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

Chapter 18

Duality

We have already given an introduction to duality theory from the view point of homotopy theory. In this chapter we present the more classical duality theory based on product structures in homology and cohomology. Since we did not introduce products for spectral homology and cohomology we will not directly relate the two approaches of duality in this book.

Duality theory has several aspects. There is, firstly, the classical Poincaré duality theorem. It states that for a closed orientable n-dimensional manifold the groups H k .M / and Hn k .M / are isomorphic. A consequence is that the cup product pairing H .M / ˝ H n .M / ! H n.M / is a regular bilinear form (say with field coefficients). This quadratic structure of a manifold is a basic ingredient in the classification theory (surgery theory). The cup product pairing for a manifold has in the context of homology an interpretation as intersection. Therefore the bilinear cup product form is called the intersection form. In the case of a triangulated manifold there exists the so-called dual cell decomposition, and the simplicial chain complex is isomorphic to the cellular cochain complex of dual cells; this is a very strong form of a combinatorial duality theorem [167].

The second aspect relates the cohomology of a closed subset K Rn with the homology of the complement Rn X K (Alexander duality). This type of duality is in fact a phenomenon of stable homotopy theory as we have explained earlier.

Both types of duality are related. In this chapter we prove in the axiomatic context of generalized cohomology theories a theorem which compares the cohomology of pairs .K; L/ of compact subsets of an oriented manifold M with the homology of the dual pair .M X L; M X K/. The duality isomorphism is constructed with the cap product by the fundamental class. We construct the cap product for singular theory.

18.1 The Cap Product

The cap product relates singular homology and cohomology with coefficients in the ring R. Let M and N be left R-modules. The cap product consists of a family of R-linear maps

H k .X; AI M / ˝ Hn.X; A [ BI N / ! Hn k .X; BI M ˝R N /; x ˝ y 7!x Z y

and is defined for excisive pairs .A; B/ in X. (Compare the definition of the cup product for singular cohomology.) If a linear map W M ˝ N ! P is given,

18.1. The Cap Product 439

we compose with the induced map; then x Z y 2 Hn k .X; BI P /. This device is typically applied in the cases M D R and W R˝N ! N is an R-module structure, or M D N D ƒ is an R-algebra and W ƒ ˝ ƒ ! ƒ is the multiplication.

We first define a cap product for chains and cochains

Sk .XI M / ˝ Sn.XI N / ! Sn k .XI M ˝ N /; ' ˝ c 7!' Z c: Given ' 2 Sp.XI M / and W pCq D Œe0; : : : ; epCq ! X, we set

' Z . ˝ b/ D . 1/pq .'. jŒeq ; : : : ; epCq ˝ b/ jŒe0; : : : ; eq

and extend linearly. (Compare in this context the definition of the cup product.) From this definition one verifies the following properties.

(1)Let f W X ! Y be continuous. Then f#.f #' Z c/ D ' Z f#c:

(2)@.' Z c/ D ı' Z c C . 1/j'j Z @c.

(3).' Y / Z c D ' Z . Z c/.

(4)1 Z c D c.

Case (3) needs conventions about the coefficients. It can be applied in the case that

(4) we assume that 1 2 S .XI R/ is the cocycle which send a 0-simplex to 1 and

Let A; B be excisive. We use the chain equivalence S .A/ C S .B/ ! S .A [ B/. After passing to cohomology we obtain the cap product as stated in the beginning. We list the

18.1.1 Properties of the cap product.

Then for x 2 H p.X; AI M /, y 2 HpCq .X; A [ BI N /,

jB x Z @B y D . 1/p@.x \ y/ 2 Hq 1.BI M ˝ N /:

440Chapter 18. Duality

(3)Let A; B be excisive, jA W .A; A \ B/ ! .X; B/ the inclusion and

We display again the properties in a table and refer to the detailed description above.

f .f x0 Z u/ D x0 Z f u

jB x Z @B y D . 1/jxj@.x Z y/

.jA/ .x Z @Ay/ D . 1/jxjC1ıx Z y 1 Z x D x

.x Y y/ Z z D x Z .y Z z/ ".x Z y/ D h x; y i

We use the algebra of the cap product and deduce the homological Thom isomorphism from the cohomological one.

(18.1.2) Theorem. Let W E ! B be an oriented n-dimensional real vector bundle with Thom class t 2 H n.E; E0I Z/. Then

tZ W HnCk .E; E0I N / ! Hk .EI N /

is an isomorphism.

Proof. Let z 2 Sn.E; E0/ be a cocycle which represents t. Then the family

SnCk .E; E0I N / ! Sk .EI N /; x 7!z Z x

is a chain map of degree n. This chain map is obtained from the corresponding one for N D Z by taking the tensor product with N . It suffices to show that the integral chain map induces an isomorphism of homology groups, and for this purpose it suffices to show that for coefficients in a field N D F an isomorphism is induced (see (11.9.7)). The diagram

is commutative (by property (6) in 18.1.1) where ˛ is the isomorphism of the universal coefficient theorem. Since Yt is an isomorphism, we conclude that tZ is an isomorphism.

Problems

1. The cap product for an excisive pair .A; B/ in X is induced by the following chain map (coefficient group Z):

1˝

S .X; A/ ˝ S .X; A [ B/ S .X; A/ ˝ S .X/=.S .A/ C S .B//

1˝D

! S .X; A/ ˝ S .X; B/ ˝ S .X; A/

1˝

! S .X; A/ ˝ S .X; A/ ˝ S .X; B/

"

! Z ˝ S .X; B/ Š S .X; B/:

D is an approximation of the diagonal, the graded interchange map, and " the evaluation.

The explicit form above is obtained from the Alexander–Whitney map D.

2. .x y/ Z .a b/ D . 1/jyjjaj.x Z a/ .y Z b/.

3. From the cap product one obtains the slant product x ˝ u 7!xnu which makes the following diagram commutative:

The properties (1)–(5) of the cap product can be translated into properties of the slant product, and the cap product can be deduced from the slant product. (This is analogous to the Y- and-product.)

18.2 Duality Pairings

We use the properties of the cap product in an axiomatic context. Let h be a cohomology theory and k ; h homology theories with values in R- MOD. A

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duality pairing (a cap product) between these theories consists of a family of linear maps

hp.X; A/ ˝ kpCq .X; A [ B/ ! hq .X; B/; x ˝ y 7!x Z y

defined for pairs .A; B/ which are excisive for the theories involved. They have the following properties:

(1) Naturality. For f W .XI A; B/ ! .X0I A0; B0/ the relation f .f x0 Z u/ D

x0 Z f u holds.

(2) Stability. Let A; B be excisive. Define the mappings jB and @B as in (18.1.1).

Then jB x Z @B y D . 1/jxj@.x Z y/.

(3) Stability. Let A; B be excisive. Define the mappings jA and @A as in (18.1.1). Then .jA/ .x Z @Ay/ D . 1/jxjC1ıx Z y.

(4) Unit element. There is given a unit element 1 2 k0.P /. The homomorphism hk .P / ! h k .P /, x 7!x Z 1 is assumed to be an isomorphism (P a point).

(In the following investigations we deal for simplicity of notation only with the case h D k .) Note that we do not assume given a multiplicative structure for the cohomology and homology theories.

As a first consequence of the axioms we state the compatibility of the cap product with the suspension isomorphisms.

(18.2.1) Proposition. The following diagrams are commutative:

(For the second diagram one should recall our conventions about the suspension isomorphisms, they were different for homology and cohomology. Again we use notations like IX D I X.)

Proof. We consider the first diagram in the case that A D ;. The proof is based on

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Proof. We use naturality and stability (2) and show that the down-right path of the diagram is . 1/k 1 times the map

hk 1.U; V / ! hk 1.U U 0; V U 0/ Z ˛1 hn k .U U 0; U V 0/ ! hn k .U 0; V 0/

and the right-down path . 1/k times the analogous map where ˛1 is replaced by ˇ1; the element ˛1 is obtained from ˛ via the morphism

! hn 1.U U 0; .U V 0/ [ .V U 0// 3 ˛1

and ˇ1 from ˇ via the analogous composition in which the primed and unprimed spaces are interchanged. Thus it remains to show ˛1 D ˇ1. This is essentially a consequence of the Hexagon Lemma. One of the outer paths in the hexagon is given by the composition

hn.U [ U 0; V [ V 0/ ! hn.U [ U 0; U [ V 0/

and the other path is obtained by interchanging the primed and unprimed objects. The center of the hexagon is hn.U [ U 0; .U [ V 0/.V [ U 0//. We then compose the outer paths of the hexagon with the excision

hn 1.U U 0; U V 0 [ V U 0/ ! hn 1..U [ V 0/.V [ U 0/; V [ V 0/I

then is mapped along the paths to ˛1 and ˇ1, respectively; this follows from the original definition of the elements by a little rewriting. The displayed morphism yields ˇ1.

18.3 The Duality Theorem

For the statement of the duality theorem we need two ingredients: A homological orientation of a manifold and a duality homomorphism. We begin with the former.

Let M be an n-dimensional manifold. For K L M we write

rKL W h .M; M X L/ ! h .M; M X K/

for the homomorphism induced by the inclusion, and rxL in the case that K D fxg.

is ˙en where en is the element which arises from 1 2 h0 under suspension. A family .oK j K M compact/ is called coherent if for each compact pair K L

the restriction relation rKLoL D oK holds. A coherent family .oK / of orientations is a (homological ) orientation of M . If M is compact, then K D M is allowed and an orientation is determined by the element oM 2 hn.M /, called the fundamental class of M .

In order to state the duality theorem we need the definition of a duality homomorphism. We fix a homological orientation .oK / of M . Given closed sets

The morphism .#/ is an excision, since M X .U X L/ D .M X U / [ L (closed) is contained in .M X K/ [ V (open).

From zKLU V we obtain the homomorphism DKLU V via the commutative diagram

We state some naturality properties of these data. They are easy consequences of the naturality of the cap product.

The naturality (18.3.1) allows us to pass to the colimit over the neighbourhoods

.U; V / of .K; L/ in M . We obtain a duality homomorphism

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We explain this with some remarks about colimits. An element x 2 hk .U; V / represents an element of the colimit. Two elements x 2 hk .U; V / and x0 2 hk .U 0; V 0/ represent the same element if and only if there exists a neighbourhood .U 00; V 00/ with U 00 U \ U 0, V 00 V \ V 0 such that x and x0 have the same restric-

tion in hk .U 00; V 00/. Thus we have canonical homomorphisms lU V W hk .U; V / !

Lk

h .K; L/. Via these homomorphisms the colimit is characterized by a univer-

restrictions h .U; V / ! h .K; L/ are compatible in this sense, and we obtain a

this homomorphism is an isomorphism; we explain this later.

(18.3.3) Duality Theorem. Let M be an oriented manifold. Then the duality homomorphism DKL is, for each compact pair .K; L/ in M , an isomorphism.

We postpone the proof and discuss some of its applications. Let M be compact and ŒM 2 hn.M / a fundamental class. In the case .K; L/ D .M; ;/ we have

(18.3.4) Poincaré DualityTheorem. Suppose the compact n-manifold is oriented by the fundamental class ŒM 2 hn.M /. Then

H p.X; AI G/ ˝ HpCq .X; A [ BI R/ ! Hq .X; BI G/

for commutative rings R and R-modules G. The Euclidean space Rn is orientable for H . I Z/. Thus we have: