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

418 Chapter 17. Cohomology

induces a pairing (sometimes called Kronecker pairing)

H n.X; AI R/ ˝ Hn.X; AI R/ ! R; x ˝ y 7! hx; y i:

This is due to the fact that the evaluations combine to a chain map, see 11.7.4.

(17.4.2) Proposition. For f W .X; A/ ! .Y; B/, y 2 H n.Y; BI R/, and x 2

Hn.X; AI R/ we have

h f .y/; x i D h y; f .x/i: For a 2 H n 1.AI R/, b 2 Hn.X; AI R/ we have

h ıa; b i C . 1/n 1h a; @b i D 0:

Proof. We verify the second relation. Let a D Œ' , ' 2 Hom.Sn 1.A/; Z/. Let

s W ! I , .t0; t1/ 7!t1. Let Œx 2 H .X/ denote the element represented by the cochain which assumes the value 1 on x 2 X and 0 otherwise.

(17.4.3) Proposition. Let e1 be the generator which is the image of Œ0 2 H 0.0/

The singular cohomology groups H n.X; AI G/ can be computed from the homology groups of .X; A/. This is done via the universal coefficient formula. We

have developed the relevant algebra in (11.9.2). The application to topology starts with the chain complex C D S.X; AI R/ D S.X; A/ ˝Z R of singular chains with coefficients in a principal ideal domain R. It is a complex of free R-modules. Note that HomR.Sn.X; A/ ˝Z R; G/ Š HomZ.Sn.X; A/; G/ where in the second

group G is considered as abelian group (D Z-module). Then (11.9.2) yields the universal coefficient formula for singular cohomology.

(17.4.4) Theorem. For each pair of spaces .X; A/ and each R-module G there exists an exact sequence

0 ! Ext.Hn 1.X; AI R/; G/ ! H n.X; AI G/ ! Hom.Hn.X; AI R/; G/ ! 0:

The sequence is natural in .X; A/ and in G and splits. In particular, we have isomorphisms H 0.XI G/ Š Hom.H0.X/; G/ Š Map. 0.X/; G/.

The statement that the splitting is not natural means that, although the term in the middle is the direct sum of the adjacent terms, this is not a direct sum of functors. There is some additional information that cannot be obtained directly from the homology functors. The topological version of (11.9.6) is:

420 Chapter 17. Cohomology

Let us begin with the special case n D 0. As a model for K.A; 0/ we take the abelian group A with discrete topology. Since A is discrete, a map X ! A is continuous if and only if it is locally constant. Moreover, all homotopies are constant. Therefore ŒX; K.A; 0/ D ŒX; A is the group of locally constant maps. A locally constant map is constant on a path component. Therefore the surjection q W X ! 0.X/ induces an injective homomorphism

0.X/ W ŒX; A ! Map. 0.X/; A/ Š Hom.H0.X/; A/ Š H 0.XI A/:

The last isomorphism is the one that appears in the universal coefficient formula. If X is locally path connected, then the homomorphism induced by q is an isomorphism. Hence we have obtained a natural isomorphism on the category of locally path connected spaces, in particular on the category of CW-complexes. If X is connected but not path connected, then 0.X/ is not an isomorphism.

Let now n 1 and write K D Kn D K.A; n/. Consider the composition

The isomorphism (1) is the universal coefficient isomorphism. The isomorphism (2)

generator. The isomorphism (3) is induced by a fixed polarization W A Š n.K/.

Let us compare n 1 and n. It suffices to consider connected CW-complexes and pointed homotopy classes. The diagram with the structure map e.n/ W †Kn 1 ! Kn of the spectrum

morphism .†X/ is a homomorphism, since the group structures are also induced by the cogroup structure of †X. In order to check the commutativity one has to arrange and prove several things: (1) The Hurewicz homomorphisms commute with suspensions. (2) The structure map e.n/ and the polarizations satisfy e.n/ ı † ın 1 D n W A ! n.Kn/. (3) The homomorphisms from the universal coefficient formula commute up to sign with suspension. Since the sign is not important for the moment, we do not go into details.

(17.5.1) Theorem. The transformation n is an isomorphism on the category of pointed CW-complexes.

Proof. We work with connected pointed CW-complexes. Both functors have value

finite-dimensional CW-complexex. The general case follows from the fact that both functors yield an isomorphism when applied to XnC1 X.

bijection

abelian group A) defined as the composition

H n.SnI A/ Š Hom.Hn.Sn/; A/ Š Hom. n.Sn/; A/ Š Hom.Z; A/ Š A:

Again we have used universal coefficients and the Hurewicz isomorphism.

(17.5.2) Proposition (Hopf). Let M D X be a closed connected n-manifold which has an n-dimensional CW-decomposition. Suppose M is oriented by a fundamental class zM . Then we have an isomorphism

H n.M I Z/ Š Hom.Hn.M /; Z/ Š Z

(17.5.3) Proposition (Hopf). Let M be a closed, connected and non-orientable n-manifold with a CW-decomposition. Then H n.M I Z/ Š Z=2; hence we have a bijection ŒM; Sn Š Z=2. It sends Œf to the degree d2.f / modulo 2 of f .

422 Chapter 17. Cohomology

Proof. The universal coefficient theorem and Hn.M I Z/ D 0 show

Ext.Hn 1.M /; Z=2/ Š H n.M I Z=2/:

We use again that Hn 1.M I Z/ Š F ˚ Z=2 with a finitely generated free abelian group F . This proves H n.M I Z=2/ Š Z=2. Naturality of the universal coefficient sequences is used to show that the canonical map H n.M I Z/ ! H n.M I Z=2/ is an isomorphism. The commutative diagram

Problems

1.is not always an isomorphism: A D Z, n D 1 and the pseudo-circle.

17.6 The Cup Product in Singular Cohomology

Let R be a commutative ring. The cup product in singular cohomology with coefficients in R arises from a cup product on the cochain level

Here W kCl D Œe0; : : : ; ekCl ! X is a singular .kCl/-simplex. Let Œv0; : : : ; vn be an affine n-simplex and W Œv0; : : : ; vn ! X a continuous map. We denote by

jŒv0; : : : ; vn the singular simplex obtained from the composition of with the map n ! Œv0; : : : ; vn , ej !7 vj . This explains the notation in (1).

(17.6.1) Proposition. The cup product is a chain map

S .XI R/ ˝ S .XI R/ ! S .XI R/; i.e., the following relation holds:

ı.' Y / D ı' Y C . 1/j'j' Y ı :

iDk

If we add the two sums, the last term of the first sum and the first term of the second

S .A C B/ D S .A/ C S .B/ S .A [ B/ is a chain equivalence. The dual maps

S .A [ BI R/ ! S .A C BI R/ and S .X; A [ BI R/ ! S .XI A C BI R/ are then chain equivalences. The last module consists of the cochains which vanish on S .A C B/; let H .XI A C BI R/ denote the corresponding cohomology group. Thus we obtain a cup product (again a chain map)

S .X; AI R/ ˝ S .X; BI R/ ! S .XI A C BI R/:

We pass to cohomology, obtain H .X; AI R/˝H .X; BI R/ ! H .XI ACBI R/ and in the case of an excisive pair a cup product

H .X; AI R/ ˝ H .X; BI R/ ! H .X; A [ BI R/:

We now prove that the cup product satisfies the axioms of Section 2. From the definition (1) we see that naturality and associativity hold on the cochain level. The unit element 1X 2 H 0.XI R/ is represented by the cochain which assumes the value 1 on each 0-simplex. Hence it acts as unit element on the cochain level. In the relative case the associativity holds for the representing cocycles in the group H .XI A C B C C I R/. In order that the products are defined one needs that the pairs .A; B/, .B; C /, .A [ B; C / and .A; B [ C / are excisive.

Commutativity does not hold on the cochain level. We use: The homomorphisms

W Sn.X/ 7!Sn.X/; 7!"nxn

424 Chapter 17. Cohomology

with "n D . 1/.nC1/n=2 and x D jŒen; en 1; : : : ; e0 form a natural chain map which is naturally chain homotopic to the identity (see (9.3.5) and Problem 1 in that section). The cochain map # induced by satisfies

#' Y # D . 1/j'jj j #.' Y /:

Since # induces the identity on cohomology, the commutativity relation follows. The stability relation (2) is a consequence of the commutativity of the next

diagram (which exists without excisiveness). Coefficients are in R.

where ı0 is the composition of H iCj .A; A \ B/ Š H iCj .A C BI B/, which is induced by the algebraic isomorphism

cocycle, the representing elements coincide. A similar verification can be carried out for stability (3).

One can deal with products from the view-point of Eilenberg–Zilber transformations. We have the tautological chain map

S .XI R/ ˝ S .Y I R/ ! Hom.S ˝ S ; R ˝ R/:

We compose it with the ring multiplication R ˝ R ! R and an Eilenberg–Zilber transformation S .X Y / ! S .X/ ˝ S .Y / and obtain a chain map

a -product on the cochain level. If the pair .A Y; X B/ is excisive, we obtain a -product

S .X; AI R/ ˝ S .Y; BI R/ ! S ..X; A/ .Y; B/I R/:

Our previous explicit definition of the cup product arises in this manner from the Alexander–Whitney equivalence and the related approximation of the diagonal. We can now apply the algebraic Künneth theorem for cohomology to the singular cochain complexes and obtain:

(17.6.2) Theorem. Let R be a principal ideal domain and H .Y; BI R/ of finite type. Assume that the pair .A Y; X B/ is excisive. Then there exists a natural short exact sequence

Problems

1. We have defined the cup product for simplicity by explicit formulas. One can use instead Eilenberg–Zilber morphisms. Let us consider the absolute case. Consider the composition

Sq .XI R/ ˝ Sq .Y I R/ D Hom.SpX; R/ ˝ Hom.Sq .Y /; R/

! Hom.Sp.X/ ˝ Sq .Y /; R/ ! Hom.SpCq .X Y /; R/

where the first morphism is the tautological map and the second induced by an Eilenberg– Zilber morphism S .X Y / ! S .X/ ˝ S .Y /. Then this composition induces the-product in cohomology. The cup product in the case X D Y is obtained by composition with S .X/ ! S .X X/ induced by the diagonal. Instead we can go directly from Hom.Sp.X/ ˝ Sq .X/; R/ to Hom.SpCq .X/; R/ by an approximation of the diagonal. Our previous definition used the Alexander–Whitney diagonal.

17.7 Fibration over Spheres

X0 D p .S /, and F D p .b0/. From the homotopy theorem of fibrations we obtain the following result.

D˙ such that '˙.b0; y/ D y for y 2 F . The h-inverses ˙ also satisfy ˙.y/ D

.b0; y/ and the fibrewise homotopies of ˙'˙ and '˙ ˙ to the identity are constant on F . Since X0 X˙ X are closed cofibrations, we have a Mayer–Vietoris sequence for .XC; X /.

We use the data of (17.7.1) in order to rewrite the MV-sequence. We work with a multiplicative cohomology theory. The embedding j W F ! D˙n F , y 7!.b0; y/ is an h-equivalence. Therefore we have isomorphisms

Problems

1. As an example for the use of the Wang sequence compute the integral cohomology ring

A multiplicative structure of this type is called a polynomial ring with divided powers. With coefficient ring Q one obtains a polynomial ring H . SnC1I Q/ Š QŒz1 .

Let n be odd. Then z1z2k D z2kC1, z1z2kC1 D 0, and z2k D kŠz2k .

Again use induction and the derivation property. Since z12 D z12, one has z12 D 0. Then ‚.z1z2k / D ‚.z1/z2k z1‚.z2k / D z2k z1z2k 1 D z2k D ‚.z2kC1/, hence z1z2k D z2kC1, since ‚ is an isomorphism. Next compute z1z2kC1 D z1.z1z2k / D z12z2k D 0. For the last formula use that ‚ ı ‚ is a derivation of even degree which maps z2k to z2k 2. The induction runs then as for even n. The elements z2k generate a polynomial algebra with divided powers and z1 generates an exterior algebra.

17.8 The Theorem of Leray and Hirsch

The theorem of Leray and Hirsch determines the additive structure of the cohomology of the total space of a fibration as the tensor product of the cohomology of the base and the fibre. We work with singular cohomology with coefficients in the ring R. A relative fibration

.F; F 0/ ! .E; E0/ ! B

consists of a fibration p W E ! B such that the restriction p0 W E0 ! B to the subspace E0 of E is also a fibration. The fibres of p and p0 over a base point 2 B are F and F 0. The case E0 D ; and hence F 0 D ; is allowed. We assume that B is path connected.

fibration. Assume that H n.F; F 0/ is for each n a finitely generated free R-module.