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;~
•••
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.
'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::
.~
,~
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)
.......
.-J
.... '... i;J "b.Q,,~ '.~ .... ..c 'l:!
.NJ
~ .lo:: .. ~
~
01;3
'1:'"' ~
R,'~.i.~ .
~,~
< ""
c
~
--
1
.Jg,
~
~l:! •. 1:10
.~
.....
...
'Q
1 ~
L---J
',~jmplex 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
' . .,"'-;:.' '
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 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 ,
~.
'""
"
'~ ..:'~ ~~ ""."".'
.~
. ..
......,.. ....
~
~
~_.-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 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 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 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
>: ~;"'''
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
;;;--.
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
.~ ..
~
'9,
~
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:
///---
---------~ -'
'---_
.... /~~
. . ·· . --~'~~-
,".
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
- _._~.- - . .•
';}!~";''''
70
9.
-~~. 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
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 : Rn+k -+
(x, t) = f(x:
' j~'~.
...
"
- ' . .>" "c .
~.
,'
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 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 ~ 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, .... .....
~
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.
;(~: 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 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 .
.~.,
.. '"' : Uo x
[0,1]
For x E Uo,
rd (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
'.. '
.~
~ 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 ( -
'\.
~
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 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 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
.
~
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\
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
:
X --+ Wffi given by
~([(x,y), (~ ~)]) = [(ox+;3y)-i(;3x+ r y)]·
(13)
Evidently 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 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
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
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
!
'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 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
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,. 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)*
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 ....
'"
~.•.. ~'. ............
..
'
' •.•
'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~
"
-
