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

178 Chapter 7. Stable Homotopy. Duality

The composition down-right sends the element .a; b; x/ to .a; b f .x// and the composition right-down to .'.b/ a x; b fQ.'.b/ a//. We use the homotopies

.'.b/ a x; b fQ.t'.b/ aC.1 t/x// and then ..1 t/aCt.'.b/ a x/; b f .x//.

(7.5.4) Proposition. Let X and Y be compact and retracts of open neighbourhoods. Then the following diagram is homotopy commutative

Proof. We unravel the definitions and deform suitable maps between pairs. The composition .1 ^ D#f /. ^ 1/.1 ^ / is induced by maps

with ˛.x; y/ D .rY .y/; x; y fQ.x//. Further investigations concern ˛. We use the next diagram

This homotopy is constant on G.f /. Hence there exists an open neighbourhood U

(7.5.5) Remark. Let X Rn be a compact ENR and f W X ! X a continuous map. From the associated n-duality we obtain a homotopy class

The degree d. f / D L.f / 2 Z is an interesting invariant of the map f , the Lefschetz fixed point index. If f is the identity, then L.id/ is the Euler characteristic

of X. [51] [54] Þ

Problems

1. Let A be a closed subset of a normal space X. Let r W W ! A be a retraction of an open

2.Verify directly that the homotopy class of the coevaluation does not depend on the choice of the retraction r W V ! X.

3.The n-dualities which we have constructed can be interpreted as representative elements for morphisms in the category ST. We obtain

" W .C.RnjX/; n/ ˝ .XC; 0/ ! .S0; 0/W .S0; 0/ ! .XC; 0/ ˝ .C.RnjX/; n/:

They satisfy the relations

.1 ^ "/. ^ 1/ D id; ." ^ 1/.1 ^ / D id

which define dualities in tensor categories.

7.6 Homology and Cohomology for Pointed Spaces

A homology theory for pointed spaces with values in the category R-MOD of

is always an isomorphism for well-pointed spaces Xj .

In Chapter 10 we define homology theories by the axioms of Eilenberg and Steenrod. They involve functors on TOP.2/. We show in Section 10.4 that they induce a homology theory for pointed spaces as defined above.

Given a homology theory Q for pointed spaces we construct from it a homology h

D Q

theory for pairs of spaces as follows. We set hn.X; A/ hn.C.X; A//. It should be clear that the hn are part of a homotopy invariant functor TOP.2/ ! R- MOD. We define the boundary operator as the composition

The isomorphism is the given suspension isomorphism of the theory Q

h . The Eilenberg–Steenrod exactness axioms holds; it is a consequence of the assumption

Q transforms a cofibre sequence into an exact sequence and of the naturality that h

of the suspension isomorphism. The excision isomorphism follows from (7.2.5). We need the additional hypothesis that the covering is numerable. Remark (7.2.6) is relevant for the passage from one set of axioms to the other.

7.7 Spectral Homology and Cohomology

In this section we report about the homotopical construction of homology and

cohomology theories. We work in the category of compactly generated spaces. A pre-spectrum consists of a family .Z.n/ j n 2 Z/ of pointed spaces and a family

.en W †Z.n/ ! Z.n C 1/ j n 2 Z/ of pointed maps. Since we only work with

pre-spectra in this text, we henceforth just call them spectra. A spectrum is called an -spectrum, if the maps "n W Z.n/ ! Z.n C 1/ which are adjoint to en are

pointed homotopy equivalences.

Let Z D .Z.n/; "n/ be an -spectrum. We define Zn.X/ D ŒX; Z.n/ 0 for a pointed space X. Since Z.n/ is up to h-equivalence a double loop space, namely Z.n/ ' 2Z.n C 2/, we see that Zn.X/ is an abelian group, and we can view Zn as a contravariant and homotopy invariant functor TOP0 ! Z- MOD. We defineW Zn.X/ Š ZnC1.†X/ via the structure maps and adjointness as

ŒX; Z.n/ 0 ."n/ ŒX; Z.n C 1/ 0 Š Œ†X; Z.n C 1/ 0:

We thus have the data for a cohomology theory on TOP0. The axioms are satisfied (Puppe sequence). The theory is additive.

We now associate a cohomology theory to an arbitrary spectrum Z D .Z.n/; en/. For k 0 we have morphisms

bnk W Œ†k X; Z.n/ 0 † Œ†.†k X/; †Z.n/ 0 .en/ Œ†kC1X; Z.n C 1/ 0 :

Let Zn k .X/ be the colimit over this system of morphisms. The bnk are compatible with pointed maps f W X ! Y and induce homomorphisms of the colimit groups. In this manner we consider Zn as a homotopy invariant, contravariant functor TOP0 ! Z- MOD. (The bnk are for k 2 homomorphisms between abelian groups.) The exactness axiom again follows directly from the cofibre sequence. The suspension isomorphism is obtained via the identity

Œ†kC1X; Z.n C k C 1/ 0 Š Œ†k .†X/; Z.n C k C 1/ 0

which gives in the colimit Zn.X/ Š ZnC1.†X/. If the spectrum is an -spectrum, we get the same theory as before, since the canonical morphisms ŒX; Z.n/ 0 ! Zn.X/ are natural isomorphisms of cohomology theories. Because of the colimit process we need the spaces Z.k/ only for k k0. We use this remark in the following examples.

7.7.1 Sphere spectrum. We define Z.n/ D S.n/ and en W †S.n/ Š S.n C 1/ the

identity. We set !k .X/ D colimnŒ†nX; SnCk 0 and call this group the k-th stable cohomotopy group of X. Þ

182 Chapter 7. Stable Homotopy. Duality

7.7.2 Suspension spectrum. Let Y be a pointed space. We define a spectrum with

7.7.3 Smash product. Let Z D .Z.n/; en/ be a spectrum and Y a pointed space. The spectrum Y ^ Z consists of the spaces Y ^ Z.n/ and the maps

id ^en W †.Y ^ Z.n// Š Y ^ †Z.n/ ! Y ^ Z.n C 1/:

(Note that †A D A^I=@I . Here and in other places we have to use the associativity of the ^-product. For this purpose it is convenient to work in the category of k-spaces.) We write in this case

Zk .XI Y / D colimnΠnX; Y ^ Z.n C k/ 0:

The functors Zk . I Y / depend covariantly on Y : A pointed map f W Y1 ! Y2

induces a natural transformation of cohomology theories Zk . I Y1/ ! Zk . I Y2/.

Þ

In general, the definition of the cohomology theory Z . / has to be improved, since this theory may not be additive.

We now construct homology theories. Let

E D .E.n/; en W E.n/ ^ S1 ! E.n C 1/ j n 2 Z/

be a spectrum. We use spheres as pointed spaces and take as standard model the

b W ŒSnCk ; X^E.n/ 0 ! ŒSnCk ^S1; X^E.n/^S1 0 ! ŒSnCkC1; X^E.nC1/ 0:

The first map is ^ S1 and the second map is induced by id ^en. For n C k 2 the morphism b is a homomorphism between abelian groups.

It should be clear from the definition that Ek . / is a functor on TOP0. We need the suspension morphisms. We first define suspension morphisms

l W Ek .Z/ ! EkC1.S1 ^ Z/:

They arise from the suspensions S1 ^

ŒSnCk ; Z ^ E.n/ 0 ! ŒSnCkC1; S1 ^ Z ^ E.n/ 0;

which are compatible with the maps b above. Then we set D . 1/k l where the map W S1 ^ Z ! Z ^ S1 interchanges the factors.

(7.7.4) Lemma. l is an isomorphism.

Proof. Let x 2 Ek .Z/ be contained in the kernel of l . Then there exists Œf 2 ŒSnCk ; Z ^ E.n/ 0 representing x such that 1 ^ f is null homotopic. Consider the diagram

In order to prove surjectivity we consider the two-fold suspension. Let x 2 EkC2.S2 ^ Z/ have the representative g W SnCkC2 ! S2 ^ Z ^ E.n/. Then

represents an element y 2 Ek .Z/. Here e2 is the composition of the spectral structure maps

E.n/ ^ S2 D .E.n/ ^ S1/ ^ S1 ! E.n C 1/ ^ S1 ! E.n C 2/:

We show l2.y/ D x. Once we have proved this we see that the second l is surjective and injective and hence the same holds for the first l .

The proof of the claim is based on the next diagram with interchange maps

; 0; 00.

The maps 0 and 00 are homotopic to the identity, since we are interchanging a sphere with an even-dimensional sphere. The composition of the left verticals represents l2.y/, and the composition of the right verticals represents x.

184 Chapter 7. Stable Homotopy. Duality

(7.7.5) Proposition. For each pointed map f W Y ! Z the sequence

is exact.

Proof. The exactness is again a simple consequence of the cofibre sequence. But since the cofibre sequence is inserted into the “wrong” covariant part, passage to the colimit is now essential. Suppose z 2 Ek .Z/ is contained in the kernel of f1 . Then there exists a representing map h W SnCk ! Z ^ E.n/ such that

.f1 ^ 1/ ı h is null homotopic. The next diagram compares the cofibre sequences of id W SnCk ! SnCk and f ^ 1 W Y ^ E.n/ ! Z ^ E.n/.

The map ' is the canonical homeomorphism (in the category of k-spaces) which makes the triangle commutative. Since .f ^ 1/1 ı h is null homotopic, there exists H such that the first square commutes. The map ˇ is induced from .h; H / by passing to the quotients, therefore the second square commutes. It is a simple consequence of the earlier discussion of the cofibre sequence that the third square is h-commutative (Problem 1). The composition .1^e/.h^1/ is another representative of z, and the diagram shows that .1 ^ e/ˇ represents an element y 2 Ek .Y / such that f y D z.

A similar proof shows that the Zk .XI Y / form a homology theory in the variable Y .

Problems

1. Consider the cofibre sequences of two maps f W A ! B and f 0 W A0 ! B0. In the diagram

assume given h and H such that the first square commutes. The map ˇ is induced from

.h; H / by passing to the quotients. Show that the third square commutes up to homotopy (use (4.6.2)).

2.Show that the homology theory defined by a spectrum is additive (for families of wellpointed spaces).

3.Show that a weak pointed h-equivalence between well-pointed spaces induces an isomorphism in spectral homology. The use of k-spaces is therefore not essential.

4.Let Z be the sphere spectrum (7.7.1). Then, in the notation of (7.7.3),

ST..X; n/; .Y; m// D Zm n.XI Y /;

the morphism set of the category ST of Section 7.1.

7.8 Alexander Duality

are compatible with the passage to the colimit and induce a homomorphism

D W Ek .A/ ! En k .B/:

The compositions

with the interchange map are compatible with the passage to the colimit if we multiply them by . 1/nt . They induce a homomorphism

D W En k .B/ ! Ek .A/:

(7.8.1) Theorem (Alexander duality). The morphisms D and D are isomorphisms. They satisfy D D D . 1/nk id and D D D . 1/nk id.

Proof. The relations of the theorem are a direct consequence of the defining properties of an n-duality. The composition " ı .A ^ / ı ı .B ^ / equals

ı†n W ŒA^St ; EkCt 0 ! ŒSn^A^St ; Sn^EkCt 0 ! ŒA^Sn^St ; Sn^EkCt 0:

no longer a quotient map. We met this problem already in the discussion of CWcomplexes. In this auxiliary section we report about some devices to deal with such problems. The idea is to construct a category with better formal properties. One has to pay a price and change some of the standard notions, e.g., redefine topological products.

A compact Hausdorff space will be called a ch-space. For the purpose of the

following investigations we also call a ch-space a test space and a continuous map f W C ! X of a test space C a test map. A space X is called weakly hausdorff or wh-space, if the image of each test map is closed.

(7.9.1) Proposition. A Hausdorff space is a wh-space. A wh-space is a T1-space. A space X is a wh-space if and only if each test map f W K ! X is proper. If X is a wh-space, then the image of each test map is a Hausdorff space. A subspace of a wh-space is a wh-space. Products of wh-spaces are wh-spaces.

A subset A of a topological space .X; T / is said to be k-closed (k-open), if for each test map f W K ! X the pre-image f 1.A/ is closed (open) in K. The k-open sets in .X; T / form a topology kT on X. A closed (open) subset is also k-closed (k-open). Therefore kT is finer than T and the identity D X W kX ! X is continuous. We set kX D k.X/ D .X; kT /. Let f W K ! X be a test map. The same set map f W K ! kX is then also continuous. For if U kX is open, then U X is k-open, hence f 1.U / K is open. Therefore X induces for each ch-space K a bijection.

Š

TOP.K; kX/ ! TOP.K; X/; f 7!X ı f:

Hence X and kX have the same k-open sets, i.e., k.kX/ D kX. A topological space X is called k-space, if the k-closed sets are closed, i.e., if X D kX. Because of k.kX/ D kX the space kX is always a k-space. A k-space is also called compactly generated. We let k-TOP be the full subcategory of TOP with objects the k-spaces.

A whk-space is a space which is a wh-space and a k-space.

The next proposition explains the definition of a k-space. We call a topology S on X ch-definable, if there exists a family .fj W Kj ! X j j 2 J / of test maps

Proof. By Zorn’s Lemma there exists a maximal ch-definable topology S. If this topology is different from kT , then there exists an S-open set U , which is not k-open. Hence there exists a test map t W K ! X such that t 1.U / is not open. If