINCARE DUALITY
U:1 Ui
= W(O)
Proof of Theorem 13.9. Consider first the special case where M an open subset. We consider the maximum norm on Rn
Ilxll ao =
max
l~1,~n
u W(1)
=
W
E U. ~
0
Rn is
IXil
whose open balls are the cubes IIi=l (ai, bi ). The argument of Proposition A.6 gives us a sequence of open II Ilao-balls Uj such that (i) W
ce (Ujj n Uj \EN' and E U. Since UinUj E U
: M as follows: h m-1
- UIj j=l
ii) implies that Wm-1 inductively that 1m is ~ to any I j with j < m 'J is the disjoint union have Wm E U (if
:ed, if the intersection l Ui i= 0, and by (8)
e following three sets
(ii) (Uj)jEN is locally finite. (iii) Each Uj is contained in at least one V;3. A finite intersection UJI n ... n Ujm is either empty or of the form rr~=l (ai, bi), so is diffeomorphic to Rn . Thus we can conclude from Lemma 13.10 that W E U. In the general case, consider first an open coordinate patch (U, h) of M, h: U -.. W a diffeomorphism onto an open subset W ~ Rn . We apply the above special case to W with cover (h- 1 (V;3));3EB' and with U h the open sets of W whose images by h- 1 belong to the given U. The conclusion is that U E U for each coordinate patch. If M is compact then we apply the argument of Lemma 13.10 to a finite cover of M by coordinate patches to conclude that M E U. In the non-compact case we make use of a locally finite cover of M by a sequence of relatively compact coordinate patches; cf. Theorem 9.11. (One may alternatively imitate the proof of Proposition A.6 using relatively compact coordinate patches instead of discs to construct the desired cover). 0
Proof of Theorem 13.5. Let U = {U ~ M n I U open, Df! an isomorphism for all
p}
and let V = (V;3) ;3EB be the trivial cover consisting only of M. Corollary 13.8 tells us that the assumptions in Theorem 13.9 are satisfied, so M E U. 0 We close the chapter with an exact sequence associated to a smooth compact manifold pair (N, M), i.e. a smooth compact submanifold M of a smooth compact manifold N. Let U be the complement U = N - M, and let i:U -.. N,
j:M
---t
N
be the inclusions. Then we have
Proposition 13.11 There is a long exact sequence
Xl
J (W
=1
= U;l Uj = U;l Uj.
m
n Wm + 1 ).
... -.. Hq-1(M) ~ Hg(U) ~ Hq(N)
1: Hq(M)
The proof of this result is based upon the following:
---t •••
...
l~~"
.
.~.'
< ~ ~
l:VJ, .:~
..
.~ ~
~J
."" . ..c::>
:g~ ~/~
..
.
~-"\
j....
,
"
- . •
-
~.
,
.'
-.
-
------ ---
13.
The chain map i*: n~(U) show that
m. 1 T E nq(N) such that on some open set in N
"'I VM ). Then (ii) follows
d extended trivially over ).
-+
-
----
137
n*(N) has image in n*(N, 1\.1), so it suffices to n*(N, M)
-+
induces an isomorphism in cohomology. Hq(N, M) in the above exact sequence. Consider an element [w] in the kernel of
Hq(i*): HZ(U)
I
-==---
POINCARE DUALITY
i*: n~(U)
(T) is exact, there exists :cally zero on some open
looth submanifold of IRk. th corresponding smooth r Nand M respectively; a smooth function such equal to 1 on some open
'
~
-+
We can then substitute HZ (U) for
Hq(N, M)
represented by a closed q-form w E n~(U). Then i*(w) = dT for some T E nq-1(N, M). Since j*(T) = 0 and sUPPN(dT) ~ U we may apply Lemma l3.12.(iii) to find IJ E n fJ - 2 (N) such that T - drJ vanishes on an open set around M. This gives us K, = (T - drJ)IU E n~-l(U) with dr" = w, so [w] = O. Let [w] E Hq(N, M) be represented by the closed q-form w E nq(N, M). We use Lemma 13.12.(iii) to find IJ E nfJ-1(N), with the property that w - dIJ vanishes on an open set containing M. Observe that
d(j*(IJ))
= j*(drJ) = j*(w) = O.
By Lemma l3.12.(ii) we may choose T E nq-1(N) with j*(rJ) = )*(T) and such that dT vanishes on a neighborhood of M. Thus rJ-T E n q- 1(N, M), and defining fi,
= (w - d(rJ - T))IU
= (w - drJ)IU + dTIl!
E n~(U),
we obtain [w] = [w - d(rJ - T)] = Hq(i*)[4
V(T) = T'N(dT) vanishes so small that di' = O.
Let us finally introduce the important signature invariant of oriented compact manifolds of dimension congruent to zero modulo 4. Given a 2k-dimensional compact smooth manifold M 2k , its intersection fonn is the bilinear form
-) = j*(T)
JL: Hk(M) x Hk(M)
that [T] = 0 in Hq(VM). ld that TINnvM is exact. ld define rJ E nq - 1 (N) laining part of N. Then
: a short exact sequence
JL([a], [,8]) =
(9)
Let us denote the leorem 4.9 gives us a
r a 1\ ,8. 1M
JL([a], [,8]) = (-l)kJL([a], [13]). We focus on k == 0 (mod 2), where JL becomes symmetric. All such bilinear forms can be diagonalized; i.e. there exists a basis e], ... , em such that 0
if i
'# j
JL(ei,ej) = { 'f" ai I Z = J .
Given a diagonalization of JL, we define the signature by
I.
-+ ....
R
This is (-1 )k-symmetric, i.e.
(10)
.... 0
-+
defined by
o
I(M)
0
rJ(JL) = #{i
I
(x,
> O} - #{i
IOi < O}.
This number is independent of the chosen diagonalization. In the case of the intersection form we know that (Xi # 0 for all i, since by Poincare duality the adjoint of JL is an isomorphism.
D'M
,
.,..; ~
'"
139
-dimensional manifold is
14.
j
THE COMPLEX PROJECTIVE SPACE epn
The set of I-dimensional complex subspaces of en+! is denoted by epn, and is called the complex projective n-dimensional space. For z = (zo, Z1, . .. ,zn) E en+! - {O} let 1r(z) = [zo, Z1, . .. , zn] denote the "point" ez E epn spanned by z. We give epn the quotient topology with respect to 1r: a set U <;;;: epn is open if and only if 1r- 1 (U) <;;;: en+! - {O} is open. In particular there are open subsets of epn,
Uj = {[zo" .. ,Zn] and the homeomorphisms hj: Uj
---t
E epn
I Zj i=0}
en,
hj([zo, ... ,zn]) = (zo/Zj, ... ,z0j, ... ,Zn/Zj)
(1)
with inverses
h";t(W1, ... ,Wn ) = [w1, ... ,1, ... ,wn].
(2)
The transition functions h k 0 h";t have coordinate functions of the form wL/wm or l/w m. The atlas 1-{ = {(Uj , hj)} gives epn the structure of a complex (analytic or holomorphic) manifold, because the transition functions are holomorphic. In the following, however, we shall mainly use the underlying structure as a smooth manifold, by interpreting 1-{ as a Coo -atlas upon identifying en with R2n .
Example 14.1 (The Riemann sphere and Hopf fibration) Since identified with 1R3 , the unit sphere S2 can be written as S2 =
Hz, t) E e x R IIzI 2 + t 2 =
e
x IR can be
I}
with north pole p+ = (0,1) and south pole p_ = (0, -1). The equatorial plane e x {O} is identified with C. The stereographic projections W±: S2 - {p±} ---t e map p to the points of intersection between the equatorial plane and the line through p± and p. A straightforward calculation gives, for (z, t) E S2 - {p±}, Z
W+(z, t) The
W±
Z
W-(z, t) = 1 + (
= 1- t'
are diffeomorphisms with inverses -1
W± The transition function
(w)
=
2
(2W 2
Iwl + 1 '
W- 0 W+. 1 : e -
{O}
W- 0 W+. 1(w)
±
---t
=
1)
Iwl Iwl 2 + 1
e-
.
{O} is easily calculated to be
(w)-1.
.. d
.,;
1j ~.
'"
"
~
~
epn
14.
the boundary of D 3 with ~rving and 1/J+ orientation ~lace 1/J+ by its conjugate between 1/J+ and 1/J- are • on 52 consisting of 1/J+ Iplex manifold (Riemann
Theorem 14.2 The cohomology of (::Ip n is
H 2j(Cpn)=R Hk(cpn) = 0
for O:::;j:::;n othelWise.
Proof. The embedding Cn C Cn+l induces an embedding of Cpn-l into Cpn, and we can use Proposition 13.11 on the pair (cpn, Cpn-l). We can assume the result for Cpn-l and that n :::: 2, since the cohomology of Cpl = 52 was given
cpn - cpn-l = Un
constant map.
.cally equivalent to Cpl. -
1
~ c'S Cpl
)=
141
in Example 9.29. The complement
).
I
TIlE COMPLEX PROJECTIVE SPACE epn
[z/(l - t), 1].
is by (l) and (2) homeomorphic with R 2n . The exact cohomology sequence takes the form ... -+
Hg (R 2n ) ~ Hq(Cpn) ~ Hq (cpn-l) ~ Hg+l (R 2n )
-+ .. '.
We know from Lemma 13.2 that Hg (R 2n ) is non-zero only for q = 2n, when it D is a copy of R, and the result follows easily.
dine a homeomorphism
;,t)#(O,-l) ;, t) # (0,1).
The complex projective [zo, Zl] E Cpl correspond :ation \11: 52 -+ C U {oo} lth pole p-, extended by
It follows from Theorem 14.2 that the general Hopf map 7[": 5 2n +l -+ Cpn cannot have a section s: Cpn -+ 5 2n+l: if s existed with 7[" 0 s = id, then
H 2(cpn) ~ H 2(5 2n + l )
£
H 2(cpn)
would be the identity, but this is impossible as H2(cpn)
E Cpn. For n = 1 this
I
ieed with the notation of
2ZOZI,
IzoJ2 -I ZI1 2).
cn+l by coordinatewise
0 and H 2(5 2n + l ) = O.
Since H 2n (Cpn) = R we know from Exercise lOA that Cpn is an oriented manifold, and hence from Theorem 13.5 that the bilinear map
Hq(Cpn) x H 2n -q(Cpn)
', ... ,zn].
#
-+
R
given by the wedge product (followed by the integration isomorphism) is a dual (non-singular) pairing. In particular, the generator of H 2p(Cpn) and the generator of H 2n - 2p(Cpn) has non-trivial product.
Theorem 14.3 The cohomology algebra H*(cpn) is a truncated polynomial algebra H*(Cpn) = R[e]j(en+l) where e is a non-zero class in degree 2, and (en+l) the ideal generated by en+l.
AZ.
recisely Cpn in the sense
n
Proof. We use induction over n, so suppose the theorem proved for cpn-l. The inclusion
j: cpn-l
-+
Cpn
...
ft. ,
~
101
.•C"P '
''';!:
,~.
, epn
14.
...
'"
,..:
THE COMPLEX PROJECTIVE SPACE epn
143
Proof. Choose U = Uj such that p E U, cf. (1). Consider the map s j: Uj given by
orphism for i
< 2n - 2.
(7)
Sj([zo, ... , Zj, ... , zn])
(
s2n+1
L I kl )-1/2 (zo, ... , Zj, ... , zn) n
=
---t
2
Z
k=O
where Zj = 1. We let S be this map composed with multiplication by the unique A E Sl such that As j (p) = v. The chain rule implies that
cpn)
o supports an orientation a map f: Cpn ---t cpn
D v7f: TvS 2n+1
---t
Tpcpn
is surjective. The kernel has real dimension 1, and since it contains iv, assertion (ii) follows. Let
TvS 2n +1
D v ¢,
DvA
T>'v s2n+1 /D>,v7r
TCpn
p ~
O. In this case there are s odd, on the other hand, 'feomorphism of cpn: ~n]
Multiplication by A can be considered as an IR-linear map C n + 1 ---t C n + 1 , and Dv
forms which represent a
Lemma 14.5 If V is a finite-dimensional C-vector space and F: V C-linear map, then
p = 7f(v). Now iv is a
det(rF)
sely the unit circle in the nent (Cv).L with respect ~n-dimensional subspace Ie real inner product on
=
IdetFl
2
---t
V is a
•
Proof. We use induction on m = dime V. If m = 1 then F is multiplication by some Z E C The matrix for rF, with respect to a basis of the form b, ib for rV, is
(~ ~y)
where x = Rez and y = Imz. Since det(rF) = x 2 + y2 = IzI the formula holds in this case. If m 2: 2 we can choose a complex line Vo C V with F(Vo) ~ Vo (generated by an eigenvector of F). F induces C-linear maps 2
s an open neighborhood r2n+ 1 such that s(p) = v )v7f
induces an R-linear
an n-dimensional C makes the isomorphisms
2S
Fo:Vo
---t
Vo,
F1:V/VO
---t
VIVo,
and we may assume the formula for both Fo and Fl. Since detF we are done.
= (detFo)(detF1) ,
detrF
= (detrFo)(detrF1),
o
~~
.,..J
C"P--.;, ,
'"
·~·1
~
....
.. ~
>;~
•••
cpn
THE COMPLEX PROJECTIVE SPACE
14.
~e
then r V has a natural over c: gives rise to a
o
145
Theorem 14.8 The W = {wp}PEcpn define a closed 2-form on c:pn and 9 = {gp} pEcpn is a Riemannian metric on c:pn (the Fubini-Study metric). Moreover, W
pply Lemma 14.5 to the Since det( r F) > 0, the
cpn
n
= n! volcpn,
where volc pn is the volume form determined by 9 and the natural orientation from Corollary 14.6.
lace with hermitian inner
Proof. Let p E OJ>n and v E s2n+l with 7f(v) = p. Choose s: U 7f 0 S = idu and s(p) = v as in Lemma 14.4. We will show that
m rV, and
(10)
,V2)
By (9) we have dWcn+1 = 0. Hence (10) will show that W is a closed 2-form on c:pn. If Wv E Tpc:pn, v = 1,2, and Dps(w v ) = tv + u v , where tv is a tangent vector to the fiber in S2 n+l over p and Uv E (C: v)...L, then
nt determined by 9 and = m! vol, where w m =
s2n+l with
Wlu = s*(wc n+!).
Wv = D v7f
0
Dps(w v ) = D v7f(t v + u v ) = D v7f(u v ).
Since Alt 2(Dv7f)(w p ) is the restriction to r(C:v)...L of WCn+!, we have
.is b1 , ... , bm of V with lOrmal basis of r V with
Wp(WI, 'W2) = WC n+1 ( U1, U2), and (10) follows from
s for Alt 1 (rV). Since other pairs of vectors,
!!.Use both sides are 1 on (See Appendix B.) 0
>duct and standard basis
)
Its of the coordinate ~
---+
Zj.
structure from Lemma 2 'I and W p E Alt r p c:pn
S*(WC n+1)(WI, W2) = WCn+! (Dps( wI), Dps(W2)) = WC n+1 (tl + Ul, t2 + U2) = WC n+1 (Ul, U2). In the final equality we used that tl and t2 are orthogonal to respectively U2 and Ul in c: n+1 , and the fact that tl and t2 are linearly dependent over R. When showing the smoothness of g, it suffices, since gp(Wl' W2) = -Wp(iWl, W2), to show for a smooth tangent vector field X on an open set U ~ c:pn that iX is smooth too. This is left to the reader. The last part of the theorem follows directly from Proposition 14.7. 0
Corollary 14.9 Let W be the closed 2-form on c:pn constructed in Theorem 14.8. The j -th exterior power wj represents a basis element of H2j (c:pn) when 1 ~ j ~ n. Proof. The class in H 2n (c:pn) ~ R determined by volcpn is non-trivial. Since [w] E H 2(c:pn) we have [wt = n![vOlcpn]
°
Therefore [wt t= and thus [w]j from Theorem 14.2.
t=
E
H 2n (c:pn).
0, for j ~ n. The assertion now follows 0
\"
~
'"
~\
..
~~
.. 147
iyv, 1/ = 0, 1. The Hopf ph: R4 -+ R3 given by
1)
FIBER BUNDLES AND VECTOR BUNDLES
Definition 15.1 A fiber bundle consists of three topological spaces E, B, F and a continuous map 7r: E -+ B, such that the following condition is satisfied: Each b E B has an open neighborhood Ub and a homeomorphism h: Ub x F
such that
will have coordinates ented real orthonormal
I by taking the matrix
2xOYO - 2X1Y1 ) 2 2 2 2 oYo + Xl - Y1 . - 2X OY1 - 2YOX1
In
15.
2
= 1) Shows that
~(v)S2 with respect to = 7r: S3 -+ 0:»1 with hain rule gives that
I) ~ (1/2,1/2). Hence
:; 9 is isometric with
7r 0
-+ 7r-
1
(Ub)
h = proh.
The space E is called the total space, B the base space and F the (typical) fiber. The pre-image 7r- 1 (x), frequently denoted by Fx , is called the fiber over x. A fiber bundle is said to be smooth, if E, Band F are smooth manifolds, 7r is a smooth map and the h above can be chosen to be diffeomorphisms. One may think of a fiber bundle as a continuous (smooth) family of topological spaces F x (all of them homeomorphic to F), indexed by x E B. The most obvious example is the product fiber bundle c:~ = (B x F, B, F, proh). In general, the condition of Definition 15.1 expresses that the "family" is locally trivial. Example 15.2 (The canonical line bundle). At the beginning of Chapter 14 we considered the action of Sl on s2n+1 with orbit space epn. We view this as an action from the right, z. A = (zo A, ... , Zn A). The circle acts also on s2n+1 X e k, (z, U)A = (ZA, A- 1U). The associated orbit space is denoted s2n+1 XSI e k. The projection on the first factor gives a continuous map 7r: s2n+1 XSI
e k -+ n~n
with fiber e k • Similarly, if Sl acts continuously (from the left) on any topological space F we get 7r: S2n+1 X Sl F -+ epn. This is a fiber bundle with fiber F. Indeed, we have the open sets Uj displayed at the beginning of Chapter 14 which cover epn, and the smooth sections Sj : Uj -+ S2n+1 from (14.7). We can define local trivializations by Sj([z], u) = (Sj([z]), u) E
7r-
1(Uj)
for [z] E Uj and U E F. If F is a smooth manifold with smooth Sl- action then we obtain a smooth fiber bundle. If we take F = e with its usual action of Sl then we obtain the dual Hopf bundle, or canonical line bundle, over epn. It will be denoted H n = s2n+1 XSI e It is a vector bundle in the sense of Definition 15.4 below.
<
...
1
(~;.·".~· '·.·
-
"Ii
~""'<.
2
--;;:.'
en ~s
15.
~ have similar bundles x SO D 1 is the Mobius
fiber bundle where the vhere the local homeo (x, -): V --t 7f-] (.r) is
ling on which category ~ being we concentrate )r bundle that is also a s the dimension of the les. If ~ is a vector bundle its fiber over b. If e. E(~I w) = 7ft(W).
;b,
Ie
a smooth manifold.
I I
defined in a neighborhood W of Po, with the property that they are orthonormal for each pEW and such that X 1 (p), ... , Xn(p) E TpM. Hence Yl(p), ... ,Yk(P) E Np(M) is a basis and the map
h: W x IRk
show that the triple lose a parametrization
--t
7f- 1(W);
k
h(p, t)
=
2:= tiYi(p) i=1
is a local trivialization.
Definition 15.7 (i) A map (j, j) between (smooth) fiber bundles (E, B, 7f) and (E', B', 7f') is a pair of (smooth) continuous maps
f: B
--t
l: E
B',
--t
E'
such that 7f' 0 j = f 0 7f. (ii) A homomorphism between (smooth) vector bundles ~ and is a (smooth) 7f-l(x) --t (7f,)-I(j(x)) is linear for all fiber bundle map such that x E B.
e
l:
Example 15.8 A smooth map f: M --t M' between smooth manifolds induces a map of tangent bundles (j, T j), where
r(p,v) =p. ~
149
FIBER BUNDLES AND VECTOR BUNDLES
j
Tpf = Dpf: TpM is the derivative of
--t
TpM'
f.
Definition 15.9 Vector bundles ~ and TJ over the same base space B are called isomorphic, if there exist homomorphisms (idE, j) and (idE, fJ) between them
x)(v).
such that j 0 fJ = id = fJ 0 j. A vector bundle which is isomorphic to a product bundle is called trivial, and a specific isomorphism is called a trivialization.
smooth manifold. Let
In the above definition the homomorphisms j and fJ are assumed to be smooth when the vector bundles are smooth. The next lemma is a convenient tool for deciding if two bundles are isomorphic.
l:
len v E Np(M).
the proof of Lemma
Lemma 15.10 A (smooth) continuous map E(O --t E(TJ) of (smooth) vector bundles over B, which map the fiber Fb( 0 isomorphically onto the fiber Fb( TJ), is a (smooth) isomorphism. Proof. Since j is a bijection, it is sufficient to show that j-l is a (smooth) homomorphism of vector b~ndles (over idE)' We need to check that j-l is continuous (smooth). Since
f
is a fiberwise isomorphism, it is enough to examine
j-l: 7f;;I(U)
--t
7fi1(U)
c
,
~.
~
~
'~'~
.~
..
1~
"
._"' .... "-,,-.~_
....-,
~J .<~
:s
15.
j
7f;;I(U)
i j ) --t
7f;;I(U). Note that
(smooth) if and only ion (- )-1: GLn(R) --t
llS
o
I
I i I ,
1
les over the same base
•7f1J(w)}
Tf) b is equal to ~b ffi Tfb.
l r
undle ~ is a (smooth) . an inner product on
FIBER BUNDLES AND VECTOR BUNDLES
151
Remark 15.14 In the above proposition we used the existence of a continuous partition of unity, i.e. continuous functions Q'i: B --t [0,1] with SUpp(Q'i) C Ui and L Q'i (b) = 1 for all b E B. When B is a smooth manifold the existence of a smooth partition of unity is proved in Appendix A. More generally, a Hausdorff space B is called paracompact if every open covering {Ua,} has an open refinement {v~} which is locally finite. For a given open cover {Ua,} of a paracompact space there exists a partition of unity subordinated to {Ua,}, i.e. continuous functions Sa: B --t [0,1] with SUPP(Sa) C Ua and such that each bE B has an open neighborhood Vb for which {Q' I sa!Vb i= O} is a finite set. Definition 15.15 A (smooth) section in a (smooth) fiber bundle (E, B, F; 7f) is a (smooth) map s: B --t E such that 7f 0 S = E. The set of sections of a vector bundle ~ is a vector space r( O. One adds sections by using the vector space structure of each fiber. The origin in f(~) is the zero section which to b E B assigns the origin in the fiber ~b. If ~ is a smooth vector bundle then we let 0°(0 c r(o denote the subspace of smooth sections. It follows from local triviality that in a neighbourhood U of each point of the base, we can find sections SI, ... , Sn E f(~IJ (or in OO(~IU)) such that {SI(X), ... , sn(x)} is a basis of ~x. We call this a frame. If ~ has an inner product we may even choose sections locally so that {SI(X)"", sn(X)} is an orthogonal basis (Gram-Schmidt). We say that {SI,"" sn} is an orthonormal frame for ~ over U. Let (j,idB) be a homomorphism from ~ to Tf, and let {sd, {ti} be frames over U. Then ix: ~x ---t Tfx is represented by a matrix, and we obtain a map (1)
ad(j): V
---t
Mn(R)
Uls an inner product.
depending on the given frames. In the smooth situation ad(j) is smooth. Note that f::: ~x ---t Tfx is an isomorphism if and only if ad(jx) E GLn(R) . If ~ and Tf have an inner product, and {Si}, {ti} are orthonormal frames, then ix is isometric precisely if ad (jx) E On, the orthogonal subgroup of GLn(R).
unity {Q'd ;==:1 with
Lemma 15.16 Let ~ and Tf be (smooth) vector bundles with inner product over the compact space B, and let i: ~ ---t Tf be an isomorphism. Then there exists an f > 0 such that every homomorphism g: ~ ---t Tf that satisfies Ilib - gbll < f for b E B is also an isomorphism.
Vi
X
Rn and hence,
o ifold is the same as
Proof. If ~ and Tf are triVial, then after choice of frames, f and 9 are represented by maps ad(j): B ---t GLn(R) and ad(g): B ---t Mn(R). Since B is compact and GLn(R) is open, some f-neighborhood of ad(j)(B) in Mn(R) is still contained in GLn(R). But then ad(g)(B) C GLn(R) when g satisfies the condition of the lemma. In general, we can cover B with a finite number of compact neighborhoods
..... :.
"'t-l. -.
1·
....
~i ..
q,..
,
~.,~
~
..
,J.'
~
.~
..•. --.,;
ILES I
15.
153
FWER BUNDLES AND VECTOR BUNDLES
Theorem 15.18 Every vector bundle E over a compact base space B has a complement TI, i.e. E~ TI ~ EN (for a suitably large N).
of the resulting epsilons.
o
Proof. Choose an open cover U 1 , . .. ,U1' of B admitting trivializations hi of EIU;' and let {O:i} be a partition of unity with SUpp(O:i) CUi. Denote by Ii the composite
wonder if there is any md smooth isomorphism. ~ same lines one may ask over a compact manifold e (cf. Exercise 15.8). t
.
7rt(UZ)
h- 1
~
. UZx
.
Rn P~2 Rn ,
and define 'er the compact manifold moothly isomorphic.
(2)
local orthonormal frames er Ui.
This is a fiberwise map and gives a homomorphism S: E -+ E nr which is an inclusion on each fiber. We give Rnr the usual inner product and let
S: E(O S (v)
=
(7r ~ ( V ),
naps
-+
o:t{ 7r~ ( v) ) l
E(TI) = {(b, 1))
B x Rn7';
(v), ... , O:r (7r~ ( v) ) r (v) ) .
I v E S(Fb(O)J..}·
It is easy to see that
h IICi(x) - ad(/~)II
1(Ui )
-+
<
TI
t
7r~1(Ui) with
= (E(7]), B, Rnr - n , proh)
is a vector bundle (cf. Example 15.6) and by definition
E~ 7] =
E
nr .
o
If Ein Theorem 15.18 is smooth then so is the constructed complement 1], provided hi and O:i are chopsen smooth. The above proof uses that B is compact to ensure r < 00. The theorem is not true without the compactness condition; see however Exercise 15.10 when Eis a smooth vector bundle - it is the finite-dimensionality which counts.
"\.1' can then use a smooth
Let Vectn(B) denote the isomorphism classes of vector bundles over B of dimension n. Direct sum induces a map Vectn(B) x Vectm(B) ~O:i(b)t = t. With t as
ler and hence a smooth
o
lundles associated to a and the normal bundle ldeed, by construction efined frame for T ~ v. over M of dimension 'Its of vector bundles.
i I
Ell -+
Vectn+m(B)
such that 00
Vect(B)
= II Vectn(B) n=O
becomes an abelian semigroup. The zero dimensional bundle EO = B x {O} is the unit element.
To any abelian semigroup (V, +) one can associate an abelian group (K(V), +)
defined as the formal differences a - b, or pairs (a, b), subject to the relation
(a
+ x)
- (b + x) = a - b
."..,;
"
£"'fJ :
:-
--...,.
15.
universal property that tors over K(V), i.e. is onstruction V ---+ K(V), to the way the integers we do not demand that B is compact, we define
't'o
Theorem 15.21 If fa and h are homotopic maps, then isomorphic. Proof. Let F: X x I
fa(x)
=
155
FIBER BUNDLES AND VECTOR BUNDLES
---+
F(x,O) and h(x)
fo(~)
and
H(O
are
B, I = [O,lJ ' be a homotopy between fa and fI, = F(x,l). When tEl we get [Jt(OJ E Vect(X). ---+
[ft(OJ
and
7]
It is sufficient to see that the function t constant.
is locally constant, and thus
Fix t and consider the bundles
rm [~J - [c:kJ, where [~J
(= projift*(O
= F*(~)
Indeed
[6 EEl 7]2] = [~] - [c:
k
]
over X x I. Since F = ft 0 proh on X x {t}, ( = 7] on X x {t}. We can choose a fiberwise isomorphism
-L
E(()
E(7])
"x /
8 2 ~ R3 is trivial, since
X x {t}
11 frame. We also know
xl, if
r(x)
[752 J
=1=
were equal to
0 for all x
The first step is to extend h to a homomorphism of vector bundles on X x [t 10, t + EJ for some 10 > 0. This can be done as follows. Since X is compact, there exists a finite cover U1 , ... , Ur of X with Ei > 0 such that both ( and Tf are trivial on Ui x [t - Ei, t + td. We can extend h to
E 52.
~
Ui
h) map and ~ a (smooth) (~) is the vector bundle
X
/
[t - Ei, t + tiJ
Let Q'1, ... , Q'r be a partition of unity on X with SUpp(Q'i) CUi. We define
'E(7])
k
E(() 1T' /*(0
• E(7])
hi
E(()
lon-zero section. We see
~
= Projl'
X x [t - 10, t
by j(x, v) = v. It is >morphic, so J* induces
where
E
KO(X),
j) become contravariant
+ EJ
= min (ti) by setting
k(v) = ~
/
L Q'i(Proj1
0
1T'((v») . hi(v).
Since hi(v) = h(v) when 1T'((v) E X x {t}, and since L:Q'i(X) == 1, we have k(v) = h(v) on X x {t}. In particular k is an isomorphism on X x {t}. We finally show that k is an isomorphism in a neighborhood X x [t - E1, t + 101] of X x {t}. Since X is compact, it suffices to show that k is an isomorphism on
.
-1:.· . ·' ~:J
",-...;
'\"
~
e-,..>:, .......
~,
~
'"
" ...... ~
'~,
LES
157
et e and s be frames of ( \.1n (R) the resulting map, ) E GLn(R), there exists s an isomorphism. CI llso KO(X)) is a homo le same map
'e base space is trivial. constant map with value y construction when f is
o
16.
OPERATIONS ON VECTOR BUNDLES AND THEIR SECTIONS
The main operations to be considered are tensor products and exterior products. We begin with a description of these operations on vector spaces, then apply them fiberwise to vector bundles, and end with the relation between the constructions on bundles and their equivalent constructions on spaces of sections. Let R be a unital commutative ring and let V and W be R-modules. In the simplest applications R = R or C and V and Ware R-vector spaces, but we present the definitions in the general setting. Denote by R[V x W] the free R-module with basis the set V x W, i.e. the space of maps from the set V x W to R that are zero except for a finite number of points in V x W. In R[V x W], we consider the submodule R(V, W) which is generated (via finite linear combinations) by elements of the form
+ V2, w) (V, WI + W2) -
(VI, w) - (V2, w) (V, wI) - (V, W2) (rv, w) - r(v, w) (v,rw) - r(v,w) (VI
~s.
There is a completely or bundles. In Definition ector spaces and h(x, - ) plex vector bundles is a ;omplex vector bundle is I hermitian inner product and 15.21 have obvious
8 of complex dimension IUp whose corresponding
:what easier to calculate
(1)
where
Vi
E V, Wi E IV and r E R.
Definition 16.1 The tensor product V 0 module R[V x W]/ R(V, W).
R
W of two R-modules is the quotient
Let 7f: R[V x W] ~ V 0R W be the canonical projection and write V0RW for the image of (v, w) E R[V x W]. It is clear from (1) that 7f: V X W ---* V ®R W is R-bilinear. Moreover, it is universal with this property in the following sense:
Lemma 16.2 Let V, Wand U be R-modules, and let f: V x W ---* U be any R-bilinear map. Then there exists a unique R-linear map 1: V 0R W ---* U, with f = 10 7f. Proof. Since the set V x W is a basis for the R-module R[V x W], f extends to an R-linear map i: R[V x W] ---* U. The bilinearity of f implies that j(R(V, W)) = 0, so that j induces a map from the quotient V ®R W to U. By construction f = 10 7f, 1 is R-linear and since 7f(V x W) generates the R-module V ®R W, is uniquely determined by f. 0
1
1
It is immediate from Lemma 16.2 that tensor product is a functor. Indeed, if
V x W 'P~'I/J Vi x Wi :::.: Vi ®R Wi
'~
:c--...
"~. ~ .. ~ ~..... ~ ~ Q...
.-j
.....,J .~
'S;:~ ~; .... 00
. 1 ~.
...
l':-~c ::t ... ~
;g ·,.n
~i._.
'"
~.
·~jk:~,..;
lSECTIONS
16.
OPERATIONS ON VECTOR BUNDLES AND THEIR SECTIONS
159
Lemma 16.4 For every R-multilinear map f: VI x ... x V k - t W there exists a unique R-linear map VI ® R . , . ® R Vk - t W such that the following diagram commutes.
1:
: cp'
0
cP ® R 7/;'
0
VI
1/) when
X '"
• W
~
ng, not every R-module c V, but if it does we
Band B', Then V ®R V'
f
x Vk
/i
VI0R'" ®R Vk
o It is easy to see that the k-variable tensor product is isomorphic to the iteration of the two-variable ones, (... ((VI ®R V2) ®R V3 ) ®R .. · 0R Vk) ~ VI 0R .. · 0R Vk·
It the stated set generates
~
Let us specialize to VI = ... = Vk = V. Consider the sub-R-module Ak(V) ~ &J~ V generated by the set {VI
0R··· 0R Vk
I Vi
E V and Vio = Vio+! for some io}.
the linear maps with Definition 16.5 The quotient module
A~(V)
k
= ®R (V)jAk(V)
composition
~fore
is called the exterior k-th power.
The image of be zero.
VI
0R ... 0R Vk in A~(V) under the canonical projection
0 k
® R (V)
7fI :
is denoted
VI
-t
A~(V)
I\R ... I\R Vk. Then for a permutation
= sign(fJ) Va(I)
fJ
13 ,
(5)
;.2. For example, scalar :0 V, and hence a map v E V into l®RV. The
as in Lemma 2.2 and Lemma 2.7, so that the composition (6)
{ariable tensor products ~mma 16.2, namely
Vi = Vj for some i
VI
I\R ... I\R Vk
p = 7fI
0
7f:
V
X .. ,
x V
I\R ... I\R Va(k)
-t
A~(V)
is an alternating map, i.e. an R-multilinear map with p( VI, ... , Vk) = 0 when < j. With these definitions we get from Lemma 16.4:
-;
C"lJ
-.
~
.
.q,.,."
..
<
, ~.
'r
~
"'"
"
L---J
: SECTIONS
---+
16.
W there is a unique D
strict attention to finite ~ dual to the alternating notation we drop the W) instead of l8lRk (V),
161
becomes multiplicative, ~(Wl /\ W2) = T,O(Wl) /\ T,O(W2), where the product on the range is the one from Definition 2.5. The composition 'IjJ : Ak(V*) ® Ak(V) y?~l Altk(V) ® Ak(V) ~ R
with ev (w, Vl/\ ... /\ Vk) = W(Vl,"" Vk) induces the isomorphism 'IjJ: Ak(V*) ~ Ak(V)* in Theorem 16.7.(iii). On the other hand, Lemma 2.13 gives that
(7) erior product introduced
OPERATIONS ON VECTOR BUNDLES AND THEIR SECTIONS
'IjJ((El/\'" /\ Ek) is> (VI /\ ... /\ Vk)) = det((ci(Vj));'j=l)'
In particular, if V has an inner product (, ), then we may identify V with V* by sending V to (v, - ), and 'IjJ becomes the map 'IjJ( WI /\ ... /\ Wk, VI /\ ... /\ Vk) = det ((Wi, Vj )7,j=1)'
Theorem 16.7.(iii) then translates into
Addendum 16.8 (Grassmann inner product) For an inner product space V, the formula (WI /\ ... /\ wk, vIA ... /\ Vk) = det( (Wi, Vj)~,j=l)
o
defines an inner product on Ak(V).
i1 < ... < id is a basis For vector spaces V and W there is an obvious linear map )*.
V* ® W
[lce dimAk(V*) ~ (~). le set in (ii) generates rnAk(V*), the common proves (i) and (ii). For map, and so defines by s a map
)th sides agree, 'IjJ is an
~
o
---+
which takes j ® W into the linear map v that this is an isomorphism V* ® W
(8)
~
Hom(V, W) 1---+
j(v)w. It follows from Lemma 16.3
dim W <
Hom(V, W);
00.
The above constructions on vector spaces induce constructions on vector bundles (over the same base space) by applying them fiberwise. So if and "1 are (smooth) vector bundles over X then we get new (smooth) vector bundles over X:
e
(9)
e ® "1,
Hom (~, "1),
l8l k (e),
Ak(~),
C,
Altk(e)
and so forth. The fiber over x E X in each case is the associated construction on ~x, "1x. To be more specific, let us run through the definition in one of the cases, say Hom (e, "1). We define
E = E(Hom(e, "1)) =
Il Hom(ex, "1x) xEX
/\ Wl/\· .. /\ WI. Then
with the obvious projection map 7r onto X, which maps the entire vector space Hom (ex , "1x) to x E X. The problem is to define a topology on E that makes this the total space of a vector bundle.
<
."...; .
~
,.
"
1 1·,,~: .· ·~.· ·: ....'. ....
"
.
~~
16.
SECTlONS
163
OPERATlONS ON VECTOR BUNDLES AND THEIR SECTIONS
rand rll uJ are trivial. J
The isomorphisms of Theorem 16.7.(iii) are again defined without reference to a basis, and so give isomorphisms
Rm .
(11)
a topology on E:
for any vector bundles ~.
We have in Theorem 16.7, Addendum 16.8, Lemma 16.9, and (11) concentrated
on real vector spaces and vector bundles, and leave the reader to formulate and
prove the corresponding results for the complex cases.
Every complex vector bundle ~ gives rise to a real vector bundle ~R upon forgetting
part of the structure. On the other hand, every real vector bundle induces a where complex vector bundle 7Jc upon complexifying each fiber, 7JC = 7J ®R denotes the trivial complex line bundle. There are the following relations between these operations.
I j.
let topology. It is left lat it is independent of , that Hom(V, W) is a
Ak(C) ~ Ak(~)* ~ Altk(~)
cb
ct
Lemma 16.10 1 ),
7
ld 7J
(ii) For a complex vector bundle ~, (~R)C ~ ~ EEl ~
X x W give
Idles when
~
(i) For a real vector bundle 7J, (7Jc)R ~ 7J EEl 7J.
Hom(V, W));
~
and 7J are
Proof. We leave (i) to the reader and prove (ii). Given a complex vector space V we get a new one V ®R C where scalar multiplication is in the second factor. The map
isomorphisms ~ ~
C*
ral isomorphisms
T)
We choose a hermitian product on D
nensions agree. How .11 below. The above V to V* defined in Jne a homomorphism
= (zv,zv)
defines an isomorphism of the underlying real vector spaces, and
f vector bundles (over
.emma 15.10.
C.
~,
V EEl V*.
and apply
D
We next consider spaces of sections. We shall primarily be interested in the space of smooth sections 0°(0 of smooth vector bundles; cf. Definition 15.15. For the tangent bundle TM, (12)
OO(Altk(TM))
=
Ok(M).
Indeed, an element of the left-hand side associates to each x E M an element W x E Altk(TxM), and since 0°(-) denotes the space of smooth sections,
"
(
~:
...t'jJ , czc,.......
.\
'"
~-
-::
"
1···••·j· •·
q".;..
... ,
.
~
..•....·· .. 1•.'.
,~
:crIONS
16.
5. Similarly nO( 7M)
F: nO (Hom (~, 77))
--+
Indeed, if it is then a ( that el (p) ... , ek(p)
lpact snwoth manifold ifree nO(M)-module. EB "rJ e:! [Hi. Then
!)
).
(sv)(x) for two sections of ~ with sv(x)
ive R-modules. Then
HOmnO(M) (no(o, nO(77))
--+
by F((j5)(s) = (j5 0 s, s E nO(O. Suppose F((j5) = O. To check injectivity of F we must show that (j5x: ~x --+ 77x is the zero homomorphism for every x E M. Fix x E M and v E ~x. There is a section Sv E nO(O with sv(x) = v. Indeed, such an Sv can always be defined in a neighborhood U of x E M where ~Iu is trivial, and we can obtain a global section upon choosing a smooth function I on M with supp (I) c U and I(x) = 1 (cf. Appendix A) and replace the local section Sv by I sv, giving it the value zero outside U. Now' F($)(sv) = 0 implies that (j5x(v) = O. Since x and v were arbitrary, F is injective. Let E HOmnO(M) (nO(~), nO(77)). We wish to define a fiberwise smooth homomorphism 0: ~ --+ 77 by setting $x (v) = ( sv) (x) where Sv E nO (~) is a section with sv(X) = v. This requires that one has
o
lodules, so the above projective R-module,
(13)
if
= (s~)(x)
= s~(x),
s(x) = 0
or in other words that
(s)(x) = O.
then
Chose sections el, ... ,e" E nO(~) such that el(p), ... ,ek(p) form a basis for~p for p in some neighborhood U of x. Again this can be done locally, and the local sections can be extended to global ones as above. Now
s(p) = L Ii (p)ei(p)
P2
165
Proof. By definition, nO (Hom (~, "rJ)) is the space of smooth fiberwise homomor phisms $: ~ --+ 77, and we define an nO(M)-linear homomorphism
r is always
a module R is a smooth map of~. We can apply l(~), n O(77), etc.
OPERATIONS ON VECTOR BUNDLES AND THEm SECTIONS
for p E U
for smooth functions Ii defined on U. Let >. E nO(M) have Supp(>.) C U and >.(x) = 1. Then
(s) = (>.s + (1 - >.)(s)) = (>.s)
free modules, where :es. The general case n 1 rt , P2 EB Q2 = R 2.
o
+ (1 -
>')(s)
so that (s)(x) = (>'s)(x). But >.s = L (>.Ji)ei and >'Ii extends to a smooth function gi defined on all of M with gi(X) = li(X) = O. Since is nO(M)-linear,
(>.s) = Lgi(ed E nO(77). But gi(X) = 0 so (>.s)(x) = O. This proves (i).
Assertion (iii) is the special case of (i) corresponding to 77 bundle, and (ii) follows from Lemma 16.9 and (i):
nO(~ 0 77) ~ nO(Hom (C, 77)) ~
HOmnO(M)
= [1, the trivial line
(nO(C), nO (77) )
~ HOmnO(M) (HomOO(M) (nO(~), nO(M)), nO(77)) ~ nO(~) 0nO(M) nO(77)
~.
.....;
,C'fJ
..
,..
~
.....~
~
~-$
q... .
";J;
.~
.... ~
";~
,
~;~
.
~'.' .. ;~ .'
,~~~S·P,> .
.
167
:nONS
nma 16.12. Finally
17.
Let
CONNECTIONS AND CURVATURE
~
be a smooth vector bundle over a smooth manifold Mn of dimension n.
Definition 17.1 A connection on
~
is an R-linear map
\J: ~o(~) - ~l(M)
01l0(M)
~o(o
see that the bottom m 16.7.(ii), and local 0 rcise.
which satisfies "Leibnitz' rule" \J(f . s) = df 0 s + f . \Js, where f E ~O(M), s E ~O(~) and d is the exterior differential. If ~ is a complex vector bundle then ~o (0 is a complex vector space and we require \J to be (:-linear.
msor products, stated an R-algebra. In our the smooth functions C). Suppose that V on V and from the
Let 7 be the tangent bundle of M. Then ~l(M) = ~0(7'"), and by Theorem 16.13 we have the following rewritings of the range for \J, (1)
~l(M) ®1l 0 (M) ~o(o ~ nO(Hom(7,O) ~ HOmIlO(M)(~0(7),~0(~)).
A tangent vector field X on M is a section in the tangent bundle X E nO (7), and induces an ~D(M)-linear map Evx: ~l(M) -+ ~O(M), and hence an ~O(M) linear map
Evx : ~l(M)
0s W is the cokemel
01l0(M)
~o(o - nO(o.
The composition Evx 0 \J is an R-linear map \J x: nO(o (2),
llications, then there r 0 S W becomes an Definition 16.1. We
fch is S -balanced in V and S E S. Then
o
nO(~) which satisfies
\Jx(fs) = dx(f)s + f \Jx (s),
where dx(f) is the directional derivative of f in the direction X, since Evxodf = dx(f). Thus a connection allows us to take directional derivatives of sections. For fixed s E ~D(~) the map X -+ \Jx(s) is nO(M)-linear in X:
\JgX+hY(S! = g \J X (s)
+ h \Jy (s)
for smooth functions g, h E nO(M) and vector fields X, Y E nO(7). Moreover, the value \J x (s ) (p) E ~p depends only on the value X p E TpM. This is clear from the second term in (1) which implies that \J can be considered as an R-linear map
\J: nO(~) - HOM(7,~)
:=
nDHom (7, 0.
Here the range is the set of smooth bundle homomorphisms from 7 to identity). If X p E TpM then \Jxp(s) = (\Js)(Xp), and (3)
-."'~"
!",-
~
(over the
\Jxp(f· s) = dXp(f)· s(p) + f(p) \Jx p (s) \JaXp+bYp(S) = a \Jxp (s) + b \JyP (s)
where Xp, Yp E TpM, and a and b are real numbers. Conversely (3) guarantees that \J x p (s) defines a connection.
-, -
'~~,
c
...t::
<
.,...; ~
~!
..
~"
-,j-f'-,:",-,
"
~'C'~"·'"
,'-~
~
.~.
,~
17.
. One can defi ne a ( T ) can be considered we set
defines a connection V on 1
n (M)
(p E TpM.
It is easy
operator in the sense lbset U ~ M then so 's induces an operator :l to open subsets. In
?) is a basis for
~p
for lO(~IU) can be written ~ction V on ~,
169
CONNECTIONS AND CURVATURE
~.
0nO(M)
Note that
nO(sn+k) ~ nO(M) EEl nO(sn+k) ~ n 1 (M) EEl
EB nO(M) ED n 1 (A1),
and that vo = dEB . .. EEl d. If ~ and 7f are complex vector bundles and sn+k is the trivial complex bundle, then the construction gives a complex connection. Example 17.3 shows that every smooth vector bundle over a compact base man ifold has at least one connection, since bundles have complements by Theorem 15.18. We observe that Example 17.2 is a special case of Example 17.3 corre sponding to ~ = TM and 7f = ]/M; see also the exercises.
Remark 17.4 After choice of a connection V on ~ one can compare the fibers ~p at different points p E M by a "parallel translation along curves". Let o:(t) be a smooth curve in M and w(t) E nO(~a(t») a section along 0:, i.e. w(t) = w(o:(t)) for some w E nO (0. There exists a unique operator ("covariant differentiation") that satisfies:
:it
. D(W1 + W2) DW1 DW2 dt = ----;rt+----;rt
.. D(J ·w) _ df fDw
(n) dt - dt w + dt
... Dw
(111) ---;It = Va'(t)w. (1)
called the connection and a frame for ~IU' IU) can be written as
Suppose first that o:(t) C U, where (U,x) is a chart on M. Let ai = O~i E nO(TU) and let e = (e1, .. " ed be a frame of nO(~IU)' Then o:(t) = X-1(Ul(t), ... , un(t)) for smooth functions Ui(t), and w(t) = EWi(t)ei(t), where ei(t) = ei(o:(t)). Conditions (i), (ii) and (iii) give
+ SiAij) 0 ej.
Dw = ' " (dwiei(t) +Wi(t) Val(t) (e i )). dt L..J dt and since a-'(t)
tk)A.
=
E~. ai, (4) implies that for certain smooth functions n
ld j: sn+k --t ~ be the . We give the trivial maps
" dIVOj(ei) dUj " dIfjiel!' dUj I! Va/(t)ei = 'L..J = 'L..J j=l j,1! This gives
~
Dw _ (dW/ dt - L..J dt [=1
I(~),
" dUj fl, z.) + 'L..J dt Jzw e/. i,j
:it
M)
0nO(M)
qi on U
nO(~)
Conversely this formula defines an operator which satisfies (i), (ii) and (iii). Since we can cover 0:( t) with coordinate patches, the assumption that 0: be contained in just one chart is irrelevant.
.~
",
.- ~
.
~
j .~ .
17.
= O. For a given Ih section, and the (1) .
.
,.~ ..... 'Ill;;;.,
171
CONNECTIONS AND CURVATURE
L
. Proof. Let 7 E OJ(M) and 8 E OO(~) and set d'V(7 0 8) = 4t'/\ 8 + (-1)3 7 /\ \78. One checks that d'V is OO(M)-ba1anced in the sense of Definition 16.14, and applies Lemma 16.15. Since 8 E 0°(0 we have dT /\ 8 = d7 0 8, and d'V = \7 when j = O. We show that (ii) is satisfied. For W E ni(M) and t = 70s E oJ (~),
s) = d(w /\ 7) @ 8 + (-1 )i+i (w /\ 7) /\ 'V 8 = (dw 1\ 7) @ 8 + (-1 w 1\ d7 0 8 + (-1 )i+i (w /\ 7) 1\ 'V 8 = dw /\ (7 @ 8) + (-1) i w /\ d'V (7 @ 8) . 0
d'V (w /\ (70 8)) = d'V ((w /\ 7) which satisfies the
@
r
We ·have now a sequence in spirit to Theorem (8)
M)-bilinear product
0--+
nO(o
.:s nl(~) ~ 02(~)
--+ '"
which when ~ is the trivial line bundle ~ = E 1 and \7 = d is precisely the de Rham complex of Chapter 9. One might expeet that (8) is a complex, i.e. that d'V 0 \7 = 0 and d'V 0 d'V = 0, but this is in general not the case. We do have however that
F'V = d'V
0
\7: nO(o
--+
n2(~)
is nO (M)-linear, since w/\ 7 is the exterior
d'V 0 \7 (J 8) = d'V (df /\ 8 + f 1\ \7 s ) bundle. In this case the nO(M)-module that w /\ 8 = W 0 8 ,the exterior product unction 1 E nO(M)
= ddf /\ 8 - df /\ \78
+ df 1\ \78 + f
/\ d'V(\78)
= f(d'V 0 \7(8)). On the other hand Theorem 16.13 gives (9)
HOmnO(M) (nO(~), n 2(0) ~ n 2(Hom (~, 0).
Indeed, there is the following string of isomorphisms 2 HOmnO(M) (nO(~), 0 2(0) ~ HOmnO(M) (nO(o, nO(~)) 0nO(M) n (M) ~ nO(Horn(~,~)) @nO(M) n 2(M)
~ n 2(Hom(E,E))·
o ) --+ ni+1(~)
that
;f) and t E nj(~).
Definition 17.7 The 2-form F'V E n2(Hom(~,~)) is called the curvaturefonn of (~, \7). A connection \7 is called flat if F'V = O. Let X, Y E nO( 7M) be two vector fields. By evaluating a 2-form we get an nO(M)-linear map EvX,y: n 2(M) --+ nO(M)
7
at (X, V),
o
"'
..ec:.J
~
"..J C)-l
,§
e; ~~
-.
~
1
.~.
~
'"
.>
.~
,~
17.
173
CONNECTIONS AND CURVATURE
Example 17.9 Let H be the canonical complex line bundle over Cpl from Example 15.2. Its total space E(H) consists of pairs (L, u) E Cpl X C 2 with
)).
U
,. (p): ~l'
E L.
Indeed, the map
~ ~l' depends
i: S3 XSl C ~ Cpl
C 2;
X
[ZI' Z2,
u]
1--+
([ZI' Z2], UZl, UZ2)
is a fiberwise monomorphism, whose image is precisely E(H). It follows that a complement to H is the bundle H..l with total space
E(H..l) 'v
=
{(L,v) I v
E
L..l}.
® ev
tjV) 0
We want to explicate the projection 1r: Cpl X C2 ~ E(H), which maps (L, Ul, U2) to the pair (L, u), where U is the orthogonal projection of (Ul, U2) onto the line L. If L = [ZI,Z2] with IZ112 + IZ212 = 1, then
ev )
tation:
1r(L,Ul,U2) = (Ul,U2) 'PL
the matrix of the linear
- A t\
Ah
where PL is the 2 x 2 matrix
Y,'
p, p
R
~i+2(0
_ (ZI ZI
[Zl,Z2] -
Z1Z2 ) Z2 Z2 .
Z2 Z1
Indeed, if L contains the unit vector z = (ZI' Z2), then orthogonal projection in C2 onto L is given by the formula
)(0°(0, 0 2 (0)
1rL(Ul, U2)
=
+ Z2 U2)(ZI, Z2) = (Ul' U2)P[Zl,Z2]'
(ZI Ul
We examine the (complex) connection from Example 17.3,
\1: nO(H) ~ 0.0 (E2) ~ 0. 1 (E 2) ~ n 1 (H), can use (9) to rewrite
) ~ Oi+2(0
g: R2 ~ VI with
=wt\F'V(s).
Z
= x + iy,
e(g(x, y)) where we also use
C Cpl;
g(x, y)
=
[1, z]
and let us consider the section e over VI
p\7.
'.6,
= (d, d),
by calculating the connection form A in (4) with respect to sections over the stereographic charts VI and V2 of Example 15.2. Let 9 be the local parametrization defined as
lundle homomorphism
maps t to t t\
\10
Z
=
(g(x, y), (1, z)) E Cp 1
X
C2
to denote the function on VI whose value at g(x, y) is x+iy.
Now
\1o(e)
o
,ely when \1 is a flat )t every vector bundle
= (g(x, y), (0, dz)),
dz
= dx + idy
and hence
\1(e) = (g(x, y), (0, dz) . Pg(x,y)) = (g(x, y), =
(g(x, y), 1 +l
lz12
(zdz, IzJ 2dz) ).
1+1'zI2 (0, dz) (~
1~2))
..
(
...
"...;
11
e-,.J
~
O;z--' ~
~. . . . .~
17.
175
CONNECTIONS AND CURVATURE
Lemma 17.10 There exists a unique connection 1*(\1) on diagram below commutes: y
nO(~)
--.5l
I*(~)
such that the
0,1 (~)
r
r
1
1
n0(J*(O) r(v) n1(J*(O)
Proof. The map f: M' -t M induces a homomorphism of rings n° (M) -t nO(M'), so that every nO(M')-module becomes an nO(M)-module. In particular n°(J*(~)) becomes an nO(M)-module, and there is a homomorphism of nO(M) modules 1*: nO(o -t n°(J*(~)),
lote that
\ dy
withf*(s)(x') = s(J(x')). We can then define a homomorphism of nO(M') modules
2"
n° (M') ®nO(M) nO(~) -t n°(J*(O)
d:r!\ dy.
by sending ¢/ ® s into ¢/ . 1*(8). This is an isomorphism; cf. Exercise 17.13. It follows that
l(g*H,g*H)):
nk(J*(~)) = nk(M')
®nO(MI)
n°(J*(~)) ~ nk(M')
®QO(M)
nO(~).
Similarly, pull-back of differential fOTITIS
jomorphism bundle section e(p) = idHp ' vature form F'V E
1*: nk(M)
-t nk(M')
is nO(M)-linear and induces a homomorphism nO(M') ®nO(M) nk(M) -t nk(M');
'2
is the parametriza
¢0w
I-'
¢I*(w).
This is not an isomorphism, but applying the functor (-) ®nO(M) nO(~) one gets a homomorphism p: n° (M') ®nO(M) nk(o -t nk(M') ®nO(M) nO(~).
The sum of the maps *(~),
C,
Horn(~,
1])
Idles equipped with Jet f: M' -t M be ction V. The map . with l*:n 1(M)-t
(J*(~)).
d®
1: n° (M') ®nO(M) nO(~) -t 0. 1 (M') ®nO(M) nO(~)
p(l ® V): nO (M')
@OO(M)
nO(o -t n 1 (M')
000(M)
0.°(0
defines the required connection
I*(v): n°(J*(O) -t n 1(J*(0)·
o
.~ ~.
.
,
.•
...•.~
~.
~.~.
17.
frame e for ~ Iu then eo! for !*(O/-l(U)' :mma 17.10 where \l
177
CONNECTIONS AND CURVATURE
Lemma 17.11 Under the identification
0::
C
@".,
~ Hom (~, ".,), \l~*®TJ =
\lHom(~,TJ)'
Proof. There is a commutative diagram of vector bundles over M
1
~
@
C @ ".,~~ @ Hom(~,,,.,)
1(,)®id'l
1(,)
c}lf @"., ~
".,
Let s E nO(~),
and a corresponding diagram of sections. s* E nO(c). Then
\lTJ((s, o:(s* @ t))) = (\l~(s), o:(s* ® t)) + (s, \lHom(~,TJ)(O:(s* d( (s, s*)) = (\l ~ (s ), s*) + (s, \l~. (s*) ) \l~.®TJ(s* @ t) = \l~*(s*) 1\ t + s* 1\ \lTJ(t).
f-j(M) -+ E 1 .
For i
t E nO(".,) and
= j = 0,
From the diagram we get that (s, o:(s*
@
~o(M).
t)))
t)) = (s, s*)t, and hence
(\l~(s), o:(s* @ t)) = (\l~(s), s*) 1\ t (s, \lC®TJ1m(s* @ t)) = (s, \lC(s*))
to the evaluation
@
1\
t
+ (s, s*) \lTJ (t).
On the other hand, using these formulas we have
(s, \lHom(CTJ) (o:(s*
n
. on
@
t))) = d((s, s*)t) - (\l~(s), s*) 1\ t = d((s, s*)) 1\ t + (s, s*) \lTJ (t) - (\l~(s), s*)
C by requiring
=
(s, \l ~* ( s*))
1\
t
t
+ (s, s*) \lTJ (t), D
and the assertion follows.
-singular. The desired deed the product from
1\
Each of the connections from (14), (15) and (16) can be extended to linear maps
ni(C) ~ ni+l(c)
I,
ni(~ @".,) ~ ni+l(~ @".,)
ni(Hom(~, ".,)) ~ ni+l(Hom(~, ".,))
and the defining formulas generalize to the following lemma, whose proof is left to the reader.
).
I Hom(~,
".,). Alterna :~) -+ nO(".,) and the )) -+ ni+j(".,) and de
Lemma 17.12 Let s E ni(~), s* E ni(c), t E n j (".,) and We have (ii)
= (d'V(s),s*) + (-l)i(s,d'V(s*)) d'V(s (fJ t) = d'V s rbt + (-l)i s 8 d'V(t) d'V((s,
(i) d((s,s*)) (iii)
E
nj(Hom(~,,,.,)).
.
...,.i
.:
....
1~ ~'~~.-.'~
e,.)
'to
... ~
. .
17.
CONNECTIONS AND CURVATURE
179
D
'espectively.
:t us state in local Let \l = \l~ be :r U. This defines
0)
The product in nO(Hom(~, product
associated to fiberwise composition induces a
/\ : ni(Hom(~,~)) ® nj(Hom(~,~))
-t
ni+j(Hom(~,O).
It is not hard to show that d\1 is a derivation with respect to this product, i.e. that
R)
d"V(Rl/\ R2)
(19)
= d"V(Rl) /\ R2 + (-l)i Rl /\ d"V(R2).
The trace homomorphism
(nn(u)).
Tr: Hom (V, V)
,
ed d\1 and d\1 then
-t
R
can be defined without reference to choice of basis as the composition Hom(V, V) ~ V* ® V ~ R
d"k (nn+l (U)).
where ev(J ® v)
= j(v).
It induces a trace Tr:Hom(~,O
/\A
-t
c1
-t
ni(M)
of vector bundles, and hence in tum a trace
)
Tr: ni(Hom (~, ~))
a follows from (5);
into the i-forms on M, and we have:
GLk(nO(U)), and ature forms F\1 (e ),
Theorem 17.14 For cjJ E ni(Hom(~, ~)), d Tr( cjJ)
= Tr( d"V cjJ)
where d\1 is associated to \l = \lHom(~,O'
ied to the equation Jm the first one and
Proof. Let s E nO(~), s* E nO(c), w E ni(M) and suppose cjJ= w ® s ® s* E ni(M)
®nO(M)
nO(~) ®nO(M) nO(~*) ~ ni(Hom(~,~)).
Then ~re
d\1 is associated
d\1cjJ
= dw ® (s ® s*) + (-l)i w ® \l(s ® s*) = dw ® (s ® s*) + (-l)i w ® (\l~(s) ® s* + s ® \le·(s*))
and we get Tr d"V cjJ = (dw)(s, s*)
+ (-l)i w /\
(( \l~(s), s*)
+ (s, \l~. (s*)))
= (s,s*)dw+ (-l)iw/\d((s,s*))
o.
o
= (s, s*)dw + d((s, s*)) /\ w = d((s, s*)w) = dTrcjJ.
D
...,.)'
j
~
1
~
tf·
~
'"
'2.
..
lSI
18. vector bundles, but ector bundles upon (Ai) by 0. i (M; C), naps to be C-linear I in Chapter 18. m \I.
Combining is {ham cohomology: that the trace of ) E
n2 (M; C)
CHARACTERISTIC CLASSES OF COMPLEX VECTOR BUNDLES
In this chapter ~ will be an n-dimensional complex smooth vector bundle over a smooth compact manifold AI, and 0.*(Al; C) will denote the de Rham complex with complex coefficients. Connections in this chapter will always be complex. Let P(A) = P( ... , Aij, ... ) be a homogeneous invariant polynomial of n2 variables displayed as an n x n matrix; cf. Appendix B. The most important examples are (1)
P(A) = <"TA,(A)
P(A) = sdA) = Tr(A k )
and
where rrdA) is the coefficient of t k in the characteristic polynomial n
det(I
+ fA) = L
<"Ti,;(A)f
k
.
k=O
is the cohomology
H 2k (M:C).
They both have degree k. Since the wedge product is commutative on even dimensional forms, we can replace the variables A = (A ij ) by differential 2 forms, Aij E 0. 2 (M; C), and thus obtain a 2k-form P(A) E 0. 2k (Af: C). More generally we define a map P: 0.2(Homc(~, 0) --t 0. 2k (j'vf; C).
(2)
Let e be a frame of
nplexified de Rham chosen so that the fhis will be proved Jhomology class is
Id a real connection
(~,
\I) is the class
~Iu;
it induces an isomorphism onto the trivial bundle, Hornc(~, 0
Iu
--t
U x M n ( C)
and hence an isomorphism 0. 2 (HOIllc(~, OW)
--t
2
0. (U; Mn(C))
--=.
1\;fn (0. 2 (U; C)).
Thus a 2-form R of Homc(~,O gives a matrix of 2-forms R(e) = (R ij ). Apply P and get an element P(R(e)) E 0. 2k (U; C). Since P is invariant, and since for any other choice of frame R(e) = gR(e')g-l with 9 E 0.°(U; Mn(C)) invertible, we have
P(R(e))
= P(R(e')).
It follows that we have defined a global 2k-form on AI which we denote P(R).
For example we have (locally) (3) 81(R) =
L Rii'
s2(R) =
Choose a complex connection get a 2k-form
(4)
L Rij v
1\
Rji'
<"T2(R) =
~(Sl(R)2 -
on ~ and apply the above with R
P(F\7)
Here are two fundamental lemmas:
E0. 2k (M;C).
82(R)).
= F\l to
~
.-i /o'~
1
::-_-~1:
"
j
l"'fJ'.
.~
~
L----.J I
18.
S
~v)
.
J2T '
I.
~___
"~7_;
CHARACTERISTIC CLASSES OF COMPLEX VECTOR BUNDLES
iii
183
Proof of Lemma 18.2. Let vo, VI be two connections on ~, and 7f: 11.1 x IR - t AI the projection onto the first factor. Let V 1/ = 7f* (v v) be the induced connections on 7f* (0; cf. Lemma 17 .10. Define a new connection on 7f* (0 by
is a closed
ependent of the
V(s)(p, t) = (1 - t)Vo(s)(p, t)
+ tVI (s)(p, t)
'I'"
where (p, t) E ]'vI x IR. Apply Lemma 17.10 to see that
Its of Appendix Inor-Stasheff]).
io(V)=vo,
it(V)=Vl
where i v : 1\11 - t 11.1 x I~ are the two inclusions in respectively top and bottom. From (17.2 I) it follows that
let V have the
~
~
io(F'V) = F'V o , and hence i:(P(P'V))~= P(F'V v f. (17.17), so
),
/!
ii(P'V) = F'V 1
=.:: 0,1. Since io '::::' i 1 and P(F'V) is closed,
we have that io([P(P'V)]) = ii([P(F'V)]).
1"\7)
1~I,
S
•
Note that isomorphic vector bundles define identical cohomology classes [P(F'V)],
since a smooth fiberwise isomorphism f: ~ - t ( induces isomorphisms between
section spaces, and since we can choose connections to make the diagram
I
DO (0 --'5:Z- r2 1 ( 0
1i.
I
I
DO(() -LD
1i. 1 (()
commute. Thus the matrices for p'V and F'V' are identical with respect to and P(P'V) = P(P'V').
corresponding frames for ~ and In particular, if ~ is a trivial vector bundle, then [P(~)] = [p(cc)] = O. Indeed,
we just use the flat connection vo on cc'
e,
I
1 i zero elsewhere.
0
I
Definition 18.3 (i) The k-th Chern class of the complex vector bundle
q(O =
E
is
H 2k (lvl', q .
(ii) The k-th Chern character class is
') F\7) = O.
[Jk ( 27fyCI -1 p'V)]
~
o
chk(O
;r
=
~[Sk( -1 P'V)] k. 27fH
E H
2k (M;C).
Here V is any complex connection on~. If k = 0 then co(O = 1 and cho(~)
=
dim~.
~
,BUNDLES
Definition 17.15. We It they determine each
18.
We integrate this form by changing to polar coordinates (x, y) Since
= cos edT
dx
= Pk(Sl(O,···, Sk(~))
we see that rix 1\ riy
£
xample we have
g*
- Tsin ede
= I'
1 1=
'l=
27r
('7) F = 21.·.
I'dI' 1\ de2 = 4Jrz
00+1'2)
0
=
= sin edT + Teos ede
dy
and
= (I' cos e, I' sin e).
dT 1\ de, and
.fF
- C2(0
185
CHARACTERISTIC CLASSES OF COMPLEX VECTOR BUNDLES
(Xl
21r1:Jo
.0
I'dI'
(1+1'2)
2
=
ds [ 1 ] (1+8)2 =-2Jrl 1+8 s=o =2Jri.
This calculation implies that
r
j: OP 1 C Cpn induces
F'V = 27rl.
JCPl
;e and for all with the
Indeed we can apply a partition of unity 1 = Po + Pl with supppo an arbitrarily large f-I-sphere in the chart g and SUPPPI a correspondingly small f-sphere in the other chart. In the limit f -----t () the integral of g*(F'V) (over all of 1R 2 ) is equal to the integral of F'V over Cpl. 0
Theorem 18.5 Let 1: N -----t 1'v1 be a smooth map and ~ a complex vector bundle on AI. For every invariant polynomial we have f*[P(O] = [P(f*(O)]. 'gy class of the volume 14.8) and if we identify 'arm of S2 in its natural
Proof. We give f*(~) the connection f*(\l) of Lemma 17.10. By formula (17.1 3), f*(F'V) = F!*('V). Hence f*(P(F'V)) = P(FI*('V)). 0
n.
For a line bundle L,
n2 (Holll(L, L)) = n2 (iV!; C) sk(F'7) = F'V
,pace
L}. pIe 15.2.
1\ ... 1\
so that F'V E
n 2(jII: C),
F'V.
This gives .
1
k
1
chdL) = 1,; ch l (L) = klc1(L) l
(9)
k
:1}) to -1.
so that chk(L) becomes the k-th term in the power series exp(cr(L)). charts~!- and 1/J+ on llated the pre-image of
Theorem 18.6 For a sum of complex vector bundles, (i) Chk(~O EB 6)
=
Chk(~O)
+ chd6)
k
(ii) cd~o ttl 6)
= L 1/=0
cv(~0)ck-v(6)·
and
··I':::3····~
'biJ:~
.......0
..
~.
~....•. ~.•'."<.?:.~.•. . . • ..t: ~ ......
·~··.···tJ·····.··~.~, .C;; ......
~
OR BUNDLFS
lentify
18.
ni (~O EB 6)
with
""
'"
~ -'~
.....;;
'~. ~.
]
.~
"...;
J."'i-l
~
CHARACTERISTIC CLASSES OF COMPLEX VECTOR BUNDLFS
187
There is a commutative diagram:
j
ni(Hom(~o, ~o)) 0 n (Hom(6, 6)) ~ni+j(Hom(~o0 ~l, ~o 06))
;1)
!
(11)
!
Tr
Tr0Tr
~
ni(M; C) 0 nj(M; C) 6))·
ni+j(M; C)
From (10) and (11) we get k
8k(F'V)
C)
=
?= (:) si(F'VO)Sk_i(F'V1) t=O
which is equivalent to the statement of the theorem. k
=
LO"v(Ao)'O"k-v(AI),
Let H 2*(M; C) denote the graded algebra
H 2*(M; C) =
v=O
EB H
D ~undles,
ch(~) =
(M; C).
~,
we define the Chern character by
L Chi(~) E H *(M; C). 2
This defines a homomorphism ch: VectC(M) -+ H 2*(M; q, which by Theorem 18.6.(ii) and the universal property of the Grothendieck construction can be extended to a homomorphism
:6)
ch: K(M)
a-wise, defines a map of
([~O]
6, ~o (6)).
tion \J on
~o
0
6 from
l(81)'
id + id 1\ F'V1 where ) id: ~p -+ ~p, It follows 1\
)A(k-i).
-+
H 2*(M; C).
An application of Theorem 18.7 shows that ch is a multiplicative map, when the product in K (M) is defined by
:1,~0 0 6)
71
2i
i~O
For a complex smooth vector bundle
n(~O 0
D
- [170])([6] - [171]) =
[~o
06]
+ [170 01]1] -
[~O
0171] - [17006]·
Without proof we state:
Theorem 18.8 The Chern character induces an isomorphism of algebras ch: K(M) 0z C
-+
H 2*(M; C).
D
Theorem 18.9 There exists precisely one set of cohomology classes Ck (0 E H 2k (M; C), depending only on the isomorphism class of~, and such that (i) I(C1(HI)) = -1, ck(Hn ) = 0 when k (ii) f*cdO = ck(f*(~)) (iii) cd~o EB 6) = 2::=0 Ci(~0)Ck-i(6).
> 1, and Co (Hn ) = 1
,.,..;
-.,
18.
ailed splitting principle,
'"
..~
.<\-..
OR BUNDLES
"
11
r,J
CHARACTERISTIC CLASSES OF COMPLEX VECTOR BUNDLES
189
Properties 18.11 (a) Ck(~) = 0 if k > dim ~ (b) Ck(~*) = (-I)k ck (O, chk(C) = (-I)kchk(~) (c) C2k+ 1(1]c) = a and ch 2k+l (1]c) = a for a real vector bundle 1].
ctor bundle ~ on M there T ---* M such that
Proof. For a line bundle, (a) follows from assertions (i) and (ii) of Theorem 18.9, because every line bundle ~ has the form 7r*(Hn ). If ~ = 1'1 EB ••• EB 1'n is a sum of line bundles, then
on 18.3 satisfy the three t.
~). Let L be an arbitrary
C(O =
II (1 + C1('Yj))
and it follows that Ck(O = 0 when k > n. For an arbitrary ~ we can apply Theorem 18.10. The proof of (b) is analogous: if dimc~ = 1 then C 0 ~ = Hom( ~,~) is trivial and Theorem 18.7 gives that Ch1(C)+ch1(~) = 0, hence C1(C) = -C1(O. For a sum of line bundles, (13)
C(C) =
II (1 + C1 ('Yj)) = II (1 -
This shows that Ck (C) = (-1) kck (0, and the splitting principle implies (b) in general. For a real vector bundle 1], 1]* ~ 1], as we can choose a metric (,) on 1] and use the isomorphism
diagram
a: TJ -- Hom(TJ,R);
(Hn )
~
C1('Yj)).
L. From (ii) it
y line bundle. Therefore
( line bundle. Inductive
es, ck(L1 EB •.. EB L n ) is
y Theorem 18.10 to see
:tor bundle. 0
a(v)
= (u, -).
Then (TJcr = (TJ*k so that Ck(TJ~) = Ck(TJc)' Now (c) follows from (b).
Note that (c) implies that ch(TJc) is a graded cohomology class, which can only
be non-zero in the dimensions 4k.
One defines Pontryagin classes and Pontryagin character classes for real vector bundles by the equations: (14)
Pk(TJ) = (_l)k c2k (TJc),
phk(TJ) = Ch2k(TJc).
We leave to the reader to check that the total Pontryagin class p( TJ) = 1+P1 (TJ )+ ... is exponential.
M;C)
C1 (L) for a line bundle.
(L 1),··· C1(Lk))
Remark 18.12 Definition 18.3 gives cohomology classes in H*(M; C), but actually all classes lie in real cohomology. This follows from Theorem 18.9.(i) for H n , and for a sum of line bundles from (ii) and (iii). The general case is a consequence of Theorem 18.10. Theorem 18.8 actually gives isomorphisms
1
q(Lk)). We have addi
ch: K(M) 0z R ~ H 2*(M) ph: KO(M) 0z R ~ H 4*(M).
.
"..;
.~
1·
,
.q......
]8.
OR BUNDLES
'"
'"
.;
~
~
CHARACTERISTIC CLASSES OF COMPLEX VECTOR BUNDLES
for any closed form w E n4n (R). But we can exhibit a closed 4n-dimensional form on R which contradicts this as follows. The tangent bundle of oR satisfies the equation T8RtBc~ = i*( TR), so when oR = CIP 2n we get after complexification
lap
aph of
(TCP,n )RC tB St
h";t
= i*( TR 0 q
and from the above together with Lemma 16.10, (TCP2n)RC
EEl c~ = (2n
+ 1)H2n EB (2n + 1)H2n.
ce The total Chern class of the right hand side is
: £1.}.
c
ldle where H = H n, and
= (1 - q(H2n ))2 n+l(1 + cl(H2n))2n+l = (1 _ cl(H)2)2 n+l
so that
::tor space Hom (L,
Ll.),
C2k ( Tcp2n )RC =
(-1) k (2n k+
1)
Now take w in (17) to be the 2n-th Chern form of
;omplex n-plane bundle r EB H 1.)
191
= (n + 1) H* .
rem 18.9 and Properties
cl(H)t+ 1,
4.3 shows that Ck ( Tcpn )
eristic classes is to the trphic to) the boundary tasheff] for the general is not the boundary of . this was the case. By
Cl (H)
TR
2k
0 C.
.
l~~ . .~
o ..c::.
.....0
~ -1::3 .~':
.~
"~
...Q.
'~
1
"..J ~.
"
1~
~.-.:
. .~
~
<~t·~
~~'n
~
(....
~
.
..
~
193
19.
THE EULER CLASS
Let ~ be a smooth real 2k-dimensional vector bundle over lYl with inner product ( , ). The inner product induces a pairing
( , ): !"i(O (9 nj (0 (WI
(9
81,W2
(9
---+
ni +j (M);
82) = Wl/\ W2
(9
(81, 82)
where (81,82) is the function that maps P E M to (81(P), 82(p)) and Wl,W2 E n*(M).
Definition 19.1 A connection \7 on
(~,
( , )) is said to be metric or orthogonal if
d (81, 82) = (\781,82)
+ (81, \782).
We express this condition locally in terms of the connection form A associated to an orthononnal frame. Let e1, ... ,ek E [2°(0 be sections over U, so that el(p), ... , ek(p) forms an orthonormal basis of ~ for p E U. Let A be the associated connection form,
\7(ei) = For every pair (i, k) we have (ei' ek) metric one gets
L Aij
(9
ej.
= 8ik (on
U), so d(ei' ek)
ej, ek) + (ei' ~Akj (9 ej) = ~j Aij(ej, ek) + ~j Akj(ei, ej) = Aik
o=
(~Aij
= O.
If \7 is
(9
+ Aki·
Thus the connection matrix with respect to an orthononnal frame is skew symmetric. If conversely A is skew-symmetric with respect to an orthonormal frame, then \7 is metric. Let F'V E n2 (Hom (~, ~)) be the curvature form associated to a metric connection. After choice of an orthonormal frame for ~IU'
n2(Hom(~'~)IU) ~ M2k(n 2(U)). In (17.10) the corresponding matrix of 2-fonns F'V (e) was calculated to be
F'V(e) = dA - A /\ A where A is the connection fonn associated to e. In particular, F'V (e) is skew symmetric, and we can apply the Pfaffian polynomial from Appendix B to F\l (e) to get (1)
Pi (F'V(e)) E [22k(U).
.
.~
~
.
11
~
-...
.q,.....
""
'-
"'
~
:":.~;'--" ->.:,~~-:;
19.
195
THE EULER CLASS
Proof. We can pull back by 71"* the metric gv and the metric connections \7 v to Let {po, pI} be a partition of unity on M x R subordinate to the cover M x (-00,3/4) and M x (1/4, (0). Then g = Po 71"*(go) + PI 71"* (gl) is a metric on which agrees with 71"* (go) over Mx (-00,1/4) and with 7I"*(gl) on M x (3/4,00). In particular i~(g) = gv. Let \7 be any metric connection on ~ compatible with g. We have connections 71"*(\70), and 71"*(\71) compatible with g over M x (-00 x 1/4), M x (1/8, 7/8) and M x (3/4,00) respectively. We use a partition of unity, subordinate..!o this cover, to glue together the three connections to construct a connection \7 over M x R. This is metric w.r.t. g, and by construction, i~V = \7v' 0
f
~(p)
and e' (p). , and that e(p) and e' (p ) S02k, and by Theorem
.-form on M. The proof
,f the choice of metric on
ns can be glued together ctions on ~ and (Pa)aEA 7as defines a connection } then \7 = L Pa \7 a is
2)
f
-
~
V
Corollary 19.3 The cohomology class [Pf(F'i7)] E H 2k (M) is independent of the metric and the compatible metric connection.
V)
Proof. Let (gO, \70) and (gl, \71) be two different choices an2 let (g, be v the metric and_connection of the previous lemma. Then i~(F'i7) = F'i7 , and hence i~ Pf (F'i7) = Pf (F'i7 v). The maps io and i 1 are homotopic, so i o = ii': Hn(M x IR) -+ Hn(M). Thus the cohomology classes of Pf (F'i7O) and Pf (F'i7 1 ) agree. 0 Definition 19.4 The cohomology class
e(O = [Pf( _~'i7)] E H 2k (M)
: (Sl, Per. \7 aS2) ~1, \7 er. S2)) I
S2).
hat contain SUPPM(Pa), e proof of Lemma 19.2
is called the Euler class of the oriented real 2k-dimensional vector bundle
Example 19.5 Suppose M is an oriented surface with Riemannian metric and that ~ = T* ~ TM is the cotangent bundle. Let el, e2 be an oriented orthonormal frame for DO(TI~) = D 1(U), such that e1/\ e2 = vol on U. Let al,a2 be the smooth functions on U determined by del = a1(e1/\ e2),
and let A 12 = aIel
+ a2e2.
We give
;) over M x R. Then
A= 'ric connections gv, \7 v ere is an inner product :h that i~ (g) = gv and
~.
TI~
de2 = a2(e1 /\ e2)
the connection with connection form
(0
-A12
A12) 0
so that \7( e1) = A 12 @e2 and \7(e2) = -A 12 @e1. This is the so-called Levi-Civita connection; cf. Exercise 19.6. By (17.10)
F'i7
.
= dA _ A /\ A =
(0
-dA12
dA 12 ) 0
i':::l > .......
.-J
.... '... i;J "b.Q,,~ '.~ .... ..c 'l:!
.NJ
~ .lo:: .. ~
~
01;3
'1:'"' ~
R,'~.i.~ .
~,~
< ""
c
~
--
1
.Jg,
~
~l:! •. 1:10
.~
.....
...
'Q
1 ~
L---J
',~j<",l":'
~
19.
called the Gauss-Bonnet I by the formula
,xercise 19.6.
jons for complex vector mnitian connections are hat ( , ) now indicates a n question. :spect to a local orthonor ic: Aik + Aki = 0 or in
197
THE EULER CLASS
when F'V is the curvature of a hermitian connection on ((, ( , )c). Thus Pf(-F~ /21r) = Cfk(iF'V /21r).
This proves (i).
The second assertion is similar to Theorem 18.6. With the direct sum connection
on (1 E9 (2,
F'l = F'll E9 F'l2
and for matrices A and B, Pf (A E9 B)
= Pf(A)Pf (B).
o
Finally assertion (iii) follows from (17.13).
>mplex dimension k with ~ (R is naturally oriented, part of ( , )C, and an
)c) with respect to an lted with the underlying the usual embedding of itian matrices into skew-
In order to prove uniqueness of Euler classes we need a version of the splitting principle for real oriented vector bundles, namely Theorem 19.7 (Real splitting principle) For any oriented real vector bundle ( over M there exists a manifold T (() and a smooth proper map f: T (() ---t M such that
1*: H*(M) H*(T) is injective. (ii) r(O =,1 E9 ... E9'n when dim ( = 2n, and r(o =,1 EEl ... E9,n EEl c: 1 when dim ( = 2n + 1, where ,1, ... are oriented 2-plane bundles, and (i)
---t
c: 1
,n
is the trivial line bundle.
I)
The proof of this theorem will be postponed to the next chapter.
e((R). Then we have
Theorem 19.8 Suppose that to each oriented isomorphism class of2n-dimensional oriented real vector bundles (2n over M we have associated a class e((2n) E H 2n (M) that satisfies (i) 1*(e(()) = e(j*(O) for a smooth map f: N ---t M (ii) e( (1 E9 (2) = e((de( (2) for oriented even-dimensional vector bundles over
(() = Ck(O·
3:16) = e(6)e(6)·
ng with Definition 18.3. by (6)
n
the same base space. Then there exists a real constant a E R such that e((2n)
= an e ((2n).
Proof. Given a complex line bundle Lover M, we can define c(L) = e(LR)' Then rc(L) = c(j*L), and the argument used at the beginning of the proof of Theorem 18.9 shows that c(L) = aCl(L). Thus e(,) = ae(r) for each oriented 2-plane bundle ,. Indeed, an oriented 2-plane bundle is of the form LR for a complex line bundle which is uniquely determined up to isomorphism. One simply defines multiplication by yCI to be a positive rotation by 1r /2.
"-> ~§t
."..;
,
-
...
~.
<::>
...s::
.,;
'"
.~.
"
1
' . .,"'-;:.' '
2
~
199
bundles then we can use ~). Finally Theorem 19.7
o
, a real vector bundle by
lows from the exponential 18.11) that the
20. COHOMOLOGY OF PROJECTIVE AND GRASSMANNIAN BUNDLES
In this chapter we calculate the cohomology of the total space of certain smooth fiber bundles, associated to vector bundles, as a module over the cohomology of the base manifold. As corollaries we obtain the splitting principles for complex and oriented vector bundles used in Chapters 18 and 19. Let 1r: E ~ M be a smooth fiber bundle over M with fiber F. There is a product
~Properties
vector bundle (, Pk(()
=
Hi(M) ® Hj(E) ~ Hi+j(E) given by the formula (1)
letric connection \J. Then where e is an orthonormal lplexified metric then e is les a hennitian connection form for (c, and C2k((c) )w follows from Theorem
o
a.e = 1r*(a)
1\
e for a E Hi(M), e E Hj (E).
Thus H*(E) becomes a (graded) module over the (graded) algebra H*(M). We shall examine this module structure in the particular simple case where we suppose given classes eet E H n ", (E) for 0: E A with the property that for every P E M, (2)
is a basis for the vector space H*(Fp ).
{i;(eet)IO:EA}
Here Fp = 1r- 1 (p) is the fiber over P and i p is the inclusion of Fp into E.
Theorem 20.1 In the above situation H*(E) is afree H*(M)-module with basis {eet I 0: E A}. Proof. The proof follows the pattern used to prove Poincare duality in Chapter 13. Let V be the cover consisting of open sets V C M, such that E is trivial over V. Let U be the cover of M by open sets, so that the theorem is satisfied with M replaced by U E U and E replaced by 1r*(U). We must verify the conditions of Theorem 13.9. We leave conditions (i), (ii) and (iv) to the reader and prove condition (iii). So suppose
U
= U1 U U2, U12
=
U1 n U2
and let E 1, E2 and E12 denote the restriction of the bundle E over Ul, U2 and U12. The classes eet E H n ", (E) restrict to classes which again satisfy condition (2), and we denote the restricted classes by the same letters. We suppose that the theorem is true for H*(E 1), H*(E2), H*(E1 2), and want to conclude it is true for H*(Eu). This employs the two Mayer-Vietoris sequences .. , ~ H*-1(E12 )
~
.
!..: H*(Ul) EB H*(U2) ~
.
£. H*(E) !..: H*(E1) EB H*(E2)
... ~ H*~1(U12) ~ H*(U)
",...
.i:"tJ .
..
~~
'!O
"
11
-
~ ....... ~
l---.J
~IAN
20.
BUNDLES
'ery element e E H*(E) Nith rna: E H*(U). We lder. By assumption we
201
COHOMOLOGY OF PROJECTIVE AND GRASSMANNIAN BUNDLES
x C n+1 is the trivial bundle then P(O = M x Cpn and H(~) = pr2(Hn). Let us give ~ an inner product. Then n*(O has an inner product, and we can form the fiberwise orthogonal complement H(O.1.. of the subbundle H(O c n*(O, i.e. ~
=M
H(~).1.. = {(p,L,1,I) I (p,L) E P(O, v E L.1..} where the orthogonal complement L.1.. is calculated in the fiber ~p. Clearly L EEl L.L = ~p = n*(~)p' So that we have an isomorphism of vector bundles
= H(O
EEl H(O.L·
Let e be the first Chern class of H (0, e = 20.1 to the classes
C1 (H (~)).
(4) i;(rn~l)) = j2(m~2») for
1T*(O
: spaces implies elements
l,e,e 2 , ... ,en E H*(P(O).
(5)
n 8* has a representation Ie theorem for H*(E I 2)
~
E the inclusion. We
Property (2) is satisfied because the fiber of n: P(~) ~ Mover p E M is the projective space P(~p) = Crn, and because the restriction of H(O to P(~p) is the canonical line bundle H n over Cpn. Now i;(e) = cI(Hn ) # 0 and the powers ei restrict by to q(Hn)i which are non-zero in H 2i (Cpn), and hence a basis as long as i ::; n, by Theorem 14.3. In the situation of Theorem 20.1 one has in particular that n*: H*(M) ~ H*(E) is injective. Indeed n*(m) = m.l, and 1 is an R-linear combination of basis elements ea:. We have proved:
i;
o
principle, as stated in M with dimc~ = n + 1. Dtal fiber space
; in the vector space
Theorem 20.2 Foranycomplexn-dimensionalvectorbundle~over M, is a free H* (M) -module with basis 1, ci (H(~)), ...
In particular, 1T*: H*(M)
;p)}. ~p.
~
,CI (H(Ot-
:ader to show that P(O There is a complex line
L}.
IleX projective n-space 15.2. If more generally
H*(P(~))
l.
o
H*(P(O) is injective.
We may now prove the splitting principle for complex vector bundles.
Proof of Theorem 18.10. Starting with consider the composition
E
We want to apply Theorem
P(~n-l) 1r~1
•••
~
over M with
dimc~
= n
+ 1,
we
~ P(6) ~ P(O ~ M
where 6 = H(~).L was defined above and where 6, H(6) are the corresponding bundles over P(6), i.e. 6EElH(6) = ni(6) etc. Thus if we let f = no o .. . onn-l, /*(0 is the sum of the pull-backs of the line bundles H(~i) over P(~i), and /* = 1T~_1 ° ... ° is injective by Theorem 20.2. 0
no
l~ "'-.0
,~.~ ,~ ;~ ,.c
~
0
-S
'"
1- ~'2
,§-
<~;l::3. .~.~ ~ < ......
.
~
·S
(1,) ...
,1M;:iiJ'l. ~~
~
20.
>NNIAN BUNDLES
class en+l =
C1 (H(Ot+l
+ An(().e n
~~<
-
c~~
...-.0""
-
II ::
-_~~
COHOMOLOGY OF PROJECTIVE AND GRASSMANNIAN BUNDLES
203
basis for H* (G 2 (R 2 n)). We now give the details, starting with a proper definition of G2 (R 2n ) and then proceeding with the somewhat cumbersome calculation of its cohomology. Let V2 (lR ffi ) denote the set of orthonormal pairs (x, y) of vectors in IR m . We view x E sm-1 and y as a unit tangent vector in T x S m - 1. Thus V2 (lR m ) becomes the unit vectors in the tangent bundle T sm-I. It is a smooth submanifold of R2m via the embedding
c TS m - 1 c T(R m ) =
V2(lR m )
x.ponentia] property of the ~nce
It is better for our own purpose however to consider the embedding (8)
c1(H(()))-1.
V2(R m )
--+ S2m-I
+l(6) = O.
ide down and use (6) to Ile bundle. One then must ast conditions of Theorem Dthendieck. It is useful in s in singular cohomology,
~
for oriented real vector ~ of complex bundles, but Ie P( 0 is replaced by the nted real vector bundle ( s replaced by the oriented :r an oriented plane in (p :() = 'Y2(() EB 'Y2(()l- as The analogue of (5) is a
r*(G 2 (()) llier class of the previous
lat the classes in (7) are a
C cm,
The manifold V2(Rm) is called the Stiefel manifold of (orthogonal) 2-frames in IR m ; it is evidently compact. The group SO(2) of rotation matrices -sin B) cos B
R = (cos () e sin ()
ff(O)i IUS Cn
1R 2m .
acts (smoothly) on V2 (R m ) by
(x, y).Re = ( (cos B)x + (sin B)y, -(sin B)x
+ (cos ())y).
The orbit space V2(R m)/SO(2) is identified with the space G2(Rm) of oriented 2-dimensional linear subspaces of Rm by associating to (x, y) E V2 (Rm) the subspace they span, oriented so as to make (x, y) a positively oriented orthonormal basis. We leave it to the reader as an exercise to specify a smooth manifold structure on G2(R m ). The resulting manifold is the Grassmann manifold of oriented 2-dimensional subspaces of Rm. It is clear from (8) that
V2(Rm)
(9)
~
s2m-1
1~o
_cm -
1~1
G2 (Rm) --!E..- cpm-l2!.... where 1fo(x, y) = spanR(x, y) and 1f:
Cm
1fI -
{O}
1~ cpm-1
is the restriction of the canonical map
{O}
--+ cpm-1.
o
~ l:.J
.
~
r:
,.,.J ...
11
't;!
..e
.~.
AN BUNDLES
'"
. .t
e-,J.
20.
."';!
COHOMOLOGY OF PROJEcrIVE AND GRASSMANNIAN BUNDLES
205
Lemma 20.5 The space Q is contractible by the homotopy F: Q x [0,1]
s a smooth submanifold re note that 7fo is a fiber mogeneous coordinates ojective space Rpm-I, 0), x and y are linearly :, so
---+
F(A, t)
Q;
=
At.
Proof. This is again a consequence of the spectral decomposition. A matrix A E Q has positive eigenvalues )., ). -1 with say ). ~ 1. Then)' depends continuously on A. The case). = 1 occurs only for the identity matrix I. If A =F I we can write A = ).P + ). -1 (I - P) = ). -1 I
+ (). _ ). -1) P,
where P is the orthogonal projection on the (I-dimensional) ).-eigenspace of A. Here P depends continuously on A E Q - {I}. We define F on (Q - {I}) x [0, 1] by F(A;t) = ).tp + ).-t(I _ P) = ).-tf + ().t _ ).-t)?
m'
. so that
1), ~al
2x2 matrices with ~roup C* on C = R2 Ip of GLt(R), where
Observing that each matrix entry in (). t - ).-t) P has numerical value at most ). - ).-1, we see that F extends continuously to Q x [0,1] by F(I, t) = f. 0
Proposition 20.6 The map
---+
cpm-l - Rpm-l
from (11) is a homotopy equivalence.
nmetric matrices with
? by conjugation.
a
b
Proof. We write Wm = cpm-l _Rpm-I, and consider the smooth map (cf. (?))
-b)
a
; positive definite, and )1/2 which commutes
"et(BB*)1/2
and A = )lar decomposition is Id Ri E 80(2) then ection. Its inverse is
o ers At for any t E R,
(~ ~))
= (ax
---+ 7f-
1
(Wm
);
+ by) - i(cx + dy).
A point in 7f- 1 (Wm ) has the form z = v - iw, where v and w are linearly independent vectors in Rm . The fiber , constructed using the Gram-Schmidt orthonormalization process. The fibers of
(x, y, B).Ro = ((x, y).Ro, Ri 1 B), so
V2 (R m
) xSO(2)
GLt(R)
---+ 7f-
1
(Wm ).
This bijection commutes with the C* -actions, if we let C* act on the domain by right multiplication in GLt(R) and on 7f- 1 (Wm ) C Cm by scalar multiplication.
.t ..
.-J
...~ ".'~-.'
. 'C)J
-
..
"
-
.. '.'
1'.$ .'':i!: ..
.q...
~
.""~',,f"'"!;<.:,,'%S
20.
IAN BUNDLES
naps are bijections
'(Wm )
1
Wm may identify the lower
COHOMOLOGY OF PROJECTIVE AND GRASSMANNIAN BUNDLES
Remark 20.7 We can offer the following more conceptual explanation of the construction in the previous proof. Consider the set Gr(Rm) of pairs (V, J) where V ~ Rm is a 2-dimensional real oriented linear subspace and J a complex structure on V compatible with the orientation, i.e. an R-linear map J: V -+ V with J2 = -id and such that (x, Jx) for x E V, x =1= 0 is a positively oriented basis for V. One forms the complexifications Ve = V 0R C C C m and Je = J 0R ide : Ve -+ Ve· Then Vc = V+ EB V_, where V± are the (±i) eigenspaces of Je. These are I-dimensional over C. In fact for x E V - {O}, V+
).
3x + ,y)]. ~,
in fact it is a homeo 4>-1 can
Vm the inverse
.) x Q x C* he global h map in . find that .s defined
p~r
X
section of the sequence the canonical on open sets
, matrix I E Q, so one
= spanc(x -
iJx),
s precisely equal to 0. homotopy equivalence. ap Q -+ {I}
leen io
0
rand idx
V_ = spanc(x + iJx)
The pair (V, J) may be recovered from V+ since V_ is complex conjugate to V+, V = (V+ + V_) n Rm and since J is the restriction of the C-linear endomorphism of V+EB V_ which acts on V± by multiplication with ±i. This gives an identification of Gr(R m) with W m in which (V, J) corresponds to V+ considered as a point in W m ~ Go m - l . The space Q ~ GLi(R)/C* parametrizes the complex structures on R2 compat ible with the standard orientation (cf. Exercise 20.5) so X = V2(Rm) XSO(2) Q is another model for Gr(Rm). The homeomorphism 4> in (13) is the natural identification of the two models. Each V E G2(R m) has a canonical complex structure Jo such that any unit vector x E V leads to a positively oriented orthonormal basis (x, Jox) for V. The inclusion G 2(R m) -+ Gr(R m), which sends V to (V, Jo), corresponds exactly to the embedding 0 in (10). We shall now use Proposition 20.6 to calculate the cohomology of G2(lR m). First we apply Poin'care duality to evaluate H*(Wm ). Consider the inclusions i: Wm -+ cpm-l ,
~
207
j:lRpm-l
-+
cpm-l.
When m = 2, 8 2 ~ Cpl by Example 14.1, and Rpl C Cpl becomes identified with a great circle in 8 2 so that W2 becomes the disjoint union of two open hemispheres in 8 2. Lemma 20.8 The map i*: Hg(Wm) -+ Hq(cpm-l) is an isomorphism except in possibly two cases: for q = 0, and, ifm is even, for q = m. In fact H2(Wm ) = 0, and for m even there is a short exact sequence
IS
0-+ R -+ H~(Wm) ~ Hm(cpm-l) -+ O.
m)
X
Q,
; well-defined because be shown with use of
o
Proof. The exact sequence of Proposition 13.11 for the pair (cpm-t, Rpm-I) takes the form
... L
Hq-l(Rpm-l).~ Hg(Wm) ~ Hq(cpm-l) ~ Hq(Rpm-l) ~.,.
< ~ 1-~
\
~
.,..J
£"to> .~~
...
"
.2
~
.l---J -
lNNIAN BUNDLES
20.
alculated in Example 9.31 , ~ R maps isomorphically : degrees. 0 Jr
if q i-
209
There is a bundle map over
'Pxi~
s2m-1 XSI C
.--.2.-.
cpm-l
V2(R m ) xSO(2) R 2
1
G2(R m)
morphism
_.~ ..... _~~,."~_ ;-0,.-.-. _
COHOMOLOGY OF PROJECTIVE AND GRASSMANNIAN BUNDLES
all p, and
. 2p:S 2m - 4
•
!
Since q (H) = e(HR) by Theorem 19.6.(ii), naturality of the Euler class gives
e(J2) = 0*(q(H)).
2m - 2,
- 2)-dimensional manifold e diagram
Let e E H2 (G 2(Rm)) be the Euler class of 12, e = e(J2). If m is even, we let
e = e(Jt) be the Euler class in Hm-2(G2(Rm)).
Theorem 20.10 With the notation above (i) For m odd and m 2 3, H*(G 2(Rm)) = R[eJl(em (ii) For m even and m 2 4,
rm )
H*(G2(R m )) (Wm )*
with dege
[sm except if p = 2m - 2, ctor space dual of i*. In
=2
= R[e, e]/(em - l , ee, e2 +
and dege
=m-
l
)
(_lr/ 2em 2)
2.
Proof. We already know the additive structure by Propositions 20.6 and 20.9, and also that
H*(0): H* (cpm-l) sequence of Lemma 20.8
-t
H*(G2(R m ))
has kernel H 2m - 2(Cpm-l) and is onto for odd m. Since
H*(Cpm-l) = R[q(H)]/(q(H)m) -t
R - t 0,
and e
o
:R
m
H*(0)(q(H)), this proves (i).
Suppose now m = 2n (15)
)
=
em -
l
2 4. We first establish the relations:
= 0,
with total spaces
ee
= 0,
e2 + (-ltem -
2
= o.
Indeed, the first one follows from Proposition 20.9, since e and hence any power of e is in the image of
;: V}
=V.L} 80(2)
R2 , the orbit space
~lv). The embedding of to (rro(x, y), XVI + YV2)
H*(cpm-l)
HJi) H*(Wm) H-sf'J H*(G 2(R m )).
The second relation is a consequence of Theorem 19.6.(ii),
ee
= e(J2)e(Jt) = e(J2 EEl It) = e(Em) = o.
l~ 2; ] i
."~.
,""'£ . lb,() 'l:3
.~
"l:3
~
~
~ 'l:3 .~ ~
--
~~
~~
'<6'
l-.r;
'"
1 i .....•
~
....c
~t:f~ .... "N
'fo
~.-'."'"
~
.J::)
~ ~
<
.,....
•••
'
. . ..
~
tJJ
.-~-=~:;~
211
IIAN BUNDLES
in class is exponential
~))
=1
v-er by Proposition 19.9
Y2 )
21. THOM ISOMORPHISM AND THE GENERAL GAUSS-BONNET FORMULA
n-l
•
2) = c2
15). From Proposition cm - 2 =I- 0, so by (15) is 2-dimensional, and h contradicts that m :t {1, c, ... , cm - 2 , e} is o a basis for the ring
o lle over M, and suppose iber bundle G2 (() over
Let ~ be an oriented vector bundle over M with total space E = E(~). The Thorn isomorphism theorem calculates the compactly supported cohomology H; (E) in terms of H*(M), namely Hq(M) ~ Hg+m(E) where m = dim~. Assuming ~ to be smooth (cf. Exercise 15.8) and M to be oriented the Thorn isomorphism theorem is a consequence of Poincare duality. Indeed, H~+q(E) ~
Hn-q(E)*
~
Hn-q(M)*
~
Hg(M)
where n = dim M. The second isomorphism is induced from the homotopy equivalence E ~ M. For M compact we give below a more direct proof of Thorn isomorphism. Suppose ~ is smooth and has an inner product, and let ~ EB 1 denote the (orthogonal) direct sum of ~ and the trivial line bundle over M. We write S(~ EB 1) for its unit sphere bundle over M, or what amounts to the same thing, for the fiberwise one-point compactification of E. Let Jr:
S(~ EB
1)
-+
M,
s=: M
-+ S(~
EB 1)
:(p)} it factor. There are two E V} E Vl.}
lch that 'Y2 ( (p) EB n 20.1 gives
'Yt((p)
be the bundle projection and the "section at infinity", respectively, that is s=(p) = (0,1) E ~p EB R. This makes S(~ EB 1) into a smooth fiber bundle over M, and hence H*(S(~ EB 1)) into a H*(M)-module (cf. Theorem 20.1). The fiber of Jr is the m-sphere sm ~ Jr- 1 (p), and the orientation of ~p induces an orientation of each fiber sm. In particular the integration homomorphism of Chapter 10 induces a fixed isomorphism (1)
lrbundle (, H*(G 2 (())
I: Hm(Jr-1(p)) ~ R.
Stereographic projection from s=(p) identifies Jr-l(p) - {s=(p)} with ~p, and globally it identifies the total space E with S(~ EB 1) - s=(M).
=2n 2: 4 = 2n + 1
2: 3
o ,Ie, Theorem 19.7, is inciple, treated earlier
Definition 21.1 An orientation class for Hm(s(~ EB 1)) that satisfies
~
is a cohomology class u E
(a) s~(u) = 0 (b) For each p E M, the restriction of u to Jr-l(p) has integral 1.
....
t~ ..~
'l:e.
~.,..
.'"", ,t3
.... ..c
"t;S
1·~·
.-J
~'
~_c_~'<.-..
.i:
~~ e ~ ~ ,.0.
~
...
"
~
.~
[)NNET FORMULA
unique orientation class
• {I, u}
n Theorem 20.1, since lple 9.29 and Corollary total space S (f, EEl 1). In ition 21.1.(b). Indeed, if has the form
ion of 1f*(x) to 1f-l(p) 1st have value 1 at each
rivialize f,u (compatible T x sm. Let
Finally consider a sequence Ul, U2, . .. of disjoint open sets in U with union U = Ui Ui- We have the isomorphism from Proposition 13.4,
Hm(Su)
'I1(Su1 nuJ,
H m (SU 1 UU2 ) with ibers 1f-l(p) ~ SUIUU2 E
--+
II Hm(SuJ·
The family of orientation classes Ui E Hm(SuJ is the image of some U E Hm(Su) with integral lover all fibers 1f-l(p) ~ Su. Hence U E U. Now Theorem 13.9 D applies to show that M E U. The above does not require M to be compact, but if it is, we may apply Proposition 13.11 to the compact manifold pair (S(~EEl1),soo(M)). Since S(f, EEl 1) - soo(M) s:! E and soo(M) s:! M we obtain a long exact sequence
.s Hq(S(f, EEl 1)) S Hq(M) ~ Hg+l(E)
--+ •••.
Now s~ 01f* = (1f 0 soo)* = id, so that s~ is an epimorphism and 8 exactness i* is a monomorphism, leading to the short exact sequence (2)
0--+ H;(E)
.s H*(S(f, EEl 1)) S H*(M)
--+
= O.
By
O.
Theorem 21.3 (Thorn isomorphism) Let f, be an oriented m-dimensional real vector bundle over a compact manifold M. There is a unique class U E H;:"(E) with integral lover each fiber f,p, and the map
: Hq(M)
-=. H~+q(E);
is an isomorphism. The class U
ave integral equal to 1. fiber, and condition (ii) iii). So suppose Ul, U2 E Hm(SuJ, /I = 1,2 In (a) and (b) for f,ulnu2' ~ the same restriction to
213
THOM ISOMORPIDSM AND THE GENERAL GAUSS-BONNET FORMULA
... ~ Hg(E)
s condition (b) is based is open, and let U be the satisfies the conclusion E U if and only if there 1 fiber 1f-l(p) for p E U. le open sets in M such y the four conditions of
!
21.
= (1)
(x)
= (1f*x) 1\ U
is called the Thom class_
Proof. The exact sequence (2) shows that the orientation class U E Hm(s(f, EEl 1)) has the form U = i*(U) for a uniquely determined U E H;:"(E). The first statement now follows from Theorem 21.2. The homomorphisms i* and s~ in (2) are H*(M)-linear and the last part of Theorem 21.2 shows that H;(E) is a D free H*(M)-module generated by U. Thus is an isomorphism. Definition 21.4 With the notation of the previous theorem, let e(f,) E Hm(M) be the class with (e(f,)) = U 1\ U. The product in H;(E) is anti-commutative (since this is the case for the rep resenting differential forms), so U 1\ U = 0 if m is odd. Thus e(f,) = 0 for odd-dimensional oriented vector bundles.
.:
·. ,.,.J
.~"~
~.
:""
.
2
..
.
21.
e(~)
=
rphism
~nts
E:
U also defines a class
(use the linear structure
w)J 1\ U
o
"
.~~
~
·BONNET FORMULA
'l!i.
. .-•. ::c... .'..
1
'ection of E. Then
\
";'
~
"'~
__
_..:-"'~"'_." ~.-.' _
""
215
mOM ISOMORPIDSM AND THE GENERAL GAUSS-BONNET FORMULA
Lemma 21.7 If 6 and 6 are oriented real vector bundles over the compact manifold M, then e(6 EB 6) = e(6)e(6). Proof. Let ~v have total space E v , bundle projection 1rv :E v ~ M and fiber dimension m v , and let V v E H;:-v(Ev ) be given by Theorem 21.3. The product map 1rl X 1r2:
El x E 2 ~ M x M
is the projection of an oriented (ml + m2)-dimensional vector bundle ~ over M x M with ~*(O = 6 EB 6, where ~: M ~ M x M is the diagonal map. If pr v : El x E2 ~ E v is the projection and Wv E n:;
W = pr;'(wl) 1\ pr2(w2)'
lditions of Theorem 19.8,
It follows from Fubini's theorem that w has integral lover each fiber Sv: M ~ E v be the zero section. Then
e(O
= [(SI
'lpact manifolds and
~
an
~
e(O
x S2)*
X
S2)*
0
pr~(wv)]
T)
o
= [p~ 0
s~(wv)]
0
pi(e(6)).~*
0
(PI 0 ~)*(e(6))'(P2 = e(6)e(6)·
=
r)
pr~(w2)]'
= p~(e(~v))
= e(~*O = ~*(e(O)
= ~*
.5 we find
0
= p;'(e(6)) 1\ p2(e(6))· Finally Lemma 21.6 yields
e(6 EB 6)
?ective1y. The map j is s back to V' = j*(V) E >riented isomorphism, V' utative diagram
pr;'(wl)] [(SI x S2)*
0
M denotes the projections, then
[(SI so
Let
= (SI x S2)*(V) = [(SI x S2)*(W)]
lpact manifolds. We shall If Pv: M x M
in~.
p~(e(6)) 0
~)*(e(6))
o
In Chapter 11 we defined the local index of a tangent vector field with isolated singularities, and in Chapter 12 we proved the Poincare-Hopf theorem, that the sum of the local indices is the Euler characteristic of the manifold. We shall now extend these notions to sections of an arbitrary oriented vector bundle ~ over a compact, oriented smooth manifold M, provided dim ~ = dim M = m. Let E = E(~) be the total space and 1r: E ~ M the bundle projection. We let so: M ~ E be the zero section of ~ that to each p E M associates the origin in the fiber ~p. Let s: M ~ E be a second smooth section. The differentials Dps and Dpso from TpM to TqE are monomorphisms since S and So have one-sided inverses.
)
~
",..,;
. r,.>
1 ~
21.
BONNET FORMULA
s, s(p)
= 80(p),
Then
8
'i
• .•....
.~
THOM ISOMORPHISM AND THE GENERAL GAUSS-BONNET FORMULA
217
In terms of homogeneous coordinates on Cpl, 8([ZO, Zl]) is the restriction of a to the fiber spanc(zo, zI) in H. The only zero of 8 is Po = [1,0]. Over the coordinate chart Uo = {[I, z] I z E C} in Cpl we have a trivialization of H defined by
Uo x C --+ H;
'os of 8.
;ent space Dp80(TpM) to to the fiber ~p which is
'"
~ ~.:;.~ .....••
~
([I, z], a)
f-4
([1, z], (a, az)).
In the dual trivialization of H* we find that 81uo corresponds to the function Uo --+ C, which maps [1, z] into z; thus in terms of local coordinates 8 is the identity. It follows that 8 is transversal to the zero section at Po, and that £(8; po) = 1. From Theorem 21.9 we conclude that I(e(O) = 1. Theorem 21.11 For oriented vector bundles over compact manifolds, e(~) = e( O.
:ot that Dp 8(Tp M) is the ector spaces are oriented : +1, if A preserves the : ~ = TM is the tangent emma 11.20). U x Rm, we can identify Then· A: TpM --+ ~p e statement that DpF is ~ orientation behavior of a local diffeomorphism of 8. If 8: M --+ E is 'S of 8 is finite, since M
zm.
to the zero section, then
) --+
e
If the dimension m is odd then we saw in the discussion following Definition 21.4 that e(O = 0, and consequently the index sum in (5) will always vanish. If ~ is even-dimensional and admits a section 8 without zeros, then the index sum is zero, and e(~) = 0. One often expresses this by saying that e(~) is the obstruction for ~ to admit a non-zero section. Note that a non-zero section is equivalent to a Indeed, we may choose an inner product on ~ and define splitting ~ ~ EB to be the orthogonal complement to the trivial subbundle of ~ consisting of lines generated by 8.
e
e c:k.
Theorem 21.12 For any oriented compact smooth manifold M,
I(e(TM))
R is the integration
'undle over Cpl, H* its I bundle. The bundle H
])2 X
Proof. We have already seen in (3) above that e(~2m) = ame(~2m) for some = (H*)R' the previous constant a; it remains to be shown that a = 1. For example shows that I ( e(~2)) = 1. On the other hand, e (~2) = Cl (H*) by Theorem 19.6.(ii), andI(cl(H*)) = 1 by Theorem 18.9.(i) and Property 18.11.(b). Since I is injective e(~2) = e(e) in this case, so a = 1. D
C2
the dual product bundle hat is the image of the linear form
= X(M).
Proof. We simply apply Theorem 21.9 to ~ = TM, taking for s a gradient-like vector field X w.r.t. some Morse function; cf. Definition 12.7. The proof of Lemma 12.8 shows that X is transversal to the zero section, and that the sum in (5) is equal to Index(X). The Poincare-Hopf theorem finishes the proof. D
We can combine the two previous theorems to give a generalization of the classical Gauss-Bonnet theorem to even-dimensional compact, oriented smooth manifolds. Theorem 21.13 (Generalized Gauss-Bonnet formula)
1M Pf( _~'l)
= x(M 2n ),
,.
.~.
~' -'."~-.".~ 1 ~~ ..
21.
onnection on the tangent
o
n 21.9. Let PI, ... ,Pk be n~ (E) which represents :lint open neighborhoods : of Vi in E, and define Ev, Vi x Rm with the
.=.
..
'.'
~
~
IIONNET FORMULA
....
'D
TROM ISOMORPffiSM AND THE GENERAL GAUSS-BONNET FORMULA
For example we can take
(6)
+ t)X
G(x, t) = ((1 - t)p(llxll)
where p: R --... R is a smooth function with
p(y) =
{o
~!
for y for y
1
~
1.
Now construct a homotopy F: M x R --... M as follows. Choose disjoint charts Wi, 1 ~ i ~ k, and diffeomorphisms <pr Wi --... Rm such that
w that
F(p, t)
=
{
l 0
~f P E Wi Ifp ¢ UiWi.
F(
Then (0:) holds for h(p) = F(p, 0) and «(3) holds for
ghborhood
Vi
of Pi, i
=
219
Vi
=
(!b m ).
D
Proof of Theorem 21.9 Pick data as in Lemma 21.14. Replacing Vi with a smaller open neighborhood of Pi, we may assume that Iio(slv,) maps Vi diffeomorphically onto an open set in Rm . Pick an open neighborhood Wi of Pi with closure Wi ~ Vi, Since M - Ui Wi and SUPPE(W) are compact, we can find a constant c > 0 such that the scaled section 8 = cs satisfies
;) is isomorphic to ,;, so orms for ~' instead of ,;.
SUpPRm (Wi)
1~i ~ k
c Ii (s(Vi)),
8( M - UWi) n SUPPE(W) = 0. i
Now e(,;) E Hm(M) is represented by the m-fonn S"(w), which is identically zero on M - Ui Wi. By Lemma 21.14
: n~( E) with integral 1 rltegral lover each fiber nted linear isomorphism ~(~p), and
~
i
Ii to h under
S*(W)/v, = (Ii
where Ii 0 (81 V;) is a diffeomorphism from Vi to an open set in Rm containing SUpPRm (Wi)' This diffeomorphism preserves or reverses orientations depending on the value of L(8;Pi) = L(S;Pi) = ±1. Fonnula (5) follows by the computation:
k
properties. To this end ;:
I(e(,;)) (7)
=
1
S*(w)
M k
k
=L
= LL(S;Pi) i==l
Dm .
(8IVJ)*(Wi)
0
i==l
i
Rm
1 V;
(Ii
0
(S1V;))*(Wi)
k
Wi = LL(S;Pi). i==l
o
<
...,;
'0
'"
'~
~,
£'10>.
~
.~
~
-
"
.
-"
~
~
..-
~
---:
:::
..... ~
-
-..;;:,-
- -~~-
221
A. SMOOTH PARTITION OF UNITY
A. SMOOTH PARTITION OF UNITY The following technical theorem is a much used tool when working with smooth maps and smooth manifolds. For a function f: U -+ R with domain U ~ Rn the support of f in U is the set suPPu(J) = {x E Ulf(x) f= O},
where the bar denotes the closure of the set in the induced topology on U. If
U is open in Rn then U - suPPu(J) is the largest open subset of U on which
f vanishes. Theorem A.1 If U ~ Rn is open and V = (Vi)iEI is a cover of U by open sets Vi, then there exist smooth functions (Pi: U -+ [0, 1] (i E 1), satisfying (i) sUPPu( cPd ~ Vi for all i E I. (ii) Every x E U has a neighborhood W on which only finitely many cPi do
not vanish. (iii) For every x E U we have
L
cPi(X) = 1
iEI
We say that (cPi)iEI is a (smooth) partition of unity, which is subordinate to the cover V. A family of functions
cPi: U
-+
R that satisfy (ii) is called locally finite. Note that
the sum LiEf cPi in this case becomes a well-defined function U -+ R. Moreover, it is smooth when all the cPi are smooth. The proof of Theorem A.I requires some preparations.
Lemma A.2 The function w: R
R defined by 0 if t < 0 w(t) = { exp( -lit) if t ; 0 -+
is smooth. Proof. It is only smoothness at t = 0 which causes difficulties. It is sufficient to see that .
hmt--+o+
w(n-l)(t) t
= 0
for all n 2: 1. By induction there exist polynomials PO,Pl,P2, ..., such that w(n)(t) = Pn(1lt) exp( -lit),
for t > 0 and n 2: 0 . The result now follows because k
limt--+o+ ((ll t )k exp (-lit)) = limx --+ oo for k 2: O.
x( . = 0
exp x
o
~
',.,oJ
"
"fJ
. --~
'.
A. SMoom PARTmON OF UNITY
I,{}othfunction '1/;; R - t [0,1]
o
The desired result is achieved by re-indexing the families (xm,j) and (tm,j), where m Z 1 and 1 ::; ) ::; dm . From (a) and ({3) follows 00
U=
< t}.
,,Yj -
Xj)
-t
00
rm
00
U B m ~ U U DEm,j(xm,j) ~ U Um ~ U, m=l
smooth function cP: Rn
223
m=l j=l
m=l
which yields (i). One obtains (ii) directly from ({3). For x E U we choose ma Z 1 with x E Umo ' Since Umo n Um = 0 when m Z ma + 3, we see that Umo can 0 intersect D2E(X m,j) only when m::; ma + 2. This proves (iii). Choose (Xj), (tj) and i(j) E I as in Proposition A.6. Apply Corollary A.4 to find smooth functions 'l/;j; Rn - t [0,00) with DEj(xj) = '1/;;1((0,00)). Condition (iii) ensures that the function 'I/; : U - t R given by
Proof of Theorem A.I.
2) .
o
disc
}.
'I/;(X) = L'I/;j(x) j
written in the form U ~ Km +1 (the interior of
·-u D 1/ 2m(X).
0
d a cover V = (Vi)iEI of sequence (tj) ofpositive
; l/iU)' ~cts
only finitely many of
is smooth, because the sum is finite on a neighborhood of an arbitrary x E U. From (i) it follows that '1/;( x) > 0 for all x E U. We introduce the modified functions ;fy U - t [0,00) given by ;J;j(x) = 'l/;j(x)'I/;(x)-l. They are smooth with DEj(xj) = ;J;;l((O, 00)) and 'L-;J;j(x) = 1 for all x E U. Set cPi = 'L- ;J;j, for i E I, summed over the set Ji of indices j for which i(j) = i (in particular cPi = 0 when Ji = 0). By Proposition A.6.(iii) it follows that cPi is smooth on U. Moreover, these functions satisfy (ii) and (iii) in the theorem. Any x E sUPPU(cPi) has a neighborhood W ~ U that satisfies Proposition A.6.(iii). Then the restriction cPilw becomes a sum of finitely many ;J;jvIW, with jl/ E J i , and there is at least one)1/ E Ji with x E suPPu(;J;jJ. Since
suppu(;J;jJ = DEjJxjJ ~ D2Ejv (XjJ ~ Vi,
Vi. Hence part (i) of the theorem is satisfied.
ditionally we set K a =
we get that x E
K m - 2.
Corollary A.3 has the following generalization to several variables:
):=1 B m. For x E B m
o
both Um and at least rIsures the existence of
Lemma A.7 If A ~ Rn is closed and U ~ Rn is open with A ~ U, then there exists a smooth function '1/;: Rn - t [0,1] with SUPPR" ('I/;) ~ U and 'I/; (x) = 1 for all x E A.
Vi.
Proof. Apply Theorem A.I to the cover of Rn consisting of the open sets Vi = U and V2 = Rn - A. Now 'I/; = cP1 has the desired properties. '\ 0
1
least one of the sets
..
"
225
A. SMOOTH PARTITION OF UNITY
et A
~
Rn there exists a
i, and use Lemma A.7 to UPPRn(7Pm) ~ K m + 1 and
Theorem A.I to the open cover of U consisting of the sets Uo and Up, p E U - A. This yields smooth functions cPo and cPP from U into [0,1], which satisfy Theorem A.l.(i), (ii) and (iii). By local finiteness, smoothness of h on Uo and the property sUPPu (cPo) ~ Uo, we can define a smooth function f: U -+ IR m by
f(x)
= cPo(x)h(x) +
L
cPp(x)h(p).
pEU-A From Theorem A.1.(iii) one obtains h(x) thus
f(x) - h(x)
irs. Let
=
L
= cPo(x)h(x) + LPEU-A cPp(x)h(x)
and
cPp(x)(h(p) - h(x)).
pEU-A Now (ii) of the lemma follows because SuPPu(cPp) follows from the calculation
he series
Ilf(x) - h(x)ll:s
L
cPp(x)llh(p) - h(x)11 =
pEU-A
:S
IPPOrt, we can find bm E I :S m. If we set Cm = lnnula (1) from the lil-th le series (1). Hence cP is get cP(x) ~ Cm7Pm(x) =
o
e Uo and U are open in Rn). Let h: U -+ W be h restriction to Uo. For loth map f: U -+ W that
7))
f: U
-+
Rm satisfies (i)
V. Hence, without loss
[} E U - A an open set ) for all x E Up, Apply
L
cPp(X)E(X) =
~
L
~
Up
U - A, and (i)
cPp(x)llh(p) - h(x)11
pEU-A,xEUp
(L cPp(x)) . E(X) :S E(X).
0
c
~
l:.;,)
;~
."Q
~~
Q",~
~
~
.~.
...
.:-.> ..
"'"
£')J ._;r-'>:~~~
~
...... '...... In' ~<M" ,SCO' ':::". tJ'~~" . . . C\
.....
227
B. INVARIANT POLYNOMIALS
B.
INVARIANT POLYNOMIALS
In this appendix we shall consider polynomials in n 2 variables. We may arrange the n 2 variables Aj as a matrix A = (Aij) and write P(A). We shall only consider homogeneous polynomials, i.e. a sum of monomials of same degree, which will be called the degree of P. The polynomial P(A) is said to be invariant when
P(gAg- 1 )
P(A)
=
for all g E GLn(C). Every polynomial P(A) determines a function
P:Mn(C)
-+
C
and this function uniquely determines the polynomial. Moreover, an invariant polynomial defines a function P: Hom(V, V)
-+
C
for every n-dimensional complex vector space V independent of choice of basis. Let us consider the characteristic polynomial n
aU) = det(I + tA) =
L ai(A)t i ,
ao(A)
= 1.
i=O
Each of the functions ai(A) is an invariant polynomial. Also consider (1) ,
d
L sk(A)t k 00
s(t) = -t dt 10g(det(I - tA))
=
k=Q
where log denotes the power series 00
log(l +x) =
(
L -I!
k-l
xk
k==l
and 10g(det(I - tA)) means that we substitute the polynomial det(I - tA) in the power series. Differentiation in (1) is performed formally,
d (
dt Conversely
t; ai t2 tt iaiC00
00
=
n
a(t) = £;ait i calculated formally too,
,)
=
,
1 .
exp(J s(~t) dt),
o
~
~
.~
u
"
<
,§
..
~~
229
B. INVARIANT POLYNOMIALS
r(A k ).
For instance Sl (A) = 0'1 (A) and S2 (A) + n~ variables identities in
ni
ag(A1,.'" An), such that
sk(A 1 EB A2) sk(A 1 0 A2)
Al EB A2
1
place A by gAg-I, and ;pace of all matrices, the
o Iult
= O.
tA' ;Ai
1=1
00
=L
det(I + t(A 1 EB A 2))
k=l
0)2 '
A
= det(I + tAl) . det(I + tA 2)
giving the first equation in (3) upon considering the coefficient of t k . The other relations are similar, and left to the reader. Let 0' i (A 1, . . . , An), i
k sk(A)t .
AI ( 0
=
and Al 0 A2 the matrix of the tensor product of the two linear maps. For the first equation it is sufficient to show that both sides of both formulas define the same functions on M n1 (C) x M n2 (C), but this is obvious from the definitions:
=
1, ... ,n be the polynomials defined by
n
n
i=l
i=O
II (1 + tAd = L O'i(A1, ... , An)ti.
ixA = diag(A1, ... , An) n
= sk(AI) . sk(A 2),
where Al EB A 2 is the matrix
k
:(A1, ... An)k).
~ 1-
= sk(AI) + sk(A 2)
t .
k=l i=l
-l)kkO'k(A)
L O'i(A 1)O'k-i(A2) i=O
(3)
f= (t A7)
20'2 (A). We have the following
k
O'k(A 1 EB A 2) =
=
= 0'1 (A) 2 -
They are the so-called elementary symmetrical polynomials in the variables (,\1, ... ,
An).
Theorem 8.3 Every polynomial P(A1, ... , An) which is invariant under permu tation of coordinates can be written in the form P(A1, ... , An) = p(O'l,'" O'n), where O'i are the elementary symmetrical polynomials and P is a polynomial. 1
(1 -
tAj~ =
-t cW dt
entity.
o
Proof. See [Lang] Chapter V.9.
-lkO'k(A)t k.
o
lines sk(A) as a poly .. ,00k(A). Conversely ... ,sk(A). We write
Theorem 8.4 Every invariant polynomial P: Mn(C) ~ C can be written in the form P(A) =p(O'l(A), ... ,O'k(A)) where P is a polynomial.
Proof. Let D n C Mn(C) be the diagonal matrices. Since P is invariant and the set
U gEGLn(C)
gDng- 1 C Mn(C)
231
B. INVARIANT POLYNOMIALS
If A E 502n and B is an arbitrary matrix then
; restriction to D n . lorphism en ----t en by ~ain denoted by 7r. If ~nts A1, . .. , An then
Theorem B.5
'I)) .
Proof. Since A is a real matrix and AAt = At A, A is normal when considered as element in M2n(C). The spectral theorem ensures the existence of an orthonormal basis of eigenvectors et, ... , e2n in e 2n and eigenvalues Ai E e, A ei = Ai ei. By conjugation we see that the complex conjugate vector ei is an eigenvector with eigenvalue "Xi. We claim that the basis can be chosen so that e2j = e2j-1 (1 j n). This is easy if all eigenvalues are zero, i.e. A = 0, so we may assume et to be an eigenvector with non-zero eigenvalue A1. By skew-symmetry
n)) )
(i) Pf(A)2 = det(A) (ii) Pf(BAB t ) = Pf(A)det(B).
:s :s
A1 = (Ae 1, e1) = (-e1, Ael) = -"Xl,
. Hence
D
geneous polynomial in in the skew-symmetric Isider Pf (A) as a map
S02n, we let
1 Vj = y'2(e2j- l
+ e2j),
1 Wj = J2H(e2j- l - e2j)
(1:S j
:s n).
Moreover for aj E R given by A2j-1 = Haj we have AVj = -ajwj and AWj = ajvj., This proves the existence of 9 E 02n such that
(0
(0
a1) 0 ' ... , -an
' ( -a1 gAg -1 = dlag
an)) 0 .
In particular
Pf(gAg- l )2 = (a1" .an )2 = det(gAg- l ) = det(A).
~n
aon ))
-1 A
e2n,
I
so Al is purely imaginary. Pick e2 = e1, and note that e2 has eigenvalue A2 = "Xl =j:. A1. Hence e2 is orthogonal to e1 in e 2n . Note that the orthogonal complement Span(e1, e1)..l is invariant under A and also under complex conjugation. This makes is possible to repeat the process in this subspace. Having arranged that e2j = e2j-l, we obtain an orthonormal basis for R2n consisting of the vectors
Since g-l = l, assertion (ii) implies that Pf(A)2 = det(A) for all A. In order to prove (ii) we consider the elements Ii = Bei E R2n . Ii = L Bviev we have that T
= L Aij Ii A!J = L Bvi Aij BIJ-j ev A elJ- = L (BABttlJ-ev
so that
T
=
elJ-'
w(BAB t ). Hence
w(BAB t )
A ... A
w(BAB t ) =
T
A ... AT =
n! Pf(A)!I
By Theorem 2.18 the map this case.
A
Since
. A 2n (B): A 2n (R 2n )
----t
A 2n (R 2n )
A ... A
hn.
.
•
233
C. PROOF OF LEMMAS 12.12 AND 12.13
C. 2n·
o
latrices. The realification S02n, denoted A f--t AR,
lal basis of eigenvectors, ,pan) with o,i E R.
PROOF OF LEMMAS 12.12 AND 12.13
In the proof of Lemma 12.13 the diffeomorphism of (iv) is obtained by applying Lemma 12.12 to a new Morse function F: M --t III which coincides with f outside small neighborhoods of the critical points Pi. The sets Ui and Vi from (ii) and (iii) together with F in a neighborhood of Pi are constructed by means of the following lemma applied to f 0 k;l, where ki is a COO-chart around Pi given by Theorem 12.6. Without loss of generality we may assume f 0 1 to be in the standard fOffil of Example 12.5.
k.;
Lemma C.1 Let W ~ Rn be an open neighborhood of the origin in Rn and let f: W --t R be the function
f(x) = a -
IX
A
n
i=1
i=A+l
I>r + L
xL
where a E R, ,\ E 7L and 0 :::; ,\ :::; n. Choose c > 0 such that W contains the set n
E = {x ER
I I>r + 2 t
o
Then there exists a Morsefunction F: W V ~ R n ->.+1 that satisfy:
xr : :;
2c }.
i=A+ 1
i=1 --t
R and contractible open sets U
(i) F(x) = f(x) when x E W - E. (ii) The only critical point of F in W is 0 and F(O) < a - c. (iii) F- 1((-00,o, + c)) = f- 1((-00,o, + c)). (iv) F- 1((-00,o, - c)) = f- 1((-00,o, - c)) U U. (v) I-I (( -00, a - c)) n U is diffeomorphic with 5 A- 1 x V.
Proof. We introduce the notation ~ = L~IX7 and TI = L~>'+IXr Then (1)
f(x)
=a-
~
+ TI
and we define F E Goo(W, R) to be
(2)
F(x) = a - ~ + 1/- f.L(~ + 2TI),
where f.L E Goo(R, R) is chosen to have the properties: (a) -1 < f.L'(t) :::; 0 for all t E R. (b) f.L(t) = 0 when t ~ 2f. (c) f.L is constant on an open interval around 0 with value f.L(O) >
f.
~
Rn ,
~.
'""
"
'~ ..:'~ ~~ ""."".'
.~
. ..
......,.. ....
~
~
~_<,<
Oy
Jill
"Ailil
,~,
235
C. PROOF OF LEMMAS 12.12 AND 12.13
x
Hence U contains the line segment with endpoints and O. Consider an arbitrary x E U and let x = (Xl, ... ,X>.,O, ... ,O). Let
!l
+ 27})
IR;
+ (1
- t)x) = F(Xl,"" x>., tX>'+1,"" tx n ).
It is increasing, because the formulas (3) and (4) give
~
2£
(~
--t
= O. This gives
=
of L Xia-:(tx + (1 i=>'+1 n
t)x) 2: O.
Xl
: ~A
l
~
~
i
If 0 ~ t ~ 1 then F(tx + (1 - t)x) =
n.
) > o. point of F. By (c), F .(0) > E. This shows that
show that F (x) ~
f (x)
It remains to construct V and prove (v). Set B Equations (5), (6) and (I) we have
B = {x E TV = {x E TV
I ~ + 27} < 2E IE < ~ <
2E
= f-l(( -00, a - E)) n U.
By
and f(x) < a - E} and Tl < min(~ -
E, E -
~/2)}.
If A = 0 then B = 0 and (v) is true (with an arbitrarily chosen V). If A > 0 we define the open set V c IR n ->.+1 by
- E)) . E)).
< a+E and f(x) 2: a+E. : 2E. This implies 7} < E, Jroves (iii). Analogously by
a - E}.
x E U be a R be the function
V = {(s, x>'+1,"" x n )
I yE < s <
~ and Tl < min(s2 -
E, E -
s2/ 2)}.
To see that V is contractible note that if q = (s,x>'+1' ... ,x n ) E V, then V contains the line segments from q to ij = (s, 0, ... , 0) and from ij to (so, 0, ... , 0), where So = !hlE + J2"E). Define finally a diffeomorphism
) E U. Let
, 1]
--t
'1': S>.-1 x V
-----+
B;
'1'(y, s, x>'+1, ... , x n ) = (s . y, x>'+1, ... , x n ),
where y E S>'-1 S;;; R>' and (s, x>'+1, ... , x n ) E V.
-E.
D
Some of the sets introduced in the proof above are indicated in Figure 2. We note that Figure 2 only displays a quarter of the constructed sets; it should be reflected in both the "f[, axis and the yfri axis for n = 2, and rotated correspondingly for
.M
t ~. ;~
:~"~
.'3
~.
"
'.
237
C. PROOF OF LEMMAS 12.12 AND 12.13
Ise between
Vi and .,fiE.
and by Lemma Cl.(iii) and (iv),
F- l (( -00, a + f))
:sJ
F-
~
l
(( -00,
= M(a + f)
a - f)) = M(a -
E) U
Ul U ... U Ur.
We know from Lemma Cl.(ii) that F has the same critical points as j and furthermore that F(Pi) < a - f (1 :S i :S r). If p is one of the other critical points, then
§l
= j(p)
F(p)
~
[a- t,a+f],
and hence [a - t, a + f] does not contain any critical value of F. Hence assertion (iv) of Lemma 12.13 follows from Lemma 12.12 applied to F. D Lemma 12.12 is a consequence of the following theorem, which will be proved later in this appendix.
Theorem C.2 Let N n be a smooth manifold of dimension 11, 2: 1 and j: N - t R a smooth function without any critical points. Let J be an open interval J ~ IR with j(N) ~ J and such that j-l([a, b]) is compact for every bounded closed interval [a, b] C J. There exists a compact smooth (11, - I)-dimensional manifold Qn-l and a diffeomorphism
Y2£ ble-hatched)
choose a smooth chart that
r x E Wi.
i to be mutually disjoint. . f, a + f] and such that all :.1 to j 0 hi 1 : vVi - t IR - t R with 1 :S i :S r, ~n contractible subset of na 12. I3.(i) and (ii), and Cl. By assertion (i) we
: VVi
~ U;=l Wi
: Q x J
vZ such that j
0
: Q x J
---+
---+
N
J is the projection onto J.
Proof of Lemma 12.12. Choose Cl < al and C2 > a2, so that the open interval J = (Cl' C2) does not contain any critical values of j. Since M is compact, we can apply Theorem C.2 to N = j-l(J). We thus have a compact smooth manifold Q and a diffeomorphism : Q x J ---+ N such that j 0 (q, t) = t for q E Q, t E J. Consider a strictly increasing diffeomorphism p: J - t J, which is the identity map outside of a closed bounded subinterval of J. Via p we can construct the diffeomorphism \II . M
---+
M·
p'
\II ( )
,p
p
= { 0 (id Q x p) 0 -l(p)
if pEN d N ~ .
'f p i p
If a E J then \II p maps M(a) diffeomorphically onto M(p(a)). It suffices to choose p so that p( at) = a2. One may choose
p(t)
(7)
= t + Itg(X)dX, CI
where 9 E C~(IR, IR) satisfies the conditions:
supp(g) ~ J, g(x)
>
~I for
x E IR,
l
al
CI
g(x)dx
=
a2 - aI,
j
C2
CI
g(x)dx
= O.
239
C. PROOF OF LEMMAS 12.12 AND 12.13
igure 3 below.
o
Lemma C.S Assume Po
E
N with f (po)
= to
E J.
Then
(i) f-l(tO) is a compact (n - I)-dimensional smooth submanifold of N. (ii) There exists an open neighborhood Wo ~ f-l(tO) of Po, a 6 > 0 with (to - 6, to + 6) ~ J and a diffeomorphism <1>0:
Wo x (to - 6, to
+ 6)
---t
W,
where W is an open neighborhood of Po in N such that the following conditions are satisfied: (a) o(p, to) = p for all p E Woo (b) f 0 <1>0 is the projection onto (to - 6, to + 6). (c) For fixed p E Wo, the function o(p, t) is an integral curve of X. (We
call W a product neighborhood of po.)
Proof. Since to is a regular value, f-l(tO) is a smooth submanifold of dimension n - 1 (see Exercise 9.6). It is compact by the assumptions of Theorem C.2. Choose a COO-chart h: U ---t U' on N with h(po) = and
°
h(U n f-l(tO)) = U' n ({O} x Rn -
imed in Lemma 12.12. which
p. is a diffeomorphism : : ;: idM to wP1 = wp ,
mrnas.
X on N n such that
. field Y on M. Now f(p). 0
xj(t) = Fj(XI(t), ... , xn(t)),
ives f
0
o(t) = t
+c
1::; j
::; n.
For y = (Y2, . .. , Yn) E Rn - l with (0, y) E U' there exists a uniquely determined solution x( t): I (y) ---t U' for (9), which is defined on a open interval I (y) around to with boundary condition x(to) = (0, y). The general theory of ordinary differential equations shows that the solution is smooth as a function of both t and y. Specifically there exists an open ball D in Rn-l with center at 0, a 6 > 0, and a smooth function x: D x (to - 6, to + 6) ---t U' such that . (0) If y E D then the map t ---t x(y, t) is a solution of (9). ((3) If y E D then x(y,to) = (O,y).
Then Wo = h-I(D) is an open neighborhood around Po in f-l(to), and we can define a smooth map <1>0:
They are smooth on
).
Let us write h*(X IU ) = (FI, ... , Fn ) where Fj E Coo(U', R). Then 0: I ---t U is an integral curve of X precisely when h 0 o(t) = (XI(t), ... , xn(t)) satisfies the system of differential equations (9)
: E [0,1]. Then Wp.(p)
l
Wo
>:<
(to - 6, to
+ 6)
---t
N;
o(p, t) = h-l(x(h(p), t)).
Now (0) implies (c) and ((3) implies (a), while (b) follows from the remarks preceeding the lemma. By (a) the differential of <1>0 at (po, to) is the identity on Tpof-l(to). By (c) X(po) is the image of %to Hence D(po,to)o is an isomorphism and the inverse function theorem implies that <1>0 is a local diffeomorphism around (po, to). For suitable sizing of Wo and 6, <1>0 becomes a diffeomorphism. 0
~
'"
.""".'i .~ -- ,
.
-
::::.p.~", ~_:
-
- -~.
C. PROOF OF LEMMAS 12.12 AND 12.13
Lemma C.S Let to E J. Define P E f-I(t). Then 1f is smooth.
~tem of equations in (9) :s a uniquely determined Dr to Lemma C.S shows
Proof. Let Wo, Wand <1>0 be given as in Lemma C.S. Then 11"lw = prwo 0 <1>0 1 , and 1f is smooth on W. A PI E N with f(pI) = tl enables us to define 1fI: N -+ f -1 (tI) by 1fI (p) = 'Pt,h (p). Then 1fI is smooth on a product neighborhood WI, of PI and since 1f = 'PtI,to 0 1fI, 11" will be smooth on WI; cf. Lemma C.7. 0
-I([to, tI]) is compact, wards it, such that the J-I(tI). We can apply
There will be some Sm y the uniqueness of the past ti. The analogous r. In total we see, again all of J. 0 f Nand a( J) intersects ., t2 E J we can define
tween
f- I (t2)
and the
-1
UT
as 'Pt2h = 'Ptl,t2' ne mines an integral curve
< Sk
= t2 and product
, such that
to cI>0 in Lemma C.S.
1,8;(a(Si-I)) = a(si).
'P So ,81 ,
o
-+
f-I(tO) by 1f(p)
241
There exists a uniquely . Moreover, f oa(t) = t
1f:
N
--..--.....-
'Pt,to (p) where
Proof of Theorem C.2. Let Q = f-I(to). By Lemma e.s \lJ: N
-+
Q x J;
\lJ(p) = (11"(p) , f(p))
is smooth. Let pEN. Consider the differential
Dp\lJ: TpN
-+
T7r (p)Q x R.
It follows from Lemmas e.7 and e.S that the subspace Tpf-I(f(p)) ~ TpN is mapped isomorphically onto T7r (p)Q x {O} and that Dp\lJ(X(p)) = (0,1). Hence D p \lJ is an isomorphism. Since \lJ is bijective by construction, we can conclude that \lJ is a diffeomorphism. The assertion follows by letting be the inverse 0 diffeomorphism.
c
~ r..l
,§
~f·
.,.;l
"'tJ.
'"
1'·
. ~
~--~-~
.... ,- ..,.
~.<
J.----.j
D.
D.
243
EXERCISES
EXERCISES
1.1. Perform the calculations of Theorem 1.7. 1.2. Let W <;;;; 1R 3 be the open set W
= {(Xl, X2, X3)
3
E 1R I either X3 =1= 0 or xi
+ x~ < I}.
Prove the existence and uniqueness of a function F E COO(W, R) such that
grad(F) is the vector field considered in Example 1.8 and F(O) = O.
Find a simple expression for F valid when xi + x~ < 1.
(Hint: First note that F is constant on the open disc in the Xl, x2-plane
bounded by the unit circle S. Then integrate along lines parallel to the
x3-axis.)
2.1. Prove the formula in Remark 2.10. 2.2. Find an W E Alt 2 1R 4 such that W /\ W 2.3. Show that there exist isomorphisms 1R 3 ~ Alt l lR 3,
=1=
1R 3
L
O.
Alt 2R3
given by
i(v)(w) = (v,w),
j(V)(Wl,W2) = det(v,Wl,W2),
where ( , ) is the usual inner product. Show that for VI, V2 E 1R3, we have
i(vI) /\ i(V2)
= j(Vl
x V2).
2.4. Let V be a finite-dimensional vector space over R with inner product ( , ), and let i: V -+ V* = Alt 1 (V) be the R-linear map given by
i(v)(w) Show that if {b l
, ... , bn }
= (w, v).
is an orthonormal basis of V, then
i(b k )
= bt,
where {bi, ... , bt} is the dual basis. Conclude that i is an isomorphism. 2.5. Assumptions as in Exercise 2.4. Show the existence of an inner product on AltP(V) such that (WI /\ ... /\ w p , Tl /\ ... /\ Tp )
= det ((Wi, Tj)),
...
~- ~c
,
-
>;l;;
'. . • '
'
"'''';''
~ ,. -
...•....•.•.... '..
-
-'
-' -
~'
._..c.' -
-
~
,,"' •
•.,.' ';•..'.".'
.-_.
.-,
-
•
.~ ..
,,~,:~
.
},;"jti:;,:,
."'~~
D.
245
EXERCISES
2.10. Let V be a 4-dimensional vector space and {EI, ... , E4} a basis of Alt l (V). Let A = (aij) be a skew-symmetric matrix and define
=
ex
.nd let fJj = i(bj). Show
j .
i<j
Show that
I) }
= 0 {:} det (A) = O.
ex /\ ex in V and let A
L aij Ei /\ E
=
(aij)
Say ex /\ ex
= ..\.€I/\
E2/\ E3/\ q. What is the relation between ..\ and det (A)?
2.11. Let V be an n-dimensional vector space with inner product ( , ) and volume element vol E Altn(V), as in Exercise 2.9. Let v E AltI(V) and
(1 ~ i ~ p) we have
p).
F v : AltP(V)
-+
AltP+l(V)
be the map
Fv(w) = v /\ w. ~(f)(W2),
Show that the map
F: = (-lynP*
0
Fv
0
*: Altp+I(V)
-+
AltP(V)
is adjoint to Fv , i.e. (Fvw, r) = (w, r;r). Let {q, , en} be an or thonormal basis of V with vol (EI, ... ,En) = 1 and {EI, ,En} the dual (orthonormal) basis of AltI(V); see Exercise 2.5. Show that
Jnal matrix} , is a finite-dimensional ner product ( , ). From ') for all p, in particular Altn(V). Hodge's star
w that * is well-defined
p+I F:(EI /\ ... /\ Ep+d =
L (_l)i+I (v, Ei) EI /\ ... /\ Ei /\ ... /\ Ep+l' i=I 2
Show that FvF: + F: Fv: AltP(V) -+ AltP(V) is multiplication by IIvll • (Hint: Suppose that v = ..\.EI and show that the general case follows from the special case.) 2.12. Let V be an n-dimensional vector space. Show for a linear map the existence of a number d(f) such that
1: V
-+
V
Altn(f)(w) = d(f)w ith vol(eI, ... , en) 1 (V). Show that
=
1 for w E Altn(V). Verify the product rule
d(g 0 1) = d(g )d(f)
. /\ Ea(n) 1) on AltP(V).
for linear maps f, g: V -+ V using the functoriality of Alt n .
Prove that d(f) = det(f).
(Hint: Pick a basis eI, ... , en for V, let I€ , ... , En be the dual basis for
Altn(V) and evaluate Alt n (f)(€ I /\ ... /\ En) on (eI, ... ,en ) in terms of
the matrix for fwith respect to the chosen basis.)
1·
.q...
J.~,.
'"
....
'1
j ..,:;
.'
~;"'''
D.
EXERCISES
(Hint: Try the case p = 1, n = 2 first. What can one say about !:J. (j . dx I )
where I = (il, ... , i p )?)
A p-form w E OP(U) is said to be harmonic if ~(w) = O. Show that
lmplex
-0
*: OP(U)
,R)
--+
~ Coo (U, R)
--+
on-p(u)
maps harmonic forms into harmonic forms.
O.
3.4. Let AltP(R m , q be the C-vector space of alternating IR-multilinear maps
Ie de Rham complex is
l usual
247
w: Rn
x ... x IR n
--+
C
(p factors). Note that w can be written uniquely
--+
0
w
constant I-forms
2.9) to define a linear
= Re w + i 1m w,
where Rew E AltP(Rm), Imw E AltP(R n ).
Extend the wedge product to a C-bilinear map
Altp(R n , C) x Altq(R n , q ~ AltP+q(Rn , C) and show that we obtain a graded anti-commutative C-algebra Alt*(lR n , q. 3.5. Introduce C-valued differential p-forms on an open set U ~ Rn by setting (see Exercise 3.4)
.. 1\
dXn and
*0 *
OP(U, q
= COO(U, AltP(R n , C)).
Note that w E OP(U, C) can be written uniquely w
= Re w + i 1m w,
where Rew E OP(U). Extend d to a C-linear operator
d: OP(U, q
d;j 1\ ... 1\ dXp
1\ ... 1\ dXip'
:or ~: OP(U)
: ~(J)dXl
--+
OP+l(U, C)
and show that Theorem 3.7 holds for C-valued differential forms. Generalize Theorem 3.12 to the case of C-valued differential forms 3.6. Take U = C - {O} = R 2 - {O} in Exercise 3.5 and let z E OO(U, q be the inclusion map U --+ C. Write x = Re z, y = 1m z. Show that
n that
d;i v
--+
Re (z- l dz)
OP(U)
where r : U
--+
R is defined by r(z)
1m
1\ ... 1\
dXp
= dlogr,
-1
(z dZ)
=
= Izl
-y 2 2 dx x +y
=
"';x 2 + y2. Show that
x
+ x 2 +y?dy.
(Observe that this is the I-form corresponding to the vector field of Example 1.2.) 3.7. Prove for the complex exponential map exp: C --+ C* that
dz exp
= exp(z)dz
and
exp*(z- l dz)
= dz.
.;,.J
•...L"tJ.•.
tnd linear maps with
..
~.
j.~
D.
EXERCISES
dn -
1
d1
° dO
..
'"
249
4.4. Let 0 ---t A ---t A l ---t ••• ---t An ---t 0 be a chain complex and assume that dimR Ai < 00. The Euler characteristic is defined by
n
X(A*) =
L
(_I)i dim Ai.
i=O
Show that X(A*) = 0 if A* is exact. Show that the sequence
injective. Show that surjective and 15 is have that if iI, 12, 14 n. (This assertion is
o
---t
Hi(A*)
---t
Ai lIm di - 1
!
Im di
---t
0
is exact and conclude that dimR Ai - dimR Im di - 1 = dimR Hi(A*) n
Show that X(A*) =
+ dimR Im i.
.
2:
(-I)~dimRHi(A*).
i=O
4.5. Associate to two composable linear maps
1: VI there exists a exact
in complexes where
11 p.
an exact sequence o ---t Ker(J)
---t
V2,
g: V2
---t
V3
1) ---t Ker(g) ---t ---t Cok(J) ---t Cok(g 0 1) ---t Cok(g) ---t O. 5.1. Adopt the notation of Example 5.4. A point (x, y) E Ul can be uniquely described in terms of polar coordinates (r, ()) E (0, (0) x (0, 21l-). Let argl E nO(Ul) be the function mapping (x, y) into () E (0,21r) (why is ---t
Ker(g
0
argl smooth?). Define similarly arg2 E nO(U2) using polar coordinates with () E (-1r, 1r) and prove the existence of a closed I-form 7 E n1 (R2 - {O}) such that
71uv
= i~(7) = darglJ
(v = 1,2).
Show that the connecting homomorphism
8°: HO(U1 n U2)
---t
HI (R 2
-
{O})
carries the locally constant function with values {O, 21r} on the upper and lower half-planes respectively into [7]. 5.2. Show that·the I-forms 7 E n1 (R 2 - {O}) of Exercise 5.1 and Im(z- l dz) of Exercise 3.6 are the same. 5.3. Can R2 be written as R2 = U U V where U, V are open connected sets such that Un V is disconnected? 5.4. (Phragmen-Brouwer property of Rn) Suppose p i= q in R n . A closed set A ~ Rn is said to separate p from q, when p and q belong to two different connected components of Rn - A. Let A and B be two disjoint closed subsets of R n . Given two distinct points p and q in Rn - (A U B). Show that if neither A nor B separates p from q, then Au B does not separate p from q. (Apply Theorem 5.2 to Ul = Rn - A, U2 = Rn - B.)
o ..t:
C'..l
...
"..J
.C"t4
. 'to
",
'". -ii~
D. EXERCISES
ce relation in the class
omotopic to a constant
> o. Show that n-1
o ~e.
lch that f(x) and g(x)
x -+ sn is homotopic
. {O}. Show that two
~-1
are homotopic.
tic to D m when m > n. ·2). Show that ,n-1
with glSn-l ~ idSn-1, ),1) be given. Suppose hat 1m f (D n ) contains l
theorem and use Exer [0,1] ---+ D 2 such that
8.3. Suppose that M ~ Rk (with the induced topology from Rk ) is an n dimensional topological manifold. Include M in Rk+n. Show that M is locally flat in Rk+ n . 8.4. Set Tn = Rn /lL n , Le. the set of cosets for the subgroup lL n of Rn with respect to vector addition. Let 1r: Rn ---+ Tn be the canonical map and equip Tn with the quotient topology (Le. W ~ Tn is open if and only if l 1r- (W) is open in Rn). Show that Tn is a compact topological manifold of dimension n (the n dimensional torus). Construct a differentiable structure on Tn, such that n 1r becomes smooth and every P E R has an open neighborhood that is mapped diffeomorphically onto an open set in Tn by 1r. Prove that T l is diffeomorphic to 51. 8.5. Define A: R2 ---+ R2 by A(x, y) = (x +~, -y). Show that there exists a smooth map A: T 2 ---+ T2 satisfying A ° 1r = 1r ° A. (Consult Exercise 8.4.) Show that A is a diffeomorphism, that A = A-I and that A(q) i= q for all q E T 2 • Let K 2 be the set of pairs {q, A(q)}, q E T 2. Show that K 2 with the quotient topology from T 2 is a 2-dimensional topological manifold (Klein's bottle). Construct a differentiable structure on K 2 • 8.6. Let Po E sn be the "north pole" Po = (0, ... ,0,1). Show that sn - {po} is diffeomorphic to Rn under stereographic projection, i.e. the map sn {po} -+ Rn that carries P E sn into the point of intersection between the line through PO and p and the equatorial hyperplane Rn ~ Rn +1. 9.1. Let M ~ Rl be a differentiable submanifold and assume the points PERL and Po E M are such that lip - poll ::; lip - qll for all q E M. Show that P - Po E TpoM 1.. 9.2. A smooth map
Dp
)oth manifold N to a : the inclusion. Show
---+
TqN,
q =
is injective. Show that there exist smooth charts (U, h) in M with p E U, h(p) = 0, and (V, k) in N with q E V, k(q) = 0, such that k o <poh- l (Xl,
larts of Theorem 7.10).
251
... ,Xm )
=
(Xl, ... ,X m
,O, ... ,O)
in a neighborhood of O. (Hint: Reduce the problem to the case where
(
8
l~ '<;0 ,to' .• ~ QJJ
,S
' ... , .....
t:.) r"\'
~Ij, """'e' ~, ~.,~ Q., 'Ij ~
~
.--J -~'-r""'
1
~,"'" \& ~ ",~".", .~.g
,
.,.F
,C"tJ
',~
~
,~
"
,
~
D.
function theorem to
I), Xm+l,
where At is the transpose of A. Note that the pre-image
. .. ,X n ).)
is an immersion, when
p is closed if
that an injecti ve closed
ltation of Exercise 8.4,
w that 0: is an injective
dense in T 2 • Conclude
1 R.)
anifolds is called sub
DA
Apply Exercise 9.6 to show that On is a differentiable submanifold of
Mn(R).
9.10. A Lie group G is a smooth manifold, which is also a group, such that both
mch that
).
mooth manifolds. Let ~ry point of the fiber (m - n )-dimensional :ult holds for all non :.
-t
G;
J.L(gl, g2)
i: G
-t
G;
i(g) = g-1
and are smooth. Show that the group On of orthogonal n x n matrices is a a Lie group. (Apply Exercise 9.9.) 9.11. Let
O*(M)
is a chain map. 9.12. The usual inner product on Rn induces an inner product on Altn(Rn) (see Exercise 2.5). Show that w E Altn(Rn) is a unit vector if and only if w( VI, ... , v n ) = ±1 for every orthonormal basis {VI, ... ,vn } of R n . 9.13. Show that Klein's bottle (Exercise 8.5) is non-orientable. 9.14. Let M n be a Riemannian manifold and f E C=(M, R). Define the gradient vector field gradf on M by demanding that gradpf E TpM satisfies (gradpf, VIp for all have
V E
= dpf(v)
TpM. Show that for a local parametrization, h: W n
I torus Tn in Rn+1. ne f: R3 -t'R4 by
+ a3 x j).
gradh(x)f =
j=1
where aj E COO(W, R), 1 equations
~
)Qth embedding. matrices we have the map
M, we
a
J
j ~ n is determined by the set of linear
af L gij(x)aj(x) = ~(x) i=1
-t
L aj(x) ax"
n
1points and therefore
= g1g2
J.L: G x G
1, h) in M with p E U,
253
EXERCISES
x~
(1 ~ i ~ n).
Show that the map grad f: M - t T M is smooth.
Let p E M with gradpf '=I O. Set c = f(p). Show that f-l(c) in a
neighborhood of p is an (n - 1)-dimensional smooth submanifold, and that
gradpf is a normal vector to f-l(c) at p.
o
~
<:..l
,g. ~
...J
. C'jJ,
~.
't!i
'"
.<>~
1~ ~. J L----J
255
D. EXERCISES
let M denote the set of wo orientations of TpM. form w E on(w) we let and op is the orientation open and 7f maps W ted set W <;::; M. Note tiable structure such that iented open set W <;::; M. air consisting of M and
where
or(M)± is the eigenspace associated to ±1 for the isomorphism A*:Or(M) ---. or(M).
Show that the de Rham complex sum of two subcomplexes
(O*(M)+, d)
(O*(M), d) decomposes into the direct (O*(M)_, d).
and
Show that 7f* maps the de Rham complex (0* (M), d) isomorphically onto (O*(M)+, d). Show that for every k E 71. we have that
H k(7f): Hk(M) ---. Hk(M) maps Hk(M) isomorphically onto the (+I)-eigenspace in Hk(M) of
at M (see Exercise 9.15) md that M is orientable
'the smooth submanifold f M (see Theorem 9.23).
I·
tion
Hk(A). 10.1. Let 7f: R 2 ---. T 2 be as in Exercise 8.4, and let Ul
= 7f(R x (0,1)),
U2
= 7f(R x (-!, -!)).
Show that U l and U2 are diffeomorphic to SI x R, and that U l n U2 has 2 connected components, which are both diffeomorphic to SI x R. Note that U l u U2 = T2. Use the exact Mayer-Vietoris sequence and Corollary 10.14 to show that
HO (T 2 )
S:!
H 2 (T 2 )
S:! R
HI (T 2 )
and
S:! R 2 .
10.2. In the notation of Exercise 10.1 we have smooth manifolds
C l = 7f(R x {a}),
the Gauss map Y. We by requiring
~
C2 = 7f({b} x R)
of T 2 which are diffeomorphic to SI. They are given the orientations induced by lFit Show that the map
0 1 (T 2 )
---.
R2 ;
w~ (r w, r w) JC
V).
,quation
of Exercise 9.15. Let lnges the two points in r 2 and that
(a,b E R)
induces an isomorphism HI independent of a and b.
(T 2 ) ---. R2.
1
JC2
Show that this isomorphism is
10.3. Using the notation of Exercise 8.4, we have smooth submanifolds in the n-dimensional torus Tn = Rn /71. n , which are diffeomorphic to SI
Cj = {7f(O, ... ,O,s,O, ... ,O) I sE R}, where s is in the j-th place, 1 ~ j ~ n. They are given the canonical orientation. Let w E 01(T n ) be a closed I-form with
JCr w = ° j
for 1
~ j ~ n.
d!.'f')
" ~''-'
~
"ll"
~
~~ ~ ";!:
-.;.•......•.' .....
....•....
..
.
.~
D.
257
EXERCISES
is an isometry for all q E M.
Show (by means of Riesz' representation theorem) that there exists a
positive measure f-LM on M such that for all f E C~(M, IR)
(w), and show that f is
r fdf-LM = ~2JMr. f
0 1r
JM
, w)
vol M .
Assume that f E C~(M, R) has support contained in an oriented open subset W ~ M. Show that
"
..:orientable smooth n
fMfdf-LM
O.
Ittle.
lentified with the map
with smooth boundary \.1. Show that for for
\ dT.
ct domain with smooth
iirected Gauss map and lopen set U ~ R3 with
: det (F(p), WI, wz).) , oriented by the Gauss t domain with smooth ield V pointing in the loth vector field on an
\.1 the oriented double iemannian metric such
10.10. Define ij:Rn-1
--+
= fwfVOlw.
Rn (1 ::; j ::; n) by
ij(XI, ... ,xn-d = (XI, ... ,O"",Xn-l) with 0 at the j-th entry. Let Yn={XERnlxi~O,
l::;i::;n}.
and w E nn-I(Rn) with supp(w) n Yn compact. Show that we have
1 t dw =
Yn
(-l)j
1
ij(w).
Yn-l
j=l
10.11. Let w E nr(M n ). Suppose that
hw=o,
d: nn-l(p)
M n that is diffeomorphic to
~ ~
for every oriented smooth manifold Show that dw = O. 10.12. Let P be a smooth manifold and --+
sr.
nn(p),
a linear operator such that
rdw= JaRr w
JR
for every w E nn-l (P) and every compact domain with smooth boundary R in an n-dimensional oriented submanifold M n ~ P. Show that d = d. 10.13. Let M be a smooth manifold. A piecewise Cl-curve on M is a continuous parametrized curve a: [a, b] --+ M for which there exists a partition
a= to < tl < ... <
tk-l
<
tk
= b,
..,. ....,l
'"
~
D.
'e continuously differen we define the path
1 (M)
ow that w is exact. on an open set U ~ Rn
\ ... 1\ dXn
d 6).
ndary R
~
U that
16g dJLn,
11 . I. Generalize the concept of degree to the case of continuous maps in such a way that Corollary 11.2 holds for f continuous. Show that Corollary 11.3, the statement deg (f) E 7L of Theorem 11.9, Corollary 11.10 and Proposition 11.11 generalize to the case of continuous maps. 11.2. Prove the formula of Theorem 11.9 under the following assumptions: (i) f is continuous. (ii) f is smooth outside a closed set A ~ N. (iii) p E M - f(A) is a regular value of 1 restricted to N - A. (Hint: Lemma 11.8 holds with U ~ M - f(A), Vi ~ N - A. Construct a homotopy F from f to a smooth map such that F t (0 ::=; t ::=; 1) is the identity near the points qi E f- l (p).) 11.3. Let N n be an oriented closed (i.e. compact) manifold. Show that every integer occurs as the degree of some smooth map N n ~ sn (Use Exercise 11.2.) 11.4. (The fundamental theorem of algebra) Suppose n-l
: the identity
r JaR
I
259
EXERCISES
P(z) = zn
+ Q(z) = zn + L
11\ *dg.
is a complex polynomial of degree n ;::: 1 without any complex root. This leads to a contradiction as follows: Regarding SI as the unit circle in C define for any T ;::: 0 a smooth map 1
1
fr:S ~S; (19) of Example 9.18 (i.e. 6p = 0). Apply n COO(lR R) and R € C'
(TW)n F(w,t) = I()n TW
=
9 1\ w, l
is its boundary
= Vol(sn-l )g(O).
fr(w) = ,
+ tQ(TW) + tQ ()I' TW
>
...
O::=;t::=;l
defines a homotopy SI x [0, 1] ~ SI between fr and the map W t--t w n and conclude from this that deg(fr) = n. 11.5. Let ~ ~ Rn be a smooth submanifold diffeomorphic to sn-l, n ;::: 2, and let UI. U2 be the open sets given by the Jordan-Brouwer separation theorem 7.10. For XO E Rn - ~ define F:~ ~
x - xo
sn-\
F(x) =
-1.) Conclude that
. side is the Lebesgue
P(TW)
Show that deg(fr) is independent of T and then (by taking T = 0) that deg(fr) = O. Prove for T chosen suitably large that the expression
tably large, to obtain
d fS n -
(Ljzj
j=O
Ilx - xoll'
Show that if Xo E U2 ±1 if Xo E Ul
deg(F) = {O
where the sign depends only on the orientation of ~.
~~
"...J
~. ')-ol;
.
....
......:i
11 ~
.' ....."<:.
''''4oi!Si!~
D.
'e to the case of equal
12.1. Show that f: R n
-+
261
EXERCISES
IR given by n
f(x) = Lsin2 (7rXj)
'rom a smooth manifold
j=l
>act subset of U. Prove boundary R such that Lemma A.7 and
c> 0
,en set U ~ IR n (n 2: 2), eld on B without zeros. allowing properties:
is a Morse function. Determine all the critical points and their indices. Show with the notation of Exercise 8.4 that there is a Morse function on Tn such that f = J 0 7r. Prove that the number of critical points of is (~). index .A for 12.2. Let f be a Morse function on a Riemannian manifold. Prove that all zeros of grad (f) are non-degenerate (see Exercise 9.14). 12.3. Show for any smooth map f: M m -+ Rn and p E M that there is a map (non-linear in general)
J
J
d~f: Ker Dpf
-+
COkDpf
such that
d~f(cl(o)) = (f
of Lemma 7.4, and (c)
b), (c).
ally depends on U and at F:B -+ IR n - {O} is 'J) = 'Y(G, U). F: U U B -+ Rn to be 'Y(F1B , U) i- 0, then F ) of Exercise 11.8 such rero of a vector field is
0)"(0) + ImDpf
0
whenever 0: (-0,0) -+ M is smooth with 0(0) = p and 0'(0) E Ker Dpf. 12.4. Let f be a Morse function on a closed connected manifold with at least two critical points of index O. Prove that f has a critical point of index 1. (Hint: With notation as in the proof of Theorem 12.16 study dim HO(Al(bj)) as a function of j.) 12.5. Is it possible for a Morse function on R2 to have only two critical points both of index O? 12.6. With the notation and assumptions of Lemma 12.13 we put v
W v = M(a - E)
U
U Ui
(O:S
1/
:S r).
i=1
Show that W v has finite-dimensional de Rham cohomology and that one of the following two cases occurs for each 1/, 1 :S 1/ :S r: Case
1: dim HP(Wv ) -
dim HP(Wv-l)
=
{O1 ifif pp #= .A.Av v
Case 2: dim HP(Wv ) _ dim HP(Wv-l) = {O
-1
vector fields both with vector field on IR n +m
~f p # .A v - 1 If P = .A v - 1.
12.7. (The Morse inequalities) For a smooth manifold Mn with finite
dimensional de Rham cohomology we define the Poincare polynomial by n
PM(t) =
L bjti , j=O
C .J::
...,.J
~
.~
,§
.J::'
-
.~ ~ ~
11 ?\
'"
~
D.
IHj(M). Let f: M
lex A (c.\
-+
R
< (0). Define
12.10. Let 7[": Mn -+ M n be a d-fold covering of closed manifolds M and M, i.e. M n can be covered by open sets U with the property that 7["-1 (U) is a disjoint union of d open sets Ul , ... , Ud such that 7["1 Vi : Ui -+ U is a diffeomorphism for 1 :S i :S d. Show that X(M) = dX(M). 12.11. Assume f and g are Morse functions on the closed manifolds Ai and N respectively. Show that a Morse function h on M x N can be defined by
h(p, q)
integral coefficients. each set W v introduced
Describe the critical points and indices for h in tenns of similar data for f and g. Derive the product formula for Euler characteristics
X(M x N) = X(M)X(N).
wi-
lSsumptions and notation ) that either C.\-1 = 0 or
:tion (see Exercise 8.4).
Hi (51) ~ R Ig sequence I: 1 :S i 1 < ~
[}
e linearly independent. CaIW lover subtori.) I
hat
= f(p) + g(q).
13.1. A symplectic space (V, w) is a real vector space equipped with an alter nating 2-form w E Alt 2 (V). A linear subspace VV ~ V is said to be non-degenerate if for every e E W - {O} we can find fEW such that w(e, f) =/=- O. Assume from now on that V is non-degenerate of finite di menSiOn. Let W ~ V be a non-degenerate subspace. Show that
) S n).
'l(R/Z)
263
EXERCISES
= {x
E VIW(.T, y)
= 0 for every
yEW}
is a non-degenerate subspace with WEB l-Vi- = V Prove that V has a basis {e1' h, e2, 12, ... , en, fn} such that w(e·t , e·) J = w(f·1., f·) J = 0,
w(ei,fj)
=
I if i = ) { 0 if i =/=-)
This is called a symplectic basis. Note that dim V must be even.
(Hint: Pick e1 =/=- 0 arbitrarily. Then find h and apply induction to Wi-,
where W is spanned by el and h.)
Let W1,T1,W2,T2, ... ,Wn ,Tn be the dual basis of Alt 1 (V) = V* to a
symplectic basis for V. Show that
n
W = LWj 1\ Tj. j=1
13.2. Let M n be an oriented closed smooth manifold of dimension n == 2 (mod 4). Show that Poincare duality organizes H n / 2 (M) as a non degenerate symplectic space (see Exercise 13.1). Prove that X(M) is even. 13.3. Consider a smooth map f: Nn -+ M n between n-dimensional oriented smooth closed manifolds, where M is connected. Prove that if Hn(j) =/=- 0 then HP(j): HP(11,1) -+ HP(N) is injective for every p.
I~< ....... ...;
'~-tj '~
~
,¢";.)'
..c,tj
,
~ ...
"~ ~ ~ ~ ,~, ,~-tj
,,~'
.
....:J
~
~ ~
~ "
.....
~ '~ ,~,.l;j '~, '.....' ..... ' 'I::
~
'~ 'tlo",
'b ~
"'>'~'
""~'
~ ;'~:'.?"-
D.
r with 0 < m < nand
265
EXERCISES
There are disjoint compact domains R l , ... , Rd with smooth boundary such that K ~ U1=1 (Rj - oRj) and f is constant on K n R j with value aj. Then
- 2). ,nces
:L aj j' d
o
w.
oRj
j=l
(Hint: Exercise 11.7 is needed.)
o.
14.1. A rational function
P(z) R(z) = Q(z)'
constructed in Exercise Find isomorphisms
A* on HJ(M). n-dimensional manifold
I
whether M is compact
md every q is at most
where P and Q are complex polynomials, is initially defined only on C with the roots of Q removed. Show that R extends to a smooth map of the Riemann sphere C U {oo} to itself. 14.2. The n-th symmetric power 5p n (X) of a topological space X is the set of orbits under the action of the symmetric group 5(n) on xn = X X ... X X (n factors) with the quotient topology from X n . Show that spn(5 2) is homeomorphic to Cpn by the map f: (Cpl ---+ C1pn taking
f
([al' ;)1],"" [an, ;)n]) into lao, al, ... ,an] E O:lln, where the ak are determined by the identity
rp(M) is either finite
'countably many copies
n
II (;)jZ -
= n~l D j , where each
i for every j. Show for 1
vector space of locally hism
f
aj) = :Lak zk , k=O
j=l
14.3. Show that HP (52 X 54) ~ HP (CP3) for every p, but that the graded algebras H* (52 X 54) and H* (CP3) are not isomorphic. 14.4. Show that any continuous f: Cpm ---+ Cpn induces the zero homomor phism
1*: HP(Cpn)
---+
HP(Cpm)
for p =f. when m > n. 14.5. Prove in the following steps that constant. Suppose
7r:
5 2n+ l
°
locally constant on an isomorphism
be evaluated on
n
F: 5 2n+ l
X
[0,1]
---+
is a homotopy from a constant Fo to Fl to g: D 2n+ 2 ---+ Cpn by
---+
Cpn is not homotopic to a
C1pn
= 7r,
and extend
E (I)
g(tz) = F(z, t),
z E 5 2n + l ,
t E [0,1].
7r
continuously
l~ ~ ~'-oJ .....
r{
....
~ -tj ".q.. .........c ~tj '.~ ~ ~ lO;: .s::.
"tj
Q;"tj . '?-:.
...... '
~
.~
.~
~
~ ......
.....tlo
~
.q...
~
'"
~ ~
D.
/
; IZjI2) 1 2]
)ijectively onto U n + l = e of 1r with the inclusion lat f 0 h = g and argue n, and pass to de Rham
To an increasing set of indices I: 1 S i l < i2 < ... < i m S nand A E Frm(IFn) is associated the m x m-submatrix AI of rows numbered by I and the (n - m) x m-matrix AI' consisting of the remaining rows. Show that UI
= {[AJ I A
E Frm(IFn), Al invertible}
is an open set in Gm(Fn) and define hr: UI (n - m) x m matrices by hI([A])
14.1.
) vectors Vj E TzC n+ 1 = Show that the hermitian 4.4 satisfies
}2)
oduct on C n + I .
fold M equipped with a iitions ~
space for every p E AI
;tic manifold.
1 dimension 2m and that
I that H* (M) contains a
2 == [w] E H (M).
nd let IF denote either R
:note by Gm(Fn) the set
>mpact subspace of the
>rthogonal projection on
ce (with countable basis
:r IF of rank m. Observe
bset of IF mn and hence
by right multiplication
/GLm(lF) with quotient
;iating to A E Frm(Fn)
(Fn).
267
EXERCISES
-+
Matn-m,m(lF) into
= A],A I I ,
Show that h] is a homeomorphism, and finally (identifying Matn-m,m(lF) with F(n-m)m) that 1i = {(U], h])} is a smooth atlas on Gm(Fn). The resulting closed smooth manifold Gm(IFn) is the Grassmann manifold of m-dimensional subspaces of F n . 15.1. Show that the pull-back 7]* (TS2) by the Hopf fibration 7]: S3 -+ S2 is a trivial vector bundle (see Example 14.10). Prove also that the tangent bundle TS3 is trivial. 15.2. Let G be any Lie group (see Exercise 9.10). Right translation by g E G
is the diffeomorphism R g:G -+ G; Rg(x) = xg. Given a tangent vector X e E TeG at the neutral element e, define X g E TgG for g E G by X g = DeRg(Xe ). Show that this extends X e to a smooth vector field X on G and moreover that X is right-invariant in the sense that
Xhg = DhRg(Xh)
for all h, g E G.
Construct a frame over G for the tangent bundle TG consisting of right invariant vector fields. 15.3. Let ~ be a smooth real line bundle over IRlP n with total space sn x sO I~, i.e. the orbit space of sn x IR under the action of Sa = {± 1}, where -1
acts by (x, t) t-+ (-x, -t). This is the canonical line bundle over IRlP n . Construct a smooth isomorphism of vector bundles TRpn
EBc l ~ (n+ 1)~,
where k~ denotes the k-fold direct sum ~ EB ... EB ~. (Hint: Both total spaces can be identified with sn x So IR n + I .) 15.4. Prove that the tangent bundle TCpn with the complex structure on TpClPn given by Lemma 14.4 is a smooth complex vector bundle over ClP n . Construct a smooth isomorphism of complex vector bundles (similar to that in Exercise 15.3) TCpn
EB c l ~ (n
+ 1)Hn ,
...
'"
~_.,..;c~;:~"
D.
bundle H n is the same : ( acts on H n .
r: Frm(Fn)
---t
269
EXERCISES
vector bundle TV over E (the tangent bundle along the fibers or vertical tangent bundle). Show that the orthogonal complements (TpFb)l.. to TpFb in TpE with respect to a Riemannian metric on E are the fibers of another smooth vector bundle T h over E (the normal bundle to the fibers or horizontal tangent bundle). Find smooth vector bundle isomorphisms
Gm(lF n )
i m a smooth section IW that the map
: SI([A])Q h", T Va:-, <::D T = TE,
T h", =
1r *( TB.)
'Oup). Define a smooth
Fffi with the induced '" that this is a smooth the assignment
]) monical vector bundle
(Fn) with V, which to )r) assigns Ax E V. ::tions S: U ---t 1f- 1 (U) 'ames (Sl, ... ,Sn) for 'Y ~ identification 'Yp ~ V
15.10. Prove Theorem 15.18 without the compactness assumption on B for smooth real vector bundles by embedding E(O in some Euclidean space (Theorem 8.11). Make use of Exercise 15.9, observing that TV ~ 1r*(O. Deduce the smooth complex case of Theorem 15.18 without the compact ness assumption. 15.11. Let ~ be a smooth vector bundle with inner product over B (not assumed to be compact). Suppose A ~ B is a closed set and U ~ B is open with A ~ U. Let s be a continuous section in ~ over B which is smooth on U. Construct for any given continuous map E: B ---t (0,00) a smooth section s over B that satisfies (i) s(b) = s(b) for b E A. (ii) Ils(b) - s(b)llb ::; E(b) for every b E B.
f: ~
---t 1] be a map of vector bundles over idB, such that the rank rk(jb) of the induced linear map of fibers is independent of b E B. Show that the subspaces
15.12. Let r
=
Imitting a complement homomorphism (I, j) ver the Grassmannian lrphically to a fiber in ~ f* ("(). Do the same nooth manifold B is ~ercise 15.7 and pick
lE, b = 1r(p) we have submanifold of E, so Show that the union Ita! space of a smooth
UIm(l~) ~ E(1])
E(Im(j)) =
in Exercise 15.5 in the canonical line bundle ~ 15.2).
bEB
E(Ker(j)) =
U Ker(jb) ~ E(~) bEB
are total sr:aces of vector bundles Im(j) and Ker(j) over B (smooth if and f are smooth).
~, 1]
1:
15.13. .Show for a map ~ ---t T/ of vector bundles over idE, that the function b t--+ rk(jb) is lower semicontinuous on B. 15.14. Suppose p: ~ ---t ~ is a vector bundle map over id B satisfying pop = p. Show that r = rk(Pb) is independent of b E B when B is connected. Prove that ~ ~ Im(p) EB Ker(p) (see Exercise 15.12). 16.1.
Show for integers n
> 0, m >
°
that
l/nl ®1 l/ml ~ l/dl where d is the greatest common divisor of nand m.
c
. .-t: o
.
~§-
;,.;. • L"'fJ .
"
.-t:
D.
16.4. ;e with Hennitian inner on AkV satisfying
F(~)~G(~) IF(g)
l~€
lVI€ h*(F(O)
npatible with pull-back, lIe isomorphism
* from the category of
lC(g)
F(T/) ~ G(T/) Prove finally with the assumptions and notation of Exercise 16.5 that the diagrams F(h*(O) 1>h'(€) G(h*(O)
'))
over B another smooth = F(~b), bE B. 1 the category of smooth ~isms over idE to itself. .aced by C in the source smooth can be changed
271
homomorphism
,W}3»)'
'finite-dimensional real the sense that the maps
EXERCISES
h'(1)€) h*(G(O)
commute. 16.7. Let V be a finite-dimensional real or complex vector space. Construct a k ,o,k linear map e: ® V ~ '01 V such that e(vi
(8) ••• (8)
1 '"' . Vk) = k! L slgn(O')vO'(I) O'ES(k)
(2) ... (8)
VO'(k)'
Show that e 0 e = e, and that the quotient homomorphism (2)kV ~ A kV induces an isomorphism Im(e) ~ AkV. Apply Exercise 15.14 to give an alternative construction of the vector bundles Ak(O. 16.8. For finite-dimensional vector spaces V and W, construct a linear map AkV 0 AIW ~ Ak+I(V EEl W)
corresponding category
carrying (VI /\ ... /\ Vk) (2) ('WI /\ ... /\ 'WI) into (VI, 0) /\ ... /\ (Vk, 0) /\ (0, 'WI) /\ ... /\ (0, 'WI). Show that these combine to give isomorphisms n
EBAkV
(2)
An-kW ~ An(V ttl W).
k=O
Extend this to vector bundle isomorphisms n
in Exercise 16.4. As ;pace V) a linear map
EB Ak~ 0 An-kT/ ~ An(~ EEl T/).
v-er B a smooth bundle
16.9. Prove Lemma 16.1O.(i). 16.10. Finish the proof of Theorem 16. 13.(iv). 16.11. Let ~ and T/ be smooth vector bundles over a compact smooth manifold M. Show that ~ and T/ are isomorphic if and only if nO(O ~ nO(T/) as nO(M) modules. Prove that every finitely generated projective nO(M)-module P is isomorphic to nO (T/) for some smooth vector bundle T/ over M. (Hint: Apply Exercise 15.14 with ~ = en.)
k=O
..J
"~
.~:_~:;;~
..,.
~'
'"
~ 1 ~~ ~
~
D.
at Hom(~, I.
0
is trivial.
[X, gY]
= j[X, Y] - (Yf)X = g[X, Y] + (Xg)Y.
Suppose X and Yare given on a chart U Remark 9.4)
,(X)).
m
X
=
a
~
M by the expressions (see
a
m
Lai ax .' 'j=l
n* M and moreover that
sian
273
Show for X, Y E nO( TM) and f, 9 E nO(M) that
[IX, Y]
mndle T M of a smooth For q E T* M (i.e. a
EXERCISES
Y
1,
=
Lbj ax" j=l
J
Show that in these coordinates [X, Y] is given on U by
[X,Y]
=
m. (m ab m aa k) a L Laiaxk' - Lb jax . aXk' k=l i=l j=l J t
1 a chart
U
~
M and
17.2. Prove for w E np(M) and Xj E nO( TM), 1
with respect to the basis
p+l
dw(X I , ... , Xp+d = ~
Exercise 14.7).
D: A
~
A that satisfies
:s: j :s: p + 1, the formula
+
L
L (-l)i-IXi(w(XI , ... , Xi, ... , X + I )) P
i=l
(-l)i+jw(([Xi , Xi], Xl,.··, Xi, ... , Xj , ... , Xp+l)).
l:S;i<j:S;p+l
In particular y smooth manifold M m
dw(X, Y) = X(w(Y)) - Y(wX) - w([X, YJ). for w E nl(M), X, Y E nO(TM)
~x
given by
17.3. Let M be a Riemannian manifold with metric ( , ). Prove the existence and uniqueness of a connection \l on TM such that \l x: nO (T) ~ nO (T) satisfies the following two conditions for smooth vector fields X, Y, Z: (a) X((Y, Z)) = (\lxY, Z) + (Y, \lxZ) (b) \lxY - \lYX = [X, Y]
Prove moreover that \l satisfies the Koszul identity
oD I of D I , D 2 E DerA [X, Y] E nO(TM) of
~t
+ Y(Z, X) - Z(X, Y) (X, [Y, ZJ) + (Y, [Z, XJ) + (Z, [X, YJ).
2(\lxY, Z) = X(Y, Z) -
[}
Ilowing conditions hold
(Hints: Let A(X, Y, Z) be the 6-term expression above. To prove unique ness, derive the Koszul identity by rewriting the six terms using the equa tions obtained from (a) and (b) by cyclic permutation of X, Y, Z. For existence verify first the identity
A(X, Y, f Z) = ) (Jacobi identity).
f A(X, Y, Z),
where f E nO(M), and construct \l x Y so that the Koszul identity holds.) This connection is the Levi-Civita connection on M.
;.), . ',}'Y,
~
q...
" ,
'IS
'"
-~,
...
.~
y::F:;
: 1 I t 1 D.
onstructed in Example 17.3, when M
~xercise
k
17.8. Let \I be a connection on ~ and consider two frames e1, ... , ek and ih, ... ,ek on the same open set U and the corresponding matrices A = (Aij), A = (Aij) of I-forms on U with
r M. Given two vector
: OO(~) ---. 0°(0 by
k
I
k
\I(ei) = LAij @ej,
Aij ® ej.
j=l
Let = (
by direct computation
~ivita connection on TM 0° (TAl) as in Exercise
\I(ei) = L
j=l
X,Y]·
rphism, and prove that
275
EXERCISES
iij = L
m=l
Show that
A
= (d-l
+ A-l.
What is the relationship between
dA - A 1\ A and dA - A 1\ A?
==0
Ls only on X p, Yp, Zp E lap R(Xp , }~): TpM ---.
= a~i the vector fields ymbols are the smooth
17.9. Prove Lemma 17.5. 17.10. Prove formula (17.18). 17.11. Given ~ with connection\l.;, a frame e1,"" ek over an open set U and the connection matrix A = (Aij). Prove that C with dual connection \If,' and dual frame ei, ... , e;; has connection matrix -At = (-Aji). Do a similar calculation for \If,01] and \lHom(f,,1]) when \11] has connection matrix B = (B r8 ) W.r.t. a frame iI, ... , fm over U. 17.12. For ~ with connection \If,' construct connections \I = \I Ai (f,) on Ai(~) for all i, such that \I Al(f,) = \If, and \lX(S 1\ t) = \lxs 1\ t
), fundamental form (see
. (gij )-1 . rse matnx ) to ai, aj, ak.)
+ (-I)i s 1\ \lxt
for s E OO(Ai(~)), t E OO(Aj(O) and X E TpM. 17.13. Let f: M' ---. M be any smooth map and ~ a smooth (real or complex) vector bundle over M. Construct an isomorphism
OO(M')
0nO(M)
0°(0 ~ O°(J*(~));
~
where 1/J E OO(M) acts on nO(M') by multiplication by !*(1/J) E nO(M'). (Hint: First handle trivial vector bundles. In general pick a complement to ~ as in Exercise 15.10.)
Lki).
18.1. Prove for the canonical line bundle Hover epn that H* is not isomorphic to H.
18.2. Show for complex line bundles ~,,,, over M that
C1(~ ®",) = C1(~)
+ Cl(Tf)·
lJ
D.
277
EXERCISES
19.1. Let M n be a Riemannian manifold with Levi-Civita connection \l on TM. Let eI, ... , en be an orthonormal frame in TM over U ~ M with connection matrix (A ij ), and denote the dual frame in TM over U by EI, ... ,En. Show that
-r
dEi(ej, ek) = -Aki(ej) ~nsional
complex vector
thatcI(An(~)) = CI(~).
16.8.)
I in Example 18.13 that
a complementary vector
rer CpI that H 1. ~ H*. that any tangent vec e written 0/(0), where lumn of a smooth curve it there is a well-defined
+ Aji(ek).
(Hint: Use Exercises 17.2 and 17.3.) Conclude in the case n = 2 that the connection described in Example 19.5 is the Levi-Civita connection. 19.2. Let V be an n-dimensionallR-vectorspace. A multilinear map F: V x V x V x V --- IR is said to be curvature-like when the following identities are satisfied: (a) F(x, y, z, w)
= -F(y, x, z, w).
(b) F(x, y, z, w)
+ F(y, z, x, w) + F(z, x, y, w) =
(c)
O.
F(x, y, z, w) = -F(x, y, w, z).
Prove that a curvature-like function on a tangent space TpM n , where 1\1 is Riemannian, is given by the expression
(Rp(Xp,Yp, Zp), Wp)p (see Exercises 17.5 and 17.6). Prove that the conditions (a), (b), (c) imply (d) F(x, y, z, w) = F(z, w, x, y).
19.3. Show that the curvature-like functions on V defined in Exercise 19.2 form a vector-space of dimension l2n2(n2 - 1).
r) ) aj(O) + V. ;n) (see Exercise 15.5). lver V is the orthogonal
19.4. Let F be curvature-like on V as defined in Exercise 19.2, and suppose V is equipped with an inner product ( , ). For a 2-dimensional subspace II ~ V, show that the expression
K(II) = _
~ En.
IUndle isomorphism
F(x, y, x, y) (x,x)(y,y) - (x,y)?'
where x, y is a basis for II, depends only on II. Show that the function K on the Grassmannian G2(V) determines F uniquely. Conclude that if K is constant with value k on G2(V), then crure). ding
F(x,y,z,w) = k((y,z)(x,w) (x,z)(y,w)). 19.5. Let II ~ TpM n be a 2-dimensional subspace, where M is a Riemannian manifold. The sectional curvature of M at II is the real number
K(p, II) = IS
.6.)
of the identification
(Rp(Xp, Yp)Xp, Yp) (Xp, X p)(Yp,Yp) (Xp, Yp) 2 '
where X p , Yp is a basis for II (see Exercises 19.2 and 19.4). (n ~ 2) with the standard Riemannian metric induced from Show that
sn
.~.
1 .......
~
'"
"
"~"'::--
_ .
II·
1:11
7
D.
J,
II) is independent of
ts by
279
EXERCISES
Verify fonnula (3) in the proof of Theorem 20.1. 20.2. Let M 1 and M2 be smooth manifolds, and assume that at least one of them have finite-dimensional de Rham cohomology. Show that the maps 20.1.
HP(Ml)
Yp ),
@
Hq(M2)
al 0 a2
I--t
-4
Hp+q(Ml x M2)
prMl (al) . prM2 (a2)
~
Exercise 19.8). and Exercise 19.1 that
combine to give isomorphisms
EB
HP(MI)
@
Hq(M2) ~ Hn(MI x M2).
p+q=n (Hint: Use Theorem 20.1.) 20.3. Construct a smooth manifold structure on G2 (R m) such that the map diagram (20.9) is a smooth embedding and 7f0 a submersion. 2004. Show that exponentiation of 2 x 2 matrices
'.
L 00
exp(A)
=
n=O
, ~ R3 as defined in ~ same expression; see
mdent of n ~re
lD
~
2. Com-
and compare con orthonormal frame
'n.) j,
1
~
i
<
'ljJ(
j ~ 2n is
(a
exp
(~
Q by
!a ).
+ ib)x = ax + bJx.
Show that GL 2n (R) acts transitively by conjugation on the set of complex structures, and that the subgroup fixing J is isomorphic to GLn(C). Prove a similar statement for the subgroup GLtn(R) of matrices with posi tive detenninant and complex structures inducing the standard orientation. 21.1.
v
a, f3) =
-4
Show that the space X defined in the proof of Proposition 20.6 can be identified with the total space of a certain complex line bundle over G2(Rm). How is this line bundle related to the canonical real vector bundle "(2 over G2(R m)? 20.5. A complex structure J on R 2n is a linear map J: R2n -4 1R 2n with J2 = -id. Given J, R2n becomes a C-vector-space by defining
:2v),
~
1
,An n.
can be used to define a homeomorphism 'ljJ: R2
Ril2 from the previous
¢ in
~
n}.
, Show that
(The Gysin sequence) Let ~ be an m-dimensional real smooth vector bundle over a compact manifold M with Riemannian metric and S(~) the unit sphere bundle. Construct an exact sequence . . . -4
HP-I(S(O)
-4
HP-m(M) ~ HP(M) ~ HP(S(O)
-4 . . .
where the labeled maps are multiplication by the Euler class e of pull-back by the projection.
~
and
< \,
'\,
'""
l'
.~ ..
~.
~.,.~--,"
281 S(~
EB 1) covered by (M), where so is the
.,4n+ 1 to the tangent bundle
)lnanifold with normal
References
[Bredon], G. E. Bredon: Topology and Geometry, Springer Verlag, New York 1993
[do Carmo] , M. P. do Carmo: Differential Geometry of Curves and Surfaces,
Prentice-Hall Inc., New Jersey 1976.
[Donaldson-Kronheimer], S. K. Donaldson and P. B. Kronheimer: The Geometry of Four-Manifolds, Oxford University Press, Oxford 1990.
Ileighborhood of M.) with the Levi-Civita in Example 10.12 and r bundle over M. Let
;.9 of S(~ EB 1) --+ M.
entation of TV.
I on TV. Show that
)] ~
with the computation nd show that so(u) =
[Freedman-Quinn], M. H. Freedman and F. Quinn: Princeton University Press, New Jersey 1990.
Topology of 4-Manifolds,
[Hirsch], M. W. Hirsch: Differential Topology, Springer-Verlag, New York 1976.
[Lang], S. Lang: Algebra, Addison-Wesley, Massachusetts 1965.
[Massey], W. S. Massey: Algebraic Topology: An Introduction, Hartcourt, Brace
& World Inc. 1967. [Milnor], J. Milnor: Morse Theory, Princeton University Press, New Jersey 1963. [Milnor-Stasheff], J. Milnor and J. Stasheff: Characteristic classes, Annals of
Math. studies, No 76, Princeton University Press, New Jersey 1974.
[Moise], E. E. Moise: Geometric Topology in Dimensions 2 and 3, Springer
Verlag, New York 1977.
[Rudin], W. Rudin: Real and Complex Analysis, McGraw-Hill, New York 1966.
[Rushing], T. B. Rushing: Topological Embeddings, Academic Press, New York
1973.
[Whitney], H. Whitney: Press, New York 1957.
Geometric Integration Theory, Princeton University
"
~
!I.
td 283
Index A Adams' theorem 10
Alexander's "homed sphere" 50
alternating
algebra 11
map 7
Altk(V) 7
antipodal map 75
B balanced product 166
base space 147
Betti number 113
Bianchi's identity 178
boundary 26
Brouwer theorem 47, 52
Brown-Sard theorem 98
c canonical line bundle 147, 267
canonical vector bundle 268
category 27
cellular set 264
chain complex 25
chain map 26
chain homotopy 31
chain rule 67
characteristic polynomial 227
Chern character 187
Chern character class 180, 183
Chern class 183
Chern total class 188
closed map 252
closed p-form 19
cohomology class 26
cokernel 25
connected component 19
connection 167
flat 171
form 168
matrix 172
contractibility 39
contravariant functor 13, 27
covariant differentiation 169
covariant functor 27
critical value 121
curvature form 171
curvature-like 277
cycle 26
D de Rham cohomology group 18
de Rham complex 17
degree 97
derivation 272
DGA 17
diffeomorphism 59
differential 17
exterior 15
graded algebra 17
p-form 15
direct image homomorphism 129
direct sum 26
directional derivative 67
divergence 4
divergence theorem 256
domain with smooth boundary 86
dual Hopf bundle 147
E embedding 59
energy 5
Euler characteristic 113, 249
o
~
<'.J
,§ ~
,
'..J
~),
!if' ~
>~.
i~,;#,...
~
.~
.~
~
't.
~~.:'. ~
~ ~
:~
.t, .' '.'
7r'c'Jil'~"i'C'::'
r£7
.$
j
285
~6
t
bundle 269
Lie group 253 linking number 102 local index 106, 108, 216 locally finite 221 long exact homology sequence 30
1
221 sets 133
;0
19 137
227
'.72 :heorem 51
~73
246 i7 ction 273
p-cycle 26 p-form 19
n-ball 47 n-sphere 47 negatively oriented basis 70 non-degenerate critical point 114 non-degenerate space 263 non-degenarate zero 109 normal bundle 148 normal vector field 76
o
R
obstruction 217 orientable manifold 70 orientation 70 standard 70 orientation form 70 orientation preserving/reversing 72 oriented chart 72 oriented double covering 254 outward directed tangent vector 86
real vector bundle 193 real splitting principle 197 regular value 98 relative compact 133 Riemann sphere 139 Riemannian manifold 73 Riemannian metric 150 Riemannian structure 73 rotation 2
maximal atlas 59 Mayer-Vietoris sequence 35 Mayer-Vietoris theorem 129 metric connection 196 Morse function 114 Morse inequalities 261-262 Morse's lacunary principle 262 multilinearity 7
N
nat
(p, q)-shuffle 8 p-boundary 26
paracompact 151 partition of unity 221 Pfaffian 230 Phragmen-Brouwer 249 PlUcker embedding 276 Poincare-Hopf theorem 113 Poincare duality 127, 131 Poincare lemma 23 Poincare polynomial 261 polynomial elementary symmetrical 229 invariant 227 Pontryagin character class 180 classes 189 positive atlas 72 positively oriented basis 70 product neighborhood 239 product orientation 76 projective module 164 projective n-space 139 projective space 58 proper map 127
M 251
p
••
o
..s::<:.J
1
.q,....
,
:;
~
"
. ";;!
- _:pi;"':
e:
)
"-='~:'
. . •,;v,C
~
\II
.~
~
Contents
~ J\ \}.
PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
The Pitt Building, Trumpington Street, Cambridge CB2 IRP, United Kingdom
~ ~
\}-. CAMBRIDGE UNIVERSITY PRESS
The Edinburgh Building, Cambridge, CB2 2RU, United Kingdom 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © 1. Madsen & J. Tornhave 1997
This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written pennission of Cambridge University Press First puhlished 1997 Printed in the United Kingdom at the University Press, Cambridge A catalogue recordfor this book is availablefrom the British Library Library of Congress Cataloguing in Publication data Madsen, 1. H. (Ib Henning), 1942 From calculus to cohomology: de Rham cohomology and chracteristic classes I Ib Madsen and J0rgen Tornehave. p. em.
Includes bibliographical references and index.
ISBN 0-521-58059-5 (he). -- ISBN 0-521-58956-8 (pbk).
1. Homology theory. 2. Differential forms. 3. Characteristic classes. 1. Tornehave, J0rgen. II. Title. QA612.3.M33 1996 514'.2--dc20 96-28589 CIP ISBN 0521 580595 headback ISBN 0521 58956 8 paperback
~
fY)
Preface .. Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter II Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Appendix A Appendix B Appendix C Appendix 0 References Index ...
• • • • • •
4
Introductiol The Altern. de Rham C Chain Com The MayerHomotopy Application Smooth M2 Differential Integration Degree, Lir The Poinca Poincare Dl The Compll Fiber Bund Operations Connection: Characterisl The Euler ( Cohomolog Thorn Isom Smooth Par Invariant p( Proof of Le Exercises
~
<~«<
.,..)<
~
.
~1
.....
~
-r{7 ~-:~~~
r
...
"'
~
III
"
;;;--.
c
i
2
1.
INTRODUCTION
Definition 1.3 A subset X c IRn is said to be star-shaped with respect to the point Xo E X if the line segment {t:ro + (1- t)xlt E [0, I]} is contained in X for all x E X.
Note that rot
Theorem 1.4 Let U C 1~2 be an open star-shaped set. For any smooth function (!I, h): U ---+ 1R 2 that satisfies (2), Question 1.1 has a solution.
Then one has
1
i
U.rl
Both Ker(rot) and Im(g able that the quotient sI We can now reformulat
[Xl!I(txl,tx2) +x2h( tx l,t:rz)]dt.
(4)
a!I a!z ] dt it (tXl' tX2) + tXl ~(tXl, tX2) + tX2~(tXl, tX2) UXI
a!I = !I(tXl, tX2) + tXl ~ UXI
On the other hand, Exarr see that HI (1~2 - {O}) i dimension of H 1 (U) is In analogy with (3) we
(tXl, tX2)
a!I UX2
+ tX2~(tXI, tX2).
Substituting this result into the formula, we get
(5)
l
r [ddtt!I(txl,tx2)+tx2 (a!z a!I (tx l, tx 2) )] dt aXl(tXI,txZ)- aX2
aF a:rl(Xl,X2) = Jo
H 1 (U)
UXI
and
d -t!I(tXI, tX2) dt
=I
1
[ .
0
grad
(3)
1
aF ~(:rl, :r2) =
0
Since both rot and grad Therefore we can cons casets Do + Im(grad) w
Proof. For the sake of simplicity we assume that Xo = 0 E 1R 2. Consider the
function F : U ---+ IR,
F(Xl,X2) =
•
J
This definition works fo
= [t!I (tx l, tX2)]Z=O = !I (Xl, X2). Analogously, g~
=
!z(Xl, X2). Theorem 1.5 An open
"Example 1.2 and Theorem 1.4 suggest that the answer to Question 1.1 depends on the "shape" or "topology" of U. Instead of searching for further examples or counterexamples of sets U and functions f, we define an invariant of U, which tells us whether or not the question has an affirmative answer (for all j), assuming the necessary condition (2). Given the open set U <;;;; 1R 2, let Coo (U, IRk) denote the set of smooth functions ¢: U ---+ IRk. This is a vector space. If k = 2 one may consider ¢: U ---+ Rk as a vector field on U by plotting ¢( u) from the point u. We define the gradient and the rotation grad: COO(U, IR) by
---+
Coo (U, 1R 2),
rot: Coo (U, 1R 2 )
---+
Proof. Assume that gra a neighborhood V (xo) , then every locally const:
{x
is closed because f is ( it is equal to U, and H( exists a smooth, surject constant, so grad(J) = (
COO(U, IR)
The reader may easily e is precisely the number <
grad(¢) =
a¢ a¢) ( aXl' aX2 '
i
a¢l a¢2 rot(¢1,¢2) = aX2 - aXl L
. <
o
~ <:..l ~ ..0 'lj •. t:'" ~
...J
•...; -lj
.~
1."1-'
.~
~
-:..., '<s '"
lj
.~.~
-:...
.
~ ...
.
';r:oo;).
• ...
,..:;}\
.
7r
""
~
.
~
• •P 1'...... :;; • . i• S
n
4
1.
INTRODUCTION
Example 1.7 Let S = {(:l in the (Xl, X2 )-plane. Cor
We next consider functions of three variables. Let U c;;: R3 be an open set. A real function on U has three partial derivatives and (2) is replaced by three equations. We introduce the notation grad: Coo(U, R) ---+ Coo(U, R3 ) rot: Coo (U, R3 )
---+
div: Coo(U, R3 )
---+
f(Xl' X2, X3)
-2XlX3 ( X~ + (xi + X~
Coo (U, R3 ) Coo(U, R)
for the linear operators defined by grad (J) =
12, h)
on the open set U = R3 One finds that rot(J) = integration along a curve "I we shall show that [I) i= (
af af af ) ' -a ' -a ( -a Xl X2 X3
(a 13 _ a12 , ah _ a13 , a12 _ ah ) aX2 aX3 aX3 aXl aX2 aX2 . ah ah ah dlv(h,h,h)=-a +-a +-a . Xl X2 X3 Note that rot 0 grad = 0 and div 0 rot = O. We define HO(U) and set Hl(U) rot (h,
=
I'(t)=(, Assume grad(F) = f as :ftF(ry(t)) in two ways. 0
as in Equations (3) and (5) and
j
H 2 (U) = Ker(div)jlm(rot).
(6)
._ . ~
Theorem 1.6 For an open star-shaped set in R3 we have that HO (U) Hl(U) = 0 and H 2 (U) = O.
R
1r-E
-1r+E
d
F (I'(t))dt dt
=
and on the other hand the
d
dtF(ry(t)) = fl(ry(t
Proof. The values of HO(U) and Hl(U) are obtained as above, so we shall restrict ourselves to showing that H 2 (U) = O. It is convenient to assume that U is star-shaped with respect to O. Consider a function F : U ---+ R3 with div F = 0, and define G : U ---+ R3 by
1
= sin 2 t -j Therefore the integral also,
1
G(x) =
Example 1.8 Let U be an smooth vector field). Recal r: [a, b] ---+ U is defined by
(F(tx) x tx)dt
where x denotes the cross product,
(h, 12, h) x
(Xl,
X2, X3)
el
h
e3
12 X2
13 X3
=1 e2 =
Xl
A.,(
(h x 3- h X2, hXl - h X3, h X2- hXl).
where (,) denotes the stan
4>,,( b), then the energy is ,
Straightforward calculations give
d
rot(F(tx) x tx) = dt(t 2 F(tx)). Hence rotG(x)
=
1
1°
d
-(t 2 F(tx))dt dt
= F(x).
(X by the chain rule; compare
o
If U c;;: R3 is not star-shaped both Hl(U) and H 2 (U) may be non-zero. i,...
-:.
~
.,...
. £"i->
.~ ..
~
'9,
~
"
.
1 ~
",,,_-,,c·;;.. p,=~_
!7.· . .
r
rap
L
8
2.
sTT]
.•
THE ALTERNATING ALGEBRA
Example 2.3 Let V = IRk and ~i = (~il"",~ik). The function w(6"",~k) det( (~ij)) is alternating, by the calculational rules for determinants.
=
We want to define the exterior product /\: AltP(V) x When p = q = 1 it is given by
Proof. 5 (k) is generated of Lemma 2.2,
Altq(V) --+ Altp+q(V).
(Wl/\ W2) = wl(6)W2(6) - w2(6)Wl(6).
Definition 2.4 A (p, q)-shuffle fJ is a permutation of {I, ... ,p + q} satisfying fJ(l) < ... < fJ(p)
fJ(p
and
+ 1) < ... <
fJ(p
Lemma 2.7 A k-linear m with ~i = ~i+l for some:
W(6"",~i,~:
Hence Lemma 2.2 holds ti It is clear from the defini
+ q).
(WI'
The set of all such permutations is denoted by 5(p, q). Since a (p, q)-shuffle is uniquely determined by the set {fJ(l), ... ,fJ(p)}, the cardinality of 5(p,q) is
(AWl WI/\
(p;q).
Definition 2.5 (Exterior product) For WI E AltP(V) and W2 E
Altq(V),
we define
Lemma 2.8 If WI E AltPC
(WI /\ W2)(6,···, ~p+q)
=
L
sign(fJ)wl (~a(I)'"'' ~a(p») . W2(~a(p+l)'"'' ~a(p+q))'
Proof. Let
aES(p,q)
It is obvious that
WI /\ W2
is a (p + q)-linear map, but moreover:
Lemma 2.6 If WI E AltP(V) and Proof. We first show that
W2 E Altq(V)
then
WI /\ W2 E
(Wl/\W2)(6,6, ... ,~p+q) =
(i) 5 12 = {fJ E 5(p, q) I fJ(l) = 1, fJ(p + 1) (ii) 521 = {fJ E 5(p, q) I fJ(l) = 2, fJ(p + 1) (iii) 50 = 5(p, q) - (512 U 521)'
Altp+q(V).
0 when ~1
= 6.
We let
= 2}
..
'~Ta(p)) . W2 (~Ta(p+l)"
S(p + q)
T(l) = T(q+ 1) = We have sign( T) = (-1)P'i
S( Note that WI
aE S12
sign(fJ)wl (~Ta(I)"
E
W2 (~aT(1
(6,6, ... , ~p+q) = sign( fJ )Wl (~a(I)' ... , ~a(p»)W2 (~a(p+l)' ... ,~a(p+q))
L -L
T
= I}
If fJ E 50 then either WI (~a(I)' ... , ~a(p») or W2(~a(p+l)' ... ,~a(p+q)) is zero, since ~a(l) = ~a(2) or ~a(p+l) = ~a(p+2)' Left composition with the transposition T = (1,2) is a bijection 512 --+ 521. We therefore have (WI /\ W2)
for wI, W~ E AltP(V) and
.. ~Ta(p+q)).
aES12
Since fJ(l) = 1 and fJ(P+ 1) = 2, while TfJ(l) = 2 and TfJ(P+ 1) = 1, we see that
TfJ( i) = rJ( i) whenever i =j:. 1, p + 1. But 6 = ~2 so the terms in the two sums
cancel. The case ~i = ~i+l is similar. Now WI /\ W2 will be alternating according
to Lemma 2.7 below. D
(~aT(q+l)"
Hence W2 /\
wI(6, ... , ~p
L
sign(fJ:
aES(q,p)
L aES(p,q)
= (-l)Pq
sign(a~
L
aES(p,q) PqW = (-l) I/\ W2
s
.,..J
.('oH
"
1~ ~ J
""
-~:;t~'~~~~~5:", ~
r:r .[
·
10
Lemma 2.9 If WI
a
2.
THE ALTERNATING ALGEBRA
E AltP(V), W2 E Altq(V)
and
then
W3 E Altr(V)
WI 1\ (W2 1\ W3) = (WI 1\ W2) 1\ W3.
Proof. Let S(p, q, r) C S(p
·'·11
al,_ltt r
I·
+ q + r)
consist of the permutations IJ with
< . . . < IJ (p),
IJ(p + 1) < ... < IJ(p + q),
IJ (p + q + 1) < . . . < IJ (p + q + r).
IJ (1)
Note that in this formula 1 ordered. There are exactly arbitrary sequence above; 1 An IR-algebra A consists of which is associative, /.l( a, algebra is called unitary if for all a E A.
Definition 2.11 We will also need the subsets S(p, q, r) and S(p, q, r) of S(p, q, r) given by IJ E S(p,q,r) <====';>IJ is the identity on {l, ... ,p} and IJ E S(p,q,r) IJ E S(p, q, f) <====';>(J is the identity on {p
+ q + 1, ... , p + q + r}
and IJ E S(p, q, r). There are bijections
-=. S(p, q, r); (IJ, T) ~ IJ x S(p, q, f) -=. S(p, q, r); (IJ, T) ~ IJ
S(p, q + r) x S(p, q, r)
(2)
S(p
+ q, r)
0
T
0
T.
L
(),w )(6,. 1\
W3)(~(T(p+I)'
...
'~(T(p+q+r))
(TES(p,q+r)
L W2
L
L
sign(IJ)
sign(T)[wI(~(T(I)'''''~(T(p))
TES(p,q,r)
(~(TT(P+l)'
... ,
~(TT(p+q) )W3 (~(TT(p+q+l)' ... , ~(TT(p+q+r))]
[sign(1l)wI (~u(I)" .. , ~u(p))W2(~u(p+l)"
.. ,
~u(p+q))
uES(p,q,r)
W3( ~u(p+q+I)' ... ,
+ w2)(6,.
~p+q+r)
sign(IJ)wI (~(T(I)" .. '~(T(p)) (W2
(TES(p,q+r)
The elements in A k are sai over I~ in the usual mann (WI
With these notations we have [WI 1\ (W2 1\ w3)](6,···,
(i) A graded R-algebr and bilinear maps (ii) The algebra A* is and if t:: R ---. Ao, (iii) The algebra A* is J.l(a, b) = (_l)kl/.l(
~u(p+q+r))]
where the last equality follows from the first equation in (2). Quite analogously one can calculate [(WI 1\ W2) 1\ W3] (6, ... ,~p+q+r), employing the second equation in (2). 0
Remark 2.10 In other textbooks on alternating functions one can often see the
The product from Definitic Altp+q(V). We set Alt°(V by using the vector space s forms can now be summa
Theorem 2.12 Alt*(V) i$ Alt* (V) is called the exter
Lemma 2.13 For ljorms
(WI 1\ ... 1\
wp )(6,.
definition wI!\w2(6,···
_1_
p!q!
""
~
(TES(p+q)
,~p+q)
sign(IJ)wI(~(T(I)'" "~(T(p))W2(~(T(p+l)"" '~(T(p+q)).
Proof. The case p to Definition 2.5,
= 2 is
_ _. .l!I!!l!!!!!!_--...._~::=:::~~"""
,
-1
{,
12
2.
THE ALTERNATING ALGEBRA
Note from Theorem 2.15 earlier, that AltP(V) = 0 i1 particular every altematin~ 2.3.
WI/\ (W2 /\ ... /\ w p )(6, ... , ~p) p
A = "L..." ( _I)J'+1 WI (~j ) (W2 /\ ... /\ wp ) ( 6,···, ~j, ... ) , ~p j=l
t
where (6, ... , j , ... , ~p) denotes the (p - I)-tuple where ~j has been omitted. 0 The lemma follows by expanding the determinant by the first row. Note, from Lemma 2. 13, that if the I-forms WI, ... , W p E Al t (V) are linearly independent then WI /\ ... /\ W p "# O. Indeed, we can choose elements ~i E V with Wi(~j) = 0 for i "# j and Wj(~j) = 1, so that det(wi(~j)) = 1. Conversely, if WI, ... ,wp are linearly dependent, we can express one of them, say wp , as a linear combination of the others. If wp = :E ~~:riwi, then p-l WI /\ ... /\ Wp-l /\ Wp = riWI /\ ... /\ Wp-l /\ Wi = 0, i=l as the determinant in Lemma 2.13 has two equal rows. We have proved
2::
WI /\ ... /\ W p
~
W
(4)
1
Lemma 2.14 For I-forms WI, ... , W p on V,
A linear map f: V
"# 0 if and only if they
)
by setting AltP(J)(w)(6,. maps we have AltP(g 0 f) properties are summarized dim V = nand f: V ~. V
,A
is a linear endomorphism 0 by a number d. From The! also be using the other m
A
0
are linearly independent.
Theorem 2.15 Let el, ... , en be a basis of V and tl, ... , En the dual basis of Altl(V). Then {Eo-(1) /\ to-(2) /\ ... /\ Eo-(p)} o-ES(p,n-p)
Let tr(g) denote the trace
I
Theorem 2.16 The characl is given by
is a basis of AltP(V). In particular
det(J
dim Altp(V) = ( dimV) p .
Proof. Since ti(ej) = 0 when i "# j, and E.Jei) = 1, Lemma 2.13 gives (3)
fi, /\ ... /\ fi,
(ej.,. .. , fj,)
~ {s~gn(0')
where n = dimV.
if {il,'''' ip} "# {jl,'" ,jp} if {i l , ... , ip} = {jl, ... ,jp}
Proof. Choose a basis el, vectors of f,
Here cr is the permutation cr( ik) = j k. From Lemma 2.2 and (3) we get W
=
z=
Let EI, ... ,En be the dual w(eo-(l)"'" ea-(p))to-(l) /\ ... /\ to-(p)
AltP(J) (to-(I) /\ ..
o-ES(p,n-p)
for any alternating p-form. Thus Ea-(l) /\ ... /\ to-(p) generates the vector space AltP(V). Linear independence follows from (3), since a relation
z=
Ao-to-(l) /\ ... /\ to-(p) = 0,
Ao- E
R
tr Alt On the other hand
o-ES(p,n-p)
evaluated on (ea-(l)"'" eo-(p)) gives Ao-
and
det(j - t) = II(Ai
= O.
0
This proves the formula w
.
r ,
14
2.
THE ALTERNATING ALGEBRA
If f is replaced by gf g-l, with 9 an isomorphism on V, then both sides of the equation of Theorem 2.16 remain unchanged. This is obvious for the left-hand side and follows for the right-hand side since
AltP(gfg-l) = AltP(g)-1
0
AltP(f)
0
AltP(g)
by the functor property. Hence tr Alt P (g 0 f 0 g-l) = tr AltP(f). Consider the set D = {gfg- 1If diagonal, 9 E GL(V)}. If V is a vector space over C and all maps are complex linear, then D is dense in the set of linear endomorphisms on V. We shall not give a formal proof of this, but it follows since every matrix with complex entries can be approximated arbitrarily closely by a matrix for which all roots of the characteristic polynomial are distinct. Since eigenvectors belonging to different eigenvalues are linearly independent, V has a basis consisting of eigenvectors for such a matrix, which then belongs to D For general f E End(V) we can choose a sequence dn E D with d n ---+ f (i.e. the (i, j)-th element in dn converges to the (i, j)-th element in f). Since both sides in the equation we want to prove are continuous, and since the equation holds for d n , it follows for f. 0
3.
DE RHAM COHOM
In this chapter U will dene and {tI",., En} the dual 1
Definition 3.1 A different. The vector space of all SUI
If p = 0 then AltO(R n ) = real-valued functions on U The usual derivative of a ~ value at x by Dxw. It is
with
(Dxw) In AltP(Rn) we have thebe with 1 ::; il < i2 < ... < '4 form w(x) = L W[(X)E[, ' The differential Dxw is th
It is not true that the set of diagonalizable matrices over R is dense in the set of
matrices over R - a matrix with imaginary eigenvalues cannot be approximated by a matrix of the form 9 f g-l, with f a real diagonal matrix. Therefore in the proof of Theorem 2.16 we. must pass to complex linear maps, even if we are mainly interested in real ones.
Dxw(t
(1)
The function x t--+ Dxw i~ maps from Rn to AltP(R1 Definition 3.2 The exterit erator
d x w(6,··· ,~P+I
with (6,··· ,~l,.'. ,~P+I) It follows from Lemrna2. p+l
L(-l 1=1
=( _l)i-l
+1 =0
;'
.·1 .•j:'~.•. .
-J
..
f"tJ
1ii.
q,..
"
~''';:i::i
,..... , -
'--r-.
t
16
3.
~
DE RHAM COHOMOLOGY
because (6, ... ,~i,' .. ,~p+l)
= (6,···, ~i+l'"
., ~P+l)'
The exterior product in Alt" defining
(WI'
Example 3.3 Let Xi: U -----* R be the i-th projection. Then dXi E Ol(U) is the constant map dXi: X -----* Ei. This follows from (1). In general, for f E OO(U), (1) shows that
af 1 dxf(() = -a (x)( + ... Xl
(2)
with ((1, ... , en)
Lemma 3.4
= (.
If w(x)
In other words: df
=
af xn
The exterior product of a ( differential (p + q)-form, so
n
+ -a (x)(
/\: 0 1
= L:: *fEi = L:: Mdxi.
f(X)E[ then dxw = dxf /\
For a smooth function
f
E (
(jWI) /\
E[.
This just expresses the bilim f /\ w = fw when f E 0°([
Proof. By (1) we have
Dxw(()
3.
af 1 af n) = (DxJ)(()E[ = ( aXl ( + ... + aX ( E[ = dxf(()E[ n
Lemma 3.6 For
WI E
OP(U
and Definition 3.2 gives
d(WI/\ W2
p+l
)
dxw(6, .. ·,~p+l) = "'"' L..,.,(-l) k-l dxf(~k)E[ ( 6,".,~k,·"~p+l k=l
Note for E[ E
AltP (Rn)
Proof. It is sufficient to sh01 then WI /\ W2 = fg EJ /\ EJ,
o
= [dxf /\ E[](6, .. ·, ~P+l)'
d(WI /\ W2) = d(j = dffj
that
Ek/\E[ =
{~-lrEJ
with r the number determined by iT
=df' =dWj
if k E I if k ~ I
< k < iT+l and J
Lemma 3.5 For p ~ 0 the composition OP(U) identically zero.
-----*
= (il,"" iT' k, . .. , ip).
OP+I(U)
-----*
OP+2(U) is c
Proof. Let w =
~
f E[ . Then
!
af af dw = df /\ E[ = -a E} /\ E[ + ... + -a En /\ E[. Xl Xn Now use Ei /\ Ei = 0 and Ei /\ Ej = -Ej /\ Ei to obtain that 2
dw
n
= "'"' L...J
i,j=l
a2 f
aX· aX· Ei /\ (E j l
/\
~ ~
E[ )
J
a2 f a 2f )
=~ a.a . - a .a . Ei . . ( X~ XJ XJ X~ t<J
I
/\
Ej /\
E[
= O.
o
l
I
~
Summing up, we have intro differential, d: S1*( and d is a derivation (satisfie (differential graded algebra).
Theorem 3.7 There is precis 0, 1, ... , such that (i) f E S10(U), df = ~
(ii) dod = 0 (iii) d(WI /\ W2) = dWI /\ I
18
3.
r
DE RHAM COHOMOLOGY
r
Proof. We have already defined d with the asserted properties. Conversely assume that d' is a linear operator satisfying (i), (ii) and (iii). We will show that d' is the exterior differential. The first property tells us that d = d' on 0° (V). In particular d' Xi = dXi for the 'l th projection X( V -> IR. It follows from Example 3.3 that d' :L'i = Ei, the constant function. Since d' 0 d' = 0 we have that d' Ei = O. Then (iii) gives d' E[ = O. Now let w = lEI = I A EI, I E COO(V, ~). Again by using (iii),
d'w = d'i Since every p-form is
+I
A
d' EI
->
+ h E2 + h E3) = dh
Oh Oh) ( -OXI - -OX2 EI
Lemma 3.9 HO(U) is the v connected component 01 U.
= d'i A EI = dl A EI = dw. the sum of such special p-forms, d = d' on all of OP(V).D A EI
For an open set V in ~3, d: 0 1 (V)
d(fIEI
components of V. A conne, ~V of U that cannot be vi subsets of W (in the topolo countably many connected I with rational coordinates.)
A E2
0 2 (V) is given as A El
+ dh A E2 + dh A
E3
+ (Oh - - -O h) E2 A E3 + (Oh OX2
Proof. A locally constant mutually disjoint open sets constant precisely when I ii
OX3
OX3
=
It follows that dimRHO(U){, the number of connected c(
Oh) E3 A EI. OXI
The elements in OP(V) win of the image d(Op-l(U)) C group thus measures wheth satisfied precisely when H cohomology class, denoted
The first equality follows from Theorem 3.7.(iii), as Ei: V -> Alt 1 (~3) is the constant map, and hence dEi ;= 0, by (1). Alternatively, we have already noted that the I-forms Ei and dXi agree, and hence dEi = do d(Xi) = 0 by Theorem 3.7.(ii). The second equality comes from the anti-commutativity, E'i AEj = -Ej AEi and Theorem 3.7.(i). Quite analogously we can calculate that
d(g3El
A E2
ogl og2 og3 ) + g1E2 A E3 + g2E.3 A Er) = ( -;-+ -;-+ -;-uX2 UX3 uXI
El A E2 A E3.
[w] and [w] = [w'] if and only closed p-forms and the vect, contrast HP(U) usually has ~-
.
~
Definition 3.8 The p-th (de Rham) cohomology group is the quotient vector space
HP(V)
In particular HP(V)
=0
Ker(d: OP(V) -> Op+1 (V)) Im(d: Op-l(V) -> OP(V)) .
(WI
->
+ dryr)
A
(W2
=
[WI/'
+ dry2) =
= We want to make V -> H' map ¢: VI -> U2 between a linear map
< 0, and HO(V) is the kernel of
d: COO (V, ~)
H
(3)
by setting [WI] [W2]
= --'---.,.-,-------'
for p
We can define a bilinear, as
f
i.
01(V),
1
f: .
,,-
and therefore is the vector space of maps I E COO (V, ~) with vanishing deriva
tives. This is precisely the space of locally constant maps.
Let be the equivalence relation on the open set V such that ql q2 if there
exists a continuous curve a: [a, b] -> V with a(a) = ql and a(b) = q2. The equivalence classes partition V into disjoint open subsets, namely the connected f'o.J
f'o.J
f' '
such that:
t
(4)
11f}. ·.'.·,· .
We first make 0* (-) into
r
t
~
HPj HPj
_c_c~~
20
3.
3.
DE RHAM COHOMOLOGY
This shows (i) when p > 0 analogous, but easier. Proper1 O. So we are left with (iii). f E nO(U2). We have that
Definition 3.10 Let U1 C IR n and U2 C IR m be open sets and ¢: U1 ---+ U2 a smooth map. The induced morphism f)P(¢): n p(U2) ---+ n p(U1) is defined by
np(¢)(w)x = Altp(Dx¢)
nO(¢)(w)x = w¢>(x)'
w(¢(x)),
0
np (¢).
Frequently one writes ¢* instead of satisfied. Indeed,
We note that the analogue of (4) is
¢*(w)x(6,·.·, ~p) = w¢>(x)(D x¢(6),···, Dx¢(~p)), and using the chain rule DxCrj!o¢) = D¢>(x)('l/J) 0 D x (¢), for ¢:U1 U2, 'l/J: U2 ---+ U3 , it is easy to see that np('l/J 0 ¢) = n p(¢) 0 np('l/J) , np(id u ) = id[lp(u),
dj when
---+
¢ *() Ei
n1 (U2)
¢*(dj)
=
we have that
=~ L...J -O¢i E k = d¢i OXk
=
1=1
=
t
1=1
O¢i EI(() OXI
=
d¢i(().
o
L o(J
0
¢) fl
:
OXI
I
I
~
= ¢~ =d(
The second last equality uses 1\
=L(-l)k-l
Conversely, if ¢': n*(U2) ---+ n*((h) is a linear operator that satisfies the three conditions, then ¢' = ¢*.
6, ... ,~p+q be vectors in Rn. Then
t
since
f
In the following it will be con'
t
..
d¢* (fik) = 0 by Exam]
t
d
¢*(w 1\ T)x (6,· .. , ~p+q)
=(w 1\ T)¢>(x)(D x ¢(6), ... , Dx¢(~p+q))
L sign(cr) [w¢>(x) (DX¢(~(7(1))"
OXk
d¢*(f[) = d(¢*(fil)
¢*(w) 1\ ¢*(T) (ii) ¢*(J) = j 0 ¢ if j E nO(U2) (iii) d¢*(w) = ¢*(dw).
=
k=11=1
¢*(dw)
=
Proof. Let x E U1 and let
c
In the more general case w : because df[ = O. Hence
Theorem 3.12 With Definition 3.10 we have the relations (i) ¢*(w 1\ T)
ft (!L 1=1
x
O¢i (I OXI
I
n
~ fi(D ¢(()) = fi (~(E ~~:(')ek)
t
=
k=1
with ¢i the i-th coordinate function. To see this, let ( E IR n . Then
=
L ¢* (!L) OXk m
k=1
¢'(fi)()
is considered as the
(i) and (ii) we obtain
It should be noted that np(i)(w) = w 0 i when i: U1 '--t U2 is an inclusion, since then Dxi = id. Example 3.11 For the constant 1-fonn Ei E
Ek
=
instead of the (constant) p-for be written as
,Dx¢(~(7(p)))
T¢>(x) (D x¢( ~(7(P+l))' ... , Dx¢ (~(7(p+q)) )]
L
= sign(cr)¢*(w)x(~(7(1)"'" ~(7(p))¢*(T)x (~(7(P+l)"'" ~(7(p+q)) =(¢*(w)x 1\ ¢*(T)x)(6,···, ~p+q).
and Example 3.11 becomes ¢~ function and ¢i = Yi 0 ¢ thei [
L:
22
3.
DE RHAM COHOMOLOGY
Example 3.13 (i) Let 1': (a, b)
---+
U be a smooth curve in U, I' w = h dXl
= (1'1, ... , 'Yn),
and let
+ ... + fndx n
Theorem 3.15 (Poincare's 11 for p > 0, and HO(U) = f
Proof. We may assume U t and wish to construct a line
be a I-form on U. Then we have that
'Y*(w) = 'Y*(h) 1\ 1'* (dxI) + + 'Y*(fn) 1\ 'Y*(dx n ) = 'Y*(h)d(l'*(xI)) + + 1'* (fn) d(l'* (x n )) = (h 0 'Y)d'Yl + + (fn 0'Y)d'Yn = [(h 0 'Yh~ + + (fn 0 'Yh~] dt = (f(l'(t)) , 'Y'(t)) dt.
such that dSp + Sp+ld = id for W E nO(U). Such an dSp ( w) = w for a closed 'J have w - w(O) = Sldw = 0. First we construct
Here ( , ) is the usual inner product. Compare Example 1.8.
Sp (ii) Let ¢: Ul
---+
U2 be a smooth map between open sets in
¢*(dXl
1\ ... 1\
dx n ) = det (Dx¢)dxl
1\ ... 1\
~n.
Then
dx n .
Indeed, from Theorem 3.12,
¢*(dXl
1\ .. . 1\
Every w E np(U x R) can t
w=Lf where I
=
(il,"" i p ) and j
dx n ) = ¢*(dxI) 1\ .. . 1\ ¢*(dx n ) = d¢*(xI) 1\ .. . 1\ d¢*(x n ) = d¢l 1\ ... 1\ d¢n = det (D x¢ )dXl 1\ ... 1\ dx n .
The last equality is a consequence of Lemma 2.13.
Sp(w) Then we have that
dSp(w) Example 3.14 If ¢: ~n X ~ ---+ ~n is given by ¢(x, t) = 7jJ(t)x, where 7jJ(t) is a smooth real valued function. Then
¢*(dXi) To a smooth map ¢: UI
---+
= xi7jJ'(t)dt + 7jJ(t)dXi.
U2 we can now associate a linear map
HP(¢) : HP(U2) ---+ HP(UI)
+ SP+ld(w) =
L( J,i
+~( =L( =LfJ
We apply this result to ¢*(v. by setting HP(¢)[w] = [np(¢)(w)] (= [¢*(w)]). The definition is independent of the choice of representative, since ¢*(w + dv) = ¢*(w)+¢*(dv) = ¢*(w)+d¢*(v). Furthermore,
Hp+q(¢)([WI][W2]) such that H*(¢): H*(U2)
---+
= (HP(¢)[WI])(HQ(¢)[W2])
H*(U I ) is a homomorphism of graded algebras.
¢: U
and 7jJ (t) is a smooth functi(
{:
;'1
,-
24
3.
DE RHAM COHOMOLOGY
Define Sp(w) = Sp(¢*(w» with Sp: 0l(U x R) ~ SlP-l(U) as above. Assume that w = L: hI(X)dxI. From Example 3.14 we have ¢*(w) =
2: hI('lj;(t)X) (d'lj;(t)Xi
1
+ 'lj;(t)dXiJ 1\ ... 1\
(d'lj;(t)Xi p
+ 'lj;(t)dXi
4.
p )
CHAIN COMPLEXES
In this chapter we present ~ which should illuminate som results will be applied later t;
In the notation used above we then get that
2: !I(x, t)dXI = 2: hI ('lj;(t)x)'lj;(t)PdXI'
A sequence of vector spaces This implies that dSp(w)
+ Sp+l d(W)
= {
L: IhI(x)dxI
w(x) - w(O)
= W
(1)
p>O p= O.
is said to be exact when 1m,
o
Ker g = {b 1m! = {II
Note that A .L B ~ 0 is exac is exact precisely when g is
~
II
(2)
.,.
~Ai~
of vector spaces and linear rn for all i. It is exact if
t
i
t-
t ~
for all i. An exact
,
I
sequenc~
(3)
,f
is called short exact. This
t
t
i~
! is injective,
The cokemel of a linear ,rna
For a short exact sequence,
Every (long) exact sequence, be used to calculate Ai) o~
l,'·
'.
~
.
.
r
...
26
4.
CHAIN COMPLEXES AND THEIR HOMOLOGY
t
Furthennore the isomorphisms
f*=H*
I
A EB B = {(a, b)la E A, bE B} A(a, b) = (Aa, Ab), A E R
+ (a2, b2)
= (al
1
+ a2, bl + b2)'
If {ail and {bj} are bases of A and B, respectively, then {(ai, 0), (O,bj)} is a basis of A ttl B. In particular
°
°
Lemma 4.1 Suppose -+ A .L B ~ C -+ is a short exact sequence of vector spaces. Then B is finite-dimensional if both A and C are, and B 9:! A ttl C. Proof. Choose a basis {ai} of A and {Cj} of C. Since 9 is surjective there exist bj E B with g(bj) = Cj. Then {f(ai), bj} is a basis of B: For b E B we have g(b) = L AjCj. Hence b - L Ajbj E Ker g. Since Ker 9 = 1m f, b - L Ajbj = f(a), so
Ajbj = f(LJLiai) = LJLi!(ai).
This shows that b can be written as a linear combination of {b j} and {f (ai)}. It is left to the reader to show that {bj, f(ai)} are linearly independent. 0 Definition 4.2 For a chain complex A * = {- .. - t AP-l we define the p-th cohomology vector space to be
dP-t
1
AP
dP -+
AP+l
-t ... }
HP(A*) = Ker dPlim dP- l . The elements of Ker dP are called p-cycles (or are said to be closed) and the elements of 1m dP- l are called p-boundaries (or said to be exact). The elements of HP(A*) are called cohomology classes. A chain map f: A* - t B* between chain complexes consists of a family JP: AP - t BP of linear maps, satisfying d~ 0 fP = JP+l 0 d~. A chain map is illustrated as the commutative diagram 1
dP dP +1 __ ... ···--AP- 1 --AP--AP 1 JPJP JP+l P- 1 P 1 d d +1 . ··-BP- --BP--BP _ ...
1
1
1
I
Proof. Let a E AP be a cy< cohomology class in HP(i needed. First, we have d~ a cycle. Second, [JP(a)] is raj. If [all = [a2J then al d~-l fP-l(x). Hence JP(al same cohomology class.
t
r
= dim A + dim B.
dim(A EB B)
b- L
CRA
Lemma 4.3 A chain map
A i - l lim di - 2 9:! A i - l jKer di - l d~l 1m di - l "'" are frequently applied in concrete calculations.
The direct sum of vector spaces A and B is the vector space
(all bI)
4.
t
l
A category C consists of ' "composition" is defined. ] there exists a morphism 9 ( ide: C -+ C is a morphism J by examples:
The category of open se the smooth maps.
The category of vector 1 The category of abelia phisms. The category of chain maps. A category with just on semigroup of morphism Every partially ordered d, when C :s d. A contravariant functor F: C E obC to an object F(C.
a morphism F(f): F(C2 )
F(g 0 f)= A covariant functor F: C - t and
F(g 0 f) =
28
4.
CHAIN COMPLEXES AND THEIR HOMOLOGY
4.
Functors thus are the "structure-preserving" assignments between categories. The contravariant ones change the direction of the arrows, the covariant ones preserve directions. We give a few examples: Let A be a vector space and F(C) = Hom(C, A), the linear maps from C to A. For ¢:CI --t C2, Hom(¢,A):Hom(C2,A) --t Horn(CI,A) is given by Hom(¢, A) ('ljJ) = 'ljJ 0 ¢. This is a contravariant functor from the category of vector spaces to itself.
F(C) = Hom(A, C),
F(¢): 'ljJ
¢ 0 'ljJ. This is a covariant functor from the category of vector spaces to itself. I--t
Let U be the category of open sets in Euclidean spaces and smooth maps, and Vect the category of vector spaces. The vector space of differential p-forms on U E U defines a contravariant functor OP: U
--t
=
bl E BP-I with gP-I(bd exist a E AP with fP(a) = b JP+ I is injective, it is suffic jP+l(d~(a)):
since b is a p-cycJe and dP c [a] E HP(A), and 1*[ aI = [,
One might expect that the Sl exact sequence, but this is DC is surjective, the pre-image ( a cycle. We shall measure v
Definition 4.5 For a short e: --t 0 we define
C*
a
Vect.
to be the linear map given 8* ([ c J)
Let Vect* be the category of chain complexes. The de Rham complex defines a contravariant functor 0*: U --t Vect*. For every p the homology HP: Vect*
--t
CHAJl
Vect is a covariant functor.
:
There are several things to b E (gP)-I(c) we have cPS a E AP+l with fP+l(a) d' [a] E HP+l (A*) is indepenc In order to prove these assel sequence in a diagram:
=
The composition of the two functors above is exactly the de Rham cohomology functor HP: U --t Vect. It is contravariant. A short exact sequence of chain complexes
o --t A * L
B* !4 C*
consists of chain maps f and 9 such that 0 for every p.
--t
--t
AP
0
-L
o BP !4 CP
--t
0 is exact
1
0- A
Lemma 4.4 For a short exact sequence of chain complexes the sequence
HP(A*)
J: HP(B*) ~ HP(C*)
o
1
-AP
1
is exact. Proof. Since gP
1
-AP
0
fP
g*
0
= 0 we have
!*([ a])
= g*([JP(a)]) = [gP(fP(a))] = 0
for every cohomology class [a] E HP(A*). Conversely, assume for [bl E HP(B) that g*[b] = O. Then gP(b) = dFI(c). Since gP-1 is surjective, there exists
The slanted arrow indicates t assertions which, when com (i) If gP(b) = c and d~ (ii) If fP+l(a) = d~(b) (iii) If gP(bl) = gP(b2)
HP+I(A*).
30
4.
CHAIN COMPLEXES AND THEIR HOMOLOGY
4.
CHID
The first assertion follows, because gp+ld~(b) = d~(c) = 0, and Ker gP+l 1m fP+l; (ii) uses the injectivity of fP+2 and that fP+2d~+l(a) = d~+l fP+l(a) = d~+ld~(b) = 0; (iii) follows since bl - b2 = JP(a) so that d~(bl) - d~(b2) = d~fP(a) = fP+ld~(a), and therefore (Jp+lrl(d~(bl)) = (Jp+lrl(d~(b2)) +
Definition 4.10 Two chain I if there exist linear maps s:
d~(a).
for every p.
Example 4.6 Here is a short exact sequence of chain complexes (the dots indicate that the chain groups are zero) with 8* i- 0:
In the form of a diagram, a ,
dBI
_AP-l_
If - /
1 l' 1 0 - 0 -R---.!.....-R-O
_BP-l_
d
1 'd lid
1
The name chain homotopy v.
O-R---.!.....-R-O-O
Lemma 4.11 For two
chain~
111 One can easily verify that 8*: R
-+
Lemma 4.7 The sequence HP(B*)
1*
R is an isomorphism. gO -+
HP(C*)
ao
HP+l(A*)
-4
Proof. If [a] E HP(A*) th
.
IS
exact.
(f* - g*)[ a]
Proof. We have 8*g*([b]) = 8*[gP(b)] = [(JP+l)-\dB(b))] = O. Conversely assume that 8* [ c] = O. Choose b E BP with gP(b) = c and a E AP, such that
=
[jP(a) - gf.
Remark 4.12 In the proof linear maps
(J
d~ (b) = fP+l (d~ a) .
Now we have d~(b - JP(a)) = 0 and gP(b - fP(a)) = c. Hence g*[b - fP(a)] = [cl. 0 Lemma 4.8 The sequence HP(C*)
aO -+
HP+l(A*)
fO -+
id*
HP+I(B*) is exact.
Proof. We have 1* 8* ([ c]) = [d~ (b)] = 0, where gP (b) = C. Conversely assume that 1*[ a] = 0, i.e. fP+I(a) = d~(b). Then d~(gP(b)) = gP+l fP+l(a) = 0, and 8*[gP(b)] = [a]. 0
Theorem 4.9 (Long exact homology sequence). Let 0 -+ A * L B* ..!!.." C* be a short exact sequence of chain complexes. Then the sequence
is exact.
HP(A*) ~ HP(B*) ~ HP(C*) ~ HP+l(A*) ~ HP+l(B*)
=0
However id* = id and 0* == I P > O.
Lemma 4.13 If A* and B* (
HP(A*
We can sum up Lemmas 4.4, 4.7 and 4.8 in the important
... -+
with dP-IS p + Sp+1dP = id 1 o (for p > 0), such that
-+
0
Proof. It is obvious that Ker
(a
Im(a
-+ ...
o
and the lemma follows.
34
5.
THE MAYER-VIETORIS SEQUENCE
5.
where j: Ul n Uz --* U is the inclusion. Hence 1m IP s:;; Ker JP. To show the converse inclusion we start with two p-forms W v E OP (Uv), WI = 2:,hdx/,
Theorem 5.2 (Mayer-Vie The re exists an exact seq,.
Wz = 2:,g/dx/.
. .. --*
°
g/(x),
1*:8
is an isomorphism.
1/=1,2
Proof. It follows from TJ
for which sUPPu(Pv) C Uv , and such that Pl(X) + pz(x) = 1 for x E U (cf. Appendix A). Let f: Ul n Uz -*Il be a smooth function. We use {PI, pz} to extend f to Ul and Uz. Since suPPu(pI) n Uz c Ul n Uz we can define a smooth function by
hex)
=
{~f(X)Pl(X)
hex)
IP:r!
is an isomorphism, and Lc: mology is also an isomorp
if x E Ul n Uz if x E Uz - sUPPU(Pl).
Analogously we define
Example 5.4 We use The spaces of the punctured p
Ul =1
'5
= { ~(x)pz(x)
if x E Ul n Uz if x E Ul - suPPu(pz).
Note that hex) - hex) = f(x) when x E Ul n Uz, because Pl(X) + pz(x) = 1. For a differential form W E OP(UI n Uz), W = "L- hdx/, we can apply the above to each of the functions fr Ul n Uz --* R. This yields the functions h,v: Uv --* R, and thus the differential forms Wv = "L- h,vdx/ E OP(Uv ). With this choice JP(Wl, wz) = w. D
Uz=1 These are star-shaped
HOCUl )
I: O*(U) --* O*(UI) 6:1 O*(Uz) J: O*(Ul) 6:1 O*(Uz) --* O*(UI n Uz )
V
is the disjoint union of the
We have proved:
= HP(O*(Ul )) EB HP(O*(Uz)).
HP(
by the Poincare lemmaaJ we have
···~HP(
are chain maps, so that Theorem 5.1 yields a short exact sequence of chain complexes. From Theorem 4.9 one thus obtains a long exact sequence of cohomology vector spaces. Finally Lemma 4.13 tells us that
HP(O*(Ul) 6:1 O*(Uz))
Opel
= HO(Uz) = R.'
(2) It is clear that
HP(i
Corollary 5.3 If Ul and I
x E Ul x E Uz .
Then IP("L- h/dx/) = (WI, wz). Finally we show that JP is surjective. To this end we use a partition of unity {PI, pz} with support in {Ul, Uz }, i.e. smooth functions
Pv:U--*[O,l],
I*
-t
Here 1*([w]) = (ii[w],i; notation of Theorem 5.1.
Since JP(Wl, wz) = we have that ji(Wl) = ji(wz), which by (1) translates into h 0)1 = g/ 0 jz or hex) = g/(x) for x E Ul n Uz. We define a smooth function hI: U --* Rn by h/(x) = {h(X),
HP(U)
HP+l(R
For p > 0,
O~H1J
36
S.
THE MAYER-VIETORIS SEQUENCE
is exact, i.e. f)* is an isomorphism, and Hq (R2 - {O}) = 0 for q ~ 2 according to (2). If p = 0, one gets the exact sequence
(3)
H-1(Ul
n U2)
HO(UI
-t
HO (~2
-
{O})
n U2) ~ HI (~2 -
~ HO(Ud EEl HO(U2) !
Since H-1(U) = 0 for all open sets, and in particular H-1(Uv ) = 0, 1° is injective. Since H1(Uv ) = 0, f)* is surjective, and the sequence (3) reduces to the exact sequence REEl~
II
a
II
a
0- HO(R 2 - {O})1. HO(Ul) EEl HO(U2) l.. HO(UI n U2) ~ H 1(R 2 - {O}) -
o.
However, R2 - {o} is connected. Hence HO(R 2 - {O}) ~ R, and since 1° is injective we must have that 1m 1° ~ R Exactness gives KerJo ~ ~, so that J O has rank 1. Therefore ImJo ~ R and, once again, by exactness f)* :
HO(U1 n U2) / ImJo
-=. HI (~2 -
{O}).
Since HO(U1 n U2) / ImJo ~ R, we have shown
W(R-10})={:
if p if p if p
~
2
=1 = o.
In the proof above we could alternatively have calculated
J O: HO(Ud EEl HO(U2)
-t
HO(U1 n U2)
by using Lemma 3.9: HO(U) consists of locally constant functions. If constant function on Ui, then
JOUd = fllulnu2 so that JO(a, b)
and
1
let V = Ul U ... U Ur-l, the exact sequence
Sl
HP-l(V n l
which by Lemma 4.1 yields
{O}) ~ H1(Ul) EEl H 1(U2).
REElR
S.
Ii
is a
JO(h) = - hlu1 nu2
= a-b.
Theorem 5.5 Assume that the open set U is covered by convex open sets
Ul," ., Ur. Then HP(U) is finitely generated. Proof. We use induction on the number of open sets. If r = 1 the assertion follows from the Poincare lemma. Assume the assertion is proved for r - 1 and
Now both V and V n Ur = I convex open sets. Therefor, H* (Ur ), and hence also for j
40
6.
HOMOTOPY
Example 6.5 Let Y ~ IR m have the topology induced by IR m . If, for the con tinuous maps fll: X --t Y, V = 0,1, the line segment in IR m from fo(x) to flex) is contained in Y for all x E X, we can define a homotopy F: X x [0,1] --t Y from fo to h by F(x, t) = (1 - t)fo(x)
Indeed, ¢*(dt 1\ dXJ) = 0 constant; see Example 3.11
In the proof of Theorem 3
+ t hex).
In particular this shows that a star-shaped set in Rm is contractible.
such that (l)
Lemma 6.6 If U, V are open sets in Euclidean spaces, then (i) Every continuous map h: U --t V is homotopic to a smooth map. (ii) If two smooth maps fll: U --t V, V = 0,1 are homotopic, then there exists a smooth map F: U x R --t V with F(x, v) = fll(X) for v = 0,1 and all x E U (F is called a smooth homotopy from fo to h).
(dSp '
Consider the composition [ between f and g. Then we
to be Sp Proof. We use Lemma A.9 to approximate h by a smooth map f: U --t V. We can choose f such that V contains the line segment from h( x) to f (x) for every x E U. Then h ~ f by Example 6.5. Let G be a homotopy from fo to fl. Use a continuous function 7jJ: R with 7jJ(t) = 0 for t ::; and 7jJ(t) = 1 for t ~ ~ to construct
!
H: U x IR
--t
V;
H(x, t)
--t
[0,11
--t
This follows from (1) applil
dSp(F*(w))
+ Sp+ldF*(w~
i,
Since H(x, t) = fo(x) for t ::; and H(x, t) = hex) for t ~ H is smooth on U x (-oo,!) u U x ( ~,oo). Lemma A.9 allows us to approximate H by a smooth map F: U x R --t V such that F and H have the same restriction on U x {O, I}. For v = 0, 1 and x E U we have that F(x, v) = H(x, v) = fll(X). 0
Theorem 6.7 Iff, g: U maps
and a
Furthermore Sp+ 1 dF* (w) =
= G(x, 7jJ(t)).
!
= Sp 0 F*,
V are smooth maps and f
~
In the situation of Theorem HP(U). For a continuous m with ¢ -:= f by (i) of Lemma that f*: HP(V) --t HP(U) is
9 then the induced chain
¢*: by setting ¢* =
1*, g*: O*(V) --t O*(U)
1*,
where
f
Theorem 6.8 For PEl am
are chain-homotopic (see Definition 4.10).
(i) If ¢o, ¢1: U
--t
Van
Proof. Recall, from the proof of Theorem 3.15, that every p-form w on U x IR can be written as
w= If ¢: U
--t
L hex, t)dx[ + L gJ(x, t)dt
U x IR is the inclusion map ¢(x) ¢*(w) =
V and ~ 7jJ*: HP(W) --t HP(l (iii) If the continuous maJ (ii) If ¢: U
1\
dXJ.
= ¢o(x) = (x,O),
L fI(x, O)d¢[ = L fI(x, O)dx[.
--t
then is an isomorphism.
42
6.
HOMOTOPY
Proof. Choose a smooth map f: U --+ V with ¢o ~ f. Lemma 6.2 gives that ¢1 ~ f and (i) immediately follows. Part (ii), with smooth ¢ and 7jJ, follows from the formula
Dl(7jJ 0 ¢) = Dl(¢)
0
Proposition 6.11 For an isomorphisms
Q
HP+l (I~n+:
HI (R n +1
Dl(7jJ).
HO(R n +J In the general case, choose smooth maps f: U --+ V and g: V --+ and 7jJ ~ g. Lemma 6.3 shows that 7jJ 0 ¢ ~ 9 0 f, and we get
~V
with ¢
~
f
Proof. Define open subsetl
(7jJ If 7jJ: V
--+
0
= (g 0 f)* = f* 0 g* = ¢* 0 7jJ*.
¢)*
U is a homotopy inverse to ¢, i.e.
1/J 0 ¢
~
id u
and
then it follows from (ii) that 7jJ*: HP(U)
--+
¢ 0 7jJ
~
id v ,
HP(V) is inverse to ¢*.
D
This result shows that HP(U) depends only on the homotopy type of U. In particular we have:
Corollary 6.9 (Topological invariance) A homeomorphism h: U --+ V between open sets in Euclidean spaces induces isomorphisms h*: HP(V) --+ HP(U) for all p.
Ul
= Rn
:
U2
=R
;
n
Then U1 U U2 = IR n +1 - A be gi ven by adding 1 to the line segments from x to ¢(: in Example 6.5 we get hom It follows that Ul is contrac described in Corollary 6.H Let pr be the projection of
i : IR n - A --+ U1 n U2 i 0 pr ~ id u1n u2' From Th
pr*: is an isomorphism for ever:
Proof. The corollary follows from Theorem 6.8.(iii), as h- 1 : V homotopy inverse to h.
--+
U is a 0
Corollary 6.10 If U ~ IRn is an open contractible set, then HP(U) = 0 when p > 0 and HO(U) = IR.
8*: H
for p 2: 1. By composition' Consider the exact sequenc
O--+HO(R J*
Proof. Let F: U x [0,1] --+ U be a homotopy from fo = idu to a constant map h with value xo E U. For x E U, F(x, t) defines a continuous curve in U, which connects x to Xo. Hence U is connected and HO(U) = IR by Lemma 3.9. If P > 0 then DP(h): DP(U) --+ DP(U) is the zero map. Hence by Theorem 6.8.(i) we get that idHP(u) and thus HP(U)
= O.
--+H
An element of HO(Ul) EB E and U2 with values al ane constant function on Ul nl
= f(; = J; = 0
o
In the proposition below, IRn is identified with the subspace Rn x {OJ of IR n+1 and R . 1 denotes the I-dimensional subspace consisting of constant functions.
and we obtain the isomoq HI (R n +1
-
A)
We also have that dim (1m (
44
6.
HOMOTOPY
Addendum 6.12 In the situation of Proposition 6.11 we have a diffeomorphism R: IR n +1 defined by R(XI, . .. ,Xn , Xn+l)
-
A _ IR n +1
= (Xl,""
-
A
~
Proof. The case n = 2 w, from induction on n, via Pr
o.
IR n +l - A - L Rn+l - A
U1
li
-.!!:l....
IR n +1
-
A-LR fI+ I
1i U2
Ij2
-
10
J
Lemma 6.14 For each 1 Hn-l(Rn - {O}) operatesl
I
-fu....
UI
Ih
Ij1
Proof. Let B be obtained j r-th row and c times the s-
UI nU2 ~ UI nU2
UI n U2 ---.l!:sl U1 n U2
An invertible real n x n m a diffeomorphism
A
1i
2
2
U2
Ij1
2
Xn , -xn+t}. The induced linear map
Proof. In the notation of the proof above we have commutative diagrams, which the horizontal diffeomorphisms are restrictions of R:
Iii
~
HP(R'
R*: HP+1 (Rn+l - A) _ HP+l (II~n+1 - A) is multiplication by -1 for p
Theorem 6.13 For n
In the proof of Proposition 6.11 we saw that
8*: HP(UI n U2) - HP+l(R n+1 - A) is surjective. Therefore it is sufficient to show that R* 0 8* ([w]) = -8*([w]) for an arbitrary closed p-form w on UI n U2 • Using Theorem 5.1 we can find Wy E f)F(Uy ), l/ = 0, 1, with w = J;(wt} - j;;(W2). The definition of 8* (see Definition 4.5) shows that 8*([w]) = [T] where T E DP+l (R n +I - A) is determined by i~ (T) = dw y for l/ = 1, 2. Furthermore we get
-R1Jw = R~ 0 j;;(W2) - R'O 0 ji(WI) = ji(RIw2) - j;;(RzwI) ii(R*T) = RI(izT) = RHdw2) = d(RIw2) iz(R*T) = Rz(iIT) = Rz(dwI) = d(RzwI). These equations and the definition of 8* give 8* ((2)
8*
0
R'O([w]) = -R'O
For the projection pr: UI n U2 - Rn the composition
-
0
[R~w])
= [R* T].
From Theorem 6.8 it foIl By a sequence of element diag (1, . " , 1, d), where d . diagonal matrices. The m, diag
8* ([w]).
5
(1
Hence
A we have that pr 0 R o = pr and therefore
HP(R n - A) ~ HP(U1 n U2)
where I is the identity matt and s-th column and zeros e by the matrices
yield a homotopy, whicl diag(1, ... ,1, ±1), so fA dum 6.12. This proves the From topological invarianci 6.13, supplemented with
HP(U I n U2)
is identical with pr*. Since pr* is an isomorphism, R~ is forced to be the identity map on HP(U I n U2), and the left-hand side in (2) is 8*[w]. This completes the ~~
0
HP(R we get
46
6.
Proposition 6.15 If n
=1=
HOMOTOPY
m then IR n and 1R 1n are not homeomorphic. 7.
°
Proof. A possible homeomorphism IR n -.. 1R 1n may be assumed to map to 0, and would induce a homeomorphism between IR n - {O} and 1R 1n - {O}. Hence HP(lR
n
{O})
-
~
HP(IR
1n
-
{O}) D
Remark 6.16 We offer the following more conceptual proof of Addendum 6.12. Let
I
---.E.- B* --.'L.- C* -
1,'
1~'
n
>
0
g~ C* 4* /; B*l~ O-"1~ 1 0
be a commutative diagram of chain complexes with exact rows. It is not hard to prove that the diagram HP(C*)
Let us introduce the standar Dn
for all p, in conflict with our calculations.
0 - A*
APPLICATIONS OF I
Sn-l
= {x E J; = {x E ~
A fixed point for a map f; J
Theorem 7.1 (Brouwer's f f: D n -.. D n has a fixed pl Proof. Assume that f(x) i= the point g(x) E sn-l as the from f(x) through x.
---L HP+1(A*)
1,'
I
n
'
HP(C;) ~ HP+l(Ai)
is commutative. In the situation of Addendum 6.12 consider the diagram 0 - f2*(U) ---.l.:.- f2*(U1 ) 8 f2*(U 2 ) -
1
R*
117
0 - f2*(U) ---.l.:.- f2*(Ud E9 f2*(U2 ) -
n U2) l- flo
0
f2*(U l n U2 ) -
0
f2*(U l
with R(W1,W2) = (Riw2,R2w1). This gives equation (2) of the proof of the addendum.
We have that g(x) = x
+ t1
t = Here x . 11 denotes the usua by solving the equation (x -+ line determined by f (x) and the solution with t ~ O. Sin follows from the lemma bel
Lemma 7.2 There is no COl Proof. We may assume that x/llxll, we get that idRn~{1 segment between x and r(x: g(t· r(x)), 0::; t::; 1 define
48
7.
7.
APPLICATIONS OF DE RHAM COHOMOLOGY
that Rn - {O} is contractible. Corollary 6.10 asserts that Hn-I(R n - {O}) = 0, which contradicts Theorem 6.13. 0
For p E Rn
-
APP
A we have a Up
The tangent space of sn in the point x E sn is Txsn = {x} 1-, the orthogonal complement in Rn+l to the position vector. A tangent vector field on sn is a continuous map v; sn -l- Rn+l such that v(x) E TxS n for every x E sn. Theorem 7.3 The sphere sn has a tangent vector field v with v(x) x E sn if and only if n is odd. Proof. Such a vector field v can be extended to a vector field by setting
w(x)
7lJ
#
=
These sets cover Rn - A partition of unity I/Jp. We
f
0 for all
g(x) =
on Rn - {O}
{
l
pE
where for p E Rn - A, a(1
=v(~). Since the sum is locally fin
We have that w(x)
# 0
and w(x) . x = O. The expression
F(x, t) = (cos 7f t)x
The only remaining probler of A. If x E Up then
+ (sin 7f t)w(x)
defines a homotopy from fo = id(Rn+l_{O}) to the antipodal map Jr, Jr (x) = -x. Theorem 6.8.(i) shows that Ii is the identity on Hn(Rn+l - {O}), which by Theorem 6.13 is I-dimensional. On the other hand Lemma 6.14 evaluates Ii to be multiplication with (-1) n+ 1. Hence n is odd. Conversely, for n = 2m - 1, we can define a vector field v with V(Xl, X2, ... ,X2m) = (-X2' Xl, -X4, X3, ... ,-X2m, X2m-I).
d(xQ,p)::; d(xo,x)+d(x Hence d(xo,p) < 2d(xo,x) we get for x E Up that d(xQ, For x E Rn - A we have
o
In 1962 J. F. Adams solved the so-called "vector field problem": find the maximal number of linearly independent tangent vector fields one may have on sn. (Tangent vector fields VI, , Vd on sn are called linearly independent if for every x E sn the vectors VI(X), , Vd(X) are linearly independent.) Adams' theorem For n = 2m - 1, let 2m = (2c + 1)2 +b, where 0 ::; b ::; 3. The maximal number of linearly independent tangent vector fields on sn is equal to 2b + Sa - 1. 4a
g(x) - g(;1
and (1)
IIg(x) - gl
where we sum over thet>< For an arbitrary E > 0 chao with d(xo, y) < 68. If x E have that d(xo, a(p)) < 68
Lemma 7.4 (Urysohn-Tietze) If A ~ Rn is closed and f: A -l- Rm continuous, then there exists a continuous map g: Rn -l- Rm with 91 A = f. Proof. We denote Euclidean distance in Rn by d( x, y) and for x E Rn we define
d(x, A) = inf d(x, y). yEA
Hg(j Continuity of 9 at
XQ
foUo
Remark 7.5 The proof a replaced by a metric space
50
7.
APPLICATIONS OF DE RHAM COHOMOLOGY
7.
Lemma 7.6 Let A ~ Rn and B ~ Rm be closed sets and let ¢: A ---+ B be a homeomorphism. There is a homeomorphism h of IR n +m to itself, such that
Proof. By induction on rr.
HP+m (IR n +rr
Hm(Rn+rr;
h(x,Om) = (On,¢(X)) for all x E A.
Proof. By Lemma 7.4 we can extend ¢ to a continuous map homeomorphism hI: IR n x IR m ---+ IR n x IR m is defined by
h: IR n
---+
Rm. A
for all m ;::: 1. Analogousl) 1R 2n - B are homeomorph they have isomorphic de R
HP(R n - A)
hl(x, y) = (x, y + h(x)).
for p >
The inverse to hI is obtained by subtracting h(x) instead. Analogously we can extend ¢-l to a continuous map 12: Rm ---+ IR n and define a homeomorphism h : IR n x IR m ---+ IR n x Rm by
API
~
HP+'
°and
HO(R n - A)/IR· 1 ~ Hni
2
h 2 (x, y) If h is defined to be h = hZ l
0
For a closed set A ~ Rn a disjoint union of at mm are open. If there are infin Otherwise the number of c'
= (x + 12(y), y).
hI, then we have for x E A that
h(x,Om) = hZl(x,h(x)) = hZl(x,¢(x)) = (x - 12(¢(x)),¢(x)) = (On,¢(x)).
o
We identify IR n with the subspace of Rn+m consisting of vectors of the form
(Xl, ... ,Xn,O,oo.,O). Corollary 7.7 If ¢: A ---+ B is a homeomorphism between closed subsets A and B of IR n, then ¢ can be extended to a homeomorphism ¢: R2n ---+ R2n . Proof. We merely have to compose the homeomorphism h from Lemma 7.6 with the homeomorphism of 1R 2n = IR n x Rn to itself that switches the two factors. 0 Note that ¢ by restriction gives a homeomorphism between 1R 2n - A and R2n - B. In contrast it can occur that Rn-A is not homeomorphic to Rn-B. A well-known example is Alexander's "horned sphere" ~ in R3 : ~ is homeomorphic to S2, but 1R 3 - ~ is not homeomorphic to R3 - S2. This and numerous other examples are treated in [Rushing]. Theorem 7.8 Assume that A i= Rn and B and B are homeomorphic, then
HP(R n - A)
~
i=
Rn are closed subsets of Rn. If A
HP(R n - B).
Corollary 7.9 IfA and B a and Rn - B have the same
Proof. If A i= IR n and B 7 remarks above. If A = Rn : components (the open halfRi cannot occur.
Theorem 7.10 (Jordan-BI homeomorphic to sn-l tht (i) Rn -
has precisl bounded and U2 is (ii) :E is the set of boUl ~
We say VI is the domain il
Proof. Since ~ is compact, 7.9, to verify it for sn-l ~
b n = {x E Rn J I By choosing r = max Ilxll xEE
J!:
'.~L--"
52
I··.~""~""
7.
APPLICATIONS OF DE RHAM COHOMOLOGY
will be contained in one of the two components in IR n - L:, and the other component must be bounded. This completes the proof of (i). Let pEL: be given and consider an open neighbourhood V of P in IRn. The set A = L: - (L: n V) is closed and homeomorphic to a corresponding proper closed subset B of sn-I. It is obvious that IRn - B is connected, so by Corollary 7.9 the same is the case for IR n - A. For PI E UI and P2 E U2, we can find a continuous curve ,: [a, b] - t IRn - A with ,(a) = PI and ,(b) = P2. By (i) the curve must intersect L:, i.e. ,-I(L:) is non-empty. The closed set ,-I(L:) ~ [a, b] has a first element CI and a last element cz, which both belong to (a, b). Hence ,(cd E L: n V and ,(C2) E En V are points of contact for ,([a, CI)) ~ UI and ,((C2, b]) ~ U2 respectively. Therefore we can find tl E [a, CI) and t2 E ( C2, b]' such that ,(tI) E UI n V and ,(t2) E U2 n V. This shows that P is a boundary point for both UI and U2, and proves (ii). 0
Theorem 7.11 If A ~ IR n is homeomorphic to D k , with k ~ n, then IRn - A is connected.
7.
A
Corollary 7.13 (Invarim Rn and is homeomorphic Proof. This follows irnm
Corollary 7.14 (Dimensj open sets. If U and V aT
Proof. Assume that m < subset of IR n via the inch contradicts that V is cont,
Example 7.15 A knot in I corresponding knot-compl
Proof. Since A is compact, A is closed. By Corollary 7.9 it is sufficient to prove the assertion for D k ~ IRk ~ Rn. This is left to the reader. 0
According to Theorem 7. SI ~ R2 C R3. First We
Theorem 7.12 (Brouwer) Let U ~ IR n be an arbitrary open set and f: U - t IR n an injective continuous map. The image f(U) is open in Rn, and f maps U homeomorphically to f(U).
(2)
Proof. It is sufficient to prove that f(U) is open; the same will then hold for f(W), where W ~ U is an arbitrary open subset. This proves continuity of the inverse function from f(U) to U. Consider a closed sphere. D
= {x
Illx - xoll ~ o} and interior D = D - S.
HP(R 2
Here D2 is star-shaped, \ Theorem 3.15 and Examp dimension 1 for P = 1, am
An analogous calculation c knot E ~ IR n, where E is he
n
E R
contained in U with boundary S It is sufficient to show that f(D) is open. The case n = 1 follows from elementary theorems about continuous functions of one variable, so we assume n ~ 2. Both Sand E = f(S) are homeomorphic to sn-I. Let UI and U2 be the two connected components of IR n - E from Theorem 7.10. They are open; UI is bounded and U2 is unbounded. By Theorem 7.11, Rn - f(D) is connected. Since this set is disjoint from E, it must be contained in UI or U2. As f(D) is compact, Rn - f(D) is unbounded. We must have Rn - f(D) ~ U2. It follows that E u UI = IR n - U2 ~ f(D). Hence
UI ~ f (D). Since D is connected, f(D) will also be connected (even though it is not known whether or not f (D) is open). Since f (D) ~ UI u U2 , we must have that UI = f (D). This completes the proof. 0
Proposition 7.16 Let E ~ U2 be the interior and exte
HP(Ud ~
{Ro
ifz otll
Proof. The case P = 0 fa p > 0 there are isomorphi
HP(UI ) EB HP(I The inclusion map i : W inverse defined by
-+
54
7.
APPLICATIONS OF DE RHAM COHOMOLOGY
7.
The two required homotopies are given by Example 6.5. From Theorem 6.8.(iii) we have that HP(i) is an isomorphism. The calculation from Theorem 6.13 yields
HP(W)
~ {~
Example 7.18 One can a i.e. the cohomology of
if p = 0, n - 1 otherwise.
We now have that HP(Ul) = 0 and HP(U2) = 0 when p r:J- {a, n - I}. On the other hand the dimensions of H n - 1 (Ul ) and H n - l (U2) are 0 or 1, so it suffices to show that Hn-l (U2) -# O. Without loss of generality we may assume that 0 E U1 and that the bounded set U1 U E is contained in Dn. We thus have a commutative diagram of inclusion maps
The "holes" Kj in ~n are d to sn-l. Hence the interi of Ej. One has
(3) ~n _
We use induction on m. the assertion is true for
W--- U2 l
JlP
{O}
;/1
and apply H n -
API
n
to get the commutative diagram
!
Hn-l(Rn - {O}) Hn-1(iY
~ ~
Let V2 = Rn - K m. Then the exact Mayer-Vietoriss
H n - l (W) _ H n - (U2) where H n - l (i) is an isomorphism. It follows that H n - l (U2)
HP(R n ) .!.
-# O.
o
Remark 7.17 The above result about H*(UI) might suggest that U1 is contractible (cf. Corollary 6.10). In general, however, this is not the case. In Topological Embeddings, Rushing discusses several examples for n = 3, where Ul is not simply connected (i.e. there exists a continuous map SI - Ul , which is not homotopic to a constant map). Hence Ul is not contractible either. Corresponding examples can be found for n > 3. If n = 2 a theorem by Schoenflies (cf. [Moise]) states that there exists a homeomorphism h:UI UE-D 2 .
By Theorem 7.12, such a homeomorphism applied to h 1u1 and h
lb2 will map U
l
homeomorphically to b 2 . A result by M. Brown from 1960 shows that the conclusion in Schoenflies' theorem is also valid if n > 2, provided it is additionally assumed that E is flat in ~n, that is, there exists a 8 > 0 and a continuous injective map ¢: sn-l X (-8,8) _ ~n with E = ¢( sn-l X {O}).
If p = 0 then HO(~n) ~ induction and HO(V2 ) s:f. HO (V) ~ R If p > 0 isomorphism H
Now (3) follows by induct
58
8.
SMOOTH MANIFOLDS
Definition 8.4 A smooth manifold is a pair (M, A) consisting of a topological manifold M and a smooth structure .A. on M.
Since chart transformatior Definition 8.7 is independ€ and M2. A composition 0
U sually .A. is suppressed from the notation and we write M instead of (M, A).
A diffeomorphism j: Ml has a smooth inverse. In p As soon as we have chosen on M are smooth. In part between an open set V C We can therefore define a smooth structure:
sn
Example 8.5 The n-dimensional sphere = {x E Rn+l Illxll = I} is an n dimensional smooth manifold. We define an atlas with 2(n + 1) charts (U±i, h±i) where
= {x E Sn I Xi > O}, n ---+ b is the map given
U+i
U-i
= {x
E
Sn I Xi < O}
and h±i: U±i by h±i(X) = (Xl, .. " Xi, ... , xn+I). The circumflex over Xi denotes that Xi is omitted. The inverse map is
A max
h±}(u) = (ul,oo.,ui-1,±V1-lluI12,1Li,oo.,Un) It is left to the reader to prove that the chart transformations are smooth.
Example 8.6 (The projective space IRpn) On sn we define an equivalence relation: X '" y {:} X = Y or X = -yo The equivalence classes [x] = {x, -x} define the set IRpn. Alternatively one can consider IRpn as all lines in Rn+l through O. Let 1r be the canonical projection 1r:
Sn
---+
Rpn;
1r(x)
= [x].
sn open.
U ~ IRpn open {:} 1r- l (U) ~ With the conventions 1r(U±i) ~ Rpn, and An equivalence class representative in U-i.
of Example 8.5, 1r(U-d = 1r(U+i). We define Ui = note that 1r- l (Ui) = U+ i U U-i with U+i n U- i = 0. [x] E Ui has exactly one representative in U+i and one Hence 1r: U+i ---+ Ui is a homeomorphism. We define
hi i
=
= h+i 0
l 1r- :
Ui
---+
b n,
Definition 8.7 Consider smooth manifolds M l and M 2 and a continuous map j: M l ---+ M2. The map j is called smooth at x E Ml if there exist charts hI: Ul ---+ U{ and h2: U2 ---+ U~ on Ml and M2 with x E Ul and j(x) E U2, such that ---+
---+
Vii \
From Remark 8.2 it follo~ open neighborhood V ~ eN.
From now on chart will me,
Definition 8.8 A subset N submanifoLd (of dimension x E N there exists a chart XE[
where R k C Rn is the star It is easy to see that a smoot manifold again. A smooth, (U, h) are charts on M sat
Example 8.9 The n-sphen charts (U±i, h±i) from Exa satisfying (l).
1, ... , n. This gives a smooth atlas on Rpn.
h2ojoh1l:hl(j-l(U2))
{f: V
The inverse diffeomorphisn
(1)
We give Rpn the quotient topology, i.e.
=
U~
is smooth in a neighborhood of hI (x). If j is smooth at all points of Ml then j is said to be smooth.
Definition 8.10 An embeda is a smooth submanifold·all Theorem 8.11 Let Mn be embedding of M n into a El This result will be proved Nn = j (M n ) satisfies the
~
"
~
--'
-
~-
~
60
8.
SMOOTH MANIFOLDS
For every pENn there exists an open neighborhood V ~ R n+ k , an open set U' ~ R n and a homeomorphism g: U'
---+
N
f(q)
Lemma 8.12 Let M n be an n-dimensional smooth manifold. For p E M there exist smooth maps ---+
IR,
fp: M
---+
IR n
such that cPp(p) > 0, and fp maps the open set M - cP;I(O) dif.{eomorphically onto an open subset of R n .
Proof. Choose a chart h: V ---+ V' with p E V. By Lemma A.7 we can find a function 1/J E Coo (R n , R) with compact support SUpPRn (1/J) ~ V', such that 1/J is constantly equal to 1 on an open neighborhood U' c V' of h(p).The smooth map fp can now be defined by if q E ~ otherwIse.
f (q) = {1/J(h(q))h(q) 0
p
On the open neighborhood U = h- 1 (U') the function fp coincides with h and therefore maps U diffeomorphically onto U'. Choose 1/Jo E coo(R n , R) with compact support SUPPRn(1/JO) ~ U' and 1/Jo(h(p)) > 0, and let
cP (q)
if q E ~ otherwIse.
= {1/Jo(h(q)) 0
p
Since M - cP;I(O) ~ U, the final assertion holds.
0
Proof of Theorem 8.11 (M compact). For every p E M choose cPP and fp as in Lemma 8.12. By compactness M can be covered by a finite number of the sets M - cP;I(O). After a change of notation we have smooth functions
cPr M
---+
R,
fr M
---+
We define a smooth map
n V,
such that 9 is smooth (considered as a map from U' to V) and such that Dxg: R n ---+ R n+ k is injective. This is the usual definition of an embedded manifold (regular surface when n = 2). Theorem 8.11 tells us that every smooth manifold is diffeomorphic to an embedded manifold. Conversely, if N ~ R n + k satisfies the above condition, then the implicit function theorem shows that it is a submanifold in the sense of Definition 8.8. A theorem by H. Whitney asserts that the codimension k in Theorem 8.]] can always be taken to be less or equal to n + 1. On the other hand k cannot be arbitrarily small. Rp2 cannot be embedded in 1R3 .
cPp: M
~-~-~-
n
R
(1 ~ j ~ d)
=
Assuming f(ql) = f(q2)
cPj(Q2) = cPj(QI) =I- 0, q2 E M is compact, f is a hon 7J"1:
IR nd+
be the projections on the fi respectively. By (ii) 7J"1 0 f 1fl maps f(U1 ) bijectively gl:
U{
---+
I
Define a diffeomorphism 1.
hI(x, y)
= (~
We see that hI maps f(U j(M), f(Ud = f(M) n 11 to be contained in 7J" l I(U{ onto an open set W{, ane required by Definition 8.8. j (M) is a smooth subm81 a diffeomorphism, nmnely the diffeomorphism f (U1 ) analogously. Hence f: M -
Remark 8.13 The general on differential topology. one uses Theorem 11.6 be manifold" to "topological "diffeomorphism" to "hom where the concept (locally Definition 8.8, but with a t Theorem 8.14 Every coml phic to a (locally flat) topoi
satisfying the following conditions: (i) The open sets Uj = M - cP;-t(O) cover M. (ii) fj\uj maps Uj diffeomorphically onto an open set Uj ~ Rn.
1
On a topological manifold functions M ---+ R. A smoe (
62
8.
SMOOTH MANIFOLDS
consisting of the maps M ~ IR, that are smooth in the structure A on M (and the standard structure on R). Usually A is suppressed from the notation, and the R-algebra of smooth real-valued functions on M is denoted by COO(M, R). This subalgebra of CO(M, R) uniquely determines the smooth structure on M. This is a consequence of the following Proposition 8.15 applied to the identity maps id M
(M, AI) ~ (M, A z ). Proposition 8.15 If g: N and M, then g is smooth
~
M is a continuous map between smooth manifolds N if the homomorphism
if and only
g*: CO(M, R) ~ CO(N, R) given by g*(7jJ) = . lj;
0
g maps COO(M,R) to COO(N,IR).
Proof. "Only if' follows because a composition of two smooth maps is smooth. Conversely if the condition on g* is satisfied, Lemma 8.12 applied to p = g(q) yields a smooth map 1: M n ~ Rn and an open neighborhood V of p in M, such that the restriction 11 v is a diffeomorphism of V onto an open subset of IRn. For the j-th coordinate function Ij E COO(M, R) we have Ij 0 g = g*(Jj) E COO(N, IR), so that 1 0 g: N ~ IRn is smooth. Using the chart lw on M, g is seen to be 0 smooth at q.
Remark 8.16 There is a quite elaborate theory which attempts to classify n dimensional smooth and topological manifolds up to diffeomorphism and home omorphism. Every connected I-dimensional smooth or topological manifold is diffeomorphic or homeomorphic to IR or 51. For n = 2 there is a complete classification of the compact connected surfaces. There are two infinite families of them: Orientable surfaces:
///---
---------~ -<-=-=-:,::-_.-
~-------~---- ._--~<
.-~'"
\
/
'-:::=-.>'
'---_
.... /~~
. . <.. .--~
~~
/--------------~
/....
'-.....--------
/'
Non-orientable surfaces: Rp2, Klein's bottle, etc. See e.g. [Hirsch], [Massey]. In dimension 3, one meets a famous open problem: the Poincare conjecture, which asserts that every compact topological 3-manifold that is homotopy equivalent to 53 is homeomorphic to 53. It is known that every topological 3-manifold M3 has a smooth structure A and that two homeomorphic smooth 3-manifolds also
are diffeomorphic. In the that are homeomorphic to . M. Kervaire he classified ~ there are exactly 28 oriente Kervaire described a topolo, the 19605 the so-called "SI classifies all manifolds of a In the early 1980s M. Freee pact topological 4-manifold aldson proved some very s which showed that there is l ical 4-manifolds. Donaldso (Yang-Mills theory). This manifolds; see [Donaldson One of the most bizarre cor that there exists a smooth sm "1R 4 " is not diffeomorphic t( every smooth structure on Ii
rt
1&
1.•··.~ ...~ • ;:.y..
~
.
~
66
9.
'I>·· . --~'~~-
,".
DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
Definition 9.2 The tangent space TpMm is the set of equivalence classes with respect to (2) of smooth curves a: I - t M, a(O) = p. We give TpM the structure of an m-dimensional vector space defined by the following condition: if (U, h) is a smooth chart in M with p E U, then
iPh: TpM
-t
Rm ,
iPh([a]) = (h
0
a)'(O),
is a linear isomorphism; here [a] E TpM is the equivalence class of a. By definition iPh is a bijection. The linear structure on TpM is well-defined. This can be seen from the following commutative diagram, where F = h 0 h- l , q = h(p)
9.
DIFFI
Suppose M m ~ Rl is a Definition 8.8 implies that usually identify TpM with a: I - t M ~ Rl is a smo( For a composite
Remark 9.4 Given a smo a basis for TpM
Rm
y TpM
~ Lemma 9.3 Let f: M m
-t
+F Rm
N n be a smooth map and p E M.
(i) There is a linear map Dpf: TpM curves by
-t
Tf(p)N given in terms of representing
Dpf([a]) = [f
0
~l~h Rm
Dpf. Dh(p)(gofo~-l)
(4)
where a = (al, ... , am)E a representing smooth cun
a].
(ii) If (U, h) is a chart around p in M and (V, g) a chart around f(p) in N,
then we have the following commutative diagram:
TpM
where (a~i)p is the image 1 vector ei = (0, ... 1, ... ,0: uniquely as
Tf(p)M
~l~g Rn
Proof. Remark 9.1.(ii) applied to F = 9 0 f 0 h- l , defined on the open set h(U n f-l(V)), shows that the bottom map in the diagram is linear and given on representing curves as stated there. Since iP h and iP g are linear isomorphisms, there exists a linear map Dpf making the diagram commutative. The formula in (i) follows by chasing around the diagram. . 0 Note that the linear isomorphism iPh in Definition 9.2 can be identified with Dph
through the identification discussed in Remark 9.1.(ii). From now on we write Dph: TpM - t Rm for this linear isomorphism, and similarly Dh(p)h- l : Rm - t TpM for its inverse.
Given
f E COO(M, R)
we
(5)
The directional derivative ~ (5), I.e. Xpf = (f 0 a)'(O),
Xpf =
~(Jj
In particular (6) . Under the assumptions of 1 ~ j :::; n, for Tf(p)N anp.
I
...-'
-~
'"
....
l' q...
68
9.
DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
and Tf(p)N is the Jacobian matrix at h(p) of 9 0 case N = lvl, f = idM we find that
( ~) =
(7)
"
aXi
P
f
alpj j=l aXi
f 0 h- 1 . Specializing this to the
(h(p))
(~) aYj
9.
DlFFEl
we can write 9 = gi 0 gi- 1 is a smooth map between ( g*(L
. p
in a neighborhood of z, and
This holds when (U, h) and (V, g) are two charts around p with transition function (lpI, ... , lpm)
= lp = go h- 1
expressing the Yj-coordinates in terms of the xi-coordinates. Suppose X is a function that to each p E lvl assigns a tangent vector X p E Tplvl. Given the chart (U, h), formula (4) holds for p E U with certain coefficient functions ai: U ~ R. If these are smooth in a neighborhood of p E U, X is said to be smooth at p. From (7), this condition is independent of the choice of smooth chart around p. If X is smooth at every point p E lvl, X is a a smooth vector field on lvl. Let us next consider families W = {w phEM of alternating k-forms on Tplvl, where wp E Alt k(Tplvl). We need a notion of w being smooth as function of p. Let g: W --t lvl be a local parametrization, i.e. the inverse of a smooth chart, where W is an open set in Rm. For x E W, Dxg: IR m
--t
Altk(Dxg): Altk(Tg(x)lvl) --t
dpw = AltA
(8)
It is not immediately obvi parametrization, but this is i any other locally has the f 6, ... , ~k+1 E TpM. We cI We must show that dyg*(w)(Wl, .
where ¢(x) = Y and Dx
where r = g*(w); see The have defined a chain comI
--t
Altk(R m ).
Alt k (lR m ) to be the function whose value at x is
g*(w)x = Altk(Dxg) (wg(x»)
can be defined via local pari is a smooth k-fonn on M
Tg(x)lvl
is an isomorphism, and induces an isomorphism
We define g*(w): W
The exterior differential
(g*(w)x = wg(x) for k = 0).
Definition 9.5 A family w = {w phEM of alternating k-forms on Tplvl is said to be smooth if g* (w) is a smooth function for every local parametrization. The set of such smooth families is a vector space nk (lvl). In particular, n° (lvl) = COO(lvl, R). Lemma 9.6 Let gi: Wi --t N be a family of local parametrizations with N U gi (Wd· If gi (w) is smooth for all i, then w is smooth.
..
We have nk(lvl) = 0 if k ::: A smooth map ¢: lvl --t N (9)
= Altk(Dp¢)(
One defines a bilinear prod (10)
A:~
One shows by choosing 10 It is equally easy to see tl (11)
Proof. Let g: W --t N be any local parametrization. and z E W. We show that g*(w) is smooth close to z. Choose an index i with g(z) E gi(Wi ). Close to z
k-'
·~12
d(w u.
for w E nk(lvl), r E 12/(.
- j,~< 1~ ~
~
"
- _._~.- - . .•
';}!~";''''
70
9.
-~~<"~
DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
Definition 9.7 The p-th (de Rham) cohomology of the manifold M, denoted HP(M), is the p-th cohomology vector space of n*(M). The exterior product induces a product HP(M) x Hq(M) ---* Hp+q(M) which makes H*(M) into a graded algebra. Note that HP(M) = 0 for p> n = dim M or p < o. The chain map ¢* induced by a smooth map ¢: M
HP(¢): HP(N)
---*
---*
9.
DlFl
Lemma 9.10 Let V = (l Suppose that all Vi have from Vi and Vj to Vin Yj ( orientation with the giver,
The proof is a typical an form:
1'1 induces linear maps
HP(M),
and the de Rham cohomology becomes a contravariant functor from the category of smooth manifolds and smooth maps to the category of graded anti-commutative R-algebras.
Theorem 9.11 Let V :;:: ( 1R 1• Then there exist smoo (i) SUPPM(¢i) ~ Vi (ii) Every p E M haj functions ¢i (i E (iii) For every p E M
Definition 9.8 (i) A smooth manifold Mn of dimension n is called orientable, if there exists an W E nn(Mn ) with wp =I 0 for all p E M. Such an W is called an orientation form on M. (ii) Two orientation forms w, T on M are equivalent if T = f . w, for some f E nO(M) with f(p) > 0 for all p E M. An orientation of M is an equivalence class of orientation forms on M.
Proof. Since M has the Vi ~ 1R 1 with Ui n M : U = UiEI Vi we get smo
On the Euclidean space IR n we have the orientation form dXl !\ .. _!\ dx n • which represents the standard orientation of IR n . Let Mn be oriented by the orientation form w. A basis bl' ... ,bn of TpM is said to be positively or negatively oriented with respect to w depending on whether the number
Let (Pi: M ---* [0, 1] be th theorem follow immediat~
(i) SUPPU(1/li) ~ Ui (ii) Local finiteness. (iii) For every x E U
Proof of Lemma 9.10. L nn(Vi), and choose
We can define w E nn(J
Wi E
wp(b 1 , ... ,bn ) E IR is positive or negative. (It cannot be 0, because w p =I 0.) The sign depends only on the orientation determined by w. If wand T are two orientation forms on Mn, then T = f . w for a uniquely determined function f E nO(M) with f(p) =I 0 for all p E M. We say that wand T determine the same orientation at p, if f(p) > O. Equivalently, wand T induce the same positively oriented bases of TpM. If M is connected, then f has constant sign on M, so we have: Lemma 9.9 On a connected orientable smooth manifold there are precisely 2 0 orientations. If V is an open subset of an oriented manifold M
n,
then an orientation of V is induced by using the restriction of an orientation form on M. Conversely we have:
where
M - SUPPM(
Wp(bl'
all terms are positive (or 2 are positively oriented w w has the desired propel
.,!
"
l' ......
72
9.
..
2
"
DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
9.
If T E on (M n ) is another orientation fonn that gives the orientation of the required type, then T = f . wand Tp(bt, ... , bn ) = f(p)wp(bt, ... , bn ). Since both Tp and wp are positive on bt, ... ,bn we have f (p) > O. Hence wand T detennine the same orientation. 0
Mr --.
Definition 9.12 Let ¢: M2' be a diffeomorphism between manifolds that are oriented by the orientation forms Wj E on(Mj). Then ¢*(W2) is an orientation We say ¢ is orientation-preserving (resp. orientation-reversing), form on as WI (resp. -wI>. when ¢*(W2) detennines the same orientation of
Mr.
Mr
Example 9.13 Consider a diffeomorphism ¢: UI --. U2 between open subsets UI, U2 of IR n , both equipped with the standard orientation of Rn . It follows from Example 3.13.(ii) that ¢ is orientation-preserving if and only if det(D x ¢) > 0 for all x E UI. Analogously ¢ is orientation-reversing if and only if all Jacobi detenninants are negative. Around any point on an oriented smooth manifold M n we can find a chart h: U --. U' such that h is an orientation-preserving diffeomorphism when U is given the orientation of M and U' the orientation of IR n . We call h an oriented chart of M. The transition function associated with two oriented charts of M is an orientation-preserving diffeomorphism. For any atlas of M consisting of oriented charts, all Jacobi detenninants of the transition functions will be positive. Such an atlas is called positive.
where el, ... , en is the sta coefficients of the first fund is symmetric and positive ( A smooth manifold equippe, manifold. A smooth subma by letting ( , )p be the restr product on RI . Proposition 9.16 If M n i~ uniquely determined orientr. for eve!)1 positively orientea VOIM the volume form on j
Proof. Let the orientation bt two positively oriented ortl: tangent space TpM. There c
and w p E AltnTpM satisfil
wp(b~,.
(12)
Proposition 9.14 If { hi: Ui --. UI liE I} is a positive atlas on M a uniquely determined orientation, so all hi are oriented charts.
n,
then M
n
has
Proof. For i E I we orient Ui so that hi is an orientation-preserving diffeomor phism. By Example 9.13, the two orientations on Ui n Uj defined by the restriction 0 from Ui and Uj coincide. The assertion follows from Lemma 9.10.
Positivity ensures that det function p: M --. (0, (0) s oriented orthonormal basis then volM = p-Iw will be Consider an orientation-pre:
Xj(q) Definition 9.15 A Riemannian structure (or Riemannian metric) on a smooth manifold M n is a family of inner products ( , )p on TpM, for all p E M, that satisfy' the following condition: for any local parametrization f: W --. M and any pair VI, V2 ERn, x --. (Dxf(vd, D x f(V2)) f(x)
= (~) = ax'J q
These form a positively Gram-Schmidt orthonorml A(q) = (aij(q)) of 'smooth
bi(q} =
(13)
is a positively oriented ortl
is a smooth function on W.
po f(q) It is sufficient to have the smoothness condition satisfied for the functions
gij(X)
= (Dxf(e'i)'
DIFFER
Dxf(ej)) f(x)'
1 ~ i,j ~ n,
(14)
= Wf(q)(bl(~ =
(detA(q))
This shows that p is smoot
Tt
.
.
.'~
i
\
~
....
1.~·
~
74
9.
.~
9.
DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
Addendum 9.17 There is the following formula for volM in local coordinates:
Now we have
D,
j*(voIM) = Vdet(gij(X)) dXl !\ ... !\ dx n ,
(15)
where
DlFl
!: W
--t
M n is an orientation-preserving local parametrization and
= (Dx!(ei), Dx!(ej))f(x)'
9ij(X)
Proof. Repeat the proof above starting with p EM. Formula (14) becomes j*(voIM)
(16)
so that Dxr(v) = Ilxll-1tJ Letting Wi be the orthog<
W
= volM, so that
p(p) = I for all
The inner product of (13) with the corresponding formula for bk(q) yields (with a Kronecker delta notation)
bik = (bi(q), bk(q)) f(q) =
I
(19) W
=
Ilxll
n
~
Example 9.19 For the a
n
LL
= Ilxll = IIxll
Hence the closed form u.
= (detA(x))-ldxl!\ ... !\ dx n .
n
Wx(Vl,"" vn-I)
aij(q)9jl(q) akl(q).
j=l 1==1
This is the matrix identity, I = A(q) G(q) A(q)t, where G(q) = (9jl(q)). In particular (detA(q))2 detG(q) = 1. Since detA(q) = ITiaii(q) > 0, we obtain (detA(q))-l
o
= VdetG(q).
Example 9.18 Define an (n - I)-form Wo E on-l(Rn) by (17)
WO x (WI, ... ,Wn-l) = det(x, WI,
for x E R. Since wox(q, ... ,ei, ... ,e n )
=
...
we have
and A is orientation-presc orientation form T on RPl map 7r: sn-l --t Rpn-l.
,wn-I) E Altn-l(l~n),
(-I)i-l xi , we have
is a linear isometry. H characterized by the requ
n
(18)
Wo
=
'~ "
. 1 Xidxl (-If-
!\ ... !\ dXi !\ ... !\
dx n .
i=l
If x E sn-l and WI, ... , wn-l is a basis of TxSn-l then x, WI, ... , Wn-l becomes a basis for R n and (17) shows that WOx 1= O. Hence WOISn-l = i*(wo) is an orientation form on sn-l. For the orientation of sn-l given by Wo, the basis WI, ... , wn-l of TxS n- 1 is positively oriented if and only if the basis n X, WI, ... , Wn-l for R is positively oriented. We give sn-l the Riemannian structure induced by Rn . Then (17) implies that volsn-l = WOlsn-l. We may construct a closed (n - I)-form on Rn - {a} with Wlsn-l = volsn-l by setting W = r*(volsn-l), where r: Rn - {a} --t sn-l is the map rex) = x/llxll. For x E R n - {O}, W x E Altn-1(Rn) is given by
Wx(Vl,"" vn-I) = WOr(x)(Dxr(vl),"" Dxr(vn_I))
=
Ilxll-1det(x, Dxr(vl),"" Dxr(vn-l)).
is an isometry for every a one gets 7r* (VOIRpn-l) = n ~ 2. Choose an orientat D x 7r is an isometry, 7r*(v by continuity the sign is constant on all of sn-l.
1
where b = ±1. We can a (-Itb volsn-l =
=
This requires that n ise, n is even.
lb
'"
76
9.
DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
Remark 9.20 For two smooth manifolds M m and N n the Cartesian product 1\1Tn X lV n is a smooth manifold of dimension m+n. For a pair of charts h: V -----. Vi and k: V -----. Vi of M and N, respectively, we can use h x k: V x V -+ V' x Vi as a chart of M x N. These product charts form a smooth atlas on M x N. For P E M and q E N there is a natural isomorphism T(p,q)(M x N) ~ TpM ttJ TqN.
If M and N are oriented, one can use oriented charts (V, h) and (V, k). The tran sition diffeomorphisms between the charts (V x V , h x k) satisfy the condition of Proposition 9.14. Hence we obtain a product orientation of M x N. If the orientations are specified by orientation forms W E om(M) and () E on(N), the product orientation is given by the orientation form pr~f (w) 1\ priv ((}), where prM and pI' N are the projections of M x N on M and N. In the following we shall consider a smooth submanifold M n ~ Rn+k of dimension n. At every point P E M we have a normal vector space TpM-' of dimension k. A smooth normal vector .field Y on an open set W ~ M is a smooth map Y: W -+ R n+ k with Y(p) E TpM-'- for every pEW. In the case k = 1, Y is called a Gauss map on W when all Y(p) have length 1. Such a map always exists locally since we have the following: Lemma 9.21 For every Po E M n ~ Rn+k there exists an open neighborhood W of PO on M and smooth normal vector .fields Yj (1 ~ j ~ k) on W such that YI(p), . .. , Yk(p) form an orthonormal basis ofTpM-'- for every pEW.
Proof. On a coordinate patch around PO EM, there exist smooth tangent vector fields Xl,' .. ,Xn , which at every point p yield a basis of TpM, cf. Remark 9.4. Choose a basis VI, . .. , Vk of Tpo M -'-. Since the (n + k) x (n + k) determinant det(XI (p), ... ,Xn(p), VI, ... , Vk ) is non-zero at Po, it also non-zero for all p in some open neighborhood W of Po on 111. Gram-Schmidt orthonormalization applied to the basis
9. DIFFE
Proposition 9.22 Let l\Ir
(i) There is a 1-1 con: M and njorms in !
Wp(Wl for p E M, Wi E (ii) This induces a 1-1 orientations of M.
Proof. If P E M then Y (p) on Y, the map Y -+ Wy IT orientation form and it can to the orientation determine from Rn + l . If 111 has a G on(M) has the form f . W covered by open sets, for holds, but then the globed 4 An orientation of M detern with Wy = VOIM. This Y i~ Theorem 9.23 (Tubular m fold. There exists an open to a smooth map r: V -+ j
(i) For x E V and yE if Y = r(x). (ii) For every p E 1I1l
P + TpM -'- with eel function on 111. If j (iii) If E: 111 -+ IR is sm£
St
i
is a smooth submai
We call V(= Vp ) the open XI(P), ... ,Xn(p),
VI, ... , Vk
(p E W)
of lI~n+k gives an orthonormal basis
Xl (p), ... Xn(p), YI (p), ... , Yk(P), 1
where the first n vectors span TpM. The formulas of the Gram-Schmidt orthonor malization show that all Xi and Yj are smooth on W, so that Y1 , ... , Yk have the desired properties. D
Proof. We first give a loca vector fields YI, ... , Yk as of Po in M for which we Let us define <1>: Rn+k -+
<1>(x, t) = f(x:
' j~'~.
...
"
- ' . .>" "c .
~.
,'<
~.~.
78
9.
DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
The Jacobi matrix of at
°
has the columns
8f,
8f
-8 (0), ... , -8 (0), Yl(po), ... , Yk(PO). Xl
Xn
The first n form a basis of Tpo M and the last k a basis of Tpo M..l. By the inverse function theorem, is a local diffeomorphism around O. There exists a (possibly) smaller open neighbourhood Wo of Po in M and an fO > 0, such that k
o(p, t)
= P + ~ tj
9.
DIFJ
Now all of M can be co smooth maps TO which sa and Tl will coincide on ~ above) we can now defin that part (i) of the theoreIJ
If in the above we always be an open ball in p + 1'., we have satisfied (i) and
The distance function frc
Yj(p)
j=l
defines a diffeomorphism from Wo x foDk to an open set Vo ~ Rn + k. The map TO = prwo 0 <1>0 1 defines a smooth map TO: Vo ---t Wo, which extends idwo' so that the fiber TOl(p) is the open ball in p + TpM..l with center at p and radius fO for every p E Woo By shrinking fO and cutting Wo down we can arrange that the following condition holds: (20)
For
X
E Vo and y E M we have Ilx - To(x)11 ::; Ilx - yll
is continuous on all of R
dJ
If p E M and x E p + 1 dM(X) = P(p). In this ca of V'. Hence the distanc d:.
with equality if and only if y = TO (x). This can be done as follows. By Definition 8.8 there exists an open neighborhood W of Po in Rn + k such that M n W is closed in W. In the above we can ensure that Vo ~ W where M n Vo remains closed in Vo. Choose compact subsets K l ~ K2 of Wo so that (in the induced topology on M) Po E intKl ~ Kl ~ intK2, where intKi denotes the interior of Ki. The set B
n k
= (IR +
-
Vo) U (M n Vo - intK2)
= {x
E Vo I TO(X) E intKl and Ilx - To(x)11
for all x E V'. In partic
Hence the restriction p = p(p) < P(p) for all p E for the constant function v
< f},
we get for x E V~ and b E M - K2 ~ B that Ilx - bll 2: lib - To(x)ll- Ilx - To(x)11 >
1111
!4 d(x'.
is closed in R n + k and disjoint from Kl. There exists an f E (0, fO] such that lib - yll 2: 2f for all b E B, y E K l . If we introduce the open set V~
:s
satisfies 0 < d(p) p(p) By Lemma A.9 the funed function 1/;: V' ---t R such
f.
Since Ilx - To(x)11 < f, the function y ---t Ilx - YII, defined on M, attains a minimum less than f somewhere on the compact set K2. Consider such a Yo E K2 with Ilx - Yoll = min Ilx - yll ::; Ilx - To(x)11 < f ::; fO· yEM
Hence x - Yo is a normal vector to M at Yo (see Exercise 9.1), but then x E Vo and Yo = TO (x). This shows that Condition (20) can be satisfied by replacing (Wo, Vo, fO) by (intKl, V~, f).
V=
both (i) and (ii) hold for 1 It remains to prove (iii). smooth submanifold of ~ <1>0 1 : Vo ---t Wo x foD k oj 0
s=
va
{(J
The projection of S onli
a 1>: a
The diffeomorphism u x Sk-l to S. This yield
.:
"&
'"
1j
~
80
9.
~
9.
DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
Remark 9.24 In Chapter 11 we will need additional information in the case where M n ~ IR n +k is compact and p > 0 is constant. To any number 1:, 0 < I: < p, we define the closed tubular neighborhood of radius I: around M by
N E = {x E V
Illx -
1'(x)11 :'S
x - 1'(x) E
(21)
Txsf,
Hd(rI)
x ESE'
We end this chapter with a few applications of the existence of tubular neigh borhoods. Let (V, i, 1') be a tubular neighborhood of M with i: M ---t V the inclusion map and 1': V ---t M the smooth retraction map such that l' 0 i = idM. In cohomology this gives
Hd(i) so that Hd(i): Hd(V) injective.
---t
0
Hd(1')
= id H d(M)l
Hd(M) is surjective and Hd(1'): Hd(M)
---t
Hd(V) is
Proposition 9.25 For any compact differentiable manifold Mn all cohomology spaces H d (M) are finite-dimensional. Proof. We may assume that M n is a smooth submanifold of IR n +k by Theorem 8.11, and that (V, i, 1') is a tubular neighborhood. Since M is compact we can find finitely many open balls VI, ... , V r in Rn +k such that their union V = VI U... UVr satisfies M ~ V ~ V. Now we have a smooth inclusion i: M ---t V and a smooth map rlU: V ---t M with 1'lu 0 i = idM. The argument above shows that
Hd(i): Hd(V)
---t
Proof. Choose tubular ne implies that i2 0 fo 0 1'1 :::= so that
I:}.
This set is the disjoint union of the closed balls in p + TpM 1- with centers at p and radius £. Note that N( is compact and that SE is the set of boundary points of N E in IR n + k . By Theorem 9.23.(i) we see for p E M that the real-valued function on SE' x ---t Ilx - pll, attains its minimum value I: at all points x E SE with 1'(x) = p. It follows that
Hd(M)
is surjective, and the assertion now follows from Theorem 5.5.
o
D1FFJi
0
Hd(
Since Hd(rI) is injective Hd(h). If ¢: MI ---t M2 is map g: VI ---t V2 with 9 :::= i Lemma 6.3 shows that f ~
Remark 9.27 As in the d mology can now be made Euclidean space and conti (with the same proofs) witl manifolds. By Theorem 8 in general. Corollary 9.28 If M n ~ tubular neighborhood, tht H d ( r) as its inverse.
Proof. We have l' 0 i = . between x and 1'(x) foral Hd(r) and Hd(i) are inve Example 9.29 For n
~
Let i: sn ---t Rn+1 - {O} r (x) = Then l' 0 i = i The result follows from •
R
Proposition 9.26 Let M I and M2 be smooth submanifolds of Euclidean spaces. (i) If fa,
h: Ah
---t
M 2 are two homotopic smooth maps, then
Hd(Jo) = Hd(h): H d(M2) ---t Hd(MI). (ii) Every continuous map M I
---t
Remark 9.30 Let VI and Using Theorem 9.11, the I significant changes. As in
M2 is homotopic to a smooth map. ---t
HP(VI U V2)
r
---t
HP(
~~"'~
.6b ....
,~
'to
.,...i .
'c::s
009'
"
£)->
~g.
..ct 't:: ..ct,
~ ;do ~.
1, .... .....
~<~~, ..~~~~\
.~
"~'-'t'~-~
82
9.
DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS
Example 9.31 We shall compute the de Rham cohomology of IRpn-1 (n ;:::: 2). With the notation of Example 9.19 we see that AltP(DxA): Alt P (T7r (x)lRpn-l)
--+
--+
INTEGRATION 01'l
Alt P (TxS n- l )
is an isomorphism for every x E sn-l. Therefore
np(71}np(Rpn-l)
10.
Let M
n
be an oriented n-d
np(sn-l)
is a monomorphism, and we find that the image of np ( 1r) is equal to the set of p on the vector space of diffc forms w on sn-l such that A*w = w. Since A* = np(A): np(sn-l) --+ np(sn-l) consider integration on su has order 2 we can decompose it into (±l)-eigenspaces Finally we calculate HnC smooth manifold M n . np(sn-l) = n~(sn-l) EEln~(sn-l) In the special case where ~ w E n~(Rn) uniquely in t where n~(sn-l) = Im(~(id ± rlP(A))). This in fact decomposes the de Rham complex of sn-l into a direct sum of two subcomplexes
n*(sn-l) =
(22)
n* (Rpn-l)
induced by 1r: sn-l
--+
~ n~ (sn-l)
IRpn-l. From (22) we get isomorphisms
HP(sn-l) ~ HP(n~ (sn-l)) EEl HP(n~ (sn-l)) ~ H~(sn-l) EEl H~(sn-l)
where H~(sn-l) is the (±l)-eigenspace of A* on HP(sn-I). Combining with
f
E coo(R n , R) has
r j(J
n~(sn-l) EEl n~(Sn-l).
There is an isomorphism of chain complexes (23)
where
JRn
where dJ.ln is the usual Lebi when w E n~(V) for an op to the whole of Rn by sett
Lemma 10.1 Let ¢: V
--+
W of Rn, and assume that (j = ±l for x E V. For u.
(23) we find that (24)
HP (Rpn-l) ~ H~ (sn-l).
There is a commutative diagram with vertical isomorphisms (See Example 9.29) Hn-1(R n - {O}) -
H n- 1(lR n - {O})
~1 i*
~1i*
Hn-l(sn-l)
---.A..:.......
Hn-l(sn-l)
where the top map is induced by the linear map x --+ -x of IR n into itself. Lemma 6.14 shows that the bottom map is multiplication by (_1)n. Using (24) and Example 9.29 we finally get (25)
HP (Rpn-l) ~
?
{IR if p = or p = n - 1 with n even
o
otherWIse.
Proof. If w is written in
with
f
E Cg"(W, R), it fo'
¢*(w) = . =1
The assertion follows frorr that
fw j(x:
'"
""
~-'",i_N.l . '
84
10.
, [ f >~.
"._
_
:.0,:,,"' _:.....
_,
H
INTEGRATION ON MANIFOLDS
Proposition 10.2 For an arbitrary oriented n-dimensional smooth manifold M n there exists a unique linear map
fM: 0,~(Mn)
->
Lemma 10.3 (i)
fM w changes sign
(ii) Ifw E
IR
f2~(Mn)
ha.
with the following property: If W E 0,~(Mn) has support contained in U, where (U, h) is a positively oriented Coo -chart, then
r
(1)
=
W
}M
r
(h- 1 )* w.
Jh(U)
Proof. First consider W E 0,~( M n ) with "small" support, Le. such that sUPP M (w) is contained in a coordinate patch. Then (U, h) can be chosen as above and the integral is determined by (1). We must show that the right-hand side is independent of the choice of chart. Assume that (U, h) is another positively oriented coo-chart with SUPPM(W) ~ U. The diffeomorphism ¢: V -> W from V = h(U n U) to W = h(U n U) given by ¢ = h 0 h- 1 has everywhere positive Jacobi determinant. Since SUPPh(u)((h- 1 )*w) ~ V,
SU PP h(u/(h- 1 )*w) ~ W
and ¢*(h- 1 )*(w) = (h-1)*(w), Lemma 10.1 shows that
r
~ _ (h- )*w. A(u) So for w E 0,~ (M) with "small" support the integral defined by (1) is independent of the chart. Now choose a smooth partition of unity (Pa\:I'EA on M subordinate to an oriented COO-atlas on M. For w E 0,~(M) we have that (h- )* W 1
1
=
Jh(U)
w =
L
PaW,
a:EA
where every term Pa:W E 0,~(M) has "small" support, and where only finitely many terms are non-zero. We define
I(w)
=L a:EA
1
Pa:W,
for w E 0,~ (M).
Proof. By a partition of SUPPM(w) is contained in consequences of Lemma II
Remark 10.4 In the abm n-forms with compact sup orientation form a E 0,n( !u, where! E CO(M, R). integral of (1) extended to
Irr:C
This linear operator is posi is sufficient to show this v Coo-chart. Then we have
Irr (
M
where the term associated to 0: E A is given by (1), applied to a Ua: with SUPPM(Pa:) ~ Ua:. It is obvious that 1 is a linear operator on f2~(M). If, in particular, SUPPM(W) ~ U, where (U, h) is a positively oriented COO-chart, the terms of the sum can be calculated by (1), applied to (U, h). This yields
I(w)
when W is given t, (iii) If ¢: N n -+ M n I have that
= fM w,
which shows that I is a linear operator with the desired properties. Uniqueness follows analogously. D
where ¢ is determined by we get 1rr (J) ~ O. According to Riesz's repre: 1a determines a positive I
fM!(
The entire Lebesgue integ use only very little of it .
...
..
.~
£"oti,.>
'-
,..;
.
'"
~ .~,E .~.'. .•
Q... ,
c
.~.
c.;
86
10.
;(~:<~
INTEGRATION ON MANIFOLDS
If M n is an oriented Riemannian manifold, the volume form volM will determine a measure I-tM on M n analogous to the Lebesgue measure on IR n . For a compact set J( the volume of J( can be defined by Vol(J() =
h(U
where IR~
=
n N)
JK volM E IR.
E IR n
{(Xl,.'" X n )
I Xl
= h(U)
n IR~
hl:U
Note that (2) is automatically satisfied when p is an interior or an exterior point of N (one can choose (U, h) with p E U, such that h(U) is contained in an open half-space in IR n defined by either Xl < 0 or Xl > 0). If p is a boundary point of N then h(p) has first coordinate equal to zero. Let (U, h) and (V, k) be smooth charts around a boundary point pEoN. The resulting transition diffeomorphism
¢ = k 0 h- l : h(U
n V)
-+
n V)
k(U
induces a map
h(U
n V) n R~
-+
k(U
n V) n R~
n V) n alR~
-+
k(U
n V) n alR~.
The Jacobi matrix at the point q = h(p) E aR~ for ¢ 011)1 ( OXl
Dq ¢ =
': (
q)
o ...
= (¢l,"" ¢n) has the form
0
Dqw
* We must have (a¢r!aXI)(q) we have that (a¢r!aXI)(q)
i= 0, as D q ¢ is invertible.
These charts have mutual smooth atlas on aN. Suppose M n is oriented. ' a positively oriented chart resulting smooth charts (U diffeomorphisms, and they property.
Remarks 10.7 (1) We want to integr boundary. In vie'
,
which restricts to a diffeomorphism
W: h(U
Proof. Every smooth char be restricted to a chart ([
,
O}.
::;
Lemma 10.6 Let N t;:;; M (n - I) -dimensional smOG Suppose M n (n ~ 2) is 01
following property: if p E vector then a basis V2," , basis VI, V2, .•. , Vn for T p
Definition 10.5 Let M n be a smooth manifold. A subset N ~ M n is called a domain with smooth boundary or a codimension zero submanifold with boundary, if for every p E M there exists a COO-chart (U, h) around p, such that (2)
100
} Since ¢ maps IR~ into R~,
> O.
A tangent vector w E TpM at a boundary point pEoN is said to be outward
directed, if there exists a CCXl-chart (U, h) around p with h(U n N) = h(U) n R~
and such that Dph( w) E Rn has a positive first coordinate. This will then also be
the case for any other smooth chart around p.
where IN is the Alternatively, onf the following ven a diffeomorphism let f be a smooth
kr:..nn (One could appro multiplying f by and converging tl (ii) In the case n = 1. orientation of 8/\ pEoN. Let VIE + if VI is a posit
.~
~
"
,.J
.~.'i
1 · •. . ~<.~~
";! .
•
88
10.
A O-fonn on we define
oN
_~~,.
'/'C'e
10.
INTEGRATION ON MANIFOLDS
is a function j: oN
l
aN
j
=
L
pEaN
IR. When
--t
I
has compact support
Let K E n~-l(lRn) be the ( rest of IR n . By diffeomorph
~Nw:
sgn(p) j(p).
= 1 of Stokes'
These conventions are used in the case n
theorem below.
and
iN dw~
Hence the proof reduces tl w E D~-l(lRn). This case i +
n
P1
i=l
and choose b > 0 such that f
o
wl&R'.:
Hence
Theorem 10.8 (Stokes' theorem) Let N ~ M n be a domain with smooth boundary in an oriented smooth manifold. Let aN have the induced orientation. For every W E nn-I(M) with N n SUPPM(W) compact we have
r
JaN --t
l:
P3
Po
where i: aN
w=
i*(w) =
~R':.
(3)
I
By Theorem 3.7 we have
r dw,
n
.IN
dw=
l: i=l
Iv! is the inclusion map.
Hence
Proof. We assume that n ;:::: 2 and leave it to the reader to make the necessary changes for the case n = 1. It is clear that i* (w) has compact support. We can choose j E n~ (M) with value constantly equal to 1 on N n SUPPM(W). Since jw coincides with w on N, both integrals are unchanged when w is replaced by jw, so we may assume that w has compact support. Choose a smooth atlas on jlvf consisting of charts of the type of Definition 10.5 and a subordinate smooth partition of unity (Pa,)aEA" The formulas
r w = L JaN r PaW, .IaN a
1
dw =
N
L Cr
1
reduce the problem to the case where w E n~-I(A1), sUPPiVl(W) ~ U and (U, h) is a smooth chart with h(U n N) = h(U) n IR~. Furthermore the chart (U, h) is assumed to be positively oriented.
d
JR':.
For 2 ::; i ::; n we get
1
00
ali
~(XI,'"
,Xi
=Ji(XI, .. . , Xi-I,
b, Xi
-00 vX~
=0,
d(Pa w )
N
f
(4)
and then by Fubini's theor
(5)
k~
5
'!Io
"
~ ~
~..... .~~ .·2
':-:-i:~
90
10.
When i
INTEGRATION ON MANIFOLDS
= 1, one gets 0 ail
1
-00
= il (0, X2, ... , Xn) - il (-b, X2, ... , Xn)
-;:;-(t, X2, ... ,xn)dt uXI
= il(O, X2,· .. , Xn),
and by Fubini's theorem
l
(6)
R~
ail dpn -;-UXI
=
n
wo=
J
=
i=l
o
Since wOlsn-1
Corollary 10.9 If M n is an oriented smooth manifold and w E n~-I(M) then fMdw = O.
h
Vol (D zrn )
for a suitably chosen smooth map f: Qd ---+ M from a d-dimensional compact oriented smooth manifold Qd. If w = dT for some T E nd-I(M), then Corollary 10.9 yields
h
d(J*(T))
Example 10.11 In Example 9.18 we considered the closed (n - I)-form on Rn - {O},
Ilxli1 n
Vol (s2m-l)
= O.
This, in essence, was the strategy from Examples 1.2 and 1.7. It can be shown (albeit in a very indirect way via cobordism theory) that [w] = 0 if and only if all integrals of the form of (7) vanish.
W=
:
This yields
j*(w) 0:1 0
j*(w) =
n - 1
By induction on m and I
Remark 10.10 Let w be a closed d-form on Mn. One way of showing that the cohomology class [w] E Hd(M) is non-zero is to show that
h
= vols
Vol (sn-:
M in Theorem 10.8 we have
(7)
L
il(O, X2,· .. , Xn) dpn-l·
By combining Equations (3-6) the desired formula follows. Taking N
Example 10.12 The vol theorem to D n with the IR n given by
We conclude this chaptel
Theorem 10.13 If Mn is
(8)
n
~
~ (-1)
i-I
xidxI/\.··/\ dXi /\ ... /\ dXn.
z=l
Since the pre-image of w under the inclusion of sn-l is the volume form volsn-l, which has positive integral over sn-l, we can conclude from Remark 10.10 that [w] 0:1 0 in H n - I (lR n - {O}). If n ~ 2 then, by Theorem 6.13, [w] is a basis of H n - I (lR n - {O}). We thus have an isomorphism
H n - I (lR n
-
is exact.
Corollary 10.14 For a c M induces an isomorph
{O}) ~ R (n ~ 2)
defined by integration over sn-l. The image of [w] under this isomorphism is the volume Vol(sn-l)
=
r
}sn-l
VOlsn-l.
In (8) it is obvious that from Corollary 10.9 that We show the converse
.(
~.
~
92
10.
j
'"
ff
.. ~
INTEGRATION ON MANIFOLDS
Lemma 10.15 Theorem 10.13 holds for M
= IR n ,
n
2: 1.
Let h E
IRn
Proof. Let w E S1~(lRn) be a differential n-fonn with w = O. We must find '" E S1~-l(Rn), such that d", = w. We can write w = I(X)dx1/\" ./\dxn , and let n
. 1
~
'l!i
'" = L...J (-1))-
C~ (IRn)
(12)
A function In E
~
Ij(X) dx 1 /\ ... /\ dXj /\ ... /\ dx n .
be the
C~(Rn)
j=l In(X1, ... ,
(13)
A simple calculation gives
dK
=
(
?= )=1
It is obvious that In is smo this end it is sufficient to s limit X n is replaced by 00,
f)lj) f)x' dXl /\ ... /\ dxn.
n
J
Hence we need to prove the following assertion:
(Pn ): Let I E C~(Rn) be a function with I l(x)df1.n fl, ... , In in C~(Rn) such that
h(X1, ... , Xn-l, t) = .
= O. There exist functions
=
~ f)lj = L...J . 1 f)x')
f.
Finally from (9) it follow1
i:
)=
We prove (Pn ) by induction. For n = 1 we are given a smooth function I E C~(R) with I~oo I(t)dt = O. The problem is solved by setting
h(xl,..
= i:/(Xl,'
fl(x)
= i~ l(t)dt.
Assume (Pn-d forn 2: 2, and let I E C~(lRn, R) be a function with I l(x)df1.n o. We choose C > 0 with supp (I) ~ [-C, and define
(9)
g(X1, ... ,Xn-l)
=
i:
ct
= o.
=
Lemma 10.16 Let (Ua)aE let p, q E M. There exist I (i) p E Ua1
I(Xl, ... ,Xn-l, xn)dxn.
q
and
(ii) Uai n Uai + 1 '=I
0
(The limits can be be replaced by -C and C, respectively).
The function Furthennore 1 supp(g) ~ [-C,ct- . Fubini's theorem yields I gdf1.n-l = Ildltn = O. Using (Pn-d we get functions gl, ... ,gn-1 in C~ (R n - 1 , R) with 9 is smooth, since we can differentiate under the integral sign.
n-l f)g.
~ _ J =g.
(10)
L...J f)Xj j=l
We choose a function p E C~(IR, IR) with I~oo p(t)dt = 1, and define Ij E C~(Rn,
R),
(11)
Ij(Xl,"" Xn-l, xn ) = gj(Xl,"" xn-dP(xn ) , 1 ~ j ~ n - 1.
Proof. For a fixed p we dl a finite sequence of indice~ It is obvious that V is bot! M is connected, we must I Lemma 10.17 Let U ~ J\, be non-empty and open. F K
E S1~-l(M) such that
Proof. It suffices to pro\! invariance it is enough to
'4'
\
",l
.
... '"
j
~
94
10.
Choose WI E n~(Rn) with SUPPWI ~ Wand
r
JRn
11
INTEGRATION ON MANIFOLDS
= 0,
(w - awI)
fRn WI
where a =
Lemma 10.15 implies that n~-1 (W) that satisfies
= 1. Then
r w.
JRn
By Lemma 10.15 we can find a '" E n~-I(Rn) with W -
Hence
W -
d", =
aWl
aWl
=
Let r E n~l-I(M) be the e Then W - d", = dr, so T +
d",.
has its support contained in W.
o
Lemma 10.18 Assume that M n is connected and let W ~ M be non-empty and open. For every W E n~(M) there exists a", E n~-I(M) with supp(w - d",) ~ W. Proof. Suppose that sUpp W ~ Ul for some open set U1 ~ M diffeomorphic to Rn . We apply Lemma 10.16 to find open sets U2, ... , Uk. diffeomorphic to IR n , such that Ui-l n Ui i- 0 for 2 :s; i :s; k and Uk ~ W. We use Lemma 10.17 to successively choose "'1, "'2,···, "'k-l in n~-I(M) such that SU Pp (
w -
~ d.
c:; Uj n Uj+1
i)
(1 'S j 'S k - 1).
1
The lemma holds for '" = 2:,7::1 "'i.
In the general case we use a partition of unity to write
m
W - L.J
-~W·
J'
j=1
where Wj E n~( M) has support contained in an open set diffeomorphic to Rn. The above gives Kj E n~-I(M), 1 :s; j :s; m, such that supp(Wj - dKj) ~ W. For m
K=
L
Kj E n~-I(M)
j=1
we have that m
W - dK
=
L
(Wj - dKj).
j=1
Hence supp(w - dK) ~
Uj=1 supp(Wj -
o
dKj) ~ W.
Proof of Theorem 10.13. Suppose given W E n~(M) with fM W = O. Choose an open set W ~ M diffeomorphic to Rn. By Lemma 10. 18 we can find a '" E n~-I(M) with supp(w - d",) ~ W. But then by Corollary 10.9,
lw
(w - d",) =
1M (w -
d",) = -
1M d", = O.
~i
,'
"" ...~' .........."
~
'"
"
~ .....•., . .. '"' ...
..•....;
~.
•..•.
~
1l:~~.,;
98
..
c,
11. DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS
11. DEGREE, I
Corollary 11.3 Suppose N n L Mn ~ pn are smooth maps between n dimensional compact oriented manifolds and that M and P are connected. Then deg(gf)
Proof. For w
E
l =L w
(gf)*(w)
= deg(f)
=
iN j*(g*(w))
iM g* (w)
= deg(f)deg(g)
I (S)
is a Lebesgue
Note that x E U belongs 1 matrix of I, evaluated at ; we can write S as a unio Theorem 11.6 thus follo' subset K of S. We shall where they follow from
= deg(f)deg(g).
nn(p),
deg(gf)
Then
l
w.
D
Remark 11.4 If I: M n ---+ M n is a smooth map of a connected compact orientable manifold to itself then deg(f) can be defined by chosing an orientation of M and using it at both the domain and range. Change of orientation leaves deg(f) unaffected.
Proposition 11.7 Let f: j and let K <:: U be a com I (K) is a Lebesgue null·
Proof. Choose a compac Let C > 0 be a constan
su:
(4)
~E.
We will show that deg(f) takes only integer values. This follows from an important geometric interpretation of deg(f) which uses the concept of regular value. In general p E M is said to be a regular value for the smooth map I: N n ---+ M m if
DqI: TqN
---+
TpM
1
is surjective for all q E I- (p). In particular, points in the complement of are regular values. Regular values are in rich supply:
Theorem 11.5 (Brown-Sard) For every smooth map regular values is dense in Mm.
f: N n
---+
I(N n )
M m the set of
When proving Theorem 11.5 one may replace M m by an open subset W s,;; M m diffeomorphic to Rn, and replace N n by I- 1 (W). This reduces Theorem 11.5 to the special case where M m = Rm. In this case one shows, that almost all points in Rm (in the Lebesgue sense) are regular values. By covering N n with countably many coordinate patches and using the fact that the union of countably many Lebesgue null-sets is again a null-set, Theorem 11.5 therefore reduces to the following result:
Here Let
Ij
is the j-th coordi
be a cube such that K ! uniformly continuous on (5)
Ilx - yll
~ 6 :::} I~
We subdivide T into a u and choose N so that (6)
diam(ll
For a small cube 'TL witt value theorem yields poi
hC
(7) Theorem 11.6 (Sard, 1942) Let set U C Rn and let
s = {x
I: U ---+ R
m
be a smooth map defined on an open Since
E U I rankDxI
< m}.
~j E T[ ~ L,
the I
~. p
.~ •.
.'
.~>.'
"',
...
.
-~
~
~
11. DEGREE, LI
100
11. DEGREE. LINKING NUMBERS AND INDEX OF VECTOR FIELDS
Proof. For each q E f-l( inverse function theorem v particular q is an isolated p
and by (6),
Ilf(y) -
(8)
f(X)11 ::; cjn diam(T1)
=
a~c.
consists of finitely many Pi neighborhoods Wi of qi il open neighborhood f(Wi)
Formula (7) can be rewritten as
f(y) = f(x) + Dxf(Y - x) + z,
(9) where
Z =
(ZI, ... ,
zn) is given by Zj
=
fj ) L (aax'fj (~j) - a ax' (x) (Yi n
~
i=l
By (6), lI~j
- xii ::;
u=
Xi)'
t
b, so that (5) gives IZj I ::;
Ilzll ::; E
(10)
Since N - U7=1 Wi is. cl is also compact. Hence l Vi = Wi n f-l(U).
anft
En
N'
Consider a smooth map f manifolds, with M connec the local index
Hence
.
(11)
Since the image of Dxf is a proper subspace of Rn, we may choose an affine hyperplane H ~ Rn with
f(x) + Im(Dxf)
~
#,.
f (x)
and II q - pr( q) II
::; E a~yn.
; :; (N C)
anyn ILn(Dt) = 2E~
an
For the Lebesgue measure ILn on Rn
n-I
Theorem 11.9 In the situ
H.
By (9) and (10) the distance from f(y) to H is less than E an Then (8) implies that f(T1) is contained in the set D 1 consisting of all points q E Rn whose orthogonal projection pr( q) on H lies in the closed ball in H with radius alP and centre we have
= {I-1
Ind(J;q)
n 1 Vol(D - ) =
In particular deg(J) is al
Proof. Let qi, Vi, and [,
hence Vi connected. The oriented, depending on v n-form with
C
E
Nn ' Then SliPPN(J*(w)) ~
where c = 2 annn+ ~ C n- 1 Vol (D n- 1 ). For every small cube T t with T t n K =F 0 we now have ILn(J(Tt )) ::; E In. Since there are at most Nn such small cubes T t , ILn (J (K)) ::; EC. This holds for every E > 0 and proves the assertion. 0
Lemma 11.8 Let p E Mn be a regular value for the smooth map f: N n --+ Mn, with N n compact. Then f-l(p) consists of finitely many points ql,· .. , qk. Moreover, there exist disjoint open neighborhoods Vi of qi in Nn, and an open neighborhood U of p in M n , such that (i) f-1(U) =
(ii) fi maps
where Wi E nn(N) and I is a consequence of the
deg(J) = deg(J)
i
k
= L1nd(J
U7=1 Vi
Vi dlffeomorphically onto U for
f
1 ::; i ::; k.
i==l
."
~
.,..i
;,. ~
;,J
t"'tJ .
.~.,
.. '"'
c':"'.:r7<~
102
11. DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS
In the special case where f-l(p) = 0 the theorem shows that deg(J) proof above we get J*(w) = 0). Thus we have Corollary 11.10
11. DEGREE, [
= 0 (in
the
Proposition 11.13 (i) lk(K 1, Jd)
= (
(ii) If J and K can bl
O.
If deg(J) ::J 0, then f is surjective.
D
Proposition 11.11 Let F: pn+l - t M n be a smooth map between oriented smooth manifolds, with M n compact and connected. Let X ~ P be a compact domain with smooth boundary N n = ax, and suppose N is the disjoint union of submanifolds Nf,···, NJ:. If Ji = FlNi ' then
(iii) Let gt and ht be h IR n +1 to smooth t all t E [0,1]. Thl (iv) Let <1>: pl+l - t f compact domain . connected compo. If Ki = <1> (Qi),
k
L deg(Ji) = O. i=1
Proof. Let
f =
FIN
so that
Proof. We look at the c k
deg(J)
L deg(Ji).
=
i=1
On the other hand, if w E nn(M) has
deg(J)
=
L
j*(w)
=
JM w =
L
dF*(w)
1, then
=
L
F*(dw)
where T interchanges fa follows from Corollary
=0 D
deg(T) == (
We shall give two applications of degree. We first consider linking numbers, and then treat indices of vector fields.
In the situation of (ii) th the assertion follows fre Assertion (iii) is a conse( a homotopy J x K x [(
where the second equation is from Theorem 10.8.
Definition 11.12 Let Jd and K 1 be two disjoint compact oriented connected smooth submanifolds of R n +1, whose dimensions d ~ 1, l ~ 1 satisfy d + l = n. Their linking number is the integer lk( J, K) = deg (WJ,I{ ) where
W = WJ,K: J x K
n
-t
S;
w(x,V)
( Finally (iv) follows from
F
v-x
= Ilv-xll'
sn
Here J x K is equipped with the product orientation (cf. Remark 9.20) and is n oriented as the boundary of D +1 with the standard orientation of Rn+ 1. We note that lk( J, K) changes sign when the orientation of eitherJ or K is reversed.
and to the domain X Ji = F\1XQi has degree
Here is a picture to i11l
<
.....J
~
~"'~:...~'.
.q......
"';!:
£¥.
.
't.
""
"'~-
.;:
...•...
l~c'.:.-..J
104
11. DEGREE,LII
11. DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS
Proof.
We apply formull w = vols2 (with integral
Figure 1
Jrg
Ik(J,K
(12)
We write W = r
0
f with
f: JxK r: R(
For x E R3
-
{O}, r*(vol: r*(vc
(cf. Example 9.18). The tal and
D f(Ql,Q2)(v( ql
If lk (J, K) i- 0 then (ii) and (iii) of Proposition 11.13 imply that J and K cannot be deformed to manifolds separated by a hyperplane. We shaH now specialize to the classical case of knots in ~3 where J and K are disjoint oriented submanifolds of 1R 3 diffeomorphic to SI. Let us choose smooth regular parametrizations 0:: ~ --.
p': IR --. K
.1,
with periods a and b, respectively, corresponding to a single traversing of J and K, respectively, agreeing with the orientation. For p E 52, consider the set
I(p) = {(ql, q2)
E J
x
J(
I q2
- ql = AP, A > O}.
Let v (ql) and w (q2) denote the positively oriented unit tangent vectors to J and J( in ql and q2, respectively.
Theorem 11.14 With the notation above we have:
Therefore (13)
\IT * (vols2)(Ql,Q2) (V(l
The integral of (12) can b period rectangle [0, a] X [ For p E 52, I (p) is exac of \IT if and only if the ( and the sign 8(Ql' q2) is . orientation. Assertions (ii
Remark 11.15 In Theofi
can be assumed to be the Xl, x2-plane may be drav e.g.
(i) (Gauss)
Ik(J, K) =
241f
r rb det (a(u) -
Jo Jo
l
;3(v), a (ul'/3 ' (v)) dl1 dv. Ila(u) - ;3('0)11
(ii) There exists a dense set of points P E 52 such that J
det (ql - q2, V(ql), W(q2)) i- 0 for (ql' q2) E I(p). (iii) For such points p, lk(J, K) = L(Qj,q2)EI(p) b(Ql' q2), where 8(ql' Q2) is the sign of the detenninant in (ii).
....
v: , ~
~
"
1~ . .' :!
~
~'~~;"~?"
106
11. DEGREE,
11. DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS
There is one element in I(p) for every place where K crosses over (and not under) 1. The corresponding sign 8 is determined by the orientation of the curves and of the standard orientation of the plane as shown in the picture
Proof. Let i: sn-l -+ R~ the retraction r(x) = xl The lemma follows fror Hn-l(r) are inverse iSG
Figure 3
J~
K
HnK
J 'l.
J
H
TI
"'\ )
,. 0=]
Given a diffeomorphism U, we can define the db
0=-1
In Fig. 2, Ik(J, K) = -1. In Fig. I, Ik(J, K2)
= Ik(J, K 4 ) = 0,
= 1,
Ik(J, K 3 )
Ik(J, K l )
=
Lemma 11.18 If FEe is a diffeomorphism to a
-1.
The sum of these linking numbers is O. This is in accordance with (iv) of Lemma 11.13. We now apply the concept of degree to study singularities of vector fields. Consider a vector field F E COO(U, Rn) on the open set U ~ IR n , n ~ 2, and let us assume that 0 E U is an isolated zero for F. A zero for F is also called a singularity for the vector field. We can choose a p > 0 with
pDn = {x E IR n
Illxll ::;
p} ~ U
and such that 0 is the only zero for F in pD n . Define a smooth map Fp : sn-l sn-l by
-+
The homotopy class of F p is independent of the choice of p, and by Corollary 11.2 and Theorem 11.9, degFp E 7L is independent of p.
Definition 11.16 The degree of Fp is caIled local index of F at 0, and is denoted
t(F; 0). Lemma 11.17 Suppose F E coo(R n , Rn ) has the origin as its only zero. Then -
{O}
-+
Rn
-
Thus it suffices to treat tl Y = (Rn , RTI Let Uo
F(px) Fp(x) = IIF(px) II
F: Rn
Proof. By shrinking U where 0 is the only zen 'IjJ: V -+ Rn. The assertic about 1/J and 'l/J 0 ¢, sinl
{O}
induces multiplication by 1.(F; 0) on Hn-l(R n - {O}) ~ R.
~
U be open ani
cI> : Uo x
[0,1]
For x E Uo,
rd
where
J:
.,1
..
,
~
~
~... "' <--~
•.
·~·i·
q...,.
108
11.
.~
DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS
11. DEGREE, L
It follows that n
L Xi¢i(tX), i=l
and in particular that has a smooth extension to an open set W with Uo x [0, 1] W ~ Uo x 1ft
~
)/
l(
For each t E [0,1], t is a diffeomorphism from Uo to an open subset of Rn . Consider the direct image under
= (t l t Y
E COO (Uo, ~n);
Xt(X)
= L(XI;O) = L(XO;O) = L((A-I)*YjO).
Since A: IR n ......... IR n is linear we have (A- I )*Y yields the commutative diagram
Rn - {a}
(A-1)·r
= A-loY 0 A: ~n .........
{a} --l.-
.I
Let X be a smooth tanget
1
~n. This
lA IR n -
{O}
L(J Proof. By shrinking U Wr open set Uo ~ Rn , which from Uo to an open set. A
Now use the functor Hn-l and apply Lemma 11.17 to both Y and (A-I)*y to get L((A-I)*y;O) = L(Y;O). Hence L(F;O) = L(Y;O). 0
Definition 11.19 Let X be a smooth tangent vector field on the manifold M n , n ::::: 2 wit Po E M as an isolated zero. The local index L(X;PO) E 7L of X is defined by L(X; po)
---
-~/;/
Lemma 11.20 If Po is a
Rn - {a}
lA IR n -
~
for a chart (U, h) with h( Po is said to be a non-de. function theorem F is a l( zero for F. Hence Po E .
0:::; t :::; 1,
and Corollary 11.2 shows that L(F;O)
/
/~~
= (Dxt)-1 Y(t(x)).
The function Xt(x) is smooth on W. Now Xl = F Wo and Xo = (A-I)*y, where A = Doc/>. Choose p > a such that pD n ~ Uo. The homotopy sn-l X [0, 1] ......... sn-l given by Xt(pX) / IIXt (px) II,
;'
//~/-
= L(h*X1u ; 0),
where (U, h) is an arbitrary smooth chart around Po with h(po) = O. We note that Lemma 11.18 shows that the local index does not depend on the choice of (U, h). One can picture vector fields in the plane by drawing their integral curves, e.g.
G: Uo x [0,1 where G can be extender Uo x [0,1]. Choose P > as sn-l ,
( between the map Fp in Dt It follows from Corollae: L(X;PO) : The map fA: Rn - {O} by multiplication by L(J! Lemma 6.14.
<
,....j
~~
~
~
"
...
~
.~
11. DEGREE, 1
110
11. DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS
Definition 11.21 Let X be a smooth vector field on M n , with only isolated singularities. For a compact set R ~ M we define the total index of X over R to be
Lemma 11.25 Suppose 1 there exists an F E C=(I with F outside a compac
Index (X; R) = I>(X;p),
Proof. We choose a func
where the summation runs over the finite number of zeros pER for X. If M is compact we write Index (X) instead of Index (X; M). Theorem 11.22 Let F E C=(U, IR n ) be a vector field on an open set U ~ IR n , with only isolated zeros. Let R ~ U be a compact domain with smooth boundary aR, and assume that F(p) # 0 for pEaR. Then
We want to define F(x) we have F(x) = F(x).
Index(F; R) = deg f, where f: aR
--t sn-l
= F(x) / IIF(x)ll.
is the map f(x)
Proof. Let PI, ... ,Pk be the zeros in R for F, and choose disjoint closed balls D j ~ R - aR, with centers Pj. Define
fj : aDj
--t
fj(x) = F(x) /
Sn-\
IIF(x)ll.
We apply Proposition 11.11 with X = R- Uj Dj. The boundary ax is the disjoint union of aR and the (n - I)-spheres aD l , ... , aDk. Here aDj, considered as boundary component of X, has the opposite orientation to the one induced from Dj. Thus
and choo~e w with Ilwll < zeros of F belong to the 0
We can pick w as a regu DpF = DpF will be il properties.
Note, by Corollary 11.2:
k
deg(j)
+L
-deg(Ji)
(14)
= O.
j=l
Finally deg(jj)
= t(F; Pj) by the definition of local index and Corollary 11.3. D
Here is a picture of F a
Corollary 11.23 In the situation of Theorem 11.22, Index(F; R) depends only on the restriction of F to aR. D \\\
Corollary 11.24 In the situation ofTheorem 11.22, suppose for every pEaR that the vector F(p) points outward. Let g: aR --t sn-l be the Gauss map which to pEaR associates the outward pointing unit normal vector to aR. Then Index(F; R)
= deg g.
Proof. By Corollary 11.2 it sufficies to show that f and g are homotopic. Since f(p) and g(p) belong to the same open half-space of Rn , the desired homotopy can be defined by
(1 - t)f(p) + tg(p) 11(1 - t)f(p) + tg(p)11
(O:::;t:::;l).
D
\\\\\
\~0 ',
\\~
-dj.
~-)J1
,W
VI!
The zero for F of inde' F, both of index -1.
--',
112
11. DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS
Corollary 11.26 Let X be a smooth vector field on the compact manifold Mn with isolated singularities. Then there exists a smooth vector field X on M having only non-degenerate zeros and with Index(X) = Index(X).
Proof. We choose disjoint coordinate patches which are diffeomorphic to Rn around the finitely many zeros of X, and apply Lemma] 1.25 on the interior of each of them to obtain
X.
The formula then follows from (14).
Theorem 11.27 Let Mn S;;; IR n+k be a compact smooth submanifold and let N E be a tubular neighborhood of radius E > 0 around M. Denote by g: oNE ~ sn+k-1 the outward pointing Gauss map. If X is an arbitrary smooth vector field on M n with isolated singularities, then
12.
THE POINCAR
In the following, M n cohomology of Mn is Betti number is given
I
(1)
The Euler characteristi,
(2)
Index(X) = deg g. This chapter's main res
Proof. By Corollary 11.26 one may assume that X only has non-degenerate zeros. From the construction of the tubular neighborhood we have a smooth projection 7r: N ~ M from an open tubular neighborhood N with N E ~ N S;;; IR n + k , and can define a smooth vector field F on N by
F(q) = X(7r(q))
(15)
+ (q - 7r(q)).
Since the two summands are orthogonal, F(q) = 0 if and only if q E M and
X(q) = O. For q E oNE, q -7r(q) is a vector normal to TqoNEpointing outwards. Hence X(7r(q)) E TqoNo and F(q) points outwards. By Corollary 11.24
Index(F; N E ) = deg g. and it suffices to show that L(X;p) = L(F;p) for an arbitrary zero of X. In local coordinates around p in M, with p corresponding to 0 ERn, X can be written in the form
a
n
(16)
X
Ii (0) = 0,
By the final result of a
such vector field X on
011 M n . Given
I
E
COO(M, R),
Proposition 12.2 SUPPI (i) There exists a q
= "~ h(x)-, ax, z
i=l
where
Theorem 12.1 (Poinca manifold M. If X has
and by Lemma 11.20 L( X; p) is the sign of
det
(17)
where 0:: ( -0,6 (ii) Let h: U ~ Rn j
oJ- ) ( ax; (0) .
By differentiating (16) and substituting 0 one gets (18)
ax
ax,J
(0) =
~ ali (O)~. ~oX' i=l
J
ax, t
It follows from (15) that DpF: Rn+k ~ IR n+k is the identity on TpMJ.., and by (18) DpF maps TpM into itself by the linear map with matrix (ali!oXj (0)) (with respect to the basis (0/ oxd o)' It follows that p is a non-degenerate zero for F and that detDpF has the same sign as the Jacobian in (17). 0
is the quadratic
'.. '
.~
<
.. ali'
,
· · 1·.<. ·.~ · · · ·
"
~ .• ~
....... .
,
c_·.
114
12. 0
U0
THE POINCARE-HOPF THEOREM
= '"((t) =
a(t) calculation yields
Proof. Set h
('"(l(t) , ... ,'"(n(t)) and ¢
12.
= f
0
h- l . A direct
,
a) (t)
= (¢ 0
, '"()
n
U0
a)" (0) =
n
LL i=lj=l
This is the value at '"('(0) Both (i) and (ii) follow.
x E IR n . By Lemma 9.21
~ 8¢ , L..J 8x. ('"((t)hi(t).
(t) =
Since P is critical, (8¢j8xi) ('"((0)) = substituting t = 0, we get
o.
l
By differentiating once again and
8x.8x. l
= Dph(a'(O)) E R of the quadratic form from (ii). D
tt
n~2!~.
=.~
(qh~(Ohj(O).
J n
a)"(O) =
1
many coordinate patches : assertion for g(U) instead We define
(4)
82¢
Consider another chart h: (j --+ IR n around P with ij = h(pJ and let F = h 0 h- l defined in a neighborhood of q. The last formula in the proof above can be compared with 0
Proof. Let g: /Rn -:; M I
1, ... ,k smooth maps, sue
i=l
u
·.··o.."..,.i~
(ij)
1~(O)1j(O),
By Sard's theorem it suffil becomes a Morse function that k becomes a Morse f (5)
k()
where ( ,) denotes the
USI
Since (8gj8xi(X), Yv(x))
i=l j=l
where J = f 0 h- l and 1 = h 0 a. Let J denote the Jacobi matrix associated with F in q. Then 1'(0) = J'"('(O) for the column vectors 1'(0) and '"('(0). By substituting this and comparing, one obtains the matrix identity
(8~
(6)
From (5) we have (7)
2
(3)
J
(ij)) J. 82¢ (q)) = Jt ( 8 ( 8Xi8xj 8Xi8Yj
and therefore
Definition 12.3 A critical point p E M of f E COCl(M, R) is said to be non degenerate, if the m~trix in Proposition 12.2.(ii) is invertible. We call f a Morse function, if all critical points of f are non-degenerate. The index of a non degenerate critical point p is the maximal dimension of a subspace V ~ TpM for which the restriction of d~f to V is negative definite.
(8)
Furthermore, by (4), (9)
For smooth submanifolds M n ~ /Rn+k one can get Morse functions by: Theorem 12.4 For almost all Po E /Rn+k the function f: M
f(p) =
--+
IR defined by
8cP 8xj
Assume that Po is a regula from (7) that g(x) - Po E
1 "2llp - poll 2 (10)
is a Morse function.
82 k
-- 8Xi8xj
ti
'" ....
--',-.~.--
""", . . >_s."§'"~ff.i!if¥
116
12.
12.
THE POINCARE-HOPF THEOREM
Theorem 12.6 Let p E M 1 There exists a Coo -chart h
and (x, t) E -I(pO)' The n + k vectors in (9) are linearly independent at the point (x,t). At this point, the equations (8), (10), (6) and (9) give _f)2_k_ f)xif)Xj
= / _f)_g , \ f)Xj
=
~ tv Yv,
(::j ,::i)
f
f)2 9 )
f)xif)Xj
_f)_g) _ / f)xi \~
'\."
, ::i) = (;~ , ::i)'
k
+ (~tv ~~;
Let A denote the invertible (n + k) x (n + k) matrix with the vectors from (9) as rows. Then ADxg takes the following form: a2/;: aXlaXl
...
a2k 8X n aXl 0
ak ... axnaX n
...
0
0
...
0
a2k aXlaX n
0
h-ll
where Di = ±1 (1::; i ::; 11 put f into the standard /01
Proof. After replacing f the problem is local and f E COO(W, IR), where W is the considered non-deg( We write f in the form
2
f(x)
=~ i=
Since
gi(O)
=
-3!:(0)
=0 n
Since
Dxg
has rank
n,
so does
ADxg.
Hence the
n
x
n
matrix
gi(X)
=
2 j:::f.
f)2k ( f)xif)Xj
)
On W we now have that
(x)
is invertible. This shows that x is a non-degenerate critical point.
Example 12.5 Let f: Rn
f( X )
---t
=C-
D where gij E COO(W, IR). If symmetric n x n matrix (
R be the function
2 2 2 Xl - x2 - ... - X-x
2 + x-X+I + ... X 2n ,
(11)
where c E R, A E 71. and 0 ::; A ::; n. Since By differentiating (11) tw gradx(J) = 2( - X l " " ,
o is
the only critical point of
f
f)2 ( f)Xif)Xj
-X-x, X-X+l,' .. , x n ),
f. We find that
(0)
)
= diag( -2, ... , -2,2, ... ,2)
with exactly A diagonal entries equal to -2. Thus the origin is non-degenerate of index A. We note that the vector field grad(J) has the origin as its only zero and that it is non-degenerate of index (-1) -x .
In particular the matrix (I Let us return to the origiI show that the Coo-chart I is given by (11) with
.,.
5
••
~.,
~
118
12.
.~
k-l
f(x)
'-
1l
THE POINCARE-HOPF THEOREM
with D a (k - 1) x (k - 1) matrix of the fonn diag(±1, ... , ±1), and E some symmetric (n k + 1) x (n k + 1) matrix of smooth functions. So suppose inductively that (12)
n
= I:>iXY +
(i) For every non-cri'
n
L L Xi Xj hij(X),
(ii) If p E M n is a cri
8i = ±1
h(U)
i=k j=k
i=l
Definition 12.7 Let f E vector field X on M is sa are satisfied:
for x in a neighborhood W of the origin. We know that the minor E is invertible at O. To start off we can perfonn a linear change of variables in Xk, ... ,X n , so that our new variables satisfy (12) with hkk(O) i= O. By continuity we may assume that hkk(X) has constant sign 8k = ±1 on the entire vV. Set
f
and
0
~
Rn with
h-1(x) = f(p h*XIU = gral
A smooth parametrized Cl
q = ~ E COO(W,R),
and introduce new variables:
Hence one gets (f 0 a)'( critical points, then f 0 a: (i).
~
hik(X)) Yk = q(x) ( Xk + L..J Xi h (x) i=k+l kk Yj = Xj for j i= k, 1 ::; j ::; n. The Jacobi detenninant for y as function of x is easily seen to be 8Yk/8xk(0) = q(O) i= O. The change of variables thus defines a local diffeomorphism W around O. In a neighborhood around 0 we have for y = w(x): f
0
w- 1 (y)
=
f(x) k-1
=
L i=1
n
8i x;
L
2
8i Xi
+ hkk(X)
hkk(X)
k
=
h j
:;~)+12
hjk(x) Xj h (x)
)
(i) Every critical poi (ii) For any a E A I one critical point in Example 12.5
n
L L
+
xixjhij(x)
i=k+1 j=k+1 2
k: n
+ i~1
Proof. We can find a CO( two conditions:
Let X a be a tangent vee h;;l)). Choose a smooth define a smooth tangent
j~1 XiXjhij(x)
n
8i
XiXj hij(x)
i=k+1 j=k+1
k
= 68iYi "" 2
hij
Xk
L y; + L L i=1
xjhjk(x)
j~~~
n
i=1
where
(n + L
(n
i=1 -
L j=k+1
k-1
=
+ X~hkk(X) + 2Xk
n
Lemma 12.8 Every Man
n
+
" 6"
n
" 6 " YiYj -h ij
0
where PaXa is taken to then, for every a E A VI
W-1 (y),
i=k+1 j=k+1
E CCXl(W, R). This completes the induction step.
o
dpf(X a ( Indeed, there is at least
We point out that with the assumptions of Theorem 12.6 p is the only critical point in U. If M is compact and f E COO(M, R) is a Morse function then f has only finitely many critical points. Among them there will always be at least one local minimum (index A = 0) and at least one local maximum (index A = n).
dp
We see that X satisfies
.-.:J
.
ii' .~
q.,..
120
12.
12.
THE POINCARE-HOPF THEOREM
If P is a critical point of j then there exists a unique a: E A with P E UO/.' It follows from (a) that X coincides with XO/. on a neighborhood of P, and condition (b) above shows that assertion (ii) in Definition 12.7 is satisfied. 0 The next lemma relates the index of a Morse function to the local index of vector fields as defined in Chapter 11.
Lemma 12.9 Let j be a Morse function on M and X a smooth tangent vector field such that dpj(X(p)) > 0 for every P E M that is not a critical point for f. Let Po E M be a critical point for j of index A. If X (po) = 0, then
i(X;PO) = (-1)-\. Proof. We choose a gradient-like vector field X. By Definition 12.7.(ii) and Example 12.5,
-
l(X; po)
=
-\
(-1) .
Let U be an open neighborhood of Po that is diffeomorphic to Rn and chosen so small that Po is the only critical point in U. The inequalities
dpj(X(p)) > 0,
dpj(X(p)) > 0,
valid for p E U - {po}, show that X (p) and X (p) belong to the same open half-space in TpM. Thus
(1 - t)X(p)
+ tX(p)
defines a homotopy between]( and Rn - {O}, and l(X; po) = i(X; po).
X considered
at maps from U - {po} to
/
D
(gradp(J) , X p) = dpj(Xp) for all X p E TpM. Then Lemma 12.9 holds for grad(J).
Theorem 12.11 Let M n be a compact differentiable manifold and X a smooth tangent vector field on M n with isolated singularities. Let j E COO(M, R) be a c-\
the number of critical points of index A for n
Index(X)
=L -\=0
(-I)-\c-\.
1
Proof. It is a consequence 01
with isolated singularities ha' gradient-like for f. The zerol claimed formula follows frorr
It is a consequence of the ab
(13)
is independent of the choice 12. 11, the Poincare-Ropf th€ the Euler characteristic; cf.1
We will give a proof of this turn involve methods from d) and will be postponed to AI
Let us fix a compact manifold
M(
Recall that a number a E F critical point.
Lemma 12.12 If there are I and M (a2) are diffeomorph
(0 ~ t ~ 1)
Remark 12.10 Given a Riemannian metric on M and j E COO(M, R), one can define the gradient vector field gradj by the equation
Morse function and have that
'"
~
f. Then we
Lemma 12.13 Suppose that points in j-I(a). Let Pi hal neighbourhoods Ui of Pi,
Sl
(i) PI,'" ,Pr are the 0 (ii) Ui is diffeomorphic (iii) Ui n M(a - E) is contractible subset ~ (iv) M(a + E) is diffeon
Proposition 12.14 In the
.~
finite-dimensional cohomoll
X(M(a·
"
122
12.
Proof. For U
12.
THE POINCARE-HOPF THEOREM
= Ul U ... U Ur , Lemma 12.13.(ii) and Corollary 6.10 imply that HP(U)
~ {~r
if p -=I 0 if p = o.
This gives X(U) = r. Condition (iii) of Lemma 12.13 shows that Ui n M(a - t) is homotopy equivalent to 5..\·-1, and Example 9.29 gives
= 1 + (_1)..\·-1.
X(Ui n M(a - t))
r
=
r
r
L(l + (_1)..\·-1) = r - L i=1
(_1)"\' = X(U) -
i=1
L
V)
=
x(U)
+ x(V) - x(U n V).
HP-l (U n V)
-+
HP(U
U
V)
-+
HP(U) EEl HP(V)
The two sums differ by the . -+
HP(U n V)
-+ ...
First we conclude that dim HP(U U V) < 00. Second, the altematidg sum of the cf. Exercise dimensions of the vector spaces in an exact sequence is equal to
zef;
~_
4A.
0
_
Theorem 12.16 If f is a Morse function on the compact manifold M
n,
=L
(-1)..\ c..\,
..\=0
where
c..\
denotes the number of critical points for
f of index 'x.
Proof. Let al < a2 < ... < ak-l < ak be the critical values. Choose real numbers bo < aI. bj E (aj, aj+l) for 1 ::; j ::; k - 1 and bk > ak. Lemma 12.12 shows that the dimensions of Hd(M(bj)) are independent of the choice of bj from the relevant interval. If M(b j - 1 ) has finite-dimensional de Rham cohomology, the same will be true for M (bj) according to Proposition 12.14, and (14)
X(M(bj)) - X(M(bj-l))
=
L pEf- 1
(aj)
(-l)..\(p)
Example 12.18 (Gauss-Bol 5 ~ 1R 3 , oriented by means of 5 at the point p is
K(p)
= dE
then
n
X(M n )
Proof. Let f be a Morse fi and - f has the same critica respect to f, then p has inde to both f and - f gives
X(M) =
Proof. We use the long exact Mayer-Vietoris sequence . . . -+
The Poincare-Hopf theorem 12.16.
Corollary 12.17 If M n is c<
i=1
Lemma 12.15 Let U and V be open subsets of a smooth manifold. If U, V and Un V have finite dimensional de Rham cohomology, the same is true for UU V, and U
X(M)
(_1)..\'.
The claimed formula now follows from Lemma 12.l3.(iv) and the lemma below, 0 applied to U and V = M(a - t).
x(U
Here the sum runs over the index of p. We can start frc dimHd(M(bj)) < 00 for a 1 ::; j ::; k gives
where p runs over the critica
Since U n M(a - t) is a disjoint union of the sets Ui n M(a - t), it has a finite-dimensional de Rham cohomology, and
X(Un M(a - t))
'"
Sard's theorem implies that both regular values of N. P assume that p± = (0,0, ±f Let f E CDO(S, R) be the critical points pES of f ~ the XI, x2-plane, i.e. N(p) = map dpN is an isomorphisrr parametrized by (u, v, f( u, '1 has the following expressic
K=(l+(~
.".
""
_._,~
124
12.
THE POINCARE-HOPF THEOREM
(see e.g. [do Carmo] page 163). does not vanish at p, so p is a the determinant is positive and is negative and p has index 1.
X(S) =
# {p
E SiN (p)
.
12
Since K(p) i= 0, the determinant in the expression non-degenerate critical point for f. If K (p) > 0 p has index 0 or 2. If K(p) < 0 the determinant We apply Theorem 12.16 to get
= P±, K (p) > o} - # {p
E SiN(p)
= P±, K (p) <
O}.
Since P+ is a regular value for N, we have by Theorem 11.9
deg(N) = #{p E N-1(p+) I K(p) >
O} -
#{p E N-1(p+) I K(p) <
Example 12.20 (Morse f1 to even functions j: sn --l
O}
for x = (XO,XI, ... ,Xn ) ( of j at x is given by
and analogously with p_ instead of P+. It follows that
d:
x(S) = 2 deg(N).
(15)
where v
The map Alt 2 (dNp): Alt 2 ( TN(p)S 2)
. = Alt 2 (TpS)
--+
= K(p),
so N*(vols2)
= K(p)vols.
is multiplication by det (dpN)
Alt 2 (TpS) Hence Thus x
rK vols = ls2r N* (vols2) = (degN) ./S2r vols2 = 411"degN. is ~
is a critical
~
(aoxo,alxl, ... ,anxn ) arc
distinct, this occurs precise actly 2n + 2 critical points. critical points lei]. In a nei
Combined with (15) this yields the Gauss-Bonnet fonnula (16)
= (Vo, VI,· .. , Vn :
rKvols = x(S).
211" ./s
Example 12.19 Consider the torus T in 1R 3 . The height function
f: T
--+
IR is a
h±I(UI, ... , t
Morse function with the four indicated critical points. The Gauss curvature of T is and in a neighborhood of p
joh±I(Ul, ... ,Un)=G The matrix of the second':'o matrix diag(2(a
s
Hence ±eo are non-degenl to the number of indices ~ holds for the other critical a2 < ... < an' Then the The induced function RI index j. We apply Theorel
l:
positive at p and s, but negative at q and r (cf. Example 12.18). Hence j is a Morse function on T. The index at p, q, rand s is 0,1,1 and 2, respectively. Theorem 12. I6 gives X(T) = O. Since we know that dim HO (T) = dim H 2 (T) = 1, we can calculate dim HI (T) = 2.
)
This agrees with Example
C"R
128
13.
POINCARE DUALITY
Remarks 13.1
defined by setting
(i) The vector space Hg (M) consists of the locally constant functions f: M -----> ~ with compact support. Such an f must be identically zero on every non-compact connected component of M. In particular Hg(M) = 0 for non-compact connected M. In contrast HO(M) = ~ for such a man ifold. (ii) If M n is connected, oriented and n-dimensional, then we have the iso morphism from Theorem 10.13,
1M: Hr:(M) --=. ~ HJ(~n) = {~
(3)
which is called the direct j: W -----> V, (i 0 j)*(w) = functor on the category of .
0-
nHUI n
is exact, where
Proof. The above remarks give the result for q = 0 and q = n, so we may assume that 0 < q < n. We identify ~n with sn - {po}, e.g. by stereographic projection, and can thus instead prove that
o.
Now the chain complex n~ (sn - {po}) is the subcomplex of of differential forms which vanish in a neighborhood of Po.
for w E n~(V). We get a
(4)
if q = n otherwise.
Hg(sn - {Po}) =
i*(w)
There is also a Mayer-Vie are open subsets of M wid inclusions, then the sequel
whereas by (i) and (2), Hn(M) = 0 if M is non-compact.
Lemma 13.2
..
...
.-i .
n* (sn)
consisting
Let W E n~(sn - {po}) be a closed form. Since Hq(sn) = 0, by Example 9.29,
w is exact in n*(sn), so there is aTE nq-1(sn) with dT = w. We must show
that T can be chosen to vanish in a neighborhood of Po. Suppose W is an open
neighborhood of {po}, diffeomorphic to ~n, where wlw = O.
If q = 1, then T is a function on sn that is constant on W, say Tlw = a. But
then ~ = T - a E n~ (sn - {po}) and d~ = w.
If 2 ~ q < n then we use that Hq-l(W) ~ Hq-l(~n) = 0, and that Tlw is a
closed form, to find a (J E q - 2 (W) with d(J = Tlw. Now choose a smooth
function [O,lJ with supp (
neighborhood U c W of {po}. The form
sn, assigning the value zero on sn - W. Let 0- be the extended form and let
~ = T - dO-. Then ~Iu = 0 and d~ = dT + ddif = w. 0
n
1q(wl, W2) = ih(Wl We leave the verification surjectivity of 1q uses a Sffii {Uv }; cf. Theorem 5.1.
Theorem 13.3 (Mayer-V quence ••• ----->
Hg(Ul
n U2) ~ H,
Proof. This follows from
In comparing Theorem 1 directions of all arrowS h For later use, let us explil as follows: write w = £.L,'1 and -dw2 agree on Ul n e., is a closed form in n~H I
Proposition 13.4 Suppm Let V cUbe open subsets of a smooth manifold M, and let i: V inclusion. There is an induced chain map
i* : n~(V) -----> n~(U),
----->
U be the
subsets of the smooth mat
(i) Hq(U) - IIo:EA (ii) EBO:EA Hl(Uo:)
~...
".J
e-p
..•...........
~".
~
'"
~--~ ."2 ;
.....•.•
. q"..
130
13.
where i a : Ua
--t
POINCARE DUALI'IY
Theorem 13.5 (Poincar D~ is an isonwrphism
U denotes the inclusion.
Proof. There are isomorphisms
q: nq(U) Wq:
EB
--t
II nq(Ua );
aEA nHUa) --t nc(U);
q(w)
The proof is based upo
= (i:(w))aEA
L ia*(wa )
wq((wa)aEA) =
aEA
which define isomorphisms of chain complexes when we give differential and view
E9 n~(Ua)
Lemma 13.6 Suppose'
IT n*(Ua )
d((ra)aEA) = (dra)aEA c IT n~(Ua) c IT n*(Ua ) as a subcomplex.
the
o
The contravariant functor that sends a vector space A to its dual vector space
HomR(A, R) is exact, i.e. if A ~ B ! C is an exact sequence of vector spaces, then C* "£ B* ~ A* is exact. It is clear that
=
be extended to a linear map on all of C. (This uses Zorn's lemma when C is infinite-dimensional.) We can therefore dualize the exact sequence of Theorem 13.3 to get the exact sequence (5) '" -> Hg+1(Ul n U2)* ~ H2(U)· ~ HHU1)* EEl H2(U2)* ~ HZ(U1 n U2) -> .. , with
(ii (a), i~(a)) where we have written ii: HZ (Ur I'(a)
=
and --t
= j1 (al) -
i(al, (2)
Hg(Ur ~
HZ (UI)* for the vector space dual of
II HZ(Uar;
(3
t->
(i~((3))aEA'
For an oriented n-dimensional manifold the exterior product defines a bilinear map
since supp (WI /\ W2)
~
SUpp(WI)
HP(M) x
n~-p(M)
--t
n~(M)
n SUpp(W2)' It induces a bilinear map H~-P(M)
--t
H~-P(M)
--t
R;
([WI], [W2])
which in tum defines a linear map D~: HP(M)
i!
--t
t->
1M WI /\ W2
H':;-p(Mr.
° Ii
I
Since sUPPu(w /\ i.(7) second integral just in1 agree on V.
Lemma 13.7 For ope,
I.
is commutative. Hen Theorem 5.2, and (), I
Proof. Let
W
E OP (I
W = ji(Wl) - j2(W2) K, E np+I(U) be the inclusions. Then
K,
1
j
11
p.
Du
H~(M)
and we may compose with integration (ef. Remark 13.1.(ii)) to obtain a bilinear pairing
HP(M) x
DvO HP(i
ih' etc.
aEA np(M) x
Proof. Let W E OP(U), classes [w] and [7]. TJ
j;(a2)
The dual of a direct sum is a direct product, so Proposition 13.4.(ii) implies the isomorphism
(6)
commutes.
It was pointed out
0. [7] 7
=
E H~-P(UI 71
+
72
I
n
with 7
c
~
~
1
.......
~
".
~
..
~
;..~.!;C7~
132
13.
POINCARE DUALITY
and let a = Ji(dTt} = -j2(dT2). Then a is a closed (n - p)-form that represents 0* ([T]). Hence
o!D~ nU ([W])([T]) ,
D~ nU ([w])[a]
=
2
,
J
=
2
u,nu 2
This is a commutative di. tions are that D~, EEl D~i by the 5-lemma (cf. Exe The proof of (ii) uses th
W /\ CT.
We must show that the two integrals are equal up to the sign (-1 )p+ 1 . We have
j
K, /\
=
T
U
j
K, /\
Tl
+
U
j
K, /\
T2
= [ dWI /\ Tl + [ dW2 /\ T2
U
Ju,
+ (-1 )P wv
since supp (Tv) ~ V v . Now d(w v /\ Tv) = dw v /\ Tv Corollary 10.9,
1 Uv
d(wv /\ Tv)
H
JU2
/\ dTv , and by
To prove (iii), we use together with Theorem
= 0,
D~ : HO(V) -+ Hr;(U
so that
(-1 )P+l [
K, /\
T
= [
Ju
+
WI /\ dTI
Ju,
is mapped to the basis
1
[ WI /\ dTI
+ [
w2 /\ dT2 =
JU2
J
W2 /\ dT2.
J
Wl/\ jh(a) -
U,
of H~(V)* ~ R.
w2 /\ j2*(CT)
u2
)i(Wl) /\ CT -
= [
Ju,nu2
=
J
[
Theorem 13.9 (Inductil manifold equipped with of open subsets of M th
j2(w2) /\ CT
J u ,nu2
o
w /\ a.
U,nu2
Corollary 13.8 (i) Let VI and V 2 be open subsets of Mn, and suppose that VI, V2 and VI n V 2 satisfy Poincare duality. Then so does V = VI U V2. (ii) Let (VaJaEA be a family of pairwise disjoint open subsets of M n . If each Va satisfies Poincare duality then so does the union V = Ua Va. (iii) Every open subset V ~ Mn that is diffeomorphic to Rn satisfies Poincare duality.
(i) 0 E U. (ii) Any open subsel (iii) If VI, V2, VI n I (iv) If VI, V2, . .. is. then their uni01
Then M n belongs to U The proof is based UpOI means that the closure
Lemma 13.10 In the si Proof. Consider the diagram W'(U)
L
-
l
L
U,
-.l.!
L
HP(U,nU2)
IDP
IDP EBDP
U
H;--P(U)"
of open, relatively comj
HP(UIlEBHP(U2)
P D
I
U2
On the other hand, dTll u, = jh (a) and dT21 U2 = - j2* (a), and we have
Ju,
The horizontal maps are vertical map is an isom(
U2
U,nu2
IDP+' U
I)P+'8! H;-P(U,)'EBH;-P(U2)" ....:!.- H;-P(U,nu2) - _ H;;-P-'(U) _ !
(
n
W+'(U) (i) Vj E
U fOl
JEJ (ii)
(Vj)jEN
is
lOCI
1· ~
134
13.
~.
"
'"
~"'-,
~-.''.'-. ...
'-~
POINCARE DUALITY
Then the union U1 U U2 U ... belongs to U.
But then Theorem 13.9.(i
Proof. First we show by induction on m that Ujl U ... U Ujm E U for every set of indices jl, .. ' ,jm. The cases m = 1,2 follow from (i) and condition (iii) of
Proof of Theorem 13.9. an open subset. We con
Theorem 13.9, so suppose m :::: 3 and that the claim is true for sets of m - 1 indices. Then setting V = Uh U ... U Ujm'
whose open balls are th gi ves us a sequence of (
m
Ujl
nV =
U Uil n Ujv E U v=2
by the induction hypothesis applied to the new sequence (Uil n Uj )j EN' and condition (iii) of Theorem 13.9 implies that UiI U .. . UUjm E U. Since UinUj E U by (i), we also have m
U (Ui v n UjJ E U
(7)
v=1
for any set of 2m indices i 1,jl, ... ,im ,jm. Inductively we define index sets 1m and open sets W m {I}, WI = Ul and for m :::: 2
~
M as follows:
h
m-l
Im={m}U{ili>m, Ui nWm-li=0}-
U Ij
j=1
(8) Wm =
U Ui iElm
If I m- 1 is finite, then W m-l is relatively compact and (ii) implies that W m-l only intersects finitely many of the sets Ui. This shows inductively that 1m is indeed finite for all m. Moreover, if m :::: 2 does not belong to any Ij with j < m then it certainly, by definition of 1m , belongs to 1m . Thus N is the disjoint union of the finite sets 1m . Since we already know that finite unions are in U we have W m E U (if 1m = 0, W m = 0 E U). Similarly, (7) shows that
W m n W m+ 1 =
U
U i n Uj E U. (i,j)EImxlm+l
Note also from (8) that W m n Wk = 0 if k :::: m + 2. Indeed, if the intersection
were non-empty, then there would exist 'i E h with W m n Ui i= 0, and by (8)
i E Ij for some j S m + 1.
One now uses condition (iv) of Theorem 13.9 to see that the following three sets
are in U:
00
W(O)
=
U W2j, j=1
00
W(1) =
U W2j-l, j=1
00
W(O)
n W(1) =
U (Wm n Wm+d· j=1
(i) W = Uj:l Uj : (ii) (Uj) jEN is local (iii) Each Uj is cont
A finite intersection Ujl ( is diffeomorphic to Rn . In the general case, consi a diffeomorphism onto a to W with cover (h- 1 (\ by h -1 belong to the gi, patch. If M is compact then ~ of M by coordinate pat we make use of a locall coordinate patches; cf. ' of Proposition A.6 usinB construct the desired co'
Proof of Theorem
u=
13.~
{U~
and let V = (V;3) ;3EB t tells us that the assumpl
We close the chapter, manifold pair (N, M), i manifold N. Let U be
be the inclusions. The
Proposition 13.11 Thf.
.. , ~ Hq-
The proof of this resul
"
~
136
13.
POINCARE DUALITY
Lemma 13.12 (i) j* = nq(j): nq(N) ----t nq(M) is an epimorphism. (ii) If w E nq(M) is closed, there exists a q-fonn T E nq(N) such that j*(T) = wand such that dT is identically zero on some open set in N
containing M. (iii) If T E nq(N) has sUPPN(dT) n M =
0 and j*(T) is exact, there exists
a fonn a E n such that T - da is identically zero on some open set in N containing M.
The chain map i*: n~(U) show that
induces an isomorphism Hq(N, M) in the above {
Consider an element [wJ i
q - 1 (N)
Proof. We can assume by Theorem 8.11 that N is a smooth submanifold of R k . Theorem 9.23 gives us tubular neighborhoods in ~k with corresponding smooth inclusions and retractions (VN, iN, rN), (VM, iM, rM) for Nand M respectively; we may arrange that VM ~ VN . Let tp: N ----t [O,IJ be a smooth function such that SUPP N (tp) ~ N n VM and such that tp is constantly equal to 1 on some open set W ~ N n VM with M ~ W. Let wE nq(M) be closed, and let = r"M(w) E nq(N n VM). Then (ii) follows upon defining T E nq(N) to be equal tpw on N n VM and extended trivially over the remaining part of N. The same argument proves (i). To prove (iii) we set
represented by a closed E nq-1(N, M). Since j l3.l2.(iii) to find a E nq M. This gives us K, = (T Let [w] E Hq(N, M) be re Lemma l3.l2.(iii) to find on an open set containing T
d(
w
The assumption that SUPP N(dT) n M = 0 implies that driv( T) = riv(dT) vanishes on a neighborhood of M. Hence VM may be chosen so small that di = O. Observe that
= (w - d«
= [w - d(a .
Let us finally introduce tl manifolds of dimension c compact smooth manifold
= (iN 0 j)*(rN(T)) = j* 0 iN 0 rN(T) = ]*(T)
so [i:M-(T)] = 0 in Hq(M). It follows from Corollary 9.28 that [T] = a in Hq(VM). Pulling this back by the inclusion N n VM ----t VM we find that TINnvM is exact. Now choose ao E nq-1(N n VM) with dao = TINnvM and define a E nq-1(N) to be tpao on N n VM and extended trivially over the remaining part of N. Then T - da vanishes on W. 0
Proof of Proposition 13.11. By Lemma l3.l2.(i) we have a short exact sequence of chain complexes
o
----t
n*(N, M)
----t
n*(N)
i. n*(M)
Hq-l(M)
----t
Hq(N, M)
----t
Hq(N)
defined by
(9) This is (- 1) k -symmetric, I
We focus on k == a (mo( forms can be diagonalized (10)
----t
0
where nq(N, M) is defined to be the kernel of nq(j). Let us denote the cohomology of (n*(N, M), d) by H*(N, M). Then Theorem 4.9 gives us a long exact sequence ••• ----t
K,
we obtain [wJ
T = riv(T)IVM E nq(VM).
i:M-(T)
By Lemma l3.12.(ii) we r: that dT vanishes on a neigh
i. Hq(M)
----t ••••
Given a diagonalization
01
a(f-t) This number is independe intersection form we knm adjoint D~ of f.l is an is!
r:.
-
'\.
~
1"~ '.'"'
q.,.....
'--
=
138
,..,~
_
~-
c>'
-
13.
-
-
~.
.-
"
~
""'~~
,
POINCARE DUALITY
Definition 13.13 The signature of an oriented closed 4k-dimensional manifold is the signature of its intersection form.
14.
THE COMPLEX P
The set of I-dimensional c called the complex projecl For z = (zo, Zl,' .. 1 Zn) E "point" Cz E Crn spanr respect to K: a set U ~ « open. In particular there ! Uj :::
and the homeomorphisms
hj([zol ...
(1)
with inverses
(2)
hjl(t
The transition functions h~ l/w m . The atlas H = {(L or holomorphic) manifold the following, however, w manifold, by interpreting ~ Example 14.1 (The Rien identified with R3 , the un
S2 = with north pole p+ = (0, C x {o} is identified with map p to the points of i through P± and p. A strai
1/1+( The
1/1± are diffeomorphi 1/1;1
The transition function 1/J.
1b
1; ~.
. ···'iilltll d ..,. .
.o
140
14.
THE COMPLEX PROJECTIVE SPACE cpn
14.
If we orient C in the usual manner and consider 8 2 as the boundary of D 3 with standard orientation from 1R 3, then 7/;- is orientation-preserving and 7/;+ orientation reversing (check at the poles!). This inspires us to replace 7/;+ by its conjugate -:;jj+: 8 2 - {p+} ---t C. Then the transisition functions between 7/;+ and 7/;- are inversion in the multiplicative group of C, so the atlas on 8 2 consisting of 7/;+ and 7/;- gives 8 2 the structure of a I-dimensional complex manifold (Riemann surface). The classical Hopf map f/: (3)
83
8
---t
f/(ZO, zd
=
2
is given by
(2ZOZ1,
/zol2 - IZlI2).
T
Theorem 14.2 The cohom H 2j
HI
Proof. The embedding C71
and we can use Propositior result for Cpn-1 and that in Example 9.29. The CO]
Hopf discovered (in 1931) that f/ is not homotopic to a constant map. The Riemann surface of Example 14.1 is holomorphically equivalent to Cpl. Indeed, by (2), the compositions 82
'l/J ~ C
{p_}
-
---t
Cp1
and
h- 1
-;;j,
82
-
{p+} -t C .2.. Cp1
are
hOI
0
7/;-(z, t)
= [1, z/(1 + t)], hI
1
0
7/;+(z, t)
= [z/(1 -
t), IJ.
is by (1) and (2) homeomo the fonn . . . ---t
HJ(1R 2n ) ~
j
We know from Lemma 13 is a copy of IR, and the re5
These expressions agree when t E (-1,1), so we can define a homeomorphism
(4)
\[J :
82
---t
\[J(z,
Cp1;
t) = { [1 + t, z], [z,1 - t],
(z,t) (z, t)
=1= =1=
(0,-1) (0,1).
Actually \[J is holomorphic with holomorphic inverse. The complex projective plane Cp1 is often identified with C U {oo} by letting [zo, Zl] E Cp1 correspond to zOl Zl, and assigning 00 to Zo = O. In this identification \[J: 8 2 ---t C U {oo} becomes the stereographic projection 7/;- from the south pole p_, extended by mapping p_ to 00.
It follows from Theorem 1 have a section s: Cpn ---t
H2 ((
would be the identity, but 1 Since H 2n(cpn) = R \\ manifold, and hence from
1
One may generalize the Hopf fibration to the map (5)
7f: 8 2n +1
---t
7f(zo, ... , zn) = [zo, . .. , zn].
cpn;
Its fiber 7f- 1 (p) is the unit circle in the complex line p E Cpn. For n = 1 this is nothing but the Hopf fibration of Example 14.1. Indeed with the notation of (3), (4) \[J-1([zO,Zl])
= W- 1 ([I, zo l z I]) = 7/;=1 ( zO l Zl ) =
The unit circle 8 1 multiplication,
(6)
(2ZOZ1,
Izol2 -lzlI2).
given by the wedge prodl (non-singular) pairing. In of H 2n - 2p(Cpn) has nO[
Theorem 14.3 The cohl algebra
~ C acts on the sphere 8 2n +1 ~ C n+1 by coordinatewise
where c is a non-zero cla.
8
1
X
8 2n+1
---t
8 2n +1;
(A,Z)
f--+
AZ.
The action is free, and the set of orbits of this action is precisely Cpn in the sense that the orbit space 8 2n +1 /8 1 is homeomorphic to Cpn.
Proof. We use induction inclusion
l~ '.~ ~ ...... .. ,..: '~ t\o.,~
'~
..c,~.
~. ~ ~ ~
:""',n <,
"~
i
..-J e-,.J.
q..
~
....
...
~
11
E: . ~ ~
~
~
~.. ;c;, U'...
~
'W·""
~
::~~~
142
14.
14.
THE COMPLEX PROJECTIVE SPACE epn
Proof. Choose U given by
induces the map j*: Hi(cpn) ~ Hi(cpn-l).
The proof of Theorem ]4.2 shows that j* is an isomorphism for i < 2n - 2. Hence en- l # 0 in H 2n - 2(cpn), and the pairing H 2(Cpn)
implies that en
X
# o.
D
It is a consequence of the above theorem that Cpn supports an orientation reversing diffeomorphism only if n is odd. Indeed, a map 1: cpn ~ Cpn induces a homomorphism
If f*(e)
~
Zj
= 1. We let
S
bet
A E Sl such that ASj(p)
H 2n - 2(Cpn) ~ H 2n (Cpn)
J*: H*(cpn)
= Uj such
Sj([zo, .. ·, Zj, ... ,
(7)
where
TIl
=
is surjective. The kernel ha (ii) follows. Let ¢: s2n+1 ~ s2n+1 be tl The chain rule gives a con
H*(cpn).
ae, then
f* (d)
= aj d (0
~ j ~ n).
For j = n we get deg(f) = an. If n is even then deg(f) ;:::: O. In this case there are no orientation-reversing diffeomorphisms of Cpn. If n is odd, on the other hand, then complex conjugation is an orientation-reversing diffeomorphism of Cpn:
1([20, Zl,"" znD
=
[zo, Zl,"
.,
znl
Multiplication by A can be D v ¢ is its restriction to Tv multiplying by A. Since thi
Let V be a (::-vector space C-linear map F: V ~ W i
has a = -1 as one sees by restricting to S2 = Cpl. We close this chapter by constructing closed differential forms which represent a basis for H2j (cpn). This requires some preparations. Consider a unit vector v E s2n+1 C C n+1 with image p = x( v). Now iv is a unit vector tangent to the Sl-orbit of v (which was precisely the unit circle in the I-dimensional C-subspace Cv). The orthogonal complement (C v with respect to the usual hermitian inner product on C n +1 is a real 2n-dimensional subspace of T v S 2n+1, and is orthogonal to iv with respect to the real inner product on T vS 2n +1 induced from C n+1 = R 2n +2.
l
Lemma 14.4 (i) Let p E Cpn and v E x-l(p) ~ s2n+1. There is an open neighborhood U around pin Cpn and a smooth map s: U ~ s2n+1 such that s(p) = v
and x 0 S = idu. (ii) Let v E s2n+1 and p = x(v). The differential Dvx induces an R-linear isomorphismjrom (Cv)..l to Tpcpn. (iii) There exists a well-defined structure on Tpcpn as an n-dimensional C vector space with hermitian inner product, which makes the isomorphisms oj (ii) into C-linear isometries.
Lemma 14.5 If V is a fit C-linear map, then
Proof. We use induction ( some z E C The matrix fo
where x = Rez and y = holds in this case.
If m ;:::: 2 we can choose a an eigenvector of F). F .
Fo:'
and we may assume the f detF
we are done.
= (detFo
c
i3
.;(
...,)
,§'
1.. . ~,.J
..e.
~
~~
~
~;;,
14.
~
•. ...
. .
144
...
".;
c,J
..
,~.,
14.
THE COMPLEX PROJECTIVE SPACE epn
rr
T
Corollary 14.6 V is an m-dimensional C-vector space then r V has a natural orientation with the property that any basis bl, .. . ,bm over C gives rise to a positive basis {bl' ib 1 , b2, ib 2 , ... ,bm , ibm} for rV.
Theorem 14.8 The w = {gp} pECpn is a Riemannian
Proof. Let b~, ... ,b~n be another basis of V. We can apply Lemma 14.5 to the C-linear map F determined by F(bj) = bj (1 ::; j ::; m). Since det(rF) > 0, the 0 assertion follows.
where VOle pn is the volume Corollary 14.6.
Proposition 14.7 Let V be an Tn-dimensional C-vector space with hennitian inner product ( , ). Then (i) g(V1' V2)
= RC(Vl, 112) defines an inner product on rV, and
W(Vl' V2) = g(iVl, V2) = -Irn(vl, V2) defini s ,m element of Alt 2(rV). (ii) vol E Alt 2m (rV) denotes the volume element detennined by 9 and the orientation from Corollary 14.6, then w m = m! vol, where w m = w /\ w /\ ... /\ w (m factors).
rr
Proof. Let p E Cpn and v 11' 0 S
=
idu and s(p)
=v
(10)
By (9) we have dWcn+1 = Crn. If W v E Tpcpn, v =
vector to the fiber in 8 271 + Wv
= Dv lr c
Since Alt 2 (D v lr)(wp ) is th
Proof. We leave (i) to the reader. An orthonormal basis b1 , ... , bm of V with respect to ( , ) gives rise to the positively oriented orthonormal basis of r V with respect to g,
and (10) follows from
b1 , ibl' b2, ib 2,···, bm , ibm.
S*(Wcn+1)(WI, W2 1
Let fl, 71, f2, 72,· .. , fm, 7 m denote the dual basis for Alt (rV). Since w(bj, ibj) = -w(ibj, bj ) = 1, and w vanishes on all other pairs of vectors, Lemma 2.13 shows that m
(8)
W
=
I:
Ej /\ 7j.
j=1
Furthermore, vol = fl /\ 71 /\ E2 /\ 72 /\ ... /\ Em /\ 7 m, because both sides are 1 on the basis above. Direct computation gives w m = m! vol. (See Appendix B.) 0 Note that if V = cn+l, with the usual hermitian scalar product and standard basis eo, ... , en, then (8) takes the form n
(9)
WCn+1
= I: dXj
/\ dYj E
n2 (rC n +1 )
j=O
where Xj and Yj are the real and the imaginary components of the coordinate Zj. We can apply Proposition 14.7 to TpClP n with the complex structure from Lemma 14.4.(iii). This gives us a real scalar product gp on TpCr n and wp E A1t 2T pCr n n . for each p E
cr
In the final equality we us Ul in C n +!, and the fact 1 When showing the smooth to show for a smooth tan is smooth too. This is Ie directly from Proposition
Corollary 14.9 Let w b, 14.8. The j -th exterior p. 1 ::; j ::; n. Proof. The class in H 2n [w] E H 2 (Cr n ) we hav4
Therefore [wt =1= 0 and from Theorem 14.2.
~.
.,..;
"
.J
~ ~,;::--.
...•
~
~
.~....•..~ q.,.,.
146
14.
~
THE COMPLEX PROJECTIVE SPACE cpn
Example 14.10 (The Hopffibration again) Let Zv = xv+iyv, v = 0,1. The Hopf fibration T7 from (3) is the restriction to 53 ~ R4 of the map h : R4 ---+ R3 given by h(xo, Yo,
Xl,
2(XOXI + YOYI) ) 2(XOYI - XIYO) ( x2 + y 2 _ x2 _ y 2 o 0 I I
=
YI)
15.
FIBER BUNDLES A
Definition 15.1 A fiber bu a continuous map 71": E ---+ J b E B has an open neighb
with Jacobian matrix Xl
2 YI (
Xo
-Xl
Xo -Yo
Yo
-Xl
YI
such that
Yo ) Xo . -YI
If v E 53 has real coordinates (xo, Yo, Xl, YI), then iv will have coordinates (-YO,xO,-YI,XI). In (Cv)..l we have the positively oriented real orthonormal basis given by
= (-Xl, YI, Xo, -Yo) ib = (-YI, -Xl, Yo, xo). b
Their images under D vT7: Tv 53 ---+ TTJ(v)5 2 can be found by taking the matrix product with the Jacobian matrix above: 1
2DvT7(b) =
(
Xo2 - Yo2 - xl2 + YI2 ) -2xoYo - 2XIYI - 2XOXI + 2YOYI
1
2 DvT7(ib)
,
=
2xoyo - 2XIYI ) 2 X - Yo2 + Xl2 - YI2 ( o - 2X OYI - 2YOXI
.
xi y0 2
A straightforward calculation (use that (x6 + Y6 + + = 1) shows that !DvT7(b) and !DvT7(ib) define an orthonormal basis of TTJ(v)5 2 with respect to the Riemannian metric inherited from R3. Since W 0 T7 = 71": 53 ---+ Cpl with \lJ: 52 ---+ Cpl the holomorphic equivalence from (4), the chain rule gives that
DTJ(v)W:TTJ(v)5 2
---+
(p
TpCpl
= 7I"(v))
with respect to the listed orthonormal bases has matrix diag (1/2,1/2). Hence
w*(w) where w
= volcpl
=
l
vols2,
71" 0
h
=
proh·
The space E is called the t The pre-image 7I"-I(x), fn fiber bundle is said to be smooth map and the h ab think of a fiber bundle as i (all of them homeomorphi The most obvious example In general, the condition c trivial.
Example 15.2 (The cano considered the action of t action from the right, z.,x (z, u),x = (z'x, ,X-Iu). Th projection on the first fae
with fiber C k • Similarly, i space F we get This is a fiber bundle wi! at the beginning of Chap from (14.7). We can del .....
by Theorem 14.8. In particular we have Vol (Cpl )
=
l
Vol(5
2
)
= 71".
It follows furthermore that Cpl with the Riemannian metric 9 is isometric with in R3. the sphere of radius
!
S'J
for [z] E Uj and U E F. we obtain a smooth fib~ If we take F = C with it or canonical line bundle It is a vector bundle in
,...J
'"
~.
~
.•. . _ "J<' . .,.
1·.····· .
~
148
15.
....
''.
.
~'
15.
FIBER BUNDLES AND VECTOR BUNDLES
I
Example 15.3 Over the real projective space IRpn we have similar bundles sn xso F, where SO = {±1}. In particular D(H) = SI xso D I is the Mobius band.
defined in a neighborhood each pEW and such that Np(M) is a basis and the
Definition 15.4 A vector bundle ~ = (E, B, V, Jr) is a fiber bundle where the typical fiber V and each Jr-I(x) are vector spaces, and where the local homeo morphism h:Vb x V --+ Jr-1(Vb) can be chosen so that h(x,-):V --+ Jr-l(x) is a linear isomorphism for each x E Vb.
h:WxR
1
is a local trivialization. Definition 15.7
Vector bundles can be real, complex or quaternion depending on which category V and h(x, -) in Definition 15.4 belong to. For the time being we concentrate on real vector bundles. A smooth vector bundle is a vector bundle that is also a smooth fiber bundle. The dimension of a vector bundle is the dimension of the fiber. Vector bundles of dimension 1 are called line bundles. We mostly denote vector bundles by small Greek letters. If ~ is a vector bundle then E(O will denote its total space and Fb(~), or just 6, its fiber over b. If WeB then we write ~Iw for the restriction of ~ to W, Le. E(~lw) = Jre-1(W). Example 15.5 (The tangent bundle) Let M n c IR n + k be a smooth manifold. Consider
TM = {(p,v) E M x IR n + k I v E TpM},
such that Jr' 0 j: (ii) A homomorphism fiber bundle map x E B.
Example 15.8 A smooth map of tangent bundles (
Jr(p,v) =p.
The fiber over p E M is the tangent space TpM. We show that the triple TM = (T M, M, Jr) is a vector bundle. Let b E M. Choose a parametrization (V, g) around b, g: W --+ V, W ~ IR n and let h:V x IR n --+ Jr-I(V);
(j, j) bet\ is a pair of (smoc
(i) A map
h(x,v) = Dgg-l(X)(V).
is the derivative of
f.
Definition 15.9 Vector b isomorphic, if there exis1 such that jog = id = 9 bundle is called trivial, al
This gives the required local triviality. Example 15.6 (The normal bundle) Let M n c Rn+k be a smooth manifold. Let
In the above definition tl when the vector bundles deciding if two bundles
Np(M) = (TpM).L be the orthogonal complement to TpM
E(VM) =
C IR n + k •
U Np(M) c M x Rn+k;
Lemma 15.10 A (smoot bundles over B, which m a (smooth) isomorphism.
Set
Jr(v) = p when v E Np(M).
pEM
We must show that VM is locally trivial. For each Po E M, the proof of Lemma 9.21 produced vector fields
Xl, ... , X n , Yl, ... , Yk : W
n k
--+ R
+
Proof. Since j is a bij homomorphism of veetc continuous (smooth). Sir
-<:.
~,
~
150
where
15.
~
,
'"
~
~
15.
FmER BUNDLES AND VECTOR BUNDLES
Remark 15.14 In the abo partition of unity, i.e. cont; and L: O:i(b) = 1 for all b of a smooth partition of 1 Hausdorff space B is calle open refinement {VI3} whi( a paracompact space there continuous functions Sa: B has an open neighborhood
and 'fI are trivial over U. Let
h: U x Rn
--t
7rt(U)
k: U x IR n
and
--t
7r;;l(U)
be isomorphisms. Then F = k0
j
0
h -1: U x IR n
--t
U x Rn
is an isomorphism of trivial bundles and it has the form
F(x, v) = (x, F2 (x, v)), The map x
--t
x
E U.
F 2 (x, -) defines a map
ad(F2): U
--t
GLn(R).
Conversely such a map induces a homomorphism ad(F2)-1: U
--t
1: 7rf:1(U)
--t
Definition 15.15 A (smool (smooth) map s: B ---t E 5
7r;;l(U). Note that
GLn(R)
The set of sections of a vee by using the vector space section which to b E B aS5 bundle then we let nO(~) ( It follows from local tri, the base, we can find sec {Sl(X), ... ,Sn(x)} is a b: product we may even chi orthogonal basis (Gram~S frame for ~ over U. Let (i, id B) be a homom( U. Then Ix: ~x --t 'fix is l'1
determines Finally, it is easy to see that f is continuous (smooth) if and only if ad(F2 ) is. The lemma now follows because matrix inversion (_)-1: GL n (lR) --t GLn(R) is a smooth map. 0
F- 1 .
Definition 15.11 The direct sum ~ EB 'fI of two vector bundles over the same base space B is the vector bundle over B with total space E(~ EB 'fI) = {(v, w) E E(~) x E('fI) 17r~(v) = 7r1](w)}
and projection 7f~EfJ1](v, w)
= 7f~(v) = 7f1](w).
The fiber (~EB 'fIh is equal to ~bEB'fIb'
Definition 15.12 An inner product on a (smooth) vector bundle ~ is a (smooth) map ¢: E(~ EB~) --t R such that ¢: Fb(~) EB Fb(O --t R is an inner product on each fiber Fb(~).
(1)
depending on the given f that Ix: ~x - 'fix is an iso have an inner product, an~ precisely if ad (ix) E On
Proposition 15.13 Every vector bundle over a compact B has an inner product. Proof. Choose local trivializations hi: Ui
X
Rn
-
7ft(Ui)
Lemma 15.16 Let ~ ant the compact space B, an t > 0 such that every hl b E B is also an isomOi
where· U1, ... , Ur cover B, and choose a partition of unity {O:i} ;=1 with supp(o:d CUi. The usual inner product in Rn induces an inner product in Ui X IR n and hence, via hi, an inner product ¢i 7ft (Ui)' Now
¢(v, w) = is an inner product in
~.
FI
L ¢i(V, W)O:i(7f(V)) o
An inner product in the tangent bundle TM of a smooth manifold is the same as a Riemannian metric on M (cf. Definition 9.15).
I
Proof. If ~ and 'fI are tri, by maps ad(j): B - GJ GLn(R) is open, some ( in GLn(R). But then aC lemma. In general, we c~
,
~!
~
""
1~
~
. .~
"
t<-...J "'-"'''''~'=?;c.,.;.
152
15.
15.
FmER BUNDLES AND VECTOR BUNDLES
FI
Theorem 15.18 Every vet complement TI, i.e. ~ EB TI ~
over which the bundles are trivial, and take the minimum of the resulting epsilons.
D
Proof. Choose an open CI ~IUi' and let {O:i} be a pa the composite
Given two smooth vector bundles ~ and 'rJ, one might wonder if there is any essential distinction between the notions of continuous and smooth isomorphism. The next result shows that this is not the case. Along the same lines one may ask if each isomorphism class of continuous vector bundles over a compact manifold contains a smooth representative. This is indeed the case (cf. Exercise 15.8).
1l'~
and define Lemma 15.17 If two smooth vector bundles ~ and 'rJ over the compact manifold B are isomorphic as continuous bundles, then they are smoothly isomorphic. Proof. We choose a cover U 1 , si = (sL ... ,s~) and t i = (tL A continuous isomorphism
,
(2)
S(v)
= (1l'.;(v
This is a fiberwise map a inclusion on each fiber. W
U r of B and smooth local orthonormal frames ,t~) for~ and 'rJ, over U i .
l: ~ -+ 'rJ gives continuous maps ad(ji): Ui
-+
E( It is easy to see that
CLn(R).
Let C i : U i -+ CLn(R) be a smooth E-approximation with IICi(x) - ad(j~)11 < E for x E Ui . Construct a smooth homomorphism gi: 1T~1 (U i ) -+ 1T,;;-1 (U i ) with ad(gi) = C i by the formula
is a vector bundle (cf. Exa If ~ in Theorem 15.18 is sn hi and O:i are chopsen sme r < 00. The theorem is nc Exercise 15.10 when ~ is which counts.
g~(L AkSk(b)) = L tk(b) . CL(b). All' k,1I
Then ad(gi) = C i , and Ilib - g~11 < E for b E Ui . We can then use a smooth partition of unity {ai}~=1 with supp(ai) C U i to define
Let Vectn(B) denote thl dimension n. Direct sum
r
g: ~
-+
'rJ,
gb
=
L ai(b) g~. i=1
Vect:
Then Ilib - gbll = Ilib - ~ai(b)g~11 :S ~ai(b)llfb - g~11 :S ~ai(b)E = E. With E as in Lemma 15.16, 9 becomes an isomorphism on every fiber and hence a smooth D isomorphism by Lemma 15.10.
such that
In Examples 15.5 and 15.6 we constructed two vector bundles associated to a submanifold M n C R n + k , namely the tangent bundle T and the normal bundle 1/. It is obvious that T EB 1/ is a trivial vector bundle. Indeed, by construction T p EB I/p = RnH for every p E M, and there is a globally defined frame for T EB 1/. Hence T EB 1/ ~ c: n + k , where c: n + k is the trivial bundle over M of dimension n + k. We now give the general construction of complements of vector bundles.
becomes an abelian semi the unit element. To any abelian semigrou: defined as the formal dif!
i'
1
~,
.,..i
oJ
L"'jJ.
,~' ' ' :'".' ' , \
"
~-z
,~
'l:---.-\' >,.' ,., .... ',, aili'II'B~ "~,
154
IS.
FffiER BUNDLES AND VECTOR BUNDLES
IS.
where x E V is arbitrary. The construction has the universal property that any homomorphism from V to an abelian group A factors over K(V), i.e. is induced from a homomorphism from K(V) to A. The construction V -+ K(V), often called the Grothendieck construction, corresponds to the way the integers are constructed from the natural numbers, except that we do not demand that cancellation "x + a = y + a ::::;. x = y" holds in V. When B is compact, we define
Theorem 15.21
Proof. Let F: X x I -+ fo(x) = F(x,O) and h(x: It is sufficient to see that 1 constant.
Fix t and consider the bu
By Theorem 15.18 every element of KO(B) has the form [~] - [c k ], where [~] denotes the isomorphism class of the vector bundle [~]. Indeed
(=
over X x I. Since F = choose a fiberwise isomOl
[6] - [6] = ([6] + [172]) - ([6] + [172]) = [6 E9 172] - [6 E9 172] = [~] - [c k ] if we choose 172 to be a complement to
6.
Example 15.19 The normal bundle to the unit sphere S2 ~ R3 is trivial, since the outward directed unit normal vector defines a global frame. We also know that 752 E9 1J52 = c 3, such that [75 2 ]
+ [c 1]
=
The first step is to extend t, t + t] for some t > O. T exists a finite cover VI,' .. on Vi x [t - ti, t + til· ,
[c3]
in Vect(S2). However, [752] i= [c 2] in Vect(S2). Indeed, if [752] were equal to [c 2], then there would exist a section s E r(752) with s(x) i= 0 for all x E S2. However, Theorem 7.3 implies that 752 does not have a non-zero section. We see that cancellation does not hold in Vect(S2).
Definition 15.20 Let f: X -+ B be a continuous (smooth) map and ~ a (smooth) vector bundle over B. The pre-image or pull-back f*(0 is the vector bundle over X given by E(f*(~)) = {(x,v) E X x E(~)lf(x)
= 7fe(v)},
7fr(e)
= proh·
We note the homomorphism (f,j): f*(~) -+ ~ given by j(x, v) = v. It is obvious that the pull-backs of isomorphic bundles are isomorphic, so f* induces homomorphisms
1*: Vect(B)
-+
Vect(X)
a1Ul
isomorphic.
KO(B) = K(Vect(B)).
(3)
If fo
F
and
1*: KO(B)
-+
KO(X),
and (g 0 f)* = f*og*, id* = id. Thus Vect(B) and KO(B) become contravariant functors.
I
,
I I II
II i,
-i.e
Let
0:1, ... ,0:T be
a partit
where t = min(ti) by s
1.(1 Since hi(V) = h(v) whe k(v) = h(v) on X x {t} We finally show that k i: of X x {t}. Since X is I
~
,.,...)
-'"
"lJ ';~,
.........
,.
..,.
l' ~
~
~~,-
156
15.
FIDER BUNDLES AND VECTOR BUNDLES
a neighborhood V(x, t) of any point (x, t) E X x {t}. Let e and 8 be frames of ( and 77 in a neighborhood W of (x, t), and ad(k:): W - t Mn(R) the resulting map, cf. (I). Since GLn(R) C Mn(R) is open and ad (k)(x, t) E GLn(R), there exists a neighborhood V(x, t) where ad(k) E GLn(R), and k is an isomorphism. 0
16.
The main operations to be We begin with a descriptio fiberwise to vector bundle on bundles and their equi, Let R be a unital commuta1 applications R = R or C l definitions in the general: basis the set V x W, i.e. zero except for a finite nu the submodule R(V, W) , elements of the form
The above theorem expresses that Vect(X) (and hence also KO(X)) is a homo topy functor: homotopic maps f ~ g: X - t Y induce the same map
r
= g*: Vect(Y)
-t
OPERATIONS ON
Vect(X).
Corollary 15.22 Every vector bundle over a contractible base space is trivial. Proof. With our assumption idE ~ f, where f is the constant map with value f(B) = {b}. Hence r(~) ~~. But 1*(0 is trivial by construction when f is constant. 0
(v In the above we have concentrated on real vector bundles. There is a completely analogous notion of complex (or even quaternion) vector bundles. In Definition 15.4 one simply requires V and 7l'-l(x) to be complex vector spaces and h(x, -) to be a complex isomorphism. The direct sum of complex vector bundles is a complex vector bundle. A hermitian inner product on a complex vector bundle is a map ¢ as in Definition 15.15 but such that it induces a hermitian inner product in each fiber. Proposition 15.13 and Theorem 15.18 and 15.21 have obvious analogues for complex vector bundles. The isomorphism class of complex vector bundles over B of complex dimension n is denoted Vect;(B). These sets give rise to a semigroup whose corresponding group (for compact B) is traditionally denoted
(4)
(v (r (t
(1)
where
'Vi E
V, Wi
E
W
~
Definition 16.1 The tens module R[V x WJ/ R(V,
Let 7l': R[V x W] - t V Q the image of (v, w) E is R-bilinear. Moreover,
Rr
K(B) = K(Vectc(B)). Lemma 16.2 Let V, W R-bilinear map. Then th f = 0 7l'.
It is a contravariant homotopy functor of B, often somewhat easier to calculate
1
than its real analogue KO(B).
1
Proof. Since the set V to an R-linear map j:
j(R(V, W)) = 0, so tl U. By construction f R-module V ®R W, i
= 7
It is immediate from 1
, -.)
'to
"
1 j~. ':~'" '.•
./.
~
.
'
.~
....~
158
16,
OPERATIONS ON VECTOR BUNDLES AND THEIR SECTIONS
16.
is bilinear, so induces a unique map
OPERATIOI
Lemma 16.4 For every R unique R-linear map VI commutes.
1:
(2)
7/J: V 0R W
rp 0R
---+
The uniqueness guarantees that (rp' 0R 7/J/) rp/: V' ---+ V", 7/-/: W' ---+ W".
0
V' 0R W'. (rp ®R
VI x··
7/J) = rp/o rp ®R 7/J' 0 7/J when
In the general setting of modules over a commutative ring, not every R-module
V has a basis, i.e. is of the form R[B] for a subset B C V, but if it does we
say that V is a free R-module.
It is easy to see that the k of the two-variable ones,
Lemma 16.3 Let V and V' befree R-modules with bases Band B ' . Then V®R V' is a free R-module with basis {b ®R b' I bE B, b' E B ' }.
(... ((VI ®R V2) Proof. The bilinearity of 1f: V V @ R V'; so suppose that
X
V'
(3)
Lrijbi®RbJ=O
---+
V ® RV' shows that the stated set generates
Let us specialize to VI = , IOIk \()I R V generated by the s,
{vI0R ... ®R is a (finite) relation. Let rpo: V
---+
R and
rp~:
V'
---+
R be the linear maps with
rpo(b i ) = 0 if i
=1=
io, rpo(bio ) = 1
rp~(bD = 0 if i
=1=
jo, rp~(bjo) = 1
Definition 16.5 The quoti
where (io, jo) is a pair of indices that appear in (3). The composition is called the exterior k-th
V 0R V' 'P~'Po R ®R R l~t R
The image of maps the left-hand side in (3) into riojo' which must therefore be zero.
VI
®R ... ®
0
We note the general relations
(i)
(4)
(ii)
(iii) (iv)
R ®R V ~ V ~ V ®R R VI ®R V2 ~ V2 ®R VI VI ®R (V2 ®R V3) ~ (VI ®R V2) ®R V3 (VI EEl V2) ®R V3 ~ VI 0R V3 EEl V2 ®R \1:3.
They all follow from the universal property of Lemma 16.2. For example, scalar multiplication defines an R-bilinear map from R x V into V, and hence a map from R ®R V into V whose inverse is the map that sends v E V into l®RV. The other cases are equally simple. There is an obvious generalization of Definition 16.1 to k-variable tensor products VI ® R ... ® R Vk and a corresponding generalization of Lemma 16.2, namely
is denoted
(5)
VI I\R ... I\R t
VI I\R ...
as in Lemma 2.2 and LeI
I t f }
I
(6)
p=
is an alternating map, i.f
Vi = Vj for some i < j. '
c
<
.~
(.;)
,.J
....
~.'~
"
""
..•..... ...
1· .~~.
~
160
16.
~
OPERATIONS ON VECTOR BUNDLES AND THEIR SECTIONS
Lemma 16.6 For any R-alternating map w: V x ... x V R-linear map w: A~(v') --+ W such that w = wop.
16. OPERAT:
W there is a unique 0
becomes multiplicative, ~ range is the one from Dc
In the special case where R = R (or C) and where we restrict attention to finite dimensional vector spaces the exterior powers A~(V) are dual to the alternating power Alt k(V) introduced in Chapter 2. To simplify notation we drop the subscript R and write ®k(V), Ak(V), V ® W, Hom(V, W) instead of ®~ (V), A~(V), V ®R H T , HomR(V, W). Let V* = HomR(V, R) be the dual vector space. The exterior product introduced in Definition 2.5 defines an alternating map
'if;: Ak(V
cp: V* x ... x V*
--+
--+
(7)
'IjJ(WI /\ ...
Theorem 16.7.(iii) then --+
Altk(V).
1
Addendum 16.8 (Grass: formula
(WI /\ ...
Theorem 16.7 (i) The map <;3 is an isomorphism. (ii) If {ed 7=1 is a basis for V then {ei 1 A ... A eik I il < for Ak(V). (iii) There is a natural isomorphism Ak(V*) ~ Ak(V)*.
defines an inner product
... < id is a basis For vector spaces V ane
Proof. The map <;3 is surjective by Theorem 2.15, and hence dim Ak (V*) 2:: (~). On the other hand Lemma 16.3 and (5) imply that the set in (ii) generates Ak(V), so that dimAk(V) ::; (~). Since dimAk(V) = dimAk(V*), the common dimension is (~). Thus <;3 must be an isomorphism. This proves (i) and (ii). For each fixed w E Altk(V), W(Vl,"" Vk) is an alternating map, and so defines by Lemma 16.6 a linear map from Ak(V) into R. This gives a map 'IjJ: Altk(V)
--+
Ak(V)*
which takes f 0 w into 1 that this is an isomorph
Consider the bilinear pairing
A:Ak(V) x AI(V) (VI A ... A vk, WI A ... A Wk)
--+
into
The above constructions (over the same base spac vector bundles over X· t
<;3: Ak(V*) --+ Altk(V)
~0
"7,
He
and so forth. The fiber on ~x, "7x. To be more cases, say Hom (~, "7).
E:
Ak+l(V) VI A ... A vk A WI A ... AWl.
V* <
(8)
(9)
which is linear and injective. Since the dimensions on both sides agree, 'IjJ is an isomorphism, and (iii) follows from (i). 0
which takes
7/'((El A ... /I
In particular, if V has an sending V to (v, -), and
AltkV
and thus by Lemma 16.6 a linear map <;3: Ak(V*)
with ev (w, VI /\ ... /\ Vk) Ak(V)* in Theorem 16.7
Then
with the obvious projec Hom(~x, "7x) to x E X this the total space of
~ , ?l
.,.J
.J:1.l .
j.
~
.;1 .......•~. ~ ..... ~. . ..":!:
.....,\
~" ..
r
T
162
16.
16.
OPERATIONS ON VECTOR BUNDLES AND THEIR SECTIONS
Let {Uj}jEJ be an open cover of X for which both Eluj and 7]luj are trivial. Choose isomorphisms hF ElUj ~ Uj x IR
n
kj: 7]IUj ~ Uj
,
H·
Uj x Hom(lR n , IR rn ) ---4 E(Hom(E, 7])) (which depend on hj and kj). We can use {Hj} to define a topology on E:
AcE open {:} H j- I (A) open for all j.
where the topology on Uj x Hom(lR n , IR rn ) is the product topology. It is left to the reader to show that (10) defines a topology and that it is independent of the choice of cover and isomorphisms hj, kj. This uses that Hom(V, W) is a continuous functor, i.e. that the maps T: Hom(V, VI) x Hom(WI, W)
-+
The isomorphisms of Thl basis, and so give isomc (11)
x IR rn .
They induce inclusions
(to)
OPERAT
for any vector bundles .~ We have in Theorem 16. on real vector spaces an( prove the corresponding Every complex vector bur part of the structure. 0 complex vector bundle 7] denotes the trivial c( between these operatiom
Et
Lemma 16.10
(i) For a real vectOI (ii) For a complex v
Hom(Hom(VI, WI), Hom(V, W));
T(j,g)(¢) = go ¢ 0 f
are continuous. In particular note that E ~ X x V and 7] ~ X x W give Hom(~, 7]) ~ X x Hom(V, W).
All of the constructions in (9) produce smooth vector bundles when E and 7] are
smooth.
Proof. We leave (i) to tl V we get a new one V Q The map i{J:
Lemma 16.9 For (smooth) vector bundles ~ and 7] there are isomorphisms E~ and C ® 7] ~ Hom (E, 7]).
C*
Proof. For finite-dimensional vector spaces there are natural isomorphisms V
-+
V**,
V*
0
W
-+
Hom(V, W)
V
Q$
defines an isomorphism complex linear if, in th conjugate vector space multiplication with the product. Then we may i, map v I---t ( - , v), and i{J
defined without any reference to basis. This gives maps of vector bundles (over the identity)
E-+ C*,
C 07]-+
We choose a hermitian] Hom(E, 7])
which are isomorphisms on each fiber, and we can apply Lemma l5.tO.
0
For finite-dimensional vector spaces V ~ V*, since the dimensions agree. How ever, it is not true in general that E ~ C; cf. Properties 18.11 below. The above proof breaks down because there is no isomorphism from V to V* defined in dependently of choice of basis. As a result one cannot define a homomorphism from E to C·
We next consider spaces of smooth sections (~ tangent bundle TM,
nO
(12)
Indeed, an element of 1 E Altk(TxM), an
Wx
~.
...,'
-:·~
NJ
.
164
16.
1 ;~1." .• ~
"
~
16.
OPERATIONS ON VECTOR BUNDLES AND THEIR SECTIONS
{W x } xEM is a smooth k-fonn in the sense of Definition 9.15. Similarly nO( TM)
is the space of smooth tangent vector fields on M. The section space nO(~) of a smooth vector bundle over M is always a module over the ring nO(M) = COO(M, R); if s E nO(~) and f: M -.. R is a smooth map then (fs)(x) = j(x)s(x), x E M is a new smooth section of~. We can apply Definitions 16.1 and 16.5 with R = nO(M) and V, W = nO(~), nO(1]), etc. The nO(M)-module nO(~) is not in general a free module. Indeed, if it is then a choice of nO(M)-basis elo ... , ek E nO(~) has the property that el(p)"" ek(p) is a basis for ~p for every P E M and ~ must be trivial.
Lemma 16.11 For every smooth vector bundle ~ over a compact smooth manifold M, nO(~) is a direct nO(M)-summand in afinitely generatedfree nO(M)-module. Proof. By Theorem 15.18 there is a complement 1] to ~, ~ EEl 1] ~ e k + l • Then nO(~) EEl nO(1]) ~ nO(~ EEl 1]) ~ nO(e k +l )
and nO (ek+l) is a free nO(M)-module (of dimension k
+ l).
o
Direct summands in free modules are called projective modules, so the above lemma tells us that nO(~) is always a finitely generated projective R-module, with R = nO(M).
Lemma 16.12 Let PI and P2 be finitely generated projective R-modules. Then there are isomorphisms P1
~ -
P** 1 ,
HomR(Pl , P2) ~ Pi @R P2
where Pi = HomR(P1 , R). Proof. One first proves the assertions for finitely generated free modules, where the argument is completely similar to the case of vector spaces. The general case follows easily upon choosing complements PI EEl Ql = Rnl, P2 EEl Q2 = R n2. Details are left as an exercise. 0 Theorem 16.13 There are the following isomorphisms (i) (ii) (iii) (iv)
",.
nO(Hom (~, 1])) ~ HOmnO(M) (nO(~), nO(1])) nO(~ @ 1]) ~ nO(~) @nO(M) nO(1]) nO(c) ~ HOmnO(M) (nO(~), nO(M)) nO(Ai~) ~ Aho(M)(nO(~)).
OPERATlOI
Proof. By definition, nO(H phisms ((5: ~ -.. 1], and we (
F: nO(Hor by F(((5)(s) = ((50 s, s E ~ we must show that ((5x: ~x Fix x E M and v E ~x. T such an Sv can always be trivial, and we can obtain on M with supp (f) c U section Sv by j sv, givirig Now F(((5)(sv) = 0 implil is injective. Let <1> E HOmnO(M) (nO( homomorphism ((5: ~ -.. 1] a section with sv(x) = v.
for two sections of ~ with if
(13)
Chose sections el, ... 1 ek for P in some neighborhoo sections can be extended 1
s(p' for smooth functions A(X) = 1. Then <1>( s)
Ii
dl
= <1>0
so that <1>(s)(x) = <1>(..\s)( function 9i defined on all 1
But 9i(X) = 0 so <1>(..\s)( Assertion (iii) is the spec bundle, and (ii) follows f nO(~
@
1]) ~ n l ~H
~nl
c
0:5
166
16.
"
"6
OPERATIONS ON VECTOR BUNDLES AND THEIR SECTIONS
where the last isomorphism is from Lemma 16.11 and Lemma 16.12. Finally there is a commutative diagram OO(®i 0
-
CONNECTIONS Al
®nO(M) 0°(0
Let ~ be a smooth vector 1
1
Definition 17.1 A connec
1
OO(Ai~) -
17.
AhO(M)(OO(€))
v:
By (ii), the upper horizontal map is an isomorphism. To see that the bottom homomorphism is also an isomorphism, one can use Theorem 16.7.(ii), and local sections as in the proof of (i). The details are left as an exercise. 0
which satisfies "Leibnitz' s E OO(~) and d is the ext 00 (€) is a complex vectol
We close with a weaker form of the universal property of tensor products, stated in Lemma 16.2. Let R be a unital commutative ring and 5 an R-algebra. In our applications in the next chapter R = Rand 5 = OO(M), the smooth functions on M, or their complex versions R = C and 5 = OO(M; C). Suppose that V and Ware R-modules and that 5 operates from the right on V and from the left on W, e.g.
Let T be the tangent bun 16.13 we have the follow
V
= Oi(M),
W
= 0°(0.
Definition 16.14 The balanced product (or tensor product) V 0 s W is the cokemel of the R-linear homomorphism V ®R 5 ®R W
f3
-+
V ®R W
given by (J(v ®R S ®R w) = vs ®R W - v ®R SW. When 5 is commutative, and this will be the case in our applications, then there are no distinctions between left and right actions of 5, and V ®s W becomes an 5-module upon defining s( v ®s w) = vs ®s w. This 5-module is obviously isomorphic to the one defined in Definition 16.1. We record for later use the obvious
Lemma 16.15 Let f: V ® R W -+ U be an R-linear map which is 5 -balanced in the sense that f(vs ®R w) = f(v ®R sw) for f E V, W E Wand s E 5. Then 0 there is an induced R-linear map 1: V ®s W -+ U.
(1)
01(M) ®nO(M) nOI
A tangent vector field X ( induces an OO(M)-linear linear map
Evx The composition Ev x (2),
0
v 'i
where dx(f) is the directi dx(f). Thus a connecti( For fixed S E 0° (€) the· V~
for smooth functions 9,' the value Vx(s)(p) E €p the second term in (1) wi
V:O Here the range is the set identity). If X p E TpM (3)
VX" VaX,,-I
where X p, Yp E TpM, : that V XI' (s) defines a 4
l~.~ .~.
··13 :
iI~~1j j;j =
_$:
.""" ...•>.O'~.
\!~
;~~'" ~~>
.
~
b't-,~ "~ '-~,~ ~i
1~
.~
=e
n,
'"
.q...(~
f .•.,_ '....•.. ~.
,~~ •....
168
17.
~~_.
>
.
•
17.
CONNECTIONS AND CURVATURE
Example 17.2 Let M n c Rn + k be a smooth manifold. One can define a connection on its tangent bundle as follows: a section S E nO(T) can be considered as a smooth function s: M --+ Rn + k with s(p) E TpM, and we set
defines a connection V
nl(M)
VXp(S) = jp(dxp(s)) E TpM
Example 17.3 shows that ~ ifold has at least one com 15.18. We observe that E sponding to ~ = TM and j
It is a consequence of the "Leibnitz rule" that V is a local operator in the sense that if S E nO(~) is a section that vanishes on an open subset U ~ M then so does V(s). A local operator between section spaces always induces an operator between the section spaces of the vector bundles restricted to open subsets. In particular a connection on ~ induces a connection on ~Iu,
Remark 17.4 After choic ~p at different points pEl a smooth curve in M and I for some w E nO(~). TheI that satisfies:
Let el"", ek E nO(O be sections such that el(p)"", ek(p) is a basis for ~p for every p E U (a frame over U). Elements of nl(U) @nO(u) nO(~lu) can be written uniquely as L: Ti @ ei for some Ti E 1 (U), so for a connection V on ~,
n
£
@
ej
..
j=l
(n)
...
where Aij E n l (U) is a k x k matrix of I-forms, which is called the connection form with respect to e, and is denoted by A.
(lll)
Suppose first that o:(t) C I and let e = (el,' .. ,ek)be for smooth functions Ui ( Conditions (i), (ii) and (i
Conversely, given an arbitrary matrix A of I-forms on U, and a frame for ~Iu, then (4) defines a connection on nO(~lu), Since S E nO(~\U) can be written as s(p) = L: si(p)ei(p), with Si E nO(U),
V(Lsiei)
= Ldsi@ei+Lsivei=L(dsj+SiAij)@ej.
Dw dt
With respect to e = (el,"" ek), V has the matrix form (5)
+ W2) = D dt Q D(f· w) df dt = dt w Dw (it = Va/(t)w.
(i) D(WI
k
v(ei) = L Aij
@no(.
and that Vo = d EfJ . , . EEl ( the trivial complex bundle,
where jp: Rn+k --+ TpM is the orthogonal projection and X p E TpM. It is easy to see that (3) is satisfied.
(4)
01
and since a'(t) = l:~. C
V(Sl, ... , Sk) = (dSl,"" ds k ) + (Sl,'" ,sk)A. Va'(t)f
Example 17.3 Suppose ~ EfJ 1] ~ c n + k , and let i: ~ --+ c n + k and j: c n + k --+ ~ be the inclusion and the projection on the first factor, respectively. We give the trivial bundle the connection Vo from (5) with A = O. There are maps
nO(~) ~ nO(c n + k) and nO(cn+k)
6
This gives
Du. dt
nO(o,
and the composition
nO(~) ~ nO(c n+ k) ~ n1(M) @nO(M) nO(c n + k) id~. nl(M) @nO(M) nO(~)
I L
Conversely this formula Since we can cover 0:( contained in just one et
o
~
(
..-J
L-,..>
~-
q""".
170
17.
~.""''~
;~
•....••...
'b
"
..•..
.~
CONNECTIONS AND CIJRVATlJRE
1~
A section w(t) in ~ along et(t) is said to be parallel, if ~~ = O. For a given w(O) E ~a(O) and smooth curve there exists a unique such section, and the assignment w(O) -+ w(l) is an isomorphism from ~Q(O) to ~a(1)'
Proof. Let l' E nj (M) ani One checks that dY' is fl applies Lemma 16.1.5. SiI when j = O. We show thai
Let us introduce the notation
d\1(w 1\ (1' ® s)) = d\1(( = (dw = dw/
. =. °
nt(O n~(M) 0nO(M) n (~).
(6)
Then a connection is an R-linear operator \7: nO(~) -+ n 1 (O which satisfies the Leibnitz rule. We want to extend \7 to an operator
dY': ni(o
-+
ni+l(o
We have now a sequence by requiring that dY' satisfy a suitable Leibnitz rule, similar in spirit to Theorem 3.7.(iii). Let ~ and fl be two vector bundles over M. There is an nO(M)-bilinear product
(7)
1\:
ni(r]) 0 nj(o -+ ni+j(fl ®
0
0-1
(8)
which when ~ is the trivial complex of Chapter 9. anI and dY' 0 dY' = 0, but this
defined by setting
1
(w ® t) 1\ (-r ® s) = w 1\ l' ® (8 ® t) is nO (M)-linear, since where wE ni(M), l' E nj(M), 8 E nO(O and t E nO(fl) and WI\1' is the exterior product; cf. Theorem 16. 13.(ii).
dY'
1
We shall first use the product when fl = c , the trivial line bundle. In this case ni(fl) = ni(M), and for i = 0 the product in (7) is just the nO(M)-module structure on n j (0. Note also for w E ni(M) and s E nO(O that w 1\ s = w ® 8 in ni(O. Given three bundles fl, () and ~ one checks from associativity of the exterior product that the product in (7) is associative, and that the constant function 1 E nO(M) acts as a unit. In particular we record (for TJ = c 1 ):
\l (J s) = ( =(
=
On the other hand Thoor
(9)
HOillnol
Indeed, there is the folIo' HOillnO(M) (nO(~),n
Lemma 17.5 The product of (7) satisfies: (i) (w 1\ 1') 1\ P = w 1\ (ii) 1 1\ P = P
0
(1' 1\ p)
where wE ni(M), l' E nj(M) and p E nk(~). Lemma 17.6 There is a unique R-linear operator d\1: nj(~) satisfies
0 -+ nj+l(~)
(i) dY' = \7 when j = 0 (ii) d\1 (w 1\ t) = dw 1\ t + (-l)i w 1\ d\1t, where w E ni(M) and t E
that
nj (0.
Definition 17.7 The 2-f(J (~, \l). A connection \7 Let X, Y E nO(1'M) be I we get an nO(M)-linear
172
17.
CONNECTIONS AND CURVATURE
Example 17.9 Let H b Example 15.2. Its total ~ U E L. Indeed, the mal
which induces a map Evx'y:n2(Hom(~,~)) _ nO(Hom(~,~)).
We write F~,y = Evx,y(F\/'). As for connections, F~,y(p):~p only on the values X p, Yp E TpM of X, Y in p. We can calculate P\7 locally by using (4),
d'V
0
~p depends
XSI
C
is a fiberwise monomorpl complement to H is the
L dAij ej - L Aij \7(ej) = L dAij (3) ej - L Aij L Ajv 0 ev
\7(ei) =
i:S 3
1\
@
1\
=
Lv (dAiV
so that F'V(ei) = Lv (dA - A 1\
F'V = dA - A
(10)
We want to explicate the f to the pair (L, u), where L. If L = [Zl, Z2] with I
ev - (L j Aij 1\ Ajv ) 0 ev) A)iv @ ev . In matrix notation: @
1\
A
where A is the connection matrix for \7. In other words, the matrix of the linear ~p - ~p in the basis el(p), ... , ek(p) is (dA - A 1\ A)xpt Y,' map F~p, y,: p p
where PL is the 2 x 2
IJ
We next consider the nO(M)-bilinear product 1\: ni(~)
which maps a pair (w
(9
x HomnO(M) (nO{o, n 2(0)
_
ni+2(~)
Indeed, if L contains the C 2 onto L is given by t
s, G), with
.
w @ s E nZ(M) @nO(M) n
0
G E Homno(M) (nO(.;), n2(~))
(~),
1fL(Ul, U2)
We examine the (comple
into
(w
(11)
@
s)
1\
G
= w 1\ G(s)
\7: OO(H)
with the right-hand side given by (7). Alternatively we can use (9) to rewrite
(11) as the composition
ni(~) 0 n 2(Hom (~, ~)) ~ ni+2(~ @ Hom (~, ~)) _ ni+2(~)
by calculating the conn< stereographic charts Ul aJ defined as
g: I
where the last map is induced from the evaluation bundle homomorphism ~(3)Hom(~,~) - f
with
Lemma 17.8 The composition d'V Proof. Let
d'V
w ® s E 0
0
d'V: ni(O _ n i+ 2(O maps t to t
.
1\
Z
= x + iy, and Ie e(g(;
F'\7.
where we also use z to d
0
nZ(M) @nO(M) n (0. By Lemma 17.6,
+ (-1) i w 1\ '\7 s ) = do d(w) ® s + w 1\ d'V 0 v(s) = w 1\ F'V(s).
i:.
Now
d'V (w ® s) = d'V (dw ® s
\7o{e
o
and hence
\7(e) = (g(x,y),(O,. We see that the sequence (8) is a chain complex precisely when \7 is a flat connection (F\l = 0). However, as will be clear later, not every vector bundle admits a flat connection.
=
(g(X,y),
1"
.r
.?
~
..
~
174
17.
~
CONNECTIONS AND CURVATURE
We have shown that V( e) = A Ag(x,y)
.~
'"
~
=,
0)
e where A is given as
Z I.I?
dz,
z(g(x, y)) = x + iy
dz,
z(x,y) = x
Lemma 17.10 There ex! diagram below commute~
or equivalently
g*(A) =, ,
z I. I?
. + ~y.
We use formula (l0) to calculate the curvature fonn. First note that
dz /\ dz = (dx + idy) /\ (dx + idy) = a dZ /\ dz = (dx - idy) /\ (dx + idy) = 2idx /\ dy so that
* (1 dg (A) = Since A /\ A =
+ Iz12) - z· z 2i 2 2 dZ /\ dz = 2 ? dx /\ dy. (1 + Izi ) (1 + Izi )
a we have the following fonnula in D2 (Hom(g* H, g* H)): 9 *( F 'V) =,
(12)
2i
'l.?
dx /\ dy.
Any complex line bundle H has trivial complex endomorphism bundle Hom(H, H), because it is a complex line bundle and has a section e(p) = id Hp ' which is a basis in every fiber. In particular the curvature fonn F'V E D2 (Hom(H, H)) is just a 2-form with complex values. It is left for the reader to calculate h*(F\7) where h: R2 -+ U2 is the parametriza tion
h(x, y) = [z, 1],
z= x
+ iy.
Proof. The map 1: M' DO(M'), so that every nOI f),0(j*(~)) becomes an DO modules withj*(s)(x') = s(j(x')) modules DO( by sending ¢' 6) s into ¢/ It follows that Dk(j*(~)) = nk(M
Similarly, pull-back of di
is DO(M)-linear and indu
nO (M')
0120 (1
This is not an isomorphisl a homomorphism p: f),°(M')
This ends the example.
(
The sum of the maps We conclude this chapter by showing that the constructions 1* (0, C, Hom (~ ,71) and ~ is) 71 can be extended to constructions on vector bundles equipped with connections. We begin with the pull-back construction. Let f: M' -+ M be a smooth map and ~ a vector bundle over M with connection V. The map j*: DO(~) -+ D0(j*(~)); j*(s)(p) = s(j(p)) can be tensored with 1*: D1(M) -+ D1(M'), to obtain a linear map j*:D1(M) ®nO(M) DO(~)
-+
D1(M') ®nO(M') D0(j*(~)).
d 01: nO(~ p(l ® V):
[2°0
defines the required conn
f
....
" . .~ ~···-".'t
41
--.-.-,
~.,
.~
~
'"""-\
$~~
...~",----==-------.--~--~
176
17.
CONNECTIONS AND CURVATURE
We note that if A(e) is the connection matrix for \l W.r.t. a frame e for ~Iu then f*(A(e)) is the connection matrix for f*('V) W.r.t. the frame eo! for J*(Ot-1(U)' There is a commutative diagram corresponding to that of Lemma 17.10 where 'V is replaced by dV': n1(~) - t n 2(0, and thus also a diagram n2(~) nO(O
--.r:...
11*
Lemma 17.11 Under
I
\lHom(~.ry)·
Proof. There is a comrn
11*
n°(J*(~)) ~ n2(J*(~)) Since f*Hom(~,~)
= Hom(J*(O, f*(~)) the
above gives and a corresponding s* E nO(e). Then
J*(FV') = Fr(V').
(13)
Consider the non-singular pairing ( , ): ni(~) ® nj(c) ~ ni+j(~ ® C)
ni+j(M)
1 where the last map is induced from the bundle map ~ ® - t 2 . For i -t
e
= j = 0,
nO(~*) 9:! HOmnO(M) (nO(~), nO(M))
-t
nO(M).
For general i and j,
(14)
T
('V~(s), o:(s* 0 {
(s, 'V1;'0rylm( s* ( On the other hand, using
(w ® s, T ® s*) = (w where w E ni(M), Given a connection
'VT/((s, o:(s* ® t) d((s, s*)) = ('V~ 'Ve0T/(S* 0 t) =
From the diagram we gel
by Theorem 16.13.(iii), and the above pairing corresponds to the evaluation ( , ): nO(~) ® HOmnO(M) (no(o, nO(M))
di~
A
T) ® (s, s*)
s E nO(~), s* E nO(O. we define the connection 'V~' on
E nj(M) and
'V~
on
~,
(s, 'VHom(~.T/)(O:(s* 0 (
e by requiring
d(s, s*) = ('V~(s), s*) + (s, 'V~'(s*)).
This specifies 'V~' uniquely because the pairing ( , ) is non-singular. The desired connection on the tensor product is defined analogously. Indeed the product from (7) induces a nO(M)-linear map j A: ni(o ®nO(M) n (1]) - t ni+j (~® 1]),
and the assertion follows. Each of the connections f
which for i = j = 0 is the isomorphism nO(o ®nO(M) nO(1]) ~ nO(~ ® 1])
!il
from Theorem 16.13.(ii). Define (15)
'V~0ry(S ®
t) = 'V~(s) At + s A 'Vry(t).
Finally we can combine (14) and (15) to define 'V~'0ry(S ® t) = 'Ve(s) At
+ s A 'V~(t).
Since C ® 1] 9:! Hom(~, 1]), this defines a connection on Hom(~, 1]). Alterna tively one can apply the evaluation nO(Hom(~, 1])) x nO(~) - t nO(1]) and the induced nO (M)-bilinear product (, ):ni(~) x nj(Hom(~,1])) - t n i + j (1]) and de fine 'VHom(Cry) by the formula (16)
'Vry((s, <1») = ('V~(s), <1» + (s, 'VHom(~.ry)(
and the defining formulas to the reader. Lemma 17.12 Let sEn We have (i) d((s, s*)) = (dV'( (ii) dV'(srt;t) = dV's (iii) dV' ((s, <1») = (dV'
. ~~
i .
'
J-••
~
...
..
:1' --.....--.;-----_.. ~.
'.'
.~
-'~
'.. . '
'
. . . ". '.' . ' .. '. .'. . "-.--- . . ' i i..
' '.
178
17.
_L.. '
k.. E.~"'-~i§,'-"<
c,c"]l'!LIi&la...4..•~
CONNECTIONS AND CURVATURE
where d'\l corresponds to V
= ve, Vrp ve*
and vHom(e,e*), respectively.
D
The definitions above may appear somewhat abstract, so let us state in local coordinates the case of most importance for our later use. Let \7 = \7e be a connection on ~, and let e = (e1,.'" ek) be a frame over U. This defines isomorphisms
~Iu ~ U x Rk ,
Hom(~, ~)Iu ~ U
The product in 0° (HOI product
/\ : Oi(Hom~
It is not hard to show th:
x Mk(R)
d'\l(R1/\J
(19)
and induces
on(~lu) ~ on(U)E&\ The connections \7 = \7e and become
V=
d'V: On(u)E&k ~ On+l(U)E&\
The trace homomorphis: On(Hom(~,Olu) ~ \7Hom(e,e)
Mk(on(u)).
and the induced d'V and dV then
d'V: Mk(on(u)) ~ M k (On+1(U)). where ev(J
They are given as (17)
d'V(Sl,"" Sk) = (dS1,"" dSk)
@
v)
= j(v:
+ (S1,"" Sk) /\ A
of vector bundles, and 1
d'V(~) = d~ - (A /\ ~ - (-It /\ A)
where A = A(e) is the connection matrix. The first formula follows from (5); the second is proved quite similarly.
If e ' is another frame for ~Iu then e ' = G· e with G E GLk(OO(U)), and
the connection matrices A = A(e), A' = A(e) and the curvature forms P'V(e),
F'V (e ' ) are related by
(18)
can be defined without r
A' = (dG)G- 1 + GAG- 1
into the i-forms on M,
Theorem 17.14 For cP
where d'V is associated
P'V(e ' ) = GP'V(e)G- 1
cf. Exercise 17.8. The first formula follows from (5) applied to the equation (Sl,.'" Sk) = (s~, ... , sUG. The second formula follows from the first one and (10).
Proof. Let
S
E nO(o,
i
cP=w@s0s*EO i Then
Theorem 17.13 (Bianchi's identity) We have d'\l p'\l to the connection \7 = \7Hom(e,e)'
= 0, where d'V is associated
d'V cP = ®.; @ (s = ®';@(8
Proof. Use the local forms (10) and (17) to get
and we get Tr d'\l cP = (dw)
p'\l = dA - A/\ A
d'\l F'\l = -d(A /\ A) + p'\l /\ A - A /\ p'\l
= -d(A /\ A) + dA /\ A -
A /\ dA = 0.
= (s,s'
o
= (s,s'
v: ,.;
.~~
"
~
1·._~ ...
.......•#.. •.•. .•. '. .
..' .•..
~
180
17.
CONNECTIONS AND CURVATURE
18. We have mostly formulated the above theorems for real smooth vector bundles, but there are of course completely analogous results for complex vector bundles upon starting with a complex connection '1. One simply replaces Di(M) by 0,i(JV]; q, and 0)f\l by (Sle and HOfllR by Home throughout, and requires maps to be (:>linear rather than IR-linear. This complex version will be used below in Chapter 18. Let ~ be a complex vector bundle with complex connection '1. Combining Theorems 17.13 and 17.14 we see that the 2-form Tr(F'V) E D 2 (i1.f; q is closed, and thus defines a cohomology class in complex de Rham cohomology: [Tr(F'V)] E H2 (JV1; q More generally, it follows from (19) that the trace of
CHARACTERISl
In this chapter ~ will be smooth compact manifol with complex coefficient!
Let P(A) = P( ... , Ai; variables displayed as al examples are (I)
P(A) =c
where ilk(A) is the caefi
p'V 1\ ... 1\ F'V E D2k(Homd~, e))
is a closed form in [22k(lvI).
Definition 17.15 The k-th Chern character class of (e, V) is the cohomology class chd('1)=
(-l)A'k
(21f.;=I) k!
[Tr(P'V
I\ ... I\P'V)] E H2k (AI;C). .
They both have degree dimensional forms, we ( forms, Ai] E n- (At; q, generally we define a m ')
p
(2)
Let e be a frame of e/U; Here H* (M; q = H* (lvI) C:9R C is the cohomology of the complexified de Rham complex. The normalizing factor in Definition 17.15 is chosen so that the cohomology <;lass is actually real, i.e. a class in H 2k (1\1). This will be proved in the following chapters, where we also show that the cohomology class is independent of the choice of connection. A real vector bundle ~ can be complexified ~e = ~
V on
e induces a complex connection '1e on ~c·
@R
cb and a real connection
Definition 17.16 The k-th Pontryagin character class phk(~' V) = Ch2k(~e, '1c) E H 4k (A1; q.
phd~,
V) is the class
and hence an isomorphi
n2 (Homde Thus a 2-form R of Hall P and get an element P any other choice of fram we have
It follows that we have For example we have C I
(3) st{R) =
L Rii'
Choose a complex cont get a 2k-form
(4) Here are two fundamen
c
~
.
~
_t
.r.J
.~c
jl
...
'l!t
L<-......J ----~,-~
182
18.
CHARACTERISTIC CLASSES OF COMPLEX VECTOR BUNDLES
18.
Lemma 18.1 For each invariant pol.ynomial and connection \7, P( F\7) is a closed form. Lemma 18.2 The cohomology class [P(F\7)] in H*(Al; choice of connection.
q
CHAR,
Proof of Lemma 18.2. the projection onto the on Jr* (~); cf. Lemma
is independent of the
v(s) where (p,t) EM x R
The first lemma follows from Theorems 17.13 and 17.14 and results of Appendix B, but there is also the following attractive alternative proof ([Milnor-Stasheff]).
Proof of Lemma 18.1. connection matrix A =
Choose a frame for (A ij ), so that
F\l
= dA -
A 1\ A
~
over U, and let \7 have the
where i v : 1'vl ---+ 1\1 x I From (17.21) it follow
= (Fij ). ~
In local terms Bianchi's identity is dF\l
= A 1\ F\l - F\l
1\
and hence i~(P(F\l)) =
A, cf. (17.17), so
we have that io([P(F\l
(5)
dP(F\l)
= '"' L
aDP (F\l) k. 1.)
1\
dFij
= Tr(P'(F\l)
1\
dF\l)
Note that isomorphic vel since a smooth fiberwis section spaces, and sinc
where P'(A.) is the transpose of the matrix of partial derivatives
p' (A) = (
aP ) aAij
t
For an invariant polynomial P one has
P'(A)A = AP'(A).
(6)
This is seen by applying the operator
P((I
+ tEij)A)
commute. Thus the IT corresponding frames fo
it to the equation = r(A(I
In particular, if ~ is a tri we just use the flat can
+ tEij))
where Eij is the basic matrix with 1 in the (i, j)-th entry and zero elsewhere. Now (6) yields the relation
Definition 18.3
(i) The k-th Chern ,
r'(F\l)
(7)
1\
F\l = F\l
1\
P'(F\l),
Ck((
and using (5) and the Bianchi identity we get
-dP(F\l)
= Tr(P'(F\l) 1\ F\l 1\ A - P'(F\l) 1\ A 1\ F\l) = Tr(F\l 1\ (P'(F\l) 1\ A) - (P'(F\l) 1\ A) 1\ F\l) =
(ii) The k-th Chern O.
D
chk(O Here v is any c cho(O = dim~.
:
:
".,
...
~..,-
~"4'C
; .,~~;::~""; ~
'"
"..J
C"'fJ
~ ..~.,~ ..... ~
.,,~.fi:
~
~ ...... ..c,~..
r~tt~~)iJ···
,.@::",. :""""~
184
18.
CHARACTERISTIC CLASSES OF COMPLEX VECTOR BUNDLES
18.
The reader may check that Definition 18.3.(ii) agrees with Definition 17.15. We shall prove some properties of these classes. First note that they determine each other. since Chk(O
1
= k/'dO,
8k(~)
= Q!c(C1(O,···, Ck(~)),
Ck(~)
We integrate this form b: Since d:r = cos edT we see that d:r
= Pk(SI (0,···, Sk(O)
12
for certain polynomials Pk and Qk; cf. Appendix B. For example we have ch 1 (0 = C1(~)
and
CHARAC
g*
1\
tty ==
(F v ) =
Ch2(0 = ~CI(02 - C2(0
=
The integration homomorphism
I: H 2 (CIP 1 ; C)
-+
This calculation implies
C
is an isomorphism by Corollary 10.14, and the inclusion j: 0])1 an isomorphism
j*: H 2 (ClP n ; C)
-+
H 2 (CIP\
C Opn
induces
Indeed we can apply a I:: large cl-sphere in the ( the other chart. In the !ir to the integral of F\J ov
c)
by Theorem 14.3. We now chose c E H 2 (CIP'\ C) once and for all with the property that
(8)
Theorem 18.5 Let f: N 011 AI. For every invaria
I(j*(c)) = -1.
It follows from Example 14.10 that -7rC is the cohomology class of the volume form of CIP 1 with the Fubini-Study metric (cf. Theorem 14.8) and if we identify 1 52 with CIP via W then -47rC corresponds to the volume form of 8 2 in its natural metric as the unit sphere and with its complex orientation.
Proof. We give 1*(0 tht
r(pV)
= F!*('V). Hene
For a line bundle L,
[22
1
Let N rl be the canonical line bundle on Cpn with total space E(H n ) = {(L, u) E Cpn
X
C n+ l
111. E
L}.
This gives
Then j*(Hn ) = HI is the canonical line bundle of Example 15.2. Theorem 18.4 The integration homomorphism maps
Cl
(H r) to -1.
Proof. Apply the two positively oriented stereographic charts '1/)- and 1J+ on 52 = CP1 from Example 14.1. In Example 17.9 we calculated the pre-image of the curvature form p\J under 9 = (7/L) -1 to be 9 * ( P 'V) =
2i --~dx 1\
(1 + Izi )
dy.
(9)
I
so that chk(L) becomes
f
Theorem 18.6 For a s
1
I
L
(i) Chk(~O t:B 6) =
(ii)
Ck (~o (fJ
Er)
o
~
.
18.
;,;
"
~
._;~
186
...< ~ .....~ .;
..-J
-~
~J
18.
CHARACTERISTIC CLASSES OF COMPLEX VECTOR BUNDLES
Proof. Choose complex connections \7v on ~v. We identify ~zi(~o EB 6) with ni(~o) EB n i (6); then \70 EB \71:no(~O EB6) ~o
is a connection on
--+
CHARA
There is a commutativ ni(Hom(~o,~o);
nl(~o EB6)
(11)
EB 6 with curvature
[li(M; C)
F'V O EB F'b E n2(Hom(~o EB 6, ~o EB 6)).
From (10) and (11)
WI
For direct sum of matrices
Ao EB Al
0)
Al
Ao
=( a
Sk E Mn+m(C) which is equivalent to t
formula (3) of Appendix B gives the equations k
sk(A o EB AI) = sk(Ao)+Sk(Al)
and
O"k(A o EB AI)
L
=
Let H 2* (M; C) denote
O"v(A O )'O"k-v(A 1),
lJ=O
o
which prove the assertions.
For a complex smooth
Theorem 18.7 For a tensor product of complex vector bundles, k
ch k (~o
@
6) =
L
ch lJ (~O)Chk-v (6)
This defines a homomo 18.6.(ii) and the univ( extended to a homomc
v=o where cho(~v) = dimc~v.
Proof. The tensor product of linear maps, applied fiberwise, defines a map of ' vector bundles
An application of Theo product in K (M) is dl
Hom(~o,~o) @ Hom(6, 6) --+ Hom(~o @ 6, ~o (6)
([~o]
and thus a product 1\:
j ni(Hom(~o, ~o)) @ n (Hom(6, 6))
For connections \70, \71 on (17.15):
\7(So
@
~o,
sd
--+
Without proof we state
ni+j(Hom(~o @ 6, ~o (6))·
6, we have the connection \7 on
~o @
Theorem 18,8 The
6 from
Theorem 18.9 There
H2k(M; C), depending
(i) I(Cl(H1)) = (ii) J*Ck(~) = Ck( (iii) Ck(~O EB 6) =
k
F'V
1\ ... 1\
F'V
ell
= \7o(so) 1\ SI + So 1\ \71(SI).
The corresponding curvature form becomes F'V = F'Vo 1\ id + id 1\ F'VI where id E nO(Hom(~v,~v)) is the section that maps p E M to id:~p --+ ~p" It follows that (10)
- [170])([6]
=
~ (~) (F'VO)Ai 1\ (F'V
1
)A(k-i).
l
1..
q.,...
/'1.'• .·•'
..
'CO
. .1Iri;.
~ i,,·
_:~,
188
18.
18.
CHARACTERISTIC CLASSFS OF COMPLEX VECTOR BUNDl.ES
Properties 18.11
The uniqueness part of Theorem 18.9 rests on the so-called splitting principle, whose proof is deferred to Chapter 20.
(a) Ck(O = 0 if ~ (b) q(C) = (-1)
Theorem 18.10 (Splitting principle) For any complex vector bundle ~ on M there exists a manifold T = T(~) and a proper smooth map f: T ~ M such that
(c) C2k+l(1]c) = 0
(i) f*: Hk(M) ~ Hk(T) is injective (ii) f*(~) ~ 1'1 EB ... EB 1'n
Proof. For a line bundl because every line bun of line bundles, then
for certain complex line bundles 1'1, ... , 1'n.
Proof of Theorem 18.9. The Chern classes of Definition 18.3 satisfy the three conditions, so it remains to consider the uniqueness part. From (i) it follows that q (HI) = C in the notation of (8). Let L be an arbitrary line bundle and L.l a complement to L, with
and it follows that Ck(~) 18.10. The proof of (b trivial and Theorem 18. For a sum of line bUll
LEB L.l = M x C n + 1 . We can define 1r:
M
~
Cpn;
X
l-+
(13)
proj2(L x )
where proj2: M x C n + 1 ~ C n + 1 . There is an obvious diagram L-L Hn
1
CHARA
C(C:
This shows that Ck(C: general. For a real vel 1] and use the isomof]
1
M~Cpn
with it p an isomorphim for every p E M. Hence 1r*(Hn ) follows that
q(L)
~
L. From (ii) it
Then (1]c)* = (1]*)c S4 Note that (c) implies I be non-zero in the dil
= 1f*(c).
Since ck(Hn ) = 0 when k > 1, the same holds for any line bundle. Therefore (i) and (ii) determine the Chern classes of an arbitrary line bundle. Inductive application of (iii) shows that for a sum of line bundles, Ck (Ll EB ... EB Ln ) is determined by cl(LI), ... ,q(Ln ). Finally we can apply Theorem 18.10 to see 0 that q(O is uniquely determined for every complex vector bundle.
One defines Pontryagi bundles by the equati (14)
We leave to the reader is exponential.
The graded class, called the total Chern class, (12)
C(O = 1 + Cl(O
+ C2(C) + ...
E H*(M; C)
Remark 18.12 Defil actually all classes lie for H n , and for a SUl consequence of Theo
is exponential by Theorem 18.9.(iii), and c(L) = 1 + Cl (L) for a line bundle. Hence k
c(L l EB ... EB L k )
=
II (1 + cl(L
v ))
=L
Pk(TJ)
O"i(q(Ld,···, q(Lk))
11=1
and it follows that ci(Ll EB ... EB Lk) = O"i(Cl (Ld, .. . , q(L k )). We have addi tional calculational rules for Chern classes: L
....
'~}
~
~ l'
-
~
••
'~
"
""
•••.••.
'2
~",
-'~1~
190
18.
18.
CHARACTERISTIC CLASSES OF COMPLEX VECTOR BUNDLES
Example 18.13 Given a line L C C n +!, consider the map
gL: Hom(L, L1-)
---*
CHARAC
for any closed fonn w E form on R which contm the equation TiJRtBE~ = i
cpn
which maps an element cP E Hom( L, L1-) into the graph of cPo Its image is the open set UL ~ Cpn of lines not orthogonal to L. The functions h j l of (14.2) are equal to n j where Lj is the line that contains the basis vector ej = (0, ... ,1,0, ... ,0). Each (UL, iiI) is a holomorphic coordinate chart on Cpn.
and from the above togt (TCP'"
)F
Let H 1- be the n-plane bundle over Cpn with total space The total Chern class of
E(H1-) = {(L,u) E Cpn x Cn +! I u E L1-}.
C
Then H tBH 1- is the trivial (n+ I)-dimensional vector bundle where H = H n , and (15)
Hom(H, H1-) ~
so that
Tcpn.
Indeed, the fiber of Hom(H, H1-) at L E Cpn is the vector space Hom(L, L1-), and the differential
(DgL)O: Hom(L, L1-)
---*
tB d: ~ Hom(H, H1-) tB Hom(H, H) ~ Hom(H, H EB H1-) = (n
+ I)H*.
Hence the total Chern class can be calculated from Theorem 18.9 and Properties 18.11, 1 1 C(Tcpn) = C(Tcpn tBc:t) = c(H*t+ = (1- cI(H)t+ , and the binomial formula gives (16)
Ck(Tcpn)
= (-1) k
(n + 1) k
k q(H).
The class q (H) E H 2 (Cpn) is a generator, and Theorem 14.3 shows that Ck ( Tcpn ) is non-zero for all k ~ n. Example 18.14 One of the main applications of characteristic classes is to the question of whether a given closed manifold is (diffeomorphic to) the boundary of a compact manifold. We refer the reader to [Milnor-Stasheff] for the general theory and just present an example. We show that cp2n is not the boundary of any 4n + I-dimensional manifold R 4n+!. Indeed, suppose this was the case. By Stokes's theorem, (17)
r
JaR
W=
r dw=O
JR
C2k(T
Now take w in (17) to t
TLcpn
defines the required fiberwise isomorphism. One can use (15) to evaluate the Chern classes of the complex n-plane bundle Tcpn. Indeed, Hom(H, H) ~ c:b so Tcpn
= (1 - cl(H21
"
""
c"",#JliJr ••-
194
19.
THE EULER CLASS
In another orthonormal frame e' over U
Proof. We can pull 1 to Let {po, PI} be !:! x (-00,3/4) and A ~ which agrees with 7f* In particular i~(9) = Let V be any metric I 7f*(Vo), and 7f*(VI and M x (3/4, (0) res cover, to glue togethe M x R. This is metric
f.
P'V(e')p = BpF'V(e)B;1
(2)
where B p is the orthogonal transisition matrix between e(p) and e' (p ). Now suppose further that the vector bundle ~ is oriented, and that e(p) and e' (p) are oriented orthonormal bases for ~p, p E U. Then B p E S02b and by Theorem B.5,
Pf(F'V (e))
(3)
V
= Pf (F'V (e') ).
It follows that Pf(P'V) becomes a well-defined global 2k-forrn on M. The proof of Lemma 18.1 shows that Pf(F'V) is a closed 2k-form.
Corollary 19.3 The
We must verify that its cohomology class is independent of the choice of metric on ~ and of the metric connection. First note that connections can be glued together by a partition of unity: if (V O,},;tEA is a family of connections on ~ and (Pa) aEA is a smooth partition of unity on M, then V S = EPaVaS defines a connection on ~. Furthermore, if each Va is a metric for g = ( , ) then V = E Pa Va is also metric. Indeed, if
(4)
d(SI, 82) = (Va Sl, 82)
0
the metric and the con
Proof. Let (gO, VO) :
the metric and cannee
hence i~Pf (F'V) = F it:Hn(M x R) ~ H Pf (p'V 1 ) agree.
+ (SI, VaS2)
Definition 19.4 The c
then ("lSI, 82)
+ (SI' V 82)
=L
(Pa V aSl, S2)
+
L (81, PaV
L PaC (V S2) + (81, V = L Pad(Sl, 82) = d(SI' S2)' a S l,
=
a
S2)
O'S2))
is called the Euler elm
Example 19.5 Suppo:
In this calculation we have only used (4) over open sets that contain suPP M(PO')' and not neccesarily on all of M. This will be used in the proof of Lemma 19.2 below.
that ~ = T* ~ TM is tt frame for T;U) = smooth functions on I
nO(
Consider the maps
df
'0
M:M x R2. M and let A l2
;1
= aIel +
with iv(x) = (x, v) and ?f(x, t) = x, and let ~ = ?f*(O over M x IR. Then i~(~) = ~ for II = 0,1 and we have:
Lemma 19.2 For any choice of inner products and metric connections gv, V v (v = .2, 1) on the smooth real vec!.pr bundle ~ over M, there is an inner product ?i on ~ and a metric connection V compatible with 9 such that i~ ("9) = gv and i~(V) = Vv'
so that v(el) = A12® connection; cf. Exerc.
I
r
t·
11.
F
.,;"'OJ
.,
"to
.£"fJ .
.....
?'..
196
19.
-K,
vol = Pf(F V
.
when FY' is the curva1
).
This definition is compatible with Example 12.18; cf. Exercise 19.6. There is also a concept of metric or hermitian connections for complex vector bundles equipped with a hermitian metric. Indeed hermitian connections are defined as above, Definition 19.1, with the sole change that ( , ) now indicates a hermitian inner product on the complex vector bundle in question. The connection form A of a hermitian connection with respect to a local orthonor mal frame is skew-hermitian rather than skew-symmetric: Aik + Aki = 0 or in matrix terms A*+A=O.
Given a hermitian smooth vector bundle ((, ( , )c) of complex dimension k with a hermitian connection, the underlying real vector bundle (R is naturally oriented, and inherits an inner product ( , )R, namely the real part of ( , )c, and an orthogonal connection. If A is the skew-hermitian connection form of ((, ( , )c) with respect to an orthonormal frame e, then the connection form associated with the underlying real situation is AR, the matrix of I-forms given by the usual embedding of Mk(C) into M 2k(R). This embedding sends skew-hermitian matrices into skew symmetric matrices, and (6)
~.
THE EULER CLASS
since A12 1\ AI2 = O. In this case Pf(F V ) = dA 12 is called the Gauss-Bonnet form, and the Gaussian curvature K, E nO (M) is defined by the formula
(5)
x·1f~!JJi2.tll'
Pf(FY'(e)R) = (_i)k det(FY'(e))
by Theorem B.6.
For a complex vector bundle ( we write e() instead of e(R). Then we have
This proves (i).
The second assertion i!
on (I EEl (2,
and for matrices A an
Finally assertion (iii) fe
In order to prove uniql principle for real orien
Theorem 19.7 (Real s: M there exists a manij
(i) f*: H*(M) ---+ (ii) f*() ='Yl EEl ,
=
when dim ( c: 1 is the trivi£
The proof of this thea
Theorem 19.8 Suppo, oriented real vector b H 2n ( M) that satisfie:;
(i) f*(e()) = e( (ii) e((1 EB (2) = c
Theorem 19.6 (i) For a complex k-dimensional vector bundle (, e() = Ck(). (ii) For oriented real vector bundles 6 and 6, e(6 EEl 6) = e(6)e(6)· (iii) e(f*(~)) = f*e(~).
Proof. The first assertion follows from (6) upon comparing with Definition 18.3. Indeed, (Jk: Mk(C) ---+ C is precisely the determinant, so by (6) Pf(-F.fj21r) = (-I)kj(21r)k pf (F:) = i kj (21r)k (Jk(F'V)
the same base Then there exists a
Tel
Proof. Given a COID] Then f*c(L) = c(f* j
Theorem 18.9 shows 2-plane bundle 'Y. 1 a complex line bund simply defines multif
."
'6
"c .-\
198
19.
THE EULER CLASS
If (2n = 1'1 EEl ... EEl I'n is a sum of oriented 2-plane bundles then we can use (ii) and Theorem 19.6.(ii) to see that e((2n) = ane((2n). Finally Theorem 19.7 0 implies the result in general.
20. COHOMOLOG' GRASSMANNIAN Bl
In (18.14) we defined the Pontryagin classes Pk(() of a real vector bundle by Pk(() = (-1/C2k((c). The total Pontryagin class (7)
p(() = 1 + P1(()
In this chapter we CalCI fiber bundles, associatel the base manifold. As and oriented vector bUT Let 11": E ~ M be a sm
+ P2(() + ...
is exponential: p((l EEl (2) = p((1)p((2). Indeed, this follows from the exponential property of the total Chern class together with the fact (Properties 18.11) that the odd Chern classes of a complexified bundle are trivial.
gi ven by the formula
Proposition 19.9 For an oriented 2k-dimensional real vector bundle (, Pk(() = e(()2.
a.e =
(1)
Thus H*(E) becomes ~ shall examine this modu given classes eo: E Hn",
Proof. We give ( a metric ( , ) and chose a compatible metric connection \1. Then e(() is represented locally by ( -1 ) k j (211" ) k Pi ( F'V (e)) where e is an orthonormal frame. If on the complexified bundle (c we use the complexified metric then e is still an orthonormal frame, and the connection \1 becomes a hermitian connection on ((c, ( , )). It follows that F'V(e) is the curvature form for (c, and C2k((c) is represented by i 2k j(211") 2k det(F'V(e)). The result now follows from Theorem 0 B.5.(i) of Appendix B.
(2)
{i;(eo:) I a
E
Here Fp = 11"-l(p) is th
Theorem 20.1 In the {en I a E A}.
Q
Proof. The proof follov Let V be the cover COl V. Let U be the cover M replaced by U E U of Theorem 13.9. We condition (iii). So su~
and let E 1, E2 and E1 U12. The classes eo: E (2), and we denote the theorem is true for H~ for H* (Eu ). This em]
... J"
~
,
L
· ..
H*
J* *_ ~H
J
.~·1~· ~
200
20.
~
COHOMOLOGY OF PROJECTIVE AND GRASSMANNIAN BUNDLES
where we write E instead of Eu. We must show that every element e E H*(E) has a unique representation of the form e = L maea with m a E H*(U). We give the existence proof and leave uniqueness to the reader. By assumption we know that .* () ~ (II) e = L...J m a .ea ,
~11
where ill: Ell
~
v = 1,2
E is the inclusion. Since J*1* = 0, ~ JI '*( m a(l l ) ea = L...J ~ h'*( m a(2 l ) ea L...J
in H*(E12), where jll: E 12
~
Ell is the inclusion.
Uniqueness of representations for H*(E12) shows that R(m~l)) = j2(m~») for each a E A, and the Mayer-Vietoris sequence for the base spaces implies elements a E H*(U) with 1*(mo J = (m~l), m~\ so that
m
I*(e
-l: maea) = 0.
It thus suffices to argue that every element of Ker 1* = 1m 8* has a representation as asserted. This in tum is an easy consequence of the theorem for H*(E12) and the formula 8*(m.ii2(e)) = 8*(m).e,
(3)
valid for any mE H*(U) and e E H*(E), with i12: E12 ~ E the inclusion. We D leave the proof of (3) as an exercise. We are now ready to prove the complex splitting principle, as stated in Theorem 18.10. Let ~ be a complex vector bundle over M with dimc~ = n + 1. We form an associated fiber bundle P(O over M with total fiber space E(P(O) = {(p, L) I p E M, L E P(~p)}.
Here P(~p) denotes the projective space of complex lines in the vector space Projection onto the first factor 'IT:
~p,
E(P(O) ~ M
makes P(O into a fiber bundle over M. We leave the reader to show that p(~) is a smooth manifold and that 'IT is a proper smooth map. There is a complex line bundle H(O over P(O with total space E(H(~)) =
{(p,L,v) I (p,L) E P(O, VEL}.
If M consists of a single point then P(~) is the complex projective n-space
C1pn and H(~) is the canonical line bundle of Example 15.2. If more generally
'It>
"
"
...
,
II
ti
202
20.
20.
COHOMOLOGY OF PROJECTIVE AND GRASSMANNIAN BUNDLES
basis for 1l* ((;2 (R 2n )) of Ch (R 2n ) and then p its cohomology.
The above discussion contains no statement about the class en+1 = cl(H(~)t+1 in H 2n+2(p(0) except of course that
q(H(Ot+1 = Ao(O.l + Al(O.e + ... + An(~).en
Let V2 (R m ) denote the : x E Sffi-l and y as a u unit vectors in the tang via the embedding
for some uniquely determined classes Ai(~) E
COHOM()
H2n+2-2i(M).
We assert that
v.
Ai(O = (-1)i+1 Cn +1_i(0·
(6)
It is better for our own
To see this we use that 7ri(~) = H(~) EB ~1 and the exponential property of the total Chern class so that c(H(0)c(6) = c(7ri(0). Hence
c(6)
= 7r*(c(~))
(8)
/\ c(H(O)-1 = c(~).(l + CI(H(O))-I.
n+l
=L
(-l)iCn+l_i(O·Cl(H(~))i
i==O
which is equivalent to (6), because dimc6 = n and thus Cn+1 (6) =
o.
Remark 20.3 One can turn the above argument upside down and use (6) to define the Chern classes, once Cl (L) is defined for a line bundle. One then must show that the Chern classes so defined satisfy the two last conditions of Theorem 18. ] O. This treatment of Chern classes is due to A. Grothendieck. It is useful in numerous situations and gives for example Chern classes in singular cohomology, in [{-theory and in etale cohomology. The rest of this chapter is about the splitting principle for oriented real vector bundles. The construction is similar in spirit to the case of complex bundles, but the details are somewhat harder. The projectivized bundle P(O is replaced by the bundle G2 (0 of fiberwise oriented 2-planes in the oriented real vector bundle ( over M, and the canonical line bundle H(~) over P(O is replaced by the oriented 2-plane bundle 1'2 = 'Y2(() over G2(() whose fiber over an oriented plane in (p is that plane itself. If ( has an inner product then 7r*(() = 1'2(0 EB 'Y2(()l.. as oriented bundles, so that the procedure may be iterated. The analogue of (5) is a set of classes in H* (G2(()), namely the classes (7)
V2 (R m
The manifold V2(R m ) i Rffi ; it is evidently com
In H2n+2(p(~)) we get the formula
cn+l(6)
'(J:
1,e('Y2),e('Y2)2, ... ,e('Y2)2n-2,e('Yt)
E
1l*((;2(())
where 1'2 = 1'2 ((), dimR( = 2n and where e( -) is the Euler class of the previous chapter. In order to apply Theorem 20.1 we must show that the classes in (7) are a
!~
t ! ~
t
acts (smoothly) on V2(
(x, y).~ : m
The orbit space V2 (R 2-dimensional linear SI subspace they span, orie basis. We leave it to structure on G2 (R ffi ). oriented 2-dimensional It is clear from (8) th
so when we identify S action of 51 on 5 2m - J
(9)
where 1fo(x, y)
= spa,]
,-.
'"
.. ····nillf. AM."~.i%V
204
20.
COHOMOLOGY OF PROJECTIVE AND GRASSMANNIAN BUNDLES
20.
If we use homogeneous coordinates on cpm-1 then
COHOM
Lemma 20.5 The spa
0(11"0(x,y)) = [x - iyJ.
(10)
It is not difficult to see that 0 is injective and that its image is a smooth submanifold of cpm-1 of (real) dimension 2m - 4; cf. Exercise 20.3. We note that 11"0 is a fiber
Proof. This is again a ( has positive eigenvalw A. The case A. = 1 OCI
bundle; cf. Example 15.2. Complex conjugation on the homogeneous coordinates for opm-1 is an involution, whose fixed set is the real projective space Rpm-I, all of whose homogeneous coordinates are real. Since in (10), x and y are linearly independent, 0(11"0 (x, y)) is never fixed under conjugation, so (11) 0: Gz(R tn ) -+ cpm-l - Rptn-1 = W .
A=
where P is the orthogl Here P depends contin by F(A; t) =
m
We show below that this map is a homotopy equivalence, so that H*(G 2(R tn )) ~ H*(Cptn-1 - Rpm-I),
and use this to calculate H*(G 2(R m )). We begin with a discussion of the group GL~(R) of real 2x2 matrices with positive determinant. The action of the multiplicative group C* on C = R 2 by complex multiplication identifies C* with a subgroup of G (R), where a + ib E C* corresponds to
Observing that each n A. - A.-I, we see that 1
Lt
a (b
(12)
-b) a
I
Proposition 20.6 The
E GL + 2 (R).
from (I I) is a homoto
GLt
Let Q c (R) be the subset of positive definite symmetric matrices with determinant 1. The subgroup 50(2) C GL~(R) acts on Q by conjugation.
Proof. We write Wm
:
Lemma 20.4 The map 'I/J: Q x C* -+ GLt(R);
(A, a + ib)
f->
A.
(~ ~b)
\
~
is a homeomorphism.
!
Proof. Let B E GLt(R) with transpose B*. Then BB* is positive definite, and by the spectral theorem it has a unique square root (B B*) l/Z which commutes with BB*. This gives the polar decomposition B = (BB*)l/Z. R ,
R = (BB*)-1/2. B ,
=
= I, so R = R o E 50(2). For d = det B = det(BB*)1/2 and A 1 d- (BB*//2 E Q, we obtain B = 'I/J(A, de iO ). The polar decomposition is unique. Indeed if B1R1 = B 2 R 2 with Bi symmetric and Ri E 50(2) then = and square roots are unique. Hence 'I/J is a bijection. Its inverse is and RR*
Bi
~(x
,
i
Bi,
continuous, since A, d and R depend continuously on B.
~.
t
I
A point in 1f 1 (Wm ) independent vectors if orthonormal bases (x as (v, w). There is Gram-Schmidt orthof the 50(2)-action,
so
~
induces a biject
0
For symmetric, positive definite matrices one can form powers At for any t E R, and we have
This bijection commt right multiplication ir
-)
'"
..,j
,ro,U
-"
~
~'.~.~~."$~"
.q.;..
206
20.
'CO
"
. . ,. •.. '
";!
20.
COHOMOLOGY OF PROJECTIVE AND GRASSMANNIAN BUNDLES
Remark 20.7 We can construction in the pre' where V ~ Rm is a 2-di structure on V compati! with J2 = -id and suc basis for V. One fon Jc = J @R ide: Ve . eigenspaces of Je. Thes
This gives a commutative diagram where the horizontal maps are bijections V2(Rm) XSO(2)
GLt(R)
-7r-
1 (V2(Rffi)
(Wm )
1
GLt(R))jC* - - -
XSO(2)
1
Wffi
Lemma 20.4 gives a bijection Q s:! GLt(R)jC*, so we may identify the lower left-hand comer of the diagram with the quotient space ffi X = V2(R ) xSO(2) Q
V+ =81
for the SO(2)-action on V2(Rffi) x Q defined by
The pair (V, J) may be 1 V = (V+ + V_)nR m an of V+EBV- which acts on of Gr(R m ) with W m in in W m ~ ClP m - 1 . The space Q s:! G Lt (R) ible with the standard 01 is another model for a identification of the twc Each V E G2(R m ) has a x E V leads to a posi inclusion G2(R m ) --+ G~ the embedding Ip in (H
(x,y,A).Ro = ((x,y).Ro,R;lARo).
Altogether we obtain a bijection
<1>:
X --+ Wffi given by
~([(x,y), (~ ~)]) = [(ox+;3y)-i(;3x+ r y)]·
(13)
Evidently <1> is continuous in the quotient topology on X, in fact it is a homeo morphism. In a neighborhood V of any given point in W ffi the inverse ~-l can be written as the composition V
~
7r- 1 (Wm )
~ V2(R ffi ) x GLt(R) id~-l V2(R m ) x Q x C* p~r X
where s is a local section given by Lemma 14.4, S the global mentioned above, and 'ljJ-I given by Lemma 20.4. Each map in is continuous, so ~-l is continuous. As a byproduct we find that map p: V2 (Rm) x Q --+ X has continuous local sections defined covering X. The conjugation action of 30(2) on Q fixes the identity matrix I has the subspace ffi ffi (14) G2(R ) ~ V2(R ) XSO(2) {I} eX
section of <1> the sequence the canonical on open sets E Q, so one
(cf. (9». Comparing (10) and (13), we see that ~ 07r01 is precisely equal to 0. It remains to be shown that the inclusion map io in (14) is a homotopy equivalence. There is an obvious retraction induced by the constant map Q --+ {I} m r: X --+ V (R ) XSO(2) {I} ~ X. 2
Finally the required homotopy H: X x [0,1] --+ X between io 0 rand idx is induced by ffi idV2(Rffl) x F: V2(R m ) x Q x [0,1] --+ V (R ) x Q,
COHOMO
~
I:, ."
l ~,
We shall now use Propo: we apply Poincare duali i:Wr;
When m = 2, 52 -=. Cf with a great circle in ~ hemispheres in 52.
Lemma 20.8 The map possibly two cases: for I and for m even there is
0--+
2
where F is defined in Lemma 20.5. Observe that H is well-defined because F(R r/ ARo, t) = R;l F(A, t)Ro. Continuity of H can be shown with use of local sections of p; 0
Proof. The exact sequc takes the form
... i: Hq-l (Rpm-l
~
~
'"
L"'tJ ;~
208
20.
',-".
20.
COHOMOLOGY OF PROJECTIVE AND GRASSMANNIAN BUNDLES
The tenus involving Rpm-l and Cpm-l have been calculated in Example 9.31 and Theorem 14.2, respectively. Note that HO(cpm-1) ~ IR maps isomorphically to HO (Rpm-I) ~ R under j*, whereas j* = 0 in other degrees. 0
COROM(
There is a bundle map canonical line bundle (
V2
Proposition 20.9 The cohomology H 2p-l(Wm ) = 0 for all p, and
112 ij2p = m - 2
H P(Wm ) ~ 2
{
i= m - 2 and 0 ~ 2p:S 2m - 4
IR
ij2p
o
ij2p~2m-2.
Moreover Hq(i): Hq (Cpm-l) and is zero if q = 2m - 2.
-->
Since
Hq(Wm ) is a monomorphism if q i= 2m - 2,
Proof. We apply Poincare duality to the oriented (2m - 2)-dimensional manifold Cpm-l and to W m . Lemma 13.6 gives a commutative diagram
HP(cpm-l)
W(il
l~
HP(Wm )
with degc = 2
H"'..:2(i)
H m - 2(wm )
H
-->
R
-->
0,
G2 (R
m
has kernel H 2m - 2 (Cp'
1 and c
o
and the proposition follows from Theorem 14.2. We have the two canonical bundles 12 and If over
Proof. We already kn< and also that
=0
but H2m- 2(Cpm-l) ~ R. In the second case, the exact sequence of Lemma 20.8 dualizes to the exact sequence
)
= H*(45)(Cl(H)
Suppose now m = 2n (15)
Cm I
with total spaces
E(2) = {(V, v) E G2(R m ) x Rm I v E V} Ehf) = ((V,v) E G2(R m ) x Rm I v E VJ..}
f
(i) For m odd and (ii) For m even an,
H*(G2(1
The previous lemma implies that HP(O is an isomorphism except if p = 2m - 2, or if p = m - 2 and m is even, because i! is the vector space dual of i*. In the first case
2(cpm-1)
Theorem 20.10 With t}
--L.. H1 m - 2 - P (Wm )'"
H 2m - 2 (wm ) ~ H~(Wm)*
= e(HR)
e(r2) = 45*(Cl(H)). Let C E H 2 (G 2 (R m )) t e = e(rf) be the Eule!
1~
H 2m - 2-p(Cpm-l)*
0--> H m -
C1 (H)
= Em. Alternatively E(r2) = V2(Rm) XSO(2) R2, the orbit space and with 12EB of the SO(2)-action ' given by (x,y,v)R() = (x,y)Re,R;l v). The embedding of 2 m m V 2 (Rm) xSO(2) R into G2(R ) x R sends [x, y, v] into (no(x, y), XVI + YV2)
where v = (VI, V2).
Indeed, the first one fa of C is in the image (
H*(C1
The second relation is
ec =
'ti
'"
"
.0····2• •• Jil:t .%~
•h ....
210
20.
COHOMOLOGY OF PROJECTIVE AND GRASSMANNIAN BUNDLES
For the third relation we use that the total Pontryagin class is exponential (cf. (18.14)) so that
21. THOM ISOMOR GAUSS-BONNET FO
(1 + PI ('Y2) ) (1 + PI ('Y;) + ... + Pn-I ('Y; )) = 1 and hence pj('Yr) = (-1)jpI('Y2)j for j:S n -1. Moreover by Proposition 19.9
e2 = e(.1)2 'Y2 = Pn-I (.1) 'Y2 = (l)n-1 PI ('Y2 )n-I . Since 'Y2 = ~*(HR) and HRC
= HEEl H*,
PI('Y2) = -~*(e2(H EEl H*))
= ~*(C1(H)2) = e 2
so e 2 = (-It- I em - 2 , which is the last equation of (15). From Proposition 20.9 and the non-triviality of el (H)m-2 we know that e m - 2 =I 0, so by (15) also that e =I O. The vector space H m - 2 (G 2 (Rm)) is 2-dimensional, and en - I = ;\e, ;\ E R - {o} gives en = ;\ee = 0, which contradicts that m 2 ~ nand em - 2 i- O. We have proved that the set {I, c, ... ,cm - 2 , e} is a vector space basis for H* (G2 (Rm)), but this is also a basis for the ring R[e, eJj(em -l, ee, e2 + (_1)n em-2). 0 Let ( be a smooth m-dimensional oriented real vector bundle over M, and suppose ( has an inner product. Consider the associated smooth fiber bundle G2 (() over M with total space
Let ~ be an oriented vee! isomorphism theorem ell terms of H*(M), namel to be smooth (cf. Exerc theorem is a consequen<
Hr;+q(E
where n = dim M. T1 equivalence E c::: M. I Thorn isomorphism.
Suppose ~ is smooth and direct sum of ~ and the unit sphere bundle over one-point compactificati 1f:
S
E(G2(()) = {(P, V) I P E M, V E G2((p)} and with 7r: G2(() --> M being the projection onto the first factor. There are two oriented vector bundles over G2(() with total spaces
E('Y2(()) = {(p, V,v) I (p, V) E G2((), v E V} E('Yr(()) = {(p, V, v) I (p, V) E G2((), v E V.1}
sm.
and 'Y2(()EEl'Y;(() ~ 7r*((). The orientation of 'Yr((p) is such that 'Y2((p) EEl 'Yr((p) has the same orientation as (p, An application of Theorem 20.1 gives
Theorem 20.11 For any oriented m-dimensional real vector bundle (, H* (G 2 (()) is a free H* (M) module with basis
I, e('Y2(()), { 1, e('Y2(()), In particular 7r*: H*(M)
, e('Y2(())m-2, e('Yr(()) ifm = 2n ~ 4 , e('Y2(())m-2 ifm = 2n + 1 ~ 3 -->
H*(G 2 (()) is injective.
be the bundle projection (0,1) E ~p EEl R. This n hence H*(S(~ ill 1)) inte the m-sphere ~ 1f each fiber In partiCl a fixed isomorphism
o
Given this result, the proof of the real splitting principle, Theorem 19.7, is precisely analogous to the proof of the complex splitting principle, treated earlier in this chapter.
sm
(1)
Stereographic projectiOl globally it identifies the
Definition 21.1 An Hm(s(~ EEl 1)) that sati (a) s~(u)
=0
(b) For each P E 11
0:0
"
212
21.
21.
THOM ISOMORPHISM AND THE GENERAL GAUSS-BONNET FORMULA
Theorem 21.2 Each oriented vector bundle ~ admits a unique orientation class u, and H*(S(~ EB 1)) is a free H*(M)-module with basis {I, u}
{,
THOM ISOMI
Finally consider a sequi U = Ui Ui. We have th
/'
Proof. The second part of the theorem follows from Theorem 20.1, since H*(Jr- 1 (p)) has basis 1 and i;(u) according to Example 9.29 and Corollary 10.14. Here i p denotes the inclusion of the fiber into the total space S(~ EB 1). In particular, it suffices to find a class v that satisfies Definition 21.1.(b). Indeed, if Hm(s(~
v is such a class then any other class in u = Jr*(x)
+ Jr*(a)
EB 1)) has the form
1\ v
for some x E Hm(M) and a E HO(M). The restriction of Jr*(x) to Jr-l(p) vanishes for all p, so the locally constant function a must have value 1 at each p E M if u is to satisfy Definition 21.1.(b). But then s~(u) = x
l
The family of orientation with integral lover all 1 applies to show that M E
t l.
The above does not rei Proposition 13.11 to th S(~ EB 1) - soc,(M) ~ E
... ~ HHE)!:.
cv;J
X
~
f
+ s~(v),
and s:"x:,(u) = 0 if and only if x = -s~(v). The existence of a class u E Hm(s(~ EB 1)) that satisfies condition (b) is based upon Theorem 13.9. Write Su = Jr- 1 (U) where U c;;;. M is open, and let U be the collection of open sets for which the restriction ~u = ~IU satisfies the conclusion of the theorem. The preceding discussion shows that U E U if and only if there exists a class in Hm(Su) with integral equal to lover each fiber Jr-l(p) for p E U. Let V = (V,a) in Theorem 13.9 be the cover with V,e the open sets in M such We must verify the four conditions of that ~1V,9 is a trivial bundle (~IV,9 ~ Theorem 13.9. The first condition is trivial. If U c;;;. V,e then we may trivialize ~u (compatible with the Riemannian metric) so as to identify Su with U x sm. Let
Su ~ U
,
(2)
o
---t
H(
Theorem 21.3 (Thorn vector bundle over a COl with integral lover eac
iJ?: Hq (
is an isomorphism. The
Sm ~ Sm
denote the resulting projection, and let u E H m (sm) have integral equal to 1. Then pr* (u) restricts to a class with integral 1 on each fiber, and condition (ii) of Theorem 13.9 is satisfied. Next we verify condition (iii). So suppose U1 , U2 and Ul n U2 belong to U. The orientation classes U v E Hm(SuJ, v = 1,2 restrict to classes in Hm(Su1 nu2) that satisfy both condition (a) and (b) for ~ulnu2' Uniqueness applied to ~ulnu2 shows that Ul and U2 have the same restriction to Su1 nu2' In the Mayer-Vietoris sequence
H m(SU1 UU2) ~ Hm(SuJ EB Hm(SuJ
Now s~ 0 Jr* = (Jr 0 Soc exactness i* is a monom
!:. H m(Sulnu2),
(Ul,U2) E KerJ* = ImI*, so we can find a class u E H m (SU1 UU2 ) with restriction Uv to SUv' This class has integral lover all fibers Jr-l(p) c;;;. SU1 UU2 and U1 U U2 belongs to U.
Proof. The exact sequer has the form u = i* (l statement now follows (2) are H*(M)-linear a free H* (M)-module ge
Definition 21.4 With t be the class with iJ?( e(,
The product in H~ (E) resenting differential f( odd-dimensional orient'
.-;
~
,
"It
....,.; • L""j->
~ ......
~
~'~'"
~
,.~, ".,-,,
214
21.
,v
THOM ISOMORPIDSM AND THE GENERAL GAUSS-BONNET FORMULA
Lemma 21.5 Let s: M
-+
E be an arbitrary smooth section of E. Then e( 0 =
s*(U).
i
21.
mOM 150M
Lemma 21.7 If 6 am manifold M, then e(6 '
I'
II
Proof. Since s is a proper map, it induces a homomorphism H~(E) -+ H~(M)
E
= H*(M),
f
cf. Chapter 13. A closed form w E n~(E) that represents U also defines a class [w] E Hm(E). Now so 7l": E -+ E is homotopic to idE (use the linear structure in the fibers), so [w] = [7l"*s*(w)]. But then
(s*(U))
= (7l"*s*(U)) /\ U = [7l"*s*(w)] /\ U =~/\U=U/\U
The next two lemmas show that and hence that
e(~)
e(~m)
(3)
0
satisfies the conditions of Theorem 19.8,
I
Proof. Let ~v have tol dimension m v , and let I map is the projection of an M x M with ~*(~) = If pr v : E1 x E2 - t Ev representing Uv , then WI
It follows from Fubini's Sy: M -+ E y be the zel
= am/2e(~m)
e(O
for even-dimensional oriented vector bundles over compact manifolds. We shall see later that a = 1.
= [(t If Pv: M x M
Lemma 21.6 Let f: N -+ M be a smooth map of compact manifolds and ( an oriented vector bundle over M. Then e(J*O = j*e(O.
M d€ X
S~
= pi(e(6)) /\ '[,
e(6 m
E,---l- E
1
-t
((S1 so e(O
Proof. We have the pull-back diagram
= (S]
1
N-l.-M
where E and E' are the total spaces of ~ and 1* ~ respectively. The map J is proper, so the class U E H;:-(E) of Theorem 21.3 pulls back to U' = J*(U) E H;:-(E'). Since maps a fiber 1*~p to ~f(p) by a linear oriented isomorphism, U' will have integral lover fibers. There is another commutative diagram
i
E,---l- E rs'
rs
N-L.M where sand s' are the zero sections, and by Lemma 21.5 we find
e(J*O = (s')*(U') = (s')* 0 J*(U) = (} 0 s')*(U) = (s 0 J)*(U) = 1* 0 s*(U) = 1*(e(~)).
o
In Chapter 11 we defi. singularities, and in a sum of the local indic now extend these noti over a compact, orlenl Let E = E(~) be the so: M -+ E be the zer the fiber ~p. Let s: M and Dpso from TpM 1 inverses.
l~
o
.-d
u:
...
21.
"
11 ~.
216
"
'ti.
.":;!
21.
THOM ISOMORPmSM AND THE GENERAL GAUSS-BONNET FORMULA
Definition 21.8 Let p E M be a zero (singularity) for s, s(p) = so(p). Then s is called transversal to So at p if i:/
(4) and
Dps(TpM) S
n Dpso(TpM)
t:
= 0
f '~
Tso(p)E ~ TpM EEl ~p. With this identification, (4) is equivalent to the statement that Dps(TpM) is the graph of a linear isomorphism A : TpM ~ ~p. Both vector spaces are oriented by assumption; we define the local index t( s; p) to be +1, if A preserves the orientations, and -1 if not. In the special case where ~ = TM is the tangent bundle this is in agreement with Definition 11.16 (cf. Lemma 11.20). Given an oriented local trivialization of ~ over U, ~Iu ~ U x Rm., we can identify the restriction of s to U with a smooth map F: U ~ Rm . Then A: TpM ~ ~p corresponds to DpF: TpM ~ Rm, and (4) becomes the statement that DpF is an isomorphism. Hence t(s;p) = ±1 depending on the orientation behavior of DpF. The inverse function theorem implies that F is a local diffeomorphism at p. In particular (4) forces p to be an isolated zero of s. If s: M ~ E is transversal to the zero section, then the number of zeros of s is finite, since M was assumed compact.
Theorem 21.9 In the situation above, if s is transverse to the zero section, then (5)
I(e(O) =
In terms of homogeneous the fiber spanc(za, zI) in chart Uo = {[l,z] I z E C
Ua x C
is called trans versal to So if this holds for all zeros of s.
The tangent space Tso(p)E is the direct sum of the tangent space Dpso(TpM) to the zero section So (M) and the tangent space at So (p) to the fiber ~p which is naturally identified with ~p itself. In other words:
THOM ISOMO
I
l
t
In the dual trivialization Uo --t C, which maps the identity. It follows t1 [,(8; po) = 1. From Theor
Theorem 21.11 Fororiel Proof. We have already
constant a; it remains to example shows that I(e( Theorem 19.6.(ii), and I( c Since I is injective e(~2)
If the dimension m is o( 21.4 that e(O = 0, and c( ~ is even-dimensional and zero, and e(O = O. One ( for Eto admit a non-zero splitting E ~ EEl In f,' to be the orthogonal ( lines generated by s.
e EL·
Theorem 21.12 For any
L t(SiP) p
where we sum over the zeros of s, and where I: Hm(M) homomorphism.
~
R is the integration
Example 21.10 Let H be the canonical complex line bundle over Cpl, H* its dual bundle, and ~ = (H*)R the underlying oriented real bundle. The bundle H
Proof. We simply apply
vector field X w.r.t. sOl Lemma 12.8 shows that. (5) is equal to Index(X)
is a subbundle of the trivial 2-dimensional bundle
E(H) = {(L;z) I L E cPl,z E L} c Cp2
X
C2
(cf. Example 17.2). Dually there is an epimorphism from the dual product bundle onto H*, and we let s: Cpl ~ E(H*) be the section that is the image of the constant section in the dual product bundle given by the linear form I7:C 2 ~ C;
I7(WO,Wl)
=
WI.
We can combine the two i Gauss-Bonnet theorem t<
Theorem 21.13 (Genen
:
.
1
!
'.
"">'0
t~/"":" ~
:
,. ilt~ll!
...
~
~
. ,',,'-...
.~.
'
...
'Q
~~ .._! ";!
1' ..•.. . -..•.•. . . •
~
.1.
.~.fi~~
218
n.
21.
mOM ISOMORPHISM AND THE GENERAL GAUSS-BONNET FORMULA
where F'V is the curvature associated to any metric connection on the tangent 0 bundle of M 2n .
THOM 180M
For example we can ta
(6) The rest of this chapter is devoted to a proof of Theorem 21.9. Let PI,." ,Pk be the zeros of s. First we construct a closed form W E n~(E) which represents U E H~(E), and local trivializations of ~ over disjoint open neighborhoods VI, ... , Vk of the zeros. Let Ev, be the inverse image of Vi in E, and define Ii: Ev, ---> R m to be the composition of a trivialization E\I; .::. Vi x Rm with the projection on Rm •
Lemma 21.14 In the above,
where Wi E n~(Rm) are forms with -+
R is a
S1
F(p, t)
1~ i ~ k
JRm Wi
Then (a) holds for h(p)
= 1.
M be a map such that
(a) h is smoothly homotopic to id M
--->
Now construct a homato] 1 ::; i ::; k, and diffeoma
Vi and Ii may be chosen so that
WIEvi = ft(Wi),
Proof. Let h: M
where p: R
.
(fJ) h is constant with value Pi on some open neighborhood 1, ... , k.
Vi
of Pi, i =
Proof of Theorem 21.9 ] open neighborhood of Pi, onto an open set in Rm . I Since M - Ui Wi and Sl that the scaled section t
It follows from (a) and Theorem 15.21 that ( = h*(~) is isomorphic to ~, so it suffices to construct the required trivializations and forms for ( instead of ~. Consider the diagram
SUl
E'i..-E
k
1~
M--!::-.M
Now e(O E Hm(M) i5 zero on M - Ui Wi· I
where E = E(~), E' = E((). Pick a closed form W E n~(E) with integral 1 over each fiber in~. Then w' = h*(w) E n~(E') has integral lover each fiber of (. Since h(Vi) = Pi, h: Ev. -+ ~Pi" We pick an oriented linear isomorphism ~Pi ~ Rm , and let Wi E n~(R"":) correspond to WI~Pi E n~(~p), and Ii to h under this isomorphism. Then
where Ii 0 (81\1;) is a di su PPR'" (Wi). This diffe on the value of ~(s; Pi) :
wiE' = ft(wd, Vi
f Wi = 1, JRm
1 S; i
~k
as required. We have left to construct h with the stated properties. To this end we use a smooth map G: Rm x R -+ Rm which satisfies:
G(x, 0) = 0 for x E lDm 2 G(x, 1)= x for x E Rm
G(x, t) = x for t E R and x E Rm - Dm.
I(e(~))
(7)
·.
,
•
\
.~ ~
.rJ,
L"'l-",
~ )
'~-~
.
"
.•..
.q,..
"2
_, . 222
tIIIIiII
A. SMOOTH PARTITION OF UNITY
Corollary A.3 For real numbers a < b there exists a smooth function 7./J: R -+ [0, 1] such that 7./J(t) = 0 for t ~ a and'l/J(t) = 1 for t 2: b. Proof. Set 'l/J(t) = w(t - a)j(w(t - a)
+ w(b -
o
t)).
The desired result is achic m 2: 1 and 1 ~ j ::; dm 00
U=
UE m=l
For x E Rand E > 0 let DE(x) = {y E Rn
Illy - xII < E}.
Corollary A.4 For x E Rn and E > 0 there exists a smooth function ¢: Rn [0,00), such that DE(x) =
-+
Proof. Define
Proof of Theorem A.I n
¢(y)
which yields (i). One obt with x E Umo ' Since L intersect D2f(Xm j) only
= W(E 2 -Ily - x11 2 ) = w(e 2 -
L (Yj -
Xj)2).
o
j=1
A.6. Apply Corollary D f ; (Xj) = 'l/Jjl((O,oo)). given by
Note that the support of ¢ is the corresponding closed disc
Df(x) = {y E R n
Illy - xII ::; e}.
Rn can be written in the form U = U:=1 K m, where the sets K m are compact and K m ~ Km+l (the interior of K m+1 ) for m 2: 1.
is smooth, because the ! From (i) it follows that
Proof. The conditions hold for K m = D2"'(O) - UXERn-u D 1/ 2m(X).
We introduce the mod ~j(x)~(x)-l. They Me
Lemma A.S An arbitrary open set U
~
0
Proposition A.6 For an arbitrary open set U ~ Rn and a cover V = CVi)iEI of U by open sets, we can find a sequence (x j) in U and a sequence (Ej) ofpositive real numbers' that satisfy the following conditions: (i) U = U~1 Dfj(xj). (ii) For every j there exists i(j) E I with D2fj (x j) ~ Viw. (iii) Every x E U has a neighborhood W that intersects only finitely many of the balls D2f; (x j ).
Proof. We choose Km(m 2: 1) as in Lemma A.5. Additionally we set Ko = K-l = 0. For m 2: 1 we introduce the sets Bm
= K m - Km- l '
Um = Km+l - K m- 2·
Here B m is compact, Um is open, B m ~ Um and U = U:=1 B m . For x E B m we can find E(X) > 0 such that D 2f (x)(X) is contained in both Um and at least one of the sets Vi, The Heine-Borel property for B m ensures the existence of Xm,j E B m and em,j > 0 (1 ~ j ~ dm ) such that (a) B m ~ U;:1 DE",,; (Xm,j).
({3) Every D2E m,j (xm,j) is contained in Um and in at least one of the sets 11,;.
for all x E U. Set
~jvIW' with jv E Ji' and
SUPPr
we get that x E \;i. Hel1 Corollary A.3 has the f,
Lemma A.7 If A ~ R exists a smooth functiOl all x E A.
Proof. Apply Theorem and V2 = Rn - A. Nov
.~
...
<
""
",..
~~.'.
1~$
......•
~
224
.
..•
.. ~
A. SMOOTH PARTITION OF UNITY
!
Proposition A.8 (Whitney) For an arbitrary closed set A smooth function ¢: Rn ---+ [0,00) with A = ¢-l (0).
~
Rn there exists a
Proof. Apply Lemma A.5 to the open set U = Rn - A, and use Lemma A.7 to find smooth functions Ibm:Rn ---+ [0,1], m ~ I, with SUPPRn(lbm) ~ Km + 1 and Ibm (x) = 1 for all x E K m . Define
Theorem A.l to the openc This yields smooth functic A.I.(i), (ii) and (iii). By II sUPPu(¢o) ~ V o, we can
f(x)
00
From Theorem A.I.(iii) or thus
¢= LCmlbm m=l
f(x)
for a suitably chosen sequence (c m ) of positive numbers. Let
Di = where i
= (il,' .. ,in)
and Iii
alii ..
ax~l ax~2
= L: i v .
.,
... ax~
We show that the series
LCmDilbm
s
'L
q
s'L¢p(:
m=l
converges uniformly on Rn. Since Ibm has compact support, we can find bm E [1,00) with sUPxERnIDilbm(x)! ~ bm for all i with Iii s m. If we set Cm = (2 m bm )-1 then L:~=l 2- m is a comparison series for formula (1) from the li\-th term onwards. This implies uniform convergence of the series (1). Hence ¢ is smooth and Di¢ is given by (1). For all x E K m we get ¢(x) ~ Cmlbm(x) = Cm > 0, and thus A = ¢-1(0). 0
Lemma A.9 Suppose that A ~ Uo ~ U ~ (In, where Uo and U are open in Rn and A is closed in U (in the induced topology from Rn ). Let h: U ---+ W be a continuous map to an open set W ~ Rm with smooth restriction to Uo. For any continuous function E: U ---+ (0,00) there exists a smooth map f: U ---+ W that satisfies
s
(i) Ilf(x) - h(x)11 E(X) for all x E U. (ii) f(x) = h(x) for all x E A.
Proof. If W =f:: Rm then E(X) can be replaced by
fl(X)
Ilf(x) - h(x)11
pEU-A
00
(1)
Now (ii) of the lemma fl follows from the calculati
= min(E(x), !d(h(x), Rn - w))
where d(y, Rn - W) = inf{lIy - zlll z E Rm - W}. If f: U ---+ Rm satisfies (i) with f} instead of E, we will automatically get f(U) ~ W. Hence, without loss of generality, we may assume that W = Rm . Using continuity of hand E, we can find for each point p E U - A an open set Up with p E Up ~ U - A, such that IIh(x) - h(p)11 f(X) for all x E Up. Apply
s
~
228
B. INVARIANT POLYNOMIALS
Lemma B.1 For every A E Mn(C) we have sk(A) Proof. Let us assume that A is a diagonal matrix, A
For instance SI (A) = 0'1 ( identities in + n~ va
= tr(A k).
= diag(Al,"
ni
., An), such that
O'k
n
det(I - tA)
= IT (1 -
(3)
t)'i)'
i=1
Sk
This yields the following equation of power series
d -t dt 10gdet(I - tA) =
n
noo
t).
~ 1 _ ~Ai
=
~
E
where Al EB A2 is the
oo(n)
k
Aft = { ;
~ Af t
k
= Af + ... + A~ = tr(diag(Al,"" Anl).
Since both sk(A) and tr(Ak) are unaltered when we replace A by gAg-l, and since the diagonalizable matrices are dense in the vector space of all matrices, the assertion follows. 0
Lemma B.2 For polynomials in n 2 variables we have that
sk(A) - Sk-l(A)O'I(A) + Sk-2(A)O'z(A) _ ... + (-l)kkO'k(A)
11
.
Hence sk(diag(Al,"" An))
Sk
and Al @Az the matrix ( equation it is sufficient tl functions on M n1 (C) X J det(I + t(
giving the first equation relations are similar, an_ Let O'i(Al,' ., ,An), ~ =
= O.
71
Ii=
Proof. It suffices to prove the equation for a diagonal matrix A = diag(Al,' .. , An) where n
= IT (1 -
7f(t)
n
tAi)
=
I:
tA' s(t) = 'L....l-t· " \ n
(-l)kO'k(A)t k and
i=1
k=O
i=1
t
They are the so-called
00
k = '" L.... sk(A)t . k=1
Theorem B.3 Every p tation of coordinates CQ where O'i are the eleme'"
Now
7f(t)s(t)
n
g
tA') n
= ( ~ 1 _ ~Ai
(1 - tAi)
n
= ~ tAi n
= I:
U
(1 - tAjl
d7f
= -t dt
Proof. See [Lang] Cha:
(-l)k-l k O'k(A)t k .
k=1
The coefficient of t k in this equation yields the desired identity.
0
Note that the equation of Lemma B.2 inductively determines sk(A) as a poly nomial with integer coefficients in the variables 0'1 (A), , O'k(A). Conversely O'k(A) is a polynomial with rational coefficients in sl(A), , sk(A). We write
(2)
sk(A) = Qk(O'I(A), O'k(A) = Pk(SI(A),
,00k(A)) , sk(A)).
(AI, ... , An).
Theorem B.4 Every i'" form
where p is a polynomi
Proof. Let D n C Mn(l
~
~ •. j
"
~2
230
B. INVARIANT POLYNOMIALS
is everywhere dense, P: Mn(C) C is determined by its restriction to D n. A permutation 1r E l:n of n elements induces an endomorphism C n C n by permuting the coordinates, i.e. an element in Mn(C), again denoted by 1r. If d(Al' ... , An) E Mn(C) is the diagonal matrix with elements AI, ... , An then
Theorem D.S If A E so
l 1rd(Al,"" An)1r- = d(A7I"(1)"'" A7I"(n))'
Proof. Since A is a real 1 element in M2n(C)· The basis of eigenvectors el· By conjugation we see with eigenvalue 'Xi. We (1 ~ j ~ n). This is ea assume el to be an eigen
-jo
-jo
In particular we have P(d(Al,"" An)) = P(d(A7I"(1)"'" A1r (n)))
for all 1r E l:n' Theorem B.3 gives that P(d(Al, ... , An)) = p(iJl,.", iJ n),
(i) Pf(A)2 = det(f (ii) Pf(BA.B t ) = P
Al
where iJi = iJi(Al, ... , An), and P is a certain polynomial. Hence P(A) = p(iJl(A), ... , iJn(A))
for all A E D n, and hence for all A E Mn(C).
D
We finally consider the Pfaffian Pf(A). This is a homogeneous polynomial in n(2n - 1) real variables, or alternatively a polynomial in the skew-symmetric 2n x 2n matrices A = (Aij). It has degree n. We can consider Pf (A) as a map
so Al is purely imaginary AI. Hence e2 is orthogc Span(el, el).l is invaria makes is possible to ref Having arranged that e consisting of the vector Vj
Pf: S02n
-jo
LAj ei 1\ ej
E
Moreover for aj E R I AWj = ajvj., This proy
A2 (R 2n )
gAg-I::
i<j
and define Pf (A) by the equation w(A)
where vol = el A
1\
e2
1\ ... 1\
= diag (
( 0 -al
1\ ... 1\
In particular w(A)
= n!Pf(A)vol,
Pf(gAg-
e2n' For the block matrix
al
o) '
(0-a2
a2 ) 0
' ... ,
(
0
-an
aon ) )
a simple calculation gives
(4)
w(A)
and one sees that
It follows that Pf(A)
Since 9 -1 = l, asserti, In order to prove (ii) Ii = 2: Bviev we haY '( =
= al el 1\ e2 + a2 e3 1\ e4 + ... + an e2n-l 1\ e2n,
L
Aij Ii
= al
1\
1\
w(A) = n!(al .. , an) vol.
an and Pf(A)2
= det(A)
in this case.
1\
f t
so that '( = w(BAB : w(BAB t )
w(A)
+
R,
from the space of skew-symmetric matrices. For an A E S02n. we let w(A) =
1 = ,j2(e2j-l
1\ ...
By Theorem 2.18 the
J
«,J
~.c'~ 1._$ .•...•..•......
""
"
.• .••..
~2
232
B. INVARIANT POLYNOMIALS
is multiplication by det (B), so that C.
II /\ ... /\ hn =
det(B)el /\ ... /\ e2n'
o
This gives (ii).
Let sUn C M n (C) denote the subset of skew-hennitian matrices. The realification map from Mn(C) to M2n(R) induces a map from SUn to S02n, denoted A J---+ AR, and we have
Theorem 8.6 Pf(AR) = (-Atdet(A). Proof. Since a skew-hennitian matrix has an orthononnal basis of eigenvectors, we may assume A is diagonal, A = diag( Hal, ... , .;=Tan) with ai E IR. Multiplication by yCTai on C corresponds to the matrix 0 ( ai
-a i
a
PROOF OF LEMJ
In the proof of Lemma 1 Lemma 12.12 to a new ~ small neighborhoods of (iii) together with F in following lemma appliec Theorem 12.6. Without standard form of Exam}
Lemma C.I Let W ~ I f: W ---'> IR be the funct
) E S02
where a E IR, ). E 7L and
so by (4) Pf(AR) = (-ltal' .. an. On the other hand
E= '
det (A) = (vCItal ... an and the result follows.
o
Then there exists a Mors V ~ Rn-A+I that satis, (i) F(x) = f(x) w
(ii) The only critica (iii) F- I (( -00, a + (iv) F-I((-oo,a (v) f-l((-oo,a -,
Proof. We introduce th (1)
and we define F E CC (2)
where f.l E COO(R, R) i (a) -1 < f.l'(t) ~ (b) f.l(t) = a wher (c) f.l is constant 0
.,.
234
~
C. PROOF OF LEMMAS 12.12 AND 12.13
Hence U contains the Ii x E U and let x = (Xl
2£
£
'P2: [0, 1]
fl
fl'
2£
-1
•
--+
'P2( t
R;
It is increasing, becaus€
Figure 1
If X E W - E, then ~ + 21] > 2E and (b) implies that /-L(~ + 21]) = O. This gives (i). A simple calculation gives: (3)
of
{ 2Xi( -1 -
OXi =
/-L'(~
+ 21]))
2xi(1 - 2J/(~ + 21]))
if 1 ::; i ::; Aif A- + 1 ::; i ::; n,
By (a) we have
(4)
-1 - Jt'(~
+ 21]) < 0,
1 - 2/-L'(~ + 21]) > O.
It follows from (3) and (4) that 0 is the only critical point of F. By (c), F coincides with f - /-L(O) on a neighborhood of 0 where /-L(O) > E. This shows that F is a Morse function and that (ii) is satisfied.
Since Jt(t) 2: 0 by (a) and (b), the formulas (1) and (2) show that F(x) ::; f(x) for all x E W. Hence f-l(( -00, a + E)) <:: F- 1(( -00, a + E)) (5)
f- 1((-00,o, - E)) <::
F- 1((-00,o, - f)).
< a+E and f(x) 2: o,+f. By Equations (]), (2) and (b) we conclude that ~ + 21] < 2f. This implies 1] < f, which contradicts (1), because f(x) ~ 0,+17 < o,+f. This proves (iii). Analogously one sees that (iv) is satisfied for the open set U defined by
If (iii) were false, there would exist an x E W with F(x)
(6)
U = {x E W
I ~ + 21] < 2f and F(x) < a -
f}.
We shall show that U is contractible. By (ii) we have 0 E U. Let i; E U be a point of the form i; = (Xl, ... , x>-, 0, ... ,0) and let 'PI: [0, 1] --+ R be the function 'Pl(t) = F(tx). By Equations (3) and (4) we have
),
of
'P~(t) = LXiax.(ti;)::; 0, i=l
If 0 ::; t::; 1 then F(tx we conclude that U con 6.5 and Lemma 6.2 irtlJ It remains to construct Equations (5), (6) and
B = {x E M
= {x If A- = 0 then B
E
M
= 0 all
define the open set V
v=
{(S,XMb""
To see that V is conti contains the line segme where So = ~(Jf + ..;:
'lJ: 5>--1 x V where y E 5>--1
<:: R),
t
so that 'PI is decreasing. If 0 ::; t ::; 1 then
F(tx) = 'P1(t) ::; 'P1(0)
'P~(
= F(O) < a -E.
Some of the sets introd that Figure 2 only dispJ in both the .,jl, axis an
...
236
n
C.
PRom'
c
OF LEMMAS 12.12 AND 12.13
> 2. In particular E is represented by the quarter ellipse between ,jE and ~.
..;r,
and by Lemma C.l.(iii)
F- 1 ((-( F- 1 (( -(
E:D U:
w
ESSl
We know from Lemma furthermore that F(Pi) < points, then
B;~
F=a+s
and hence [a - E, a + E] d (iv) of Lemma 12.13 foll
yF.,
Lemma 12.12 is a conse later in this appendix.
Theorem C.2 Let N n b, a smooth function withol with f(N) ~ J and SU( interval [a, b] C J. Ther; Qn-l and a dijfeomorpJ So
~
Figurc 2 (U hatched: r1((-oo, a - c)) n U double-hatched)
such that
f
a <1>:
Q
X
J
Proof of Lemma 12.12. Proof of Lemma 12.13. By Theorem 12.6 we can choose a smooth chart h( vVi - 7 Wi ~ IR n with Pi E TVi, hi(p) = 0 and such that fa hi1(x) = a. -
Ai
71
j=l
j=Ai+l
2:::1:; + 2::
xJ
for x E Wi.
The coordinate patches Wi (1 :::; i :::; r) can be assumed to be mutually disjoint. Choose E > 0 such that a is the only critical value in [a - E, a + E] and such that all W,: contain the closed ball "fiED 71 • We apply Lemma C.l to fa hi 1 : Wi - 7 I~ and this E. We obtain new Morse functions F i : Wi - 7 IR with 1 :::; i :::; r, and an open set Ui ~ Wi such that hi (Ui) is the open contractible subset of IR n given by Lemma c.l. Thus we have satisfied Lemma 12.13.(i) and (ii), and Lemma 12.13.(iii) follows from assertion (v) of Lemma c.l. By assertion (i) we get a Morse function
F:M
-7
R
F(q) = {Fi a hi(q) f(q)
if q E Wi if q ~ U~=l Wi
J = (C1, C2) does not ( we can apply Theorem manifold Q and a diffe< q E Q, t E J. Conside is the identity map outs construct the diffeomor
\lip: M
--+
M;
If a E J then '1' p maI choose p so that p(al) (7)
where 9 E C~(R, R)
supp(g) ~ J, g(x) >
!
'b
~
':--;".
j.;::i'i"
.'-~- >::_~j .•- .~-,.. . . . . .
~ -".-:.
238
.-.
c
C. PROOF OF LEMMAS 12.12 AND 12.13
Now 9 can be constructed easily via Corollary A.4. See Figure 3 below.
o
Lemma C.S Assume PO .
(i) f-l(tO) is a comj (ii) There exists an c (to - 8, to + 8) ~ f~'
where W is an ( conditions are sa
I
i
(a) o(p, to) = j (b) f 0 <1>0 is the (c) For fixed p E
I
call W a pre Cl
~2
11 8 \
\
l
·1+
(
I I I
C2
Proof. Since to is a regul n - 1 (see Exercise 9.6: Choose a Coo -chart h: V
) ,---
-~
h(L Figure 3
Remark C.3 In the proof above, we proved more than claimed in Lemma 12.12. Indeed, we found a diffeomorphism III p of M to itself for which
Let us write h*(X IU ) = I an integral curve of X I system of differential eq (9)
IlIp(M(ad) = M(a2)' Let Ps: J ---t J be the map Ps(t) = sp(t)+(l - s)t, where s E [0,1]. Then IlIp.(p) is smooth as a function of both sand p. Moreover, every III p. is a diffeomorphism of M onto itself. This gives a so-called isotopy from III po = idM to III Pl = Wp' It remains to prove Theorem C.2. We first prove a few lemmas. Lemma C.4 There exists a smooth tangent vector field X on N n such that dpf(X(p)) = 1 for all p E M. Proof. We use Lemma 12.8 to find a gradient-like vector field Y on M. Now p(p) = dpf(Y(p)) > a and we can choose X(p) = p(p)-ly(p). 0
xj(t)
For Y = (Y2,' .. ,Yn) E Ii solution x(t): I(y) ---t V' to with boundary condit The general theory of 0 smooth as a function of R n - 1 with center at 0, a such that
(a) If Y E D then tt
((3) If y E D then :z
Then Wo = h-1(D) is ~ define a smooth map
o/(t)
=
By Lemma C.4 and (8) we have -itf for a constant c.
0
---t
N for X. They are smooth on
X(a(t)). a(t)
= 1, which gives
f
0
a(t)
=t+c
Now (a) implies (c) al preceeding the lemma. Tpof-1(to). By (c) X(
Hence D(po,to)
'G
'-"
240
C. PROOF OF LEMMAS 12.12 AND 12.13
Lemma C.6 Let Po be a point of N such that j(po) = to. There exists a uniquely determined integral curve 0;: J ----t N ofX with O;(to) = Po. Moreover, j oa(t) = t (t E J). Proof. The existence and uniqueness theorem for the system of equations in (9) shows that for suitable open interval I around to there exists a uniquely determined integral curve 0': I ----t N with a(to) = Po. The remark prior to Lemma C.S shows that j 0 a( t) = t if tEl. Hence I ~ J. Assume that I has a right endpoint tl E J. Since j-l([to, tID is compact, we can find a sequence Sm in [to, tr) that converges towards tl, such that the sequence (a(sm)) in N converges towards a point PI E j-l(ir). We can apply Lemma C.5 to find a product neighborhood WI of Pl. There will be some Sm with a(sm) E WI, but now we are in a position to apply the uniqueness of the integral curve, which starts at a( 8 m ), to extend a slightly past tl. The analogous extension can be performed if I has a left endpoint t2 E J. In total we see, again 0 via local uniqueness, that 0; can be extended uniquely to all of J. The image a( J) is a I-dimensional smooth submanifold of Nand a( J) intersects every fiber j-l (t) in exactly one point. For two points tlo t2 E J we can define
----t
j-l(t2)
by mapping P E j-l (tr) to the point of intersection between j-l(t2) and the integral curve through p. We have
Lemma C.7 The maps
are dijfeomorphisms.
Proof. It is sufficient to show the smoothness of il. An arbitrary PI E j-I(il) determines an integral curve a: J ----t N with a(tl) = Pl. We can find a subdivision tl = So < Sl < S2 < ... < Sk-l < Sk = t2 and product neighborhoods Wi, 1 :S i :S k, of the type of Lemma C.S, such that
a([Si-l, siD ~ Wi
(1:S i :S k).
For each of these we have a diffeomorphism, analogous to
----t
j-l(Si)
is smooth in a neighborhood of a(si-r). Note that
o
Lemma e.S Let to E P E j-l(t). Then 11' is , Proof. Let Wo, Wand l and 11' is smooth on W. j-l (tl) by 11'1 (p) = tpt,t1 of PI and since 11' =
j
is smooth. Let pEN.
It follows from
Lemma~
mapped isomorphically c D p W is an isomorphism that W is a diffeomorph diffeomorphism.
o
.,
l~ ~...I::'J • ...i .0
·N
: :
1:10 ,~ '.~
;rJ
,
e,.>'
..c ~'~' .
--.~-
. '..c::: ~. ~. ~
,,~
~
'l::l.,'~ n. ~ '<~ ...I::'J ~'.~ ....
~
~ti' .'Q6 -J!'IQ,
--,.~
et;l
244
D.
whenever
Wi, Tj E
.
.~
EXERCISES
Alt 1 (V), and (W,T)
Let {bt, that
1· ~·"'· ·
""
"
2.10. Let V be a 4-dim Let A ::::: (aij) be 1
1
= (C (w),C (T)).
... ,bn } be an orthonormal basis of V, and let {(3O'(l) 1\ ... 1\ (3O'(p) 10" E
(3j
= i(b j ). Show Show that
S(p, n - p) }
is an orthonormal basis of AltP(V). 2.6. Suppose wE AltP(V). Let V1,""V p be vectors in V and let A = (aij) be a p x p matrix. Show that for Wi =
P
I: aijVj
(1 :=:; i :=:; p) we have
j=l
W(W1,""
wp ) = det AW(V1,"" v p ). be the map
(Try p = 2 first.) 2.7. Show for f: V ---t W that p Alt +q (J)(W11\ W2) ::::: AltP(J)(wI) 1\ Alt q (J)(W2) ,
where WI E AltP(W), 2.8. Show that the set
Say a: 1\ a: ::::: .\.f1 2.11. Let V be an n-dil1 element vol E Al
W2
Show that the
It
F:::::: (
E Altq(W).
{f E End(V) 13g E GL(V): gfg-1 a diagonal matrix} is everywhere dense in End(V), assuming that V is a finite-dimensional complex vector space. 2.9. Let V be an n-dimensional vector space with inner product ( ,). From Exercise 2.5 we obtain an inner product on AltP(V) for all p, in particular on Altn(V). A volume element of V is a unit vector vol E Altn(V). Hodge's star operator *: AltP(V) ---t Altn-p(V) is defined by the equation (*W, T)vol = W 1\ T. Show that * is well-defined and linear. Let {ell"" en} be an orthonormal basis of V with vol(q, ... , en) ::::: 1 and {fl, ... , fn} the dual orthonormal basis of Alt 1 (V). Show that
is adjoint to Fv thonorma! basis (orthonormal) ba
F:
(f1 1\ ... 1\ tp
Show that FvF; (Hint: Suppose I the special case 2.12. Let V be an n-di the existence of
for w E Altn(V
*(q 1\ ... 1\ f p ) = f p +1 1\ ... 1\ f n and in general that
*(f a (l) 1\ ... 1\ fa(p))
= sign( 0") f a (p+1) 1\ ... 1\ fO'(n)
with 0" E S(p, n - p). Show that
* 0 * : : : (_l)p(n- p) on
AltP(V).
for linear maps Prove that d(f) (Hint: Pick a 1:: Altn(V) and e' the matrix for
..,.
246
D.
EXERCISES
(Hint: Try the case where I = (il,'" A p-form w E S1P(
3.1. Show for an open set in RZ that the de Rham complex
o ---+ nO(U) ---+ nl(U) ---+ nZ(U) ---+ 0 is isomorphic to the the complex
maps harmonic fo 3.4. Let Altp(R m , C) b
0---+ COO(U,~) g~d COO(U, RZ ) ~ COO(U,~) ---+ O.
Analogously, show that for an open set in ~3 the de Rham complex is isomorphic to
0---+ COO(U, R) g~ COO(U, R3) ~ COO(U, ~3) ~ COO(U,~)
(p factors). Note
---+ 0
defined in Chapter 1. 3.2. Let U ~ Rn be an open set and dXI, . .. ,dxn the usual constant I-forms (dXi = Ei). Let vol = dXI 1\ ... 1\ dX n E nn(u). Use *: Altp(R n ) ---+ Altn-p(Rn) (from Exercise 2.9) to define a linear operator (Hodge's star operator)
where Re w E AI: Extend the wedge
AltP(R and show that we
3.5. Introduce C-valut
*: np(U) ---+ nn-p(U)
(see Exercise 3.4
and show that *(dXI 1\ ... 1\ dXp) = dXp+l 1\ ... 1\ dX n and (-It(n- p). Define d*:Ql(U) ---+ np-I(U) by
* *
!
0
Note that wEn
d* (w) = (-1 P+n- 1 * 0 d 0 *(w ).
t
where Rew
Show that d* 0 d* = O. Verify the formula
d*(JdxI
1\ ... 1\
~ ·af dxp) = L.,,( -1)) ax' dXI j=l
1\ ... 1\
1\ ... 1\
dXip)
=
t
1\ .. . 1\
dxp
)
and more generally for 1 :::; il < iz < ... < i p
d*(Jdxil
dXj
(-It
v=l
:::;
n that
:~ dXil 1\ ... 1\ d:l:iv 1\ ... 1\ dXip'
and show that T Generalize Thea
3.6. Take U = C - . the inclusion m~ Show that
v
3.3. With the notation of Exercise 3.2, the Laplace operator~: np(U) ---+ np(U) is defined by ~ =
En
d 0 d*
+ d* 0
d.
Let f E nO(U). Show that ~(JdXI 1\ ... 1\ dX p) = ~(J)dXI 1\ ... 1\ dx p where
-~(J) =
aZf axr
+ ... +
aZf ax~'
where r : U ---+
In (Observe that Example 1.2.)
3.7. Prove for the c
dz e)(
~.
:,,;
~1
~
~
'" ",<
~
?,j;;)b _ 3
248
D.
EXERCISES
4.] . Consider a commutative diagram of vector spaces and linear maps with exact rows
dO
4.4. Let 0 --t AO --t Al that dimRAi < 00.
A 1- A 2 - A3 - A 4 - A 5
l~
lh
lh
lh
lh
B 1 - B 2 - B 3 - B4 - B 5 Suppose that 14 is injective, h is surjective and 12 is injective. Show that 13 is injective. Suppose that 12 is surjective, 14 is surjective and 15 is injective. Show that 13 is surjective. In particular we have that if h, 12, 14 and 15 are isomorphisms, then 13 is an isomorphism. (This assertion is called the 5-lemma.) 4.2. Consider the following commutative diagram
0 - A 1 - A 2 - A3 - O
. lil
lh
Cok h --t Cok 12 --t Cok 13 --t O.
(Hint: Try the long exact cohomology sequence).
4.3. In the commutative diagram
--t
000
0
! ! A! ,o _ ... ! ! !
A 1,1 _ A ,1 _ A ,1 - ' .. ! A! A! ,2 _ ... A
0 - AO,o _ A 1,o _ A2,o _
,2 _
!
0 - A O,3 _
!
2
1,2 _
!
dimR Ai - dil
Show that X(A*) =
2,2 _
!
A 2 ,3 _
!
!
3
!
A 3 ,3 - "
•
!
the horizontal (A*,q) and the vertical (AP'*) are chain complexes where AP,q = 0 if either p < 0 or q < O. Suppose that
HP(A*,q) = 0 for q FO and all p Hq(AP'*) = 0 for p FO and all q. --t
an exact sequence o --t Ke --t
Co
5.1. Adopt the notation described in terms arg1 E nO(V1) be arg1 smooth?). Define similarly al and prove the exis
3
3
A 1,3 _
Construct isomorphisms HP(A*'O)
is exact and conch
lh
where the rows are exact sequences. Show that there exists a exact sequence o --t Ker h --t Ker 12 --t Ker 13 --t
,l _
o--t R
4.5. Associate to two c
0 - B 1 - B2 - B3 - 0
! ! 0 - AO ! 0 - AO
Show that X(A*) =
HP(Ao,*) for all p.
'TIU,
Show that the con
aD
carries the locally lower half-planes 5.2. Show that the 1-f( of Exercise 3.6 aJ 5.3. Can R2 be writtel such that V n V . 5.4. (Phragmen-Brou\\ A ~ Rn is said to connected compol Let A and B be points p and q in p from q, then A I VI = Rn - A, L
.-
~ ,.J
..,
~ .. ~ )~. ~
"
.
.,<:
~.~
250
D.
EXERCISES
6.1. Show that "homotopy equivalence" is an equivalence relation in the class of topological spaces. 6.2. Show that all continuous maps f: U ---+ V that are homotopic to a constant map induce the O-map 1*: HP(V) ---+ HP(U) for P > O. 6.3. Let PI. ... ,Pk be k different points in IR n , n ~ 2. Show that Hd(lRn_{pI. ... ,Pk})~
{
IRk
for d = n - 1
IR
ford=O
o
otherwise.
6.4. Suppose f, g: X ---+ sn are two continuous maps, such that f(x) and g(x) are never antipodal. Show that f ~ g. Show that every non-surjective continuous map f: X ---+ sn is homotopic to a constant map. 6.5. Show that sn-l is homotopy equivalent to Rn - {O}. Show that two continuous maps
fa,
II: Rn
-
{O}
---+
IR n
-
{O}
are homotopic if and only if their restrictions to sn-l are homotopic. 6.6. Show that sn-l is not contractible. 7.1. Show that Rn does not contain a subset homeomorphic to V m when m > n.
7.2.
Let I; <:: IRn be homeomorpic to Sk (1 ~ k ~ n - 2). Show that HP(lRn _ I;) ~ {IR
o
for P ~ 0, n - k - 1, n - 1 otherwIse.
7.3. Show that there is no continuous map g: V n ---+ sn-l with glsn-l ~ idsn-l. 7.4. Let f: V n ---+ IR n be a continuous map, and let r E (0,1) be given. Suppose for all x E sn-l that Ilf(x) - xii::; 1 - r. Show that 1m f(V n ) contains the closed disc with radius r and center O. (Hint: Modify the proof of Brouwer's fixed point theorem and use Exer cises 6.4 and 7.3.) 7.5. Assume given two injective continuous maps 0:, [3: [0, 1] ---+ V 2 such that
0:(0) = (-1,0), [3(0)
=
(0, -1),
0:(1) = (1,0) [3(1) = (0,1).
Prove that the curves 0: and [3 intersect (apply both parts of Theorem 7.10). 8.1. Fill in the details of Remark 8.2. 8.2. Let
8.3. Suppose that M dimensional topl locally flat in R 8.4. Set Tn = Rn II respect to vecto equip Tn with t 1/"-l(W) is opel Show that Tn i: dimensional tOll 1/" becomes smo mapped diffeorn diffeomorphic tl ,
2
8.5. Define A: R
---l
smooth map A: 8.4.) Show that for all q E T 2 Let K 2 be the the quotient tOI (Klein's bottle) Construct a difJ 8.6. Let Po E sn be is diffeomorphi {po} ---+ Rn tha line through po 1
9.1. Let M <:: R pERL and Po Show that p
9.2. A smooth map sive at p E M
is injective. Sh h(p) = 0, and ko
in a neighborl (Hint: Reduce open neighbor
<
. ,."J
~.•~
~
;'--'"
"'
252
D.
~
~
.~
EXERCISES
is an invertible m x m matrix. Apply the inverse function theorem to F: W x IR n -
m
---t
IR n ;
where At is the tra identity matrix is ex A E M and B E J
F(xI, ... , x n ) = ('PI (XI, ... , x m ), ... , 'Pm(XI, ... , x m ), Xm+l,"" x n ).)
9.3. The smooth map 'P: M m ---t N n from Exercise 9.2 is an immersion, when it is immersive at every p E M. We say that 'P is closed if 'P(A) is closed in N n for every closed set A ~ Mm. Show that an injective closed immersion is a smooth embedding. 9.4. Let w be an irrational real number. Using the notation of Exercise 8.4, define the map a: IR ---t T2 by a( t) = 7r( t, wt). Show that a is an injective immersion and that the image a(R) is everywhere dense in T2. Conclude that a is not a smooth embedding. (Hint: The additive group 1. + 1.w ~ IR is dense in R.)
(Hint: Use the CUI" Apply Exercise 9.6
Mn(R). 9.10. A Lie group G is a: IL:C
and
are smooth. Show 1 Lie group. (Apply
9.5. A smooth map 'P: Mm ---t N n between smooth manifolds is called sub 9.11. Let 'P: M mersive at p E M, when
Dp'P: TpM
---t
-t
N be
I
TqN, q = 'P(p) is a chain map.
is surjective. Show that there exist smooth charts (U, h) in M with p E U, h(p) = 0, and (V, k) in N with q E V, k(q) = 0, such that
k 0 'P
0
h-l(xI, ... , x m )
= (XI, .. . , x n ).
(Hint: Imitate Exercise 9.2.) 9.6. Suppose 'P: M m ---t N n is a smooth map between smooth manifolds. Let q E Nn and assume that 'P is submersive at every point of the fiber 'P-I(q) (see Exercise 9.5). Show that 'P-I(q) is an (m - n)-dimensional differentiable submanifold of M. Note that the result holds for all non empty fibers 'P- I (q), when 'P is a submersion, i.e. 'P is submersive at every p E M. 9.7. Construct a smooth embedding of the n-dimensional torus Tn in Rn+l. 9.8. Let aI, a2, a3 be three distinct real numbers and define f: R3
---t'R4
by
n·
f(XI,X2,X3) = (X2 X3,XIX3,xlx2,alxi +a2x~ +a3 X
The restriction fl52 takes the same values at antipodal points and therefore f induces a map IRp2 ---t 1R4 . Show that j is a smooth embedding. 9.9. In the vector space M = Mn(R) of real-valued n x n matrices we have the subspace of symmetric matrices Sn. Define a smooth map 'P: M ---t Sn by
f:
'P(A) = AtA,
9.12. The usual inner pro Exercise 2.5). She W(Vl,"" vn ) = ±J 9.13. Show that Klein's
9.14. Let M n be a Riema vector field grad}
for all v E TpM. ~ have
where aj E COO (1 equations n
2:9 i=l
Show that the rna Let p E M with neighborhood of p gradpf is a norm
.~ ~
.".J .~.
;-....
.....
~
,
" .....
'-~~.-'''
.~
';";'~~'<
254
D.
EXERCISES
9.15. Let M be a smooth n-dimensional manifold and let if denote the set of pairs (p, op), where p E M and op is either of the two orientations of TpM. The projection 7f: if -> M sends (p, op) to p. For an open oriented set W <;;; M with orientation form w E nn(w) we let W <;;; if be the set of pairs (p, op), where pEW and op is the orientation of TpM determined by wp E AltnTpM. Show that if has a topology such that W is open and 7f maps W homeomorphically onto W for every open oriented set W <;;; M. Note that if is a topological manifold. Show that if has a uniquely determined differentiable structure such that 7f maps W diffeomorphically onto W for every oriented open set W ~ M. Show that if has a canonical orientation. The pair consisting of if and 7f is called the oriented double covering of M. 9.16. Let M be a connected smooth manifold. Show that if (see Exercise 9.15) consists of at most two connected components, and that M is orientab1e if and only if iiI is not connected. 9.17. Let V <;;; Rn+k be an open tubular neighborhood of the smooth submanifold M n ~ R n + k with the associated projection r; V -> M (see Theorem 9.23). Define a smooth map f : V - M -> R by
f(x) Show that
= Ilx -
r(x)11
= yEM min IIx - yll·
f is a solution to the differential equation n+k
L
(~)
2
(x
cp(x)Y(r(x)) = x - r(x)
L
j=1
(
Ocp)
2
=
(n Show that 7f*
map~
(n*(M)+l d).
She
maps Hk(M) iso Hk(A).
10.1. Let 7f: R2
-4
T2 b
VI
= 7f(
Show that VIand 2 connected camp that VI U V2 = '] Use the exact Ma~ HO(T 2)
of T 2 which are induced by R. SI
n1 (
= 1.
induces an isomo independent of a 10.3. Using the notatio n-dimensional tOI
OXj
Cj
9.18. Let 7f: if -> M be the oriented double covering of Exercise 9.15. Let A: if -> if be the map that for p E M interchanges the two points in 7f-1(p). Show that A is a diffeomorphism of order 2 and that
nr(M)
Show that the de ] sum of two subco
E V).
Show that cp is a smooth solution to the eikonal equation 1
is
01 = 7f(R
(the eikonal equation from geometrical optics). Suppose that k = 1, and that M n is oriented by the Gauss map Y. We can define the signed distance from M, cp: V -> R by requiring
n+
nr(Mh
10.2. In the notation of
= 1
aXj
j=1
where
nr(M)+
EEl nr(ifL,
where s is in th~ orientation. Let (
..
< '" ~ ~ ,i~ ....
'-.
"
.
~2
:~§~~
256
D.
EXERCISES
Show that W is exact.
(Hint: Find an f E coo(R n , R) such that df = 7r*(w), and show that f is
periodic with period 1 in all n variables.)
Also show that the map
Ql(T n )
-+
Rn ;
i"~""J.\s~
~ (f
W
JCI
induces an isomorphism H1(Tn)
-+
f
W,o .• ,
JCn
is an isometry for al Show (by means of positive measure J-LM
JM
w)
Assume that f E C~ subset W ~ M. She
Rn .)
lOA. Show that for every connected compact, non-orientable smooth n dimensional manifold M we have that Hn(M) = O. (Hint: Use Exercise 9.18.) 10.10. Define ir Rn - 1 ----+ ~ 10.5. Calculate the de Rham cohomology of Klein's bottle. (Hint: The oriented double covering can be identified with the map ij(Xl, .. T 2 -+ K2 from Exercise 8.5.) with 0 at the j-th er 10.6. (Partial integration). Let R be a compact domain with smooth boundary
in an oriented n-dimensional smooth manifold M. E QP-l(M), T E Qn-p(M) we have
Show that for for
Yn =
W
rdw JR
1\
T
=
rw JaR
1\
T
+ (-l)P
r JR
W
and 1\ dT.
10.7. (Divergence theorem) Suppose that R is a compact domain with smooth boundary in R3 . Let N: oR -+ 52 be the outward directed Gauss map and let F E Coo (U, R3) be a smooth vector field on an open set U ~ R3 with R ~ U. Show that
r
JR
div F dJ-LR3 =
f
JaR
(F, N)dJ-LaR.
(Hint: Consider W E Q2(U) given by Wp(Wl' W2) = det (F(p),
WI, W2).)
10.8. (Classical Stokes) Let 5 ~ be a regular surface, oriented by the Gauss map N: 5 -+ 52, and let R ~ S be a compact domain with smooth boundary oR. Along oR we have a unit vector field V pointing in the positive direction. Let F E Coo (U, R3 ) be a smooth vector field on an open set U ~ R3 with S ~ U. Show that
r (rot F, N)dJ-LS = JaR r {F, V)dJ-LaR'
JR
10.9. Let M n be a Riemannian manifold and 7r: if -+ M the oriented double covering from Exercise 9.15. Let M be given a Riemannian metric such that -+
T-rr(q)M
E nn-l(Rn )
"
in 10.11. Let w E ff(M n ).
:
for every oriented Sl Show that dw = O.
R3
D q7r: TqM
W
10.12. Let P be a smooth
a linear operator su
for every w E nn-I R in an n-dimensic 10.13. Let M be a smooth parametrized curve
a=
..~
.~
0
J~! ~. ... q ....~.,~.~. '..-' ..~ . . .'.'.1:) .. . ,"ot:l ~
;~
'1.. . •.~.
· . . ~i~\~
~. . .
.a
"
~
.,"",!,,:~·~n~~
258
D.
EXERCISES
11.1. Generalize the COl a way that Coroll Show that Coroll. Corollary 11.10 at maps.
such that the restrictions 0Wi-l,til (1 :S i :S k) are continuously differen tiable (from one side at the endpoints). For W E Ql(M) we define the path integral of w along 0: by
1
w=
Q
L i==l
kit.
'w(o:'(t))dt.
ti-l
Suppose that
lw=o for every closed piecewise e1-curve
on M. Show that
0:
11.2. Prove the formula (i) j is conti (ii) j is smoc (iii) p E M (Hint: Lemma 11 a homotopy F fn identity near the 1 n 11.3. Let N be an ori integer occurs as t 11.2.) 11.4. (The fundamental
wis exact.
10.14. (Green's identity) Show for real-valued functions on an open set U S;;; IR n that d(J /\ *dg) = ((gradj,gradgl- j6.g)dXl/\"'/\ dX n (see Exercises 3.2 and 3.3 for definitions of * and 6.).
Conclude for a compact domain with smooth boundary R S;;; U that
1 1
1 1
-1 1 -1
j /\ *dg = (gradj, gradgldf.Ln j 6.g df.Ln, M M R
where f.Ln is the Lebesgue measure on IR n. Derive the identity
j 6.g df.Ln -
R
10.15. Define p: IR n - {O}
g6.j df.Ln =
R
--+
9 /\ *dj
8R
j /\ *dg.
8R
I
R for n 2: 2 by p(x) = ¢(llxll)
where ¢(r) for r > 0 is given by logr ¢(r) = { _1_ 2_nr 2-n
if n = 2 Of > 3 1 n _ .
Show for the closed form w defined by equation (19) of Example 9.18 that *dp = w, and conclude that p is harmonic (i.e. 6.p = 0). Apply the last identity of Exercise 10.14 to j = p, 9 E ego (IR n , R) and R E = {x E IR n I E:S Ilxll :S a}, where E > 0 and a is suitably large, to obtain
f p6.g = JRn-EDn
1
ES n - 1
p /\
where ED is the open disc of radius sphere. Show that n
lim E-->O+
1
p /\ *dg = 0
and
ESn-1
E
-1
ES n
1
lim! ESn-1 n 1 ES is E1 - n
n 1
ES -
Show that deg(j deg(fr) = O. Pre
F(w
defines a homot< and conclude ff(
9 /\ w,
around 0 and
is a complex pol) leads to a contrac define for any r
n
is its boundary
11.5. Let ~ S;;; IR be and let Ul, U2 t theorem 7.10. I
9 /\ w = Vol(sn-l )g(O).
c-->O+
(Hint: The pull-back of w to
-1 \ g(O) = .. 7_' (rt",_l for every 9 integral.
* dg
p
f
E Cgo(lR n ,
q,
l
Rn
VoIEsn-l.) Conclude that Show that
p6.g df.Ln
where the right hand side is the Lebesgue where the sign
. Fj,<, 'ir!·.~ :a:•• '~:;;,""' : " ..
'l::3.
• -:
~
~.:;,
·'01~~~~l1~~'
"'.
~
- .:" ·1·.·· .. '.···········.·.· . .· · · · · · · ·.· · ..
.
13;
--
.....
~
~2
260
D.
11.6. Prove the case n
,
)
-I .. ~
EXERCISES
< m of Theorem
11.5 (reduce to the case of equal
12. 1. Show that f: Rn
dimensions). Apply this to show that every map f: N n -;. from a smooth manifold of dimension n < m is homotopic to a constant.
sm
11.7. Let U
~ ~n
be a bounded open set and A a compact subset of U. Prove the existence of a compact domain with smooth boundary R such that A ~ R ~ U.
(Hint: Try R = 1jJ-l ([C, 00 )) where 1jJ is given by Lemma A. 7 and c > 0
is a regular value of 1jJ.)
11.8. Let B be the set of boundary points of a bounded open set U ~ ~n (n ~ 2),
and let F: B -;. ~n - {O} be a continuous vector field on B without zeros. Show the existence of X: U u B -;. ~n with the following properties: (a) X is a continuous extension of F. (b) X is smooth on U. (c) X has only finitely many zeros in U. (Hint: (a) and (b) can be achieved by the proof of Lemma 7.4, and (c) from the proof of Lemma 11.25.)
Prove that the integer
,(F, U) =
L
i(X;p)
pEU,X(p)=o
is independent of the choice of X satisfying (a), (b), (c).
(Hint: Use Exercise 11.7 and Theorem 11.22.)
Find an example in ~2 to show that ,( F, U) actually depends on U and
not only on Band F.
11.9. Suppose under the assumptions of Exercise 11.8 that F: B -;. ~n homotopic to G: B -;. ~n - {O}. Prove that ,(F, U) = ,(G, U).
is a Morse functi< Show with the nc on Tn such that index ;\ for f is
f be a Morse of grad(J) are nc 12.3. Show for any sm, (non-linear in gel
12.2. Let
such that
dP2 whenever a: (-8,
12.4. Let f be a MOl least two critical of index l. (Hint: With n dimH°(1Vl(bj)) , 12.5. Is it possible for both of index 07 12.6. With the notation
{O} is
Wv
11.10. With U and B given as in Exercise 11.8 assume F: U U B -;. IR n to be
Show that W v h: of the following
-
continuous without any zeros on B. Prove that if ,(FIB, U) i 0, then F has at least one zero in U. (Hint: Otherwise find X with properties (a) and (b) of Exercise 11.8 such that X has no zeros in U.) 11.11. Show that the condition defining a non-degenerate zero of a vector field is
independent of the choice of chart. 11.12. Let F E C oo (1R 1\Rn) and G E coo(lRm,~m) be vector fields both with the origin as the only zero. Show that F x G is a vector field on Rn + m
with the same property, and that
i(F
X
G; 0) = i(F; O)i(G; 0).
Case 1: dim HPI Case 2: dim HPI
12.7. (The Morse in
dimensional de by
J 1~- 4
,
't;
~.
262
D.
EXERCISES
where bj is the j-th Betti number of M, bj = dimRHj(M). Let f: M ---* IR be a Morse function with c>. critical points of index>. (c>. < 00). Define the polynomial n
CM(t)
= Clvf,j(t) =
L c>.t>'.
12.10. Let 7r: Mn - M~ i.e. 1\In can be c( a disjoint union ( diffeomorphism fc 12.11. Assume f and 9 respectively. Shol
>'=0
Prove for any closed manifold M that
CM(t) - PM(t)
=
(t + l)RM(t),
where RM(t) is a polynomial with non-negative integral coefficients.
(Hint: Prove by induction a similar statement for each set W v introduced
in Exercise 12.6.)
Derive the Morse inequalities
j
L
j
(-l)j-k Cj ;:::
k=O
L
(-l)j-k bj
(0::; j ::; n).
k=O
Observe that the Morse inequalities imply Cj ~
bj
(0::; j ::; n).
= Wi 1
J\ ... J\ Wi p E
13.1. A symplectic spa, nating 2-form W non-degenerate il w( e, f) -I- O. Ass mension. Let W S;;; V be a W..L=
12.8. (Morse's lacunary principle) Continuing with the assumptions and notation of Exercise 12.7, suppose for each>. (1::; >. ::; n) that either C>'-l = 0 or c>. = O. Prove that bj = Cj for every j. 12.9. Let pri: Tn = Rn jzn ---* IRjZ be the i-th projection (see Exercise 8.4). Pick W E D 1 (lRjZ) representing a generator of H 1 (lRjZ) ~ H1(SI) ~ IR and define Wi = pr;(w) E D1(T n ). To an increasing sequence I: 1 ::; i 1 < i2 < ... < ip ::; n we associate the closed p-form WI
Describe the critit f and g. Derive I
DP(T n ).
Prove that the resulting classes [WI] E HP(T n ) are linearly independent. (Hint: Consider integrals of linear combinations L: aIwI over subtori.)
is a non-degenera {e1,h,e2,h,···
w(ei,ej)
=
This is called a ~ (Hint: Pick e1 f. where W is spa! Let WI,TI,W2,T2 symplectic basis
I
Prove with the help of Exercises 12.1 and 12.7 that dim HP(T
n
)
=
(~),
and conclude that there is an isomorphism
H*(T n ) of graded algebras.
~
Alt*(R n )
13.2. Let M n be an 2 (mod 4). She degenerate symp 13.3. Consider a smo smooth closed II then HP(f): HPI
;;;
.;,J
'l:>
~
-;,J
. . f"tJ.
~
~ 1 ~ t
•.
~ ..
".~
'.
"~.:''G'
264
D.
EXERCISES
13.4. Let (SrL, ]I;!rn) be a smooth compact manifold pair with 0 U = sn - A1 m . Construct isomorphisms
HP(U)
~
Hn-p-l(M)*
<
m
<
There are disjoint such that K ~ aj. Then
nand
u1:
(1::; P ::; n - 2).
Show that Hn(u) = 0 and find short exact sequences
o ---+ Hn-l(U) ---+ HO(M)* ---+ IR ---+ 0 o ---+ IR ---+ HO(U) ---+ Hn-l(M)* ---+ O.
(Hint: Exercise l' 14.1. A rational functio
13.5. Let 1f: M ---+ M be the oriented double covering constructed in Exercise 9.15 and define A: M ---+ ]1;1 as in Exercise 9.18. Find isomorphisms
HP(Af)
---+
where P and Q a with the roots of . the Riemann sphe
(H:;-P(M)_I)*
where Hg(M)-l denotes the (-I)-eigenspace of A* on Hg(M). 13.6. Compute Hn(M n ) for every smooth connected n-dimensional manifold Af n . (Hint: Use Exercise 13.5. The answer depends on whether M is compact or not, and also whether lvf is orientable or not,) 13.7. Prove that Hg (M) for every smooth manifold and every q is at most countably generated. (Hint: Induction on open sets.)
13.8. Show that any de Rham cohomology space HP(M) is either finite dimensional or isomorphic to a product n~=llR of countably many copies of IR. 13.9. A compact set K ~ IR n is said to be cellular if K = D j , where each n Dj ~ IR is homeomorphic to Dn and Dj+l ~ Dj for' every j. Show for K cellular that
14.2. The n-th symmetri orbits under the ac (n factors) with tl Show that Spn(s taking
ern
into [aD, al,··· 1 a·
n;:l
HP(lRn _ K) ~ {IR
o
?'
if p = n - 1 otherwIse.
14.4. Show that any c( phism
13.10. For K ~ IR n compact denote by fIe (K, R) the vector space of locally constant functions K ---+ If'!. Construct an isomorphism 1>1: fIO(K, IR)
---+
H;(If'!n - K)
such that 1>1(1) = [dj] whe.!e j E cgo(J~n, If'!) is locally constant on an open set containing K and f extends f. Find an isomorphism 1>: H n - l (lR n
-
K)
---+
14.3. Show that HP(S algebras H* (S2 >
for p i= 0 when 14.5. Prove in the foll< constant. Suppo:
fIo(K, IR)*
such that 1>([w]) for w E Dn - l (lR n - K) can be evaluated on fIO(K, IR) by the following procedure:
is a homotopy fr to g: D 2n + 2 ---+ (
f E (1)
g(tz
...
~.
~ ~.-.~
'"
.
1_$ ............ ..
~
•...
~
l:---..i
266
D.
Define h: D 2n +2
---+
EXERCISES
cpn+1 by
h(ZO,ZI, ... ,Zn) =
[
ZO,ZI, ... ,Zn,
(1- ~Izjl n
2
)
1/2]
J=O
and observe that h maps the open disc iJ2n+2 bijectively onto Un+1 = Cpn+l - Cpn. Moreover h lS 2n+l is the composite of 1r with the inclusion j: Cpn ---+ cpn+ 1. Find f: cpn+ 1 ---+ cr n so that f 0 h = 9 and argue
that f is continuous. Observe that f 0 j = idcpn, and pass to de Rham
cohomology to obtain a contradiction.
This proves Hopf's result mentioned in Example 14.1.
14.6. Given z E C n+ 1 - {O}, p = 1r(z) E cpn, and two vectors Vj E T z C n + 1 = C n+l, j = 1,2. Let Wj = D z 1r(vj) E Tpcpn. Show that the hermitian inner product (( , ))p on Tpcpn from Lemma 14.4 satisfies
((Wl,W2))p = (1J1,V2) _ (Vl,Z)(Z,V2) (z,z) , where ( , ) denotes the usual hermitian inner product on c n+ 1 . 14.7. A symplectic manifold (AI,w) is a smooth manifold M equipped with a 2-form w E [22(M) satisfying the following conditions (i) dw = O. (ii) (TpM, wp ) is a non-degenerate symplectic space for every p E M (see Exercise 13.1).
To an increasin! A E Frm(Fn) i~ by I and the (n' Show that U[O
is an open set (n - m) x m IT
Show that h[ is with F(n-m)m)
resulting closed m-dimensional :
15.1. Show that the I trivial vector bi bundle TS3 is tr
15.2. Let G be any L is the diffeomor X e E TeG at 1 X g = DeRg(X~ on G and more~ J
Show that Cpn admits the structure of a symplectic manifold.
Show that a symplectic manifold (M, w) has even dimension 2m and that
w m = w 1\ ... 1\ w is an orientation form on !'vI.
Show for a closed symplectic manifold (M 2m , w) that H* (M) contains a
subalgebra isomorphic to lR[cJl(cm+ 1 ), where c = [wJ E H 2 (M).
14.8. (Grassmann manifolds) Fix integers 0 <m < n and let F denote either IR or C Equip F n with the usual inner product. Denote by Gm(Fn) the set of m-dimensional linear subspaces F ~ Fn. Show that Gm(Fn) can be identified with a compact subspace of the n x n-matrices over F by associating to F the orthogonal projection on F. This makes Gm(Fn ) a compact Hausdorff space (with countable basis for the topology). Denote by Frm(Fn) the set of n x m matrices over F of rank m. Observe that Frm(Fn) can be identified with an open subset of F mn and hence is a smooth manifold. The group GLm(F) acts by right multiplication on Frm(Fn). Show that the orbit space Frm(Fn)/GLm(F) with quotient topology can be identified with Gm(Fn) by associating to A E Frm(F n ) the span of its column vectors denoted [AJ E Gm(F n ).
Construct a frar invariant vector
15.3. Let ~ be a smo i.e. the orbit sp: acts by (x, t) 1 Construct a sm
where k~ deno! (Hint: Both tot
15.4. Prove that the 1 given by Lemn Construct a SIT that in Exercis
..
~jo
,....J
;,;.
J""t-> > . ~",~
1·
. q......
268
D.
;~
'\.
-"' ~ .• "
..
....
'
" ~
EXERCISES
where H n is conjugate to H n, i.e. as a real vector bundle H n is the same as H n, but z E C acts on H n the same way as z E C acts on H n. 15.5. Continuing Exercise 14.8, define the smooth map 7r: Frm(Fn) ---+ Gm(f n ) by 7r(A) = [Aj. Construct for each I: 1 ~ il < i2 < ... < i m a smooth section SI: UI ---+ 7r- I (UI) such that SI([A]) = AA[I. Show that the map kr: UI x GLm(f)
---+ 7r-
1
(UI );
vector bundle TV ( tangent bundle). Show that the ortJ respect to a Riem: vector bundle T h ( tangent bundle). Find smooth vecto
kI([A], Q) = SI([A])Q T
is a diffeomorphism (note that GLm(f) is a Lie group). Define a smooth action of GLm(F) on Frm(f n ) x pn:
(A, x)Q = (AQ, Q-Ix) and fonn the orbit space E = Fr(fn)m x GL", (F) pn with the induced projection 7f: E ---+ Gm(Fn), 7f([A, x]) = [Aj. Show that this is a smooth m-dimensional f-vector bundle in such a way that the assignment
[A]
15.6.
15.7.
15.8.
15.9.
t->
([SI([A]), el], ... , [SI([A]), enD
defines a smooth frame over UI. This is the canonical vector bundle 'Y = 'Y~'" over Gm(fn). Give an identification of the fiber 'YV over V E Gm(fn) with V, which to [A,x] E 'YV (with [Aj = V, x E fn, a column vector) assigns A.T E V. Establish a 1-1 correspondence between smooth sections S: U ---+ 7r- I (U) over a given open set U <;;;; Gm(fn) and smooth frames (SI, ... ,Sn) for 'Y over U such that (Sl(p), ... , sn(P)) for P E V via the identification 'Yp ~ V corresponds to the column vectors of S(p). Show that the canonical vector bundle constructed in Exercise 15.5 in the case m = 1 can be identified (smoothly) with the canonical line bundle over IRlP n- 1 or cpn-l (Exercise 15.3 and Example 15.2). Let ~ be an m-dimensional vector bundle over B admitting a complement 'fl, i.e. ~ ED TJ ~ EN. Construct a vector bundle homomorphism (j, f) to the m-dimensional canonical vector bundle 'Y over the Grassmannian Gm = Gm (IR N ), such that the fiber ~b maps isomorphically to a fiber in 'Y by including ~b in E{; ~ IR N . Conclude that ~ s::: 1* ('Y). Do the same for complex vector bundles. Show that any vector bundle over a compact smooth manifold B is isomorphic to a smooth vector bundle. (Apply Exercise 15.7 and pick . a homotopy from f to a smooth map.) Let 7r: E ---+ B be a smooth fiber bundle. For pEE, b = 7r(p) we have the fiber Fb through p. Observe that Fb is a smooth submanifold of E, so that TpFb can be identified with a subspace of TpE. Show that the union of these subspaces TpFb as p runs through E is the total space of a smooth
15.10. Prove Theorem IS. real vector bundles 8.11). Make use 0 Deduce the smoott ness assumption. 15.11. Let ~ be a smooth to be compact). S, A <;;;; U. Let s be a Construct for any s over B that sati
(i) s(b) = s(i
(ii) Ils(b) - s(
15.12. Let f: ~ ---+ TJ be r = rk(/b) of the] Show that the sut E E
are total sI:aces 0 and f are s 15.13. Show for a map b t-> rk(/b) is 10\ 15.14. Suppose p: ~ ---+ ~ Show that r = rk Prove that ~ s::: II ~ , TJ
16.1. Show for integer:
where d is the gI
<
".J c-,.}
"
,:
>....
'"
~.•.. ~'. ............
..
'
' •.•
'2
~
~
270
D.
EXERCISES
16.2. Prove the isomorphisms (4) listed above Lemma 16.4. 16.3. Let V be a finite-dimensional complex vector space with Hermitian inner product ( , ). Construct a hermitian inner product on Ak V satisfying (VI 1\ ... 1\
Vk, WI 1\ ... 1\ Wk)
= Jet ( (V
Q ,
Wf3) ).
16.4. Let F be a covariant functor from the category of finite-dimensional real vector spaces to itself. Assume F to be smooth in the sense that the maps induced by F HomR(V, W)
are smooth.
Construct for a given smooth real vector bundle ~ over B another smooth
real vector bundle F(O over B with fibers F(Ob = F(~b), b E B.
Show that this extends F to a covariant functor from the category of smooth
real vector bundles over B and smooth homomorphisms over idB to itself.
(Note: Many variations are possible: R can be replaced by C in the source
and/or target category, F can be contravariant, and smooth can be changed
to continuous.)
16.5. Prove that the construction of Exercise 16.4 is compatible with pull-back, i.e. construct for h: B' - 4 B a smooth vector bundle isomorphism
7/Jr;: F(h*(~))
-4
h*(F(~)).
Show that pull-back by h extends to a functor h* from the category of vector bundles over B (see Exercise 16.4) to the corresponding category over B', and that the diagram
F(h*(~)) ~ h*(F(~))
IF (h*(!l))
lh*(F(!l))
F(h*(T))) ~ h*(F(T))) commutes for every §: ~ - 4 T) over idB. 16.6. Consider two smooth covariant functors F, G as in Exercise 16.4. As sume given (for each finite-dimensional vector space V) a linear map cPv: F(V) - 4 G(V) such that the diagram
...1x- G(V)
IF(h)
F(W)
Prove finally witl diagrams
HOillR(F(V), F(W))
-4
F(V)
homomorphism vector bundle he diagram
lC(h)
...:bY- G(W)
is commutative for every h E HOillR (V, W).
Construct for a given smooth real vector bundle over B a smooth bundle
commute. 16.7. Let V be a finite linear map e:
&/
e(VI 0'" Q
Show that e 0 e : induces an isom( Apply Exercise bundles Ak(O. 16.8. For finite-dimens
carrying
(0, WI)
(VI 1\ ..
1\ ... 1\
(C
E k
Extend this to v'
16.9. Prove Lemma 1 16.10. Finish the proof 16.11. Let ~ and T) be SI Show that ~ and modules. Prove is isomorphic te (Hint: Apply E
272
D.
EXERCISES
16.12. Prove for any line bundle ~ (real or complex) that Hom (~, 0 is trivial. Prove for any real line bundle that ~ &> ~ is trivial. 16.13. Let T* M be the total space of the dual tangent bundle T Mof a smooth manifold Mn, and 71": T* M ---4 M the projection. For q E T* M (i.e. a linear form on T7r (q)M) define
Bq E Altl(Tq(T* M));
Show for X, Y E l
Suppose X and Y Remark 9.4)
Bq(X) = q(D'171"(X)).
X=
Show that this defines a differential I-form f) on T* M and moreover that B can be given in local coordinates by the expression
Show that in these
n
[X,Y] =
Liidxi' i=l
where Xl, ... , Xn are the coordinate functions on a chart U ~ M and
iI, ... ,in are linear coordinates on T; M, p E U with respect to the basis
dpXI' ... ,dpxn .
17.2. Prove for wE [;lP('
dW(Xl,"" Xp+l) =
Show that (T* M, dB) is a symplectic manifold (see Exercise 14.7).
17.1. A derivation on an R-algebra A is an R-linear map D: A the identity
---4
A that satisfies
D(xy) = (Dx)y + x(Dy).
[;lO(TM) ~ Der[;lo(M) which to a vector field X assigns the derivation Lx given by
dw(X,Y for w E [;ll(M),
17.3. Let M be a Riem. and uniqueness of satisfies the follo~
(a) X( (Y, Z)) (b) \7x Y - '\
Lx(f) = Xf = df(X). (Hint: Derivations are local operators.) Show that the commutator [Dl, D z ] = D 1 oDz-DzoD l of Dl' D z E DerA also belongs to DerA, and define the Lie bracket [X, Y] E [;lO( TM) of smooth vector fields X, Y on M by the condition L[x,Yj = [Lx, Ly].
Prove that [;lO( TM) is a Lie algebra, i.e. that the following conditions hold
+ [Z, [X, Y]]
L (-: l~i<j~p+l
In particular
These form an R-vector space Der A. Show for any smooth manifold Mm that there is a linear isomorphism
(L 1) [ , ] is bilinear.
(L2) [X, X] = O.
(L3) [X, [Y, Z]] + [Y, [Z, X]]
+
= 0 (Jacobi identity).
Prove moreover tl
2(\7x Y,Z)
(Hints: Let A(X, ness, derive the K tions obtained fn existence verify 1
where f E [;lO(M This connection i
~
"
i~~Ji1i.·.
274
D.
EXERCISES
17.4. Prove for lvI n ~ Rn+k that the connection on TM constructed in Example 17.2 is the same as the Levi-Civita connection of Exercise 17.3, when M is given the Riemannian metric induced from Rn+k. 17.5. Let \7 be any connection on the vector bundle ~ over A1. Given two vector fields X, Y E OO( TM) define the operator R(X, Y): 0°(0 ---t 0°(0 by R(X, Y) = \7x
0
\7y - \7y
0
\7(ei) =
\7x - \7[X,YJ'
Verify that R(X, Y) is a OO(M)-module homomorphism, and prove that R(X, Y) = FJ y.
(Hint: Work loc~lly using the formula (17.4). Show by direct computation
that the two operators agree on fi.)
17.6. Let [\lIn be a Riemannian manifold and \7 the Levi-Civita connection on TM from Exercise 17.3. Define R(X, Y):OO(TM) ---t OO(TM) as in Exercise 17.5. Prove the Bianchi identity R(X, Y)Z
17.8. Let \7 be a conn el, ,.. ,ele on the s ") A -- (A tJ.. ) ~ (A tJ'
+ R(Y, Z)X + R(Z, X)Y
Let ep = (¢jm) be
Show that
A=
dA - A 1\ A and
1
(Q
a
17.9. Prove Lemma 17.5
= 0
17.10. Prove formula (17. for X, Y, Z E DU(TM).
Show that the value of R(X, Y)Z at p E M depends only on X p, Yp, Zp E
TpM. Hence R defines for X p, }; E TpM a linear map R(Xp , Yp): Tplvl ---t
TpM.
17.7. Let h: V ---t V' ~ Rn be a chart in Mn and Oi = a~i the vector fields on V considered in Remark 17.4. The Christoffel symbols are the smooth functions r7j on V' detemlined by \7ai
aj =
L (rt h) Ok·
17.11. Given ~ with conn the connection mat with Prove that connection matrix Do a similar calcui: matrix B = (B rs )
e
17.12. For ~ with connecti all i, such that \7].
0
Ie
\7X(i
Prove the formula
rr.] = ~2 '""" II (ag ~
jl
for s E OO(Ai(~))
+ agil _ OYi j ),
O,'Ci
OXj
17.13. Let f: M' ---t M t vector bundle over
OXI
where (gij) is the matrix of coefficients to the first fundamental form (see below Definition 9.15) and where (glel) is the inverse matrix (9ij )-1. (Hint: Apply the Koszul identity (cf. Exercise 17.3) to ai, aj , ad Define functions RiJle on V' by
R(Oj, OIe)8i =
L
(RiJle
0
h) am
OO(M') ®nO(
where'ljJ E OO(M) (Hint: First handIt to ~ as in Exercis
rn
18.1. Prove for the cano' to H.
Show that
arm
Rr:n:/c = -'Ei t] ax . J
ar~ _J_t
ox Ie
+ '""" (r11e.rr:n1- rl..r mlel ) ~ I
t]
Jt
18.2. Show for .
comple~
"
-
<
.
~.
~.
~
\
~
'.
1>~
'"
~
276
D.
EXERCISES
Use the splitting principle to derive the formula k
Ck(~ ® TJ) = ~ (~ =~) Cr(OCl(TJ)k-r, where TJ is a complex line bundle and bundle over M.
~
any n-dimensional complex vector
18.3. Show for a complex n-dimensional vector bundle ~ that cl(An(~)) = Cl(O. (Hint: Apply the splitting principle and Exercise 16.8.) 18.4. Show for the vector bundle H.L over epn defined in Example 18.13 that
= (-l)k cl (H)k.
ck(H.L)
19.1. Let Mn be a Riemanni Let el, ... ,en be an ortl matrix (Aij), and den01 that dEi(ej,
(Hint: Use Exercises : Conclude in the case n is the Levi-Civita com
19.2. Let V be an n-dimensi, V x V ---t R is said to satisfied:
F(x,y,z,w) = (b) F(x, y, z, w) + (c) F(x,y,z,w) =
(a)
Show that n is the smallest possible dimension of a complementary vector bundle over cpn to H. 18.5. Show for the complex line bundles Hand H.L over Cpl that H.L ~ H*. (Hint: Consider H ® H.L.)
Prove that a curvature, is Riemannian, is give
18.6. Adopt the notation of Exercise 14.8. Show that any tangent vec n tor to Gm(Fn) at the "point" V ~ F can be written a'(O), where a(t) = [al(t), ... , am(t)] and aj(t) is the j-th column of a smooth curve A: (-8,8) ---t Frm(Fn) with V = [A(O)]. Prove that there is a well-defined R-linear isomorphism
(see Exercises 17.5 anl
Dv: TvGm(Fn)
---t
n
HomF(V, IF IV)
such that Dv( a' (0) ): V ---t F I V sends aj (0) into aj (0) + V.
Let, be the canonical vector bundle over Gm(Fn) (see Exercise 15.5).
Construct another vector bundle whose fiber over V is the orthogonal
n complement to V in F , and show that, EB ~ en.
Prove that the maps Dv define a smooth vector bundle isomorphism
n
,.L
'TGm(Fn)
~ HomF(r, ,.L)
18.7. (pliicker embedding) Prove that a smooth embedding of Gm(F n ) into the copy Gl(AmlF n) of RpN or CpN, N = 1, can be defined by
(;:J -
=
19.3. Show that the curvatuI a vector-space of dim
19.4. Let F be curvature-lik is equipped with an i II ~ V, show that thl
,.L
(in particular 'TGm(Cm) has a natural complex structure).
(V)
(d) F(x, y, z, w) =
K(II:
where x, y is a basis f, on the Grassmannian is constant with valm:
F(x,y,z,u 19.5. Let II ~ TpM n be a manifold. The sectior,
[al 1\ ... 1\ am],
K(p, II) where al," ., am is any basis of V.
(Hint: Tangent maps can be computed in terms of the identification
TvGm(IFn) ~ HomF(V, F n IV) from Exercise 18.6.)
=
where X p , Yp is a bas Show that (n 2':
sn
2:
'b
"
...J
.r,.;
~k~
278
D.
~-,~
EXERCISES
Rn + 1 has constant sectional curvature, Le. that K(p, IT) is independent of p E sn and the plane IT ~ TpS n . Show that F'Jp,Yp = Rp(Xp, Yp): Tpsn . . . . . TpS n acts by
20.1. Verify fonnula (3 20.2. Let M1 and M2 be have finite-dimem
HP(1 F'Jp, y,p (Zp) = k((Yp, Zp)Xp - (X p, Zp)Yp), where k is the sectional curvature (in fact k = 1, see Exercise 19.8). 19.6. Show for the connection described in Example 19.5 and Exercise 19.1 that the Gaussian curvature K, E nO (M 2 ) satisfies
K,(p) = K(p, Tp M 2 ).
combine to give·
Eej p+q=n (Hint: Use TheOl
19.7. Show with the notation of Exercise 17.7 that
20.3. Construct a smo01 diagram (20.9) is
m
K(p, Span(81, 8 2)) = - "wm R 1122 gllg22 - g12
20.4. Show that expom
Show for dim M = 2 that this reduces to
K(p, Tp M
2
)
=
-ill Rr12'
(Hint: Use that (R(81, 82)81, 8 1 ) = 0 to eliminate R~12 from the previous expression.) Remark: Gaussian curvature K(p) of a surface S ~ R3 as defined in Example 12.18 is given in local coordinates by the same expression; see e.g. [do Carmo] page 234. 19.8. Show that the constant k in Exercise 19.5 is independent of n 2: 2. Com pute it for n = 2. (Hint: Embed sn ~ sn+l as an equatorial sphere and compare con nection matrices by applying Exercise 19.1 to an orthonormal frame e1, . .. , en, en+l, where en+l E (Tpsn)J. for p E sn.) 19.9. Prove that the Pfaffian polynomial in variables Xij 1 1 given by
Pf(Xij) =
L
~
i
<
n
sign(O")
uET(n)
IT X
U
(2v-l)a(2v) ,
v=l
where
T(n) = {O" E S(2n) I 0"(2v - 1) < 0"(2v) for 1 ~ v ~ n}. 19.10. Let
€ 1 ,""
€2n be the standard basis
Pf(€i /\
for Alt
1
(R 2n ).
€j) = 1 ·3·5··· (2n - 1)
vol.
Show that
j
~
can be used to dl
Show that the s be identified witl G2(R m ). How bundle "12 over I
20.5. A complex Strul J2 = - id. Givel
2n is
Show that GL2n structures, and tl Prove a similar s tive determinant
21.1. (The Gysin sequ over a compact sphere bundle.
. , ........... HP-1(S
where the label, pull-back by th
..:~
11 .....
280
D.
u
" -"""----,-
2
EXERCISES
(Hint: Consider the Mayer-Vietoris sequence for S(~ EB 1) covered by Uoo = S(~ EB 1) - soo(M) and Uo = S(~ EB 1) - so(M), where So is the zero section M ---+ E = E(~) ~ S(~ EB 1).) 21.2. Show for n 2: 1 that
HP (V2(R 2n +1))
~ {~
HP(V2 (R 2n+2))
~ {~
for p = 0, 4n - 1 otherwise for p = 0, 2n, 2n + 1, 4n + 1 otherwise
(Hint: Apply the Gysin sequence of Exercise 21.1 to the tangent bundle of a sphere.) 21.3. Let Mn ~ Rn+k be a compact orientable smooth submanifold with normal bundle v. Show that v is orientable with e(v) = O. (Hint: Identify the total space E(v) with a tubular neighborhood of M.) 21.4. Verify Theorem 21.13 directly for the sphere s2n with the Levi-Civita connection on TS2n by using the information given in Example 10.12 and Exercises 19.5 and 19.10. be an even-dimensional smooth real vector bundle over M. Let 21.5. Let '11/ be the vertical tangent bundle from Exercise 15.9 of S(~ EB 1) ---+ M. Show that the orientation of ~ induces a natural orientation of '11/. Let be the curvature of some metric connection on '11/. Show that
em
F\l
U
=
1[
2"
(-F\l) -
PI ~
(-F\l)]
7f*s~PI 2;
is an orientation class. 21.6. Give an alternative proof of Theorem 21.13 beginning with the computation in Exercise 21.4. Construct u as in Exercise 21.5 and show that So (u) = e(~).
References
[Bredon], G. E. Bredon: 1993 [do Canno], M. P. do ( Prentice-Hall Inc., New [Donaldson-Kronheimer: of Four-Manifolds, Oxfo
[Freedman-Quinn], M.] Princeton University Pre [Hirsch], M. W. Hirsch:
[Lang], S. Lang: Algebr
[Massey], W. S. Massey & World Inc. 1967.
[Milnor], 1. Milnor: MOl
[Milnor-Stasheffl, 1. Mi Math. studies, No 76, P [Moise], E. E. Moise: Verlag, New York 197~ [Rudin], W. Rudin: Ret<
[Rushing], T. B. Rushill 1973.
[Whitney], H. Whitney Press, New York 1957.
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284
Euler class 195
exact p-form 19
exact sequence 25
exterior algebra 11
exterior differential 15
exterior k'th power 159
exterior product 8
functor 156
inverse 39
type 39
Hopf fibration 146
horizontal tangent bundle 269
Lie group 253
linking number 102
local index 106, 108, 216
locally finite 221
long exact homology sequt
I
M
F fiber bundle 147
fixed point theorem 47
flat connection 171
flat manifold 54
frame 151
free R-module 158
Fubini-Study metric 145
functor
contravariant 13, 27
covariant 27
fundamental form, first 73
fundamental theorem of algebra 259
immersion 252
immersive map 251
index 114
local 216
induced topology 221
induction on open sets 133
inner product 150
integral curve 119
intersection form 137
invariance
dimension 53
domain 53
topological 42
invariant polynomial 227
isotopy 238
G
J Gauss map 76
Gauss-Bonnet formula 124
generalized 217
gradient 2
gradient-like 119
Grassmann inner product 161
Grassmann manifold 203
Grothendieck construction 154
Gysin sequence 279
k-linearity 7
knot 53
Koszul identity 273
H
L
hermitian connection 196
Hodge's star operator 244
homogeneous polynomial 227
homotopy 39
equivalence 39
Laplace operator 246
Leibnitz' rule 167
Levi-Civita connection 273
Lie algebra 272
Lie bracket 272
Jacobi identity 272
Jordan-Brouwer theorem 51
maximal atlas 59
Mayer-Vietoris sequence '
Mayer-Vietoris theorem 1:
metric connection 196
Morse function 114
Morse inequalities 261-2t
Morse's lacunary principle
multilinearity 7
N
n-ball 47
n-sphere 47
negatively oriented basis
non-degenerate critical poi I
non-degenerate space 263
non-degenarate zero 109
normal bundle 148
normal vector field 76
K
o
obstruction 217
orientable manifold 70
orientation 70
standard 70
orientation form 70
orientation preserving/revel
oriented chart 72
oriented double covering : outward directed tangent Vl
286
s Sard theorem 98
section 151
parallel 170
sectional curvature 277
sequence
exact 25
Mayer-Vietoris 35
short exact 25
signature 137
signature invariant 137
singularity 106
smooth fiber bundle ]47
smooth submanifold 59
splitting principle 188
real 197
standard orientation 70
star-shaped set 2
Stiefel manifold 203
Stokes' Theorem 88
classical 256
submersive map 252
support 221
symmetric power 265
symplectic basis 263
symplectic manifold 266
symplectic space 263
topological invariance 42
total space ]47
transversality 2] 6
trivial bundle ]49
trivialization ]49
tubular neighborhood 47
closed 80
typical fiber 147
u unitary algebra 1]
Urysohn-Tietze lemma 48
v
Vect 28
vector bundle 148
complement ]52
complex 156
isomorphism 149
real 156
vector field, normal 76
vertical tangent bundle 269
volume form 73
w T tangent bundle 148
tangent space 66
tangent vector, outward directed 86
tensor product ]57
Thorn class 213
Thorn isomorphism 2] 3
Whitney 224
Whitney's embedding theorem 61
z
t
1
I
zero section 15 ]
