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

428 Chapter 17. Cohomology

Let cj 2 H .E; E0/ be a family of elements such that the restrictions i .cj / form an R-basis of H .F; F 0/. Then

is an isomorphism of R-modules. Thus H .E/ is a free graded H .B/-module with basis fcj g.

We explain the statement of the theorem. The source of L is a direct sum of modules H k .B/ ˝ H l .F; F 0/. The elements i c which are contained in H l .F; F 0/ are a finite R-basis of the R-module H l .F; F 0/. A basic property of the tensor product says that each element has a unique expression of the form

P b ˝ i .c /; b 2 H k .B/:

By the conventions about tensor products of graded modules, L is a map of degree zero between graded modules.

Proof. Let A B be a subspace. We have the restricted fibrations .F; F 0/ !

.EjA; E0jA/ ! A with EjA D p 1.A/ and E0jA D E0 \ EjA. The elements cj yield by restriction elements cj jA 2 H .EjA; E0jA/ which again restrict to a basis of H .F; F 0/.

We first prove the theorem for CW-complexes B by induction over the skeleta Bn. If B0 D f g, then L has the form H 0.B0/ ˝ H k .F; F 0/ ! H k .F; F 0/ and

it is an isomorphism by the unit element property of the cup product.

Suppose the theorem holds for the .n 1/-skeleton Bn 1. We write Bn D U [V , where U is obtained from Bn by deleting a point in each open n-cell and V is the union of the open n-cells. We use the MV-sequence of U; V and EjU; EjV and obtain a commutative diagram

The left column is the tensor product of the MV-sequence for U; V with the graded module M D H .F; F 0/. It is exact, since the tensor product with a free module preserves exactness. We show that LU ; LV and LU \V are isomorphisms. The Five Lemma then shows that LU [V is an isomorphism. This finishes the induction step.

Case U . We have the commutative diagram

H .Bn 1/ ˝ H .F; F 0/ H .EjBn 1; E0jBn 1/:

Since Bn 1 U , EjBn 1 EjU , and E0jBn 1 E0jU are deformation retracts, the vertical maps are isomorphisms. We use the induction hypothesis and see that LU is an isomorphism.

(1) and (2) are isomorphisms by the additivity of cohomology. The map (3) is the direct sum of homomorphisms of the type Q Mj ˝ N ! Q.Mj ˝ N / with a finitely generated free module N and other modules Mj . In a situation like this the tensor product commutes with the product. Hence (3) is an isomorphism. The homomorphisms L.ejn/ are isomorphisms, since ejn is pointed contractible.

Case U \V . We combine the arguments of the two previous cases. By additivity and finite generation we reduce to the case of U \ejn, a cell with a point deleted. This space has the .n 1/-sphere as a deformation retract. By induction, the theorem holds for an .n 1/-sphere.

From the finite skeleta we now pass to arbitrary CW-complexes via the lim– lim1-sequence (17.1.6). For general base spaces B we pull back the fibration along a CW-approximation.

(17.8.2) Example. Consider the product fibration p D prB W B .F; F 0/ ! B. Let H .F; F 0/ be a free R-module with homogeneous basis .dj j j 2 J /, finite

430 Chapter 17. Cohomology

is an isomorphism (of graded algebras). This is a special case of the Künneth formula. Þ

(17.8.3) Remark. The methods of proof for (17.8.1) (induction, Mayer–Vietoris

sequences) gives also the following result. Suppose H k .F; F 0/ D 0 for k < n. Then H k .E; E0/ D 0 for k < n. Þ

Let now h be an arbitrary additive and multiplicative cohomology theory and

.F; F 0/ ! .E; E0/ ! B a relative fibration over a CW-complex. We prove a LerayHirsch theorem in this more general situation. We assume now that there

(17.8.4) Theorem (Leray–Hirsch). h .E; E0/ is a free left h .B/-module with basis .tj /.

Proof. Let us denote by h .C /h t i the free graded h .C /-module with (formal) basis tj in degree n.j /. We have the h .C /-linear map of degree zero

'.C / W h .C /h t i ! h .EjC; E0jC /

which sends tj to tj jC . These maps are natural in the variable C B. We view h . /h t i as a cohomology theory, a direct sum of the theories h . / with shifted degrees. Thus we have Mayer–Vietoris sequences for this theory. If U and V are open in B, we have a commutative diagram of MV-sequences.

We use this diagram as in the proof of (17.8.1). We need for the inductive proof that '.e/ is an isomorphism for an open cell e. This follows from two facts:

(1) '.P / is an isomorphism for a point P D fbg B, by our assumption about the tj .

(2) .EP ; EP0 / ! .Eje; E0je/ induces an isomorphism in cohomology, since EP Eje is a homotopy equivalence by the homotopy theorem for fibrations.

The finiteness of the set ftj g is used for the compatibility of products and finite sums. The passage from the skeleta of B to B uses again (17.1.6).

There is a similar application of (17.8.4) as we explained in (17.8.2).

17.9 The Thom Isomorphism

We again work with a cohomology theory which is additive and multiplicative. Under the obvious finiteness conditions (e.g., finite CW-complexes) additivity is

not needed.

Let .p; p0/ W .E; E0/ ! B be a relative fibration over a CW-complex. A Thom class for p is an element t D t.p/ 2 hn.E; E0/ such that the restriction to each fibre

tb 2 hn.Fb; Fb0 / is a basis of the h -module h .Fb; Fb0 /. We apply the theorem of Leray–Hirsch (17.8.4) and obtain:

(17.9.1) Theorem (Thom Isomorphism). The Thom homomorphism

Let us further assume that p induces an isomorphism p W H .B/ ! H .E/. We use the Thom isomorphism and the isomorphism p in order to rewrite the exact sequence of the pair .E; E0/; we set D ˆ 1ı.

We call e the Euler class of p with respect to t. From the definitions we verify that J is the cup product with e, i.e., J.x/ D x Y e.

(17.9.2) Theorem (Gysin Sequence). Let .E; E0/ ! B be a relative fibration as above such that p W h .B/ Š h .E/ with Thom class t and associated Euler class e 2 hn.B/. Then we have an exact Gysin sequence

! :

We discuss the existence of Thom classes for singular cohomology H . I R/. In this case it is not necessary to assume that B is a CW-complex (see (17.8.1)). We have for each path w W I ! B from b to c a fibre transport w# defined as follows: Let q W .X; X0/ ! I be the pullback of p W .E; E0/ ! B along w. Then we have

432 Chapter 17. Cohomology

The homomorphism w# only depends on the class of w in the fundamental groupoid. In this manner we obtain a transport functor “fibre cohomology”.

(17.9.3) Proposition. We assume that H .Fb; Fb0 I R/ is a free R-module with a basis element in H n.Fb; Fb0 I R/. A Thom class exists if and only if the transport functor is trivial.

Proof. Let t be a Thom class and w W I ! B a path from b to c. Then w# ıic D ib , where ib denotes the inclusion of the fibre over b. Thus w# sends the restricted Thom class to the restricted Thom class and is therefore independent of the path. (In general, the transport is trivial on the image of the ic .)

Let now the transport functor be trivial. Then we fix a basis element in a particular fibre H n.Eb; Eb0 / and transport it to any other fibre uniquely (B path connected). A Thom class tC 2 H n.EjC; E0jC / for C B is called distinguished, if the restriction to each fibre is the specified basis element. This requirement determines tC . By a MV-argument and (17.8.3) we prove by induction that H k .EjBn; E0jBn/ D 0 for k < n and that a distinguished Thom class exists. Then we pass to the limit and to general base spaces as in the proof of (17.8.1).

The preceding considerations can be applied to vector bundles. Let W E ! B be a real n-dimensional vector bundle and E0 the complement of the zero section. Then for each fibre H n.Eb; Eb0/ Š R and is a homotopy equivalence. A Thom class t. / 2 H n.E; E0I R/ is called an R-orientation of . If it exists, we have a Thom isomorphism and a Gysin sequence. We discuss the existence of Thom classes and its relation to the geometric orientations.

(17.9.4) Theorem. There exists a Thom class of with respect to singular cohomology H . I Z/ if and only if the bundle is orientable. The Thom classes with respect to H . I Z/ correspond bijectively to orientations.

Proof. Let us consider bundles over CW-complexes. Let t be a Thom class. Consider a bundle chart ' W U Rn ! 1.U / over a path connected open U . The image of tjU in H n.U .Rn; Rn X0// Š H 0.U / under ' and a canonical suspension isomorphism is an element ".U / which restricts to ".u/ D ˙1 for each point u 2 U , and u 7!".u/ is constant, since U is path connected. We can therefore change the bundle chart by an automorphism of Rn such that ".u/ D 1 for each u. Bundle charts with this property yield an orienting bundle atlas.

Conversely, suppose has an orienting atlas. Let ' W U Rn ! 1.U / be a positive chart. From a canonical Thom class for U Rn we obtain via ' a local Thom class tU for 1.U /. Two such local Thom classes restrict to the same Thom class over the intersection of the basic domains, since the atlas is orienting. We can now paste these local classes by the Mayer–Vietoris technique in order to obtain a global Thom class.

Here is a geometric property of the Euler class:

(17.9.5) Proposition. Suppose has a section which is nowhere zero. Then the Euler class is zero.

Proof. Let s0 W B ! E0 be a map such that ı s0 D id. The section s0 is homotopic to the zero section by a linear homotopy in each fibre. Therefore e. / is the image of t. / under a map

The Thom classes and the Euler classes have certain naturality properties. Let f W ! be a bundle map. If t. / is a Thom class, then f t. / is a Thom class for and f e. / is the corresponding Euler class. If W X ! B and W Y ! C are bundles with Thom classes, then the -product t. / t. / is a Thom class forand e. / e. / is the corresponding Euler class.

In general, Thom classes are not unique. Let us consider the case of a trivial bundle D pr2 W Rn B ! B. It has a canonical Thom class pr2 en. If t. / is an arbitrary Thom class, then it corresponds under the suspension isomorphism h0.B/ ! hn.Rn B; Rn0 B/ to an element v. / with the property that its restriction

to each point b 2 B is the element ˙1 2 h0.b/. Under reasonable conditions, an

element with this property (call it a point-wise unit) is a (global) unit in the ring h0.B/. We call a Thom class for a numerable bundle strict if the restrictions to

the sets of a numberable covering correspond under bundle charts and suspension isomorphism to a unit in h0.

(17.9.6) Proposition. Let U be a numerable covering of X. Let " 2 h0.X/ be an element such that its restriction to each U 2 U is a unit. Then " is a unit.

Proof. Let X D U [ V and assume that .U; V / is excisive. Let "jU D "U and "jV D "V be a unit. Let U ; V be inverse to "U ; "V . Then U and V have the same restriction to U \ V . By the exactness of the MV-sequence there exists

a unit. Restrictions of units are units. By additivity, if X is the disjoint union of U and "jU is a unit for each U 2 U, then " is a unit. We finish the proof as in the proof of (17.9.7).

The Thom isomorphism is a generalized (twisted) suspension isomorphism. It is given by the product with a Thom class. Let W E ! B be an n-dimensional

real vector bundle and t. / 2 hn.E; E0/ a Thom class with respect to a given multiplicative cohomology theory. The Thom homomorphism is the map

where W hk .B; A/ ! H k .E. /; E0. // is the homomorphism induced by . The Thom homomorphism defines on h .E; E0/ the structure of a left graded h .B/-module. The Thom homomorphism is natural with respect to bundle maps. Let

E. / F E. /

f

BC

be a bundle map. Let t. / be a Thom class. We use t. / D F t. / as the Thom class for . Then the diagram

ˆ. /

is commutative. We assume that f W .B; A/ ! .C; D/ is a map of pairs.

The Thom homomorphism is also compatible with the boundary operators. Let t. A/ be the restriction of t. /. Then the diagram

is commutative.

The Thom homomorphisms are also compatible with the morphisms in the MVsequence. We now consider the Thom homomorphism under a different hypothesis.

(17.9.7) Theorem. The Thom homomorphism of a numerable bundle with strict Thom class is an isomorphism.

Proof. Let be a numerable bundle of finite type. By hypothesis, B has a finite numerable covering U1; : : : ; Ut such that the bundle is trivial over each Uj and the Thom class is strict over Uj . We prove the assertion by induction over t. For t D 1 it holds by the definition of a strict Thom class. For the induction step consider C D U1 [ [ Ut 1 and D D Ut . By induction, the Thom homomorphism is an isomorphism for C , D, and C \ D. Now we use that the Thom homomorphism is compatible with the MV-sequence associated to C , D. By the Five Lemma we see that ˆ. / is an isomorphism over C [ D.

Suppose the bundle is numerable over a numerable covering U. Assume that for each U 2 U the Thom class t. U / is strict. In that case ˆ. U / is an isomorphism. For each V U the Thom class t. V / is also strict. There exists a numerable covering .Un j n 2 N/ such that jUn is numerable of finite type with strict Thom

tained in a finite number of Vj and therefore the bundle over such a set is numerable of finite type. The sets Cn D f 1 2n 1; 2n C 1Πare open and disjoint. Over Cn

(17.9.8) Example. Let 1.n/ W H.1/ ! CP n be the canonical line bundle introduced in (14.2.6). A complex vector bundle has a canonical orientation and an associated Thom class (17.9.4). Let c 2 H 2.CP n/ be the Euler class of1.n/. The associated sphere bundle is the Hopf fibration S2nC1 ! CP n. Since

Z Š H 0.CP n/ Š H 2.CP n/ Š Š H 2n.CP n/

and similarly H k .CP n/ D 0 for odd k. We obtain the structure of the cohomology

ring

H .CP n/ Š ZŒc =.cnC1/:

In the infinite case we obtain H .CP 1I R/ Š RŒc where R is an arbitrary commutative ring.

436 Chapter 17. Cohomology

A similar argument for the real projective space yields for the cohomology ring

H .RP nI Z=2/ Š Z=2Œw =.wnC1/ with w 2 H 1.RP nI Z=2/ and for the infinite projective space H .RP 1I Z=2/ Š Z=2Œw . Þ

(17.9.9) Example. The structure of the cohomology ring of RP n can be used to

give another proof of the Borsuk–Ulam theorem: There does not exist an odd map

F W Sn ! Sn 1.

u D F v from F .x/ to F . x/ D F .x/ yields a loop that generates 1.RP n 1/. Hence f W 1.RP n/ ! 1.RP n 1/ is an isomorphism. This fact implies (universal coefficients) that f is an isomorphism in H 1. I Z=2/. Since wn D 0 in

Problems

1.A point-wise unit is a unit under one of the following conditions: (1) For singular cohomology H . I R/. (2) B has a numerable null homotopic covering. (3) B is a CW-complex.

2.Prove the Thom isomorphism for vector bundles over general spaces and for singular cohomology.

3.Let W E. / ! B and W E. / ! B be vector bundles with Thom classes t. / and t. /. Define a relative Thom homomorphism as the composition of x 7!x t. /,

hk .E. /; E0. /[E. A// ! hkCn.E. / E. /; .E0. /[E. A// E. /[E. / E0. // with the map induced by

.E. ˚ /; E0. ˚ / [ E. A ˚ A//

! .E. / E. /; .E0. / [ E. A/ E. / [ E. / E0. //;

a kind of diagonal, on each fibre given by .b; v; w/ 7!..b; v/; .b; w//. This is the previously defined map in the case that dim D 0. The product t. / t. / is a Thom class and also its restriction t. ˚ / to the diagonal. Using this Thom class one has the transitivity of the Thom homomorphism ˆ. /ˆ. / D ˆ. ˚ /.

4.Given i W X ! Y , r W Y ! X such that ri D id (a retract). Let W E ! X be a bundle over X and r D the induced bundle. Let t. / be a Thom class and t. / its pullback. If ˆ. / is an isomorphism, then ˆ. / is an isomorphism.

5.Let Cm S1 be the cyclic subgroup of m-th roots of unity. A model for the canonical map pm W BCm ! BS1 is the sphere bundle of the m-fold tensor product m D ˝ ˝ of the canonical (universal) complex line bundle over BS1.

6.Let R be a commutative ring. Then H .BS1I R/ Š RŒc where c is the Euler class of .

7. We use coefficients in the ring R. The Gysin sequence of pm W BCm ! BS1 splits into short exact sequences

This implies H 2k .BCm/ Š R=mR, H 2k 1.BCm/ Š mR for k > 0, where mR is the

m-torsion h x 2 R j mx D 0i of R. In even dimensions we have the multiplicative isomorphism H 2 .BS1/=.mc/ Š H 2 .BCm/ induced by pm.

8. The sphere bundle of the canonical bundle ECm Cm C ! BCm has a contractible

total space. Therefore the Gysin sequence of this bundle shows that the cup product Yt W H j .BCmI R/ ! H j C2.BCmI R/ is an isomorphism for j > 0; here t D pmc.

9. The cup product

H 1.BCmI R/ H 1.BCmI R/ ! H 2.BCmI R/ is the R-bilinear form

mR mR ! R=mR; .u; v/ 7!m.m 1/=2 uv:

Here one has to take the product of u; v 2 mR R and reduce it modulo m. Thus if m is odd, this product is zero; and if m is even it is .u; v/ 7!m=2 uv.