топология / Tom Dieck T. Algebraic topology (EMS, 2008)

458 Chapter 18. Duality

We take now R D K as coefficients and assume n D 4t. In that case

H 2t .M / H 2t .M / ! R; .x; y/ 7!x ˇ y

is a regular symmetric bilinear form. Recall from linear algebra: Let .V; ˇ/ be a real vector space together with a symmetric bilinear form ˇ. Then V has a decomposition V D VC ˚ V C V0 such that the form is positive definite on VC, negative definite on V and zero on V0. By Sylvester’s theorem the dimensions of VC and V are determined by ˇ. The integer dim VC dim V is called the signature of ˇ. We apply this to the intersection form and call

.M / D dim H 2t .M /C dim H 2t .M /

the signature of the closed oriented 4t-manifold M . We also set .M / D 0, if the dimension of M is not divisible by 4. If M denotes the manifold with the

opposite orientation, then one has . M / D .M /. If M D M1 C M2 then H 2t .M / D H 2t .M1/C H 2t .M2/, the forms on M1 and M2 are orthogonal, hence

.M1 C M2/ D .M1/ C .M2/.

(18.7.2) Proposition. The signature of CP 2n with its natural orientation induced by the complex structure is 1.

implies p cn D nŠ t1t2 : : : tn. The map p has degree nŠ. These facts imply

nŠ D h p cn; ŒCP 1 n i D h cn; p ŒCP 1 n i D h cn; nŠ ŒCP n i D nŠ h cn; ŒCP n i;

hence h cn; ŒCP n i D 1.

(18.7.3) Proposition. Let M and N be closed oriented manifolds. Give M N the product orientation. Then .M N / D .M / .N /.

Proof. Let m D dim M and n D dim M . If m C n 6 mod4 then .M N / D 0 D .M / .N / by definition. In the case that m C n D 4p we use the Künneth isomorphism and consider the decomposition

H 2p.M N / D H m=2.M / ˝ H n=2.N /

L2i<m H i .M / ˝ H 2p i .N / ˚ H m i .M / ˝ H n 2pCi .N / :

The first summand on the right-hand side is zero if m or n is odd. The form on H 2p.M N / is transformed via the Künneth isomorphism by the formula

.x ˝ y/ ˇ .x0 ˝ y0/ D . 1/jyjjx0 j.x ˇ x0/.y ˇ y0/. Products of elements in different summands never contribute to the top dimension m C n. Therefore the signature to be computed is the sum of the signatures of the forms on the summands.

Consider the first summand. If m=2 and n=2 are odd, then the form is zero. In the other case let A be a basis of H m=2.M / such that the form has a diagonal matrix

with respect to this basis and let B be a basis of H n=2.N / with a similar property. Then .a ˝ b j a 2 A; b 2 B/ is a basis of H m=2.M / ˝ H n=2.N / for which the

form has a diagonal matrix. Then

Now consider the summand for 2i < m. Choose bases A of H .M / and B of H 2p i .N / and let A , B be the dual bases of H m i .M /, H n 2pCi .N / respec-

tively. Then

.a ˝ b C a ˝ b ; a ˝ b a ˝ b j a 2 A; b 2 B/

is a basis of the summand under consideration. The product of different basis elements is zero, and .a ˝ b C a ˝ b /2 D .a ˝ b a ˝ b /2 shows that the number of positive squares equals the number of negative squares. Hence these summands do not contribute to the signature.

There exists a version of the intersection form for cohomology with integral coefficients. We begin again with the bilinear form

We denote by A} the quotient of the abelian group A by the subgroup of elements of finite order. We obtain an induced bilinear form

s} W H k .M /} H n k .M /} ! Z:

(18.7.4) Proposition. The form s} is regular, i.e., the adjoint homomorphism H k .M /} ! Hom.H n k .M /}; Z/ is an isomorphism (and not just injective).

Proof. We use the fact that the evaluation H k .M I Z/} Hk .M I Z/} ! Z is a regular bilinear form over Z. By the universal coefficient formula, the kernel of H k .M I Z/ ! Hom.Hk .M I Z/; Z/ is a finite abelian group; hence we have isomorphisms

460 Chapter 18. Duality

We return to field coefficients. Let M be the oriented boundary of the compact oriented manifold B. We set Ak D Im.i W H k .B/ ! H k .M // with the inclusion

is Ak . The isomorphism is x 7!.y 7! hy Yx; ŒM i/; the second map is the restriction to An k . In particular dim H k .M / D dim Ak Cdim An k , and in the case n D

2t we have dim H t .M / D 2 dim Ak and dim Ht .M / D 2 dim Ker.i W Ht .M / ! Ht .B//.

Proof. Consider the diagram

By stability of the cap product, the square commutes up to the sign . 1/k . By commutativity and duality

x 2 Ak , ı.x/ D 0 , ı.x/ Z ŒB D 0 , i .x Z ŒM / D 0:

The regularity of the pairing H j .M / Hj .M / ! K, .x; y/ 7! hx; y i says that i .x Z ŒM / D 0 is equivalent to h H n k .B/; i .x Z ŒM /i D 0. Properties of pairings yield

h H n k .B/; i .x Z ŒM i D h i H n k .B/; x Z ŒM i D h An k ; x Z ŒM i

D h An k Y x; ŒM i

and we see that x 2 Ak is equivalent to h An k Y x; ŒM i D 0, and the latter

(18.7.7) Proposition. Let M be the boundary of a compact oriented .4k C 1/- manifold B. Then the signature of M is zero.

Proof. It follows from Proposition (18.7.5) that the orthogonal complement of A2k

We generalize the preceding by taking advantage of the general duality isomorphism (coefficients in K). Let M be an n-manifold and K L a compact pair in M . Assume that M is K-oriented along K. We define a bilinear form

H iCj .M; W [ .M X K// ! H iCj .M; M X K/:

The colimit over the neighbourhoods .V; W / yields Yy in . /.

(18.7.8) Proposition. Let M be an n-manifold and K L compact ENR in M . Then

(18.7.9) Example. Let M be a compact oriented n-manifold for n 2 mod .4/. Then the Euler characteristic .M / is even.

The intersection form H n=2.M / H n=2.M / ! Q with coefficients in Q is skew-symmetric and regular, since n=2 is odd. By linear algebra, a form of this type only exists on even-dimensional vector spaces.

we have used dim H .M / D dim H .M /, and this holds because of H .M / Š Hom.Hi .M /; Q/ and hence dim H i .M / D dim Hi .M / D dim H n i .M /. Þ

18.8 The Euler Number

Let W E. / ! M be an n-dimensional real vector bundle over the closed connected

orientable manifold M . The manifold is oriented by a fundamental class ŒM 2 Hn.M I Z/ and the bundle by a Thom class t. / 2 H n.E; E0I Z/. Let s W M !

462 Chapter 18. Duality

E. / be a section of and assume that the zero set N.s/ is contained in the disjoint sum D D D1 [ [ Dr of disks Dj . The aim of this section is to determine the Euler number e. / D h s t. /; ŒM i by local data. We assume given positive charts 'j W Rn ! Uj with disjoint images of M such that 'j .Dn/ D Dj . The bundle is trivial over Uj . Let

be a trivialization. We assume that ˆj is positive with respect to the given orientation of . These data yield a commutative diagram

image of t. / in H n.E. D /; E0. Dj // is the Thom class t. Dj /. The vertical maps have their counterpart in homology

464 Chapter 18. Duality

(2) There exists a cell D which contains every zero. Hence one has to consider a single local

7. Consider the bundle W Sn Z=2 Rn ! RP n, .x; z/ 7!Œx . Then W Œx0; : : : ; xn 7!

..x0; : : : ; xn/; .x1; : : : ; xn// is a section with a single zero. The sections correspond to maps f W Sn ! Rn such that f . x/ D f .x/. One form of the theorem of Borsuk–Ulam says that maps of this type always have a zero. We would reprove this result, if we show that the Euler class mod (2) is non-zero. The tautological bundle over RP n has as Euler class the non-zero element w of H 1.RP nI Z=2/. The Euler classes are multiplicative and D n . Hence e. / D wn D6 0.

18.9 Euler Class and Euler Characteristic

Let M be a closed orientable n-manifold. We define in a new manner the Thom class of the tangent bundle. It is an element t.M / 2 H n.M M; M M X D/

such that for each x 2 M the restriction of t.M / along

H n.M M; M M X D/ ! H n.x M; x .M X x//

is a generator (integral coefficients, D the diagonal). The image of t.M / under the composition

n.M M; M M D/ H n.M M / d H n.M / H X ! !

(where d is the diagonal map) is now called the associated Euler class e.M / of M . Let us use coefficients in a field K. We still denote the image of the fundamental class ŒM 2 Hn.M I Z/ in Hn.M I K/ by ŒM . We use the product orientation ŒM M D ŒM ŒM . Let W E. / ! M be the normal bundle of the diagonal d W M ! M M with diskand sphere bundle D. / and S. / and tubular map j W D. / ! M M . The fundamental class of ŒM M 2 H2n.M M / induces a fundamental class ŒD. / 2 H2n.D. /; S. // via

H2n.M M / ! H2n.M M; M M X D/ Š H2n.D. /; S. //:

Let z D j .ŒD. / . The diagram

commutes (naturality of the cap product). Suppose M is connected. From the isomorphisms

and therefore a Thom class. We define t.M / 2 H .M M; M M X D/ by j t.M / D t. /. The image .M / 2 H n.M M / of t.M / is characterized by

image of t. / under H n.D. /; S. // ! H n.D. // ! H n.M /, hence the Euler

elements of degree p. Then

h .˛ ˇ0/ Y ; ŒM M i

Dh ˛ ˇ0; Z ŒM M i D h ˛ ˇ0; d ŒM i

Dh d .˛ ˇ0/; ŒM i D h ˛ Y ˇ0; ŒM i

D. 1/p.n p/h ˇ0 Y ˛; ŒM i D . 1/p.n p/ı˛ˇ :

466 Chapter 18. Duality

A second computation gives

h .˛ ˇ0/ Y ; ŒM M i

Dh .˛ ˇ0/ Y P A. ; ı/ 0 ı; ŒM M i

DP A. ; ı/. 1/jˇ 0jj 0jh .˛ Y 0/ .ˇ0 Y ı/; ŒM ŒM i

DP A. ; ı/. 1/jˇ 0jj 0j. 1/n.jˇ 0jCjıj/h ˛ Y 0; ŒM ih ˇ0 Y ı; ŒM i:

Only summands with D ˛ and ı D ˇ are non-zero. Thus this evaluation has the

Chapter 19

Characteristic Classes

Characteristic classes are cohomological invariants of bundles which are compatible with bundle maps. Let h . / be a cohomology theory. An hk -valued characteristic class for numerable n-dimensional complex vector bundles, say, assigns to each such bundle W E. / ! B an element c. / 2 hk .B/ such that for a bundle map! over f W B ! C the naturality property f c. / D c. / holds.

An assignment which has these properties is determined by its value c. n/ 2 hk .BU.n// on the universal bundle n, and this value can be prescribed in an arbitrary manner (Yoneda lemma). In other words, the elements of hk .BU.n// correspond to this type of characteristic classes.

It turns out that in important cases characteristic classes are generated by a few of them with distinguished properties, essentially a set of generators of the cohomology of classifying spaces.

We work with a multiplicative and additive cohomology theory h and bundles are assumed to be numerable. A C-orientation of the theory assigns to each n-dimen-

sional complex vector bundle (numerable, over a CW-complex,…) W E. / ! B a Thom class t. / 2 h2n.E. /; E0. // such that for a bundle map f W E. / !

E. / the naturality f t. / D t. / holds and the Thom classes are multiplicative

t. / t. / D t. /. If an assignment of this type is given, then the theory is called C-oriented. In a similar manner we call a theory R-oriented, if for each n-

dimensional real vector bundle W E. / ! B a Thom class t. / 2 hn.E. /; E0. // is given which is natural and multiplicative. It is a remarkable fact that structures of this type are determined by 1-dimensional bundles.

(19.0.1) Theorem. A C-orientation is determined by its value

t. 1/ 2 h2.E. 1/; E0. 1//

on the universal 1-dimensional bundle 1 over CP 1. Each Thom class t of 1 determines a C-orientation. A similar bijection exists between Thom classes of the universal 1-dimensional real vector bundle over RP 1 and R-orientations.

An example of a C-oriented theory is H . I Z/; a complex vector bundle has a canonical Thom class and these Thom classes are natural and multiplicative. One can use an arbitrary commutative ring as coefficient ring.

An example of an R-oriented theory is H . I Z=2/; a real vector bundle has a unique Thom class in this theory. One can use any commutative ring of characteristic 2 as coefficient ring.