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

168 Chapter 7. Stable Homotopy. Duality

7.3 Euclidean Complements

This section is devoted to the proof of (7.1.3). We need an interesting result from general topology.

(7.3.1) Proposition. Let A Rm and B Rn be closed subsets and let f W A ! B be a homeomorphism. Then there exists a homeomorphism of pairs

F W .Rm Rn; A 0/ ! .Rn Rm; B 0/

such that F .a; 0/ D .f .a/; 0/ for a 2 A.

Proof. By the extension theorem of Tietze (1.1.2) there exists a continuous extension ' W Rm ! Rn of f W A ! B Rn. The maps

ˆ˙ W Rm Rn ! Rm Rn; .x; y/ 7!.x; y ˙ '.x//

are inverse homeomorphisms. Let G.f / D f.a; f .a// j a 2 Ag denote the graph

The desired homeomorphism F is the composition ‰ ı ıˆC where interchanges

Let X Rn and Y Rm be closed subsets and f W X ! Y a homeomorphism. The induced homeomorphism F from (7.3.1) can be written as a homeomorphism

F W .Rn; Rn X X/ .Rm; Rm X 0/ ! .Rm; Rm X Y / .Rn; Rn X 0/:

We apply the mapping cone functor to F and use (7.2.2) and (7.2.7). The result is a homotopy equivalence

C.Rn; Rn X X/ ^ Sm ' C.Rm; Rm X Y / ^ Sn:

If X 6DRm and Y 6DRm we obtain together with (7.2.3)

†mC1.Rn X X/ ' †nC1.Rm X Y /:

If X 6DRn then we have C.Rn; Rn X X/ ' †.Rn X X/, and if X D Rn then we have C.Rn; Rn X X/ ' S0.

Suppose X 6DRn but Y D Rm. Then †mC1.Rn X X/ ' Sn. Since n m the homotopy group n.†mC1.Rn X X// D 0 and n.Sn/ Š Z. This contradiction shows that Y 6DRm.

Suppose X D Rn and Y 6DRm. Then n D m is excluded by the previous proof. Thus

Sm ' C.Rn; Rn X X/ ^ Sm ' C.Rm; Rm X Y / ^ Sn ' †mC1.Rm X Y /:

If X D Rn and Y D Rm, then

Sn ' †nC.Rn; Rn X X/ ' †mC.Rm; Rm X Y / ' Sm

and therefore m D n.

This finishes the proof of (7.1.3).

7.4 The Complement Duality Functor

The complement duality functor is concerned with the stable homotopy type of Euclidean complements Rn X X for closed subsets X Rn. We consider an associated category E. The objects are pairs .Rn; X/ where X is closed in Rn. A morphism .Rn; X/ ! .Rm; Y / is a proper map f W X ! Y . The duality functor is a contravariant functor D W E ! ST which assigns to .Rn; X/ the

object † nC.Rn; Rn X X/ D .C.Rn; Rn X X/; n/. The associated morphism D.f / W † mC.Rm; Rm X Y / ! † nC.Rn; Rn X X/ will be constructed via a

representing morphism D.f /mCn. Its construction needs some preparation. Given the data X Rn; Y Rm and a proper map f W X ! Y . Henceforth

we use the notation AjB D .A; A X B/ for pairs B A. Note that in this notation AjB C jD D A BjC D. The basic step in the construction of the functor will be an associated homotopy class

D#f W RnjDn RmjY ! RnjX Rmj0:

Here Dn again denotes the n-dimensional standard disk. A scaling function for a proper map f W X ! Y is a continuous function ' W Y ! 0; 1Πwith the property

'.f .x// kxk; x 2 X:

The next lemma shows the existence of scaling functions with an additional property.

(7.4.1) Lemma. There exists a positive continuous function W Œ0; 1Œ ! 0; 1Œ such that the inequality .kf xk/ kxk holds for x 2 X. A scaling function in the sense of the definition is then y 7! .kyk/.

Proof. The set f 1D.t/ D maxfx 2 X j kf xk tg is compact, since f is proper. Let Q .t/ be its norm maximum maxfkxk j x 2 X; kf xk tg. Then Q .kf xk/ D

170 Chapter 7. Stable Homotopy. Duality

The set of scaling functions is a positive convex cone. Let '1; '2 be scaling functions and 0 1; then '1 C .1 /'2 is a scaling function. Let 'Q '; if ' is a scaling function then also 'Q.

Let ' be a scaling function and set M.'/ D f.x; y/ j '.y/ kxkg. Then we have a homeomorphism

RnCmjDn Y ! RnCmjM.'/; .x; y/ 7!.'.y/ x; y/:

The graph G.f / D f.x; f x/ j x 2 Xg of f is contained in M.'/. We thus can continue with the inclusion and obtain a map D1.f; '/ of pairs

RnCmjDn Y ! RnCmjG.f /; .x; y/ 7!.'.y/ x; y/:

The homotopy class of D1.f; '/ does not depend on the choice of the scaling function: If '1; '2 are scaling functions, then

.x; y; t/ 7!..t'1.y/ C .1 t/'2.y// x; y/

is a homotopy from D1.f; '2/ to D1.f; '1/. A continuous map f W X ! Y has a Tietze extension fQ W Rn ! Rm. The homeomorphism .x; y/ 7!.x; y fQ.x// of RnCm sends .x; f .x// to .x; 0/. We obtain a homeomorphism of pairs

D2.f; fQ/ W RnCmjG.f / ! RnCmjX 0:

The homotopy class is independent of the choice of the Tietze extension: The homotopy .x; y; t/ 7!.x; y .1 t/fQ1.x/ tfQ2.x// proves this assertion. The duality functor will be based on the composition

D#.f / D D2.f; fQ/ ı D1.f; '/ W RnjDn RmjY ! RnjX Rmj0:

We have written D#.f /, since the homotopy class is independent of the choice of the scaling function and the Tietze extension. The morphism Df W † mC.RmjY / !

† nC.RnjX/ is defined by a representative of the colimit:

.Df /nCm W †nC.RmjY / ! †mC.RnjX/:

Consider the composition

and CD#f is obtained by applying the mapping cone to D#f ; finally, we multiply

the homotopy class of the composition by . 1/nm. We take the freedom to use

.Df /nCm W C.RmjY / ^ C n ! C.RnjX/ ^ C m as our model for Df , i.e., we do not compose with the h-equivalences of the type C n ! Sn obtained in (7.2.2).

(7.4.2) Lemma. Let f be an inclusion, f W X Y Rn. Then Df has as a representative the map C.RnjY / ! C.RnjX/ induced by the inclusion. In particular the identity of X is send to the identity.

Proof. We take the scaling function y 7! kyk C 1 and extend f by the identity. Then D#.f / is the map .x; y/ 7!..kyk C 1/ x; y .kyk C 1/ x/. The map D1 is .x; y/ 7!..kyk C 1/ x; y/ and the homotopy

.x; y; t/ 7!..1 t/.kyk C 1/ x C t.x C y/; y/

is a homotopy of pairs from D1 to .x; y/ 7!.xCy; y/. Hence D2ıD1 is homotopic to .x; y/ 7!.x C y; x/ and the homotopy .x; y; t/ 7!..1 t/x C y; x/ shows it to be homotopic to .x; y/ 7!.y; x/. Now interchange the factors and observe

(7.4.3) Lemma. The maps .x; y/ 7!.'.kyk x; y/ and .x; y/ 7!.x CgQ.y/; y/ are as maps of pairs RnjDn RmjY ! RnCmjG.f / homotopic. Here W Œ0; 1Œ !0; 1Œ is a function such that .kf xk/ kxk and '.r/ D 1 C .r/.

Proof. We use the linear homotopy ..1 t/.x C gQ.y// C t'.kyk/ x; y/. Suppose this element is contained in G.f /. Then y 2 Y and hence g.y/ D gQ.y/, and the first component equals g.y/. We solve for x and obtain

In the situation of the previous lemma the map D#.f / is homotopic to the restriction of the homeomorphism Rnj0 RmjY ! RnjX Rmj0 obtainable from (7.3.1). Another special case is obtained from a homeomorphism h of Rm and X Rm; Y D h.X/ Rm. In this case h and h 1 are Tietze extensions.

For the verification of the functor property we start with the following data:

.Rn; X/, .Rm; Y /, .Rp; Z/ and proper maps f W X ! Y , g W Y ! Z. We have the inclusion G.g/ Rm Rp and the proper map h W X ! G.g/, x 7!.f x; gf x/.

is homotopy commutative. Here nm are the appropriate interchange maps.

Proof. For the proof we use the intermediate morphism D#.h/. In the sequel we skip the notation for the scaling function and the Tietze extension. If 'f is a scaling function for f and 'gf a scaling function for gf , then

1 W Y Z ! 0; 1Π; .y; z/ 7!'f .y/; 2 W Y Z ! 0; 1Π; .y; z/ 7!'gf .z/

are scaling functions for h. We have a factorization D2.h/ D D22.h/D21.h/ where

commutes. The notation f.x; 0; gf x/g means that we take the set of all element of the given form where x 2 X. We verify that the diagram

with D D21h ı D1h ı .1 D1g/ and W RmjDm Rmj0 commutes up to homotopy. The map is, with the choice 'h D 'gf , the assignment

.x; y; z/ 7!.'gf .z/ x; 'g .z/ y fQ.'gf .z/ x/; z/:

We use the linear homotopy .'gf .z/ x; s.'g .z/ y fQ.'gf .z/ x/ C .1 s/y; z/. We verify that this is a homotopy of pairs, i.e., an element fx;Q 0; gf .x/Q g only occurs as the image of an element .x; y; z/ 2 Dn Dm Z. Thus assume

(i) xQ D 'gf .z/ x 2 X;

(ii)s'g .z/ y sfQ.'gf .z/ x/ C .1 s/y D 0;

(iii)z D gf .x/Q .

Since 'gf .z/ x 2 X, we can replace in (ii) fQ by f . We apply 'gf to (iii) and obtain

'gf .z/ D 'gf .gf .'gf .z/ x// 'gf .z/ kxk;

hence kxk 1. The equation (ii) for s D 0 says y D 0, hence y 2 Dm. Thus assume s 6D0. Then f .'gf .z/ x/ D .'g .z/ C s 1 1/ y. We apply g to this equation and use (ii):

z D gf .'gf .z/ x/ D g..'g .z/ C s 1 1/ y/:

Finally we apply 'g to this equation and obtain

'g .z/ D 'g g..'g .z/ C s 1 1/ y/ .'g .z/ C s 1 1/kyk 'g .z/kyk;

and therefore kyk 1.

Finally we show that the diagram

RnjDn RmCpjG.g/ D# .h/ RnCmCpjfx; 0; 0g

1 D2g

D# .f / 1 RnjDn RmjY Rpj0

commutes up to homotopy. In this case we use for h the scaling function 1. Then

D1h W .x; y; z/ 7!.'f .y/ x; y; z/ and

.D#f 1/ ı .1 ı D2g/.x; y; z/ D .'f .y/ x; y fQ.'f .y/ x/; z gQ.y//; D#.h/.x; y; z/ D .'f .y/ x; y fQ.'f .y/ x/; z gQfQ.'f .y/ x//:

Again we use a linear homotopy with z gQ..1 t/y C tfQ.'f .y/ x// as the third component and have to verify that it is a homotopy of pairs. Suppose the image is contained in fx; 0; 0g. Then

(i) 'f .y/ x 2 X;

Q.i/

(ii)y D f .'f .y/ x/ D f .'f .y/ x/ 2 Y ;

.ii/

(iii) z D gQ..1 t/y C tf .'f .y/ x// D g.y/.

(iii) shows that .y; z/ 2 G.g/. We apply 'f to (ii) and see that kxk 1. The three

diagrams in this proof combine to the h-commutativity of the diagram in (7.4.4).

(7.4.5) Proposition. Suppose that h W X I ! Y is a proper homotopy. Then

D.h0/ D D.h1/.

174 Chapter 7. Stable Homotopy. Duality

Proof. Let j0 W X ! X I , x 7!.x; 0/. The map is the composition of the homeomorphism a W X ! X 0 and the inclusion b W X 0 X I . Thus Da is an isomorphism and Db is induced by the inclusion RnC1jX I RnC1jX 0; hence Db is an isomorphism, since induced by a homotopy equivalence (use (7.4.2)). Thus Dj0 is an isomorphism. The composition pr ıj0 is the identity; hence D.pr/ is inverse to D.j0/. A similar argument for j1 shows that Dj0 D Dj1. We conclude that the maps ht D h ı jt have the same image under D.

(7.4.6) Remark. The construction of the dual morphism is a little simpler for a map between compact subsets of Euclidean spaces. Let X Rn be compact. Choose a disk D such that X D. Then the dual morphism is obtained from

Rnj0 RmjY RnjD RmjY RnCmjG.f / ! RnjX Rmj0

where the last morphism is as before .x; y/ 7!.x; y fQ.x//. Also the proof of the functoriality (7.4.4) is simpler in this case.

The composition .D#f 1/.1 D#g/ is .x; y; z/ 7!.x; y fQ.x/; z gQ.y//. The other composition is .x; y; z/ 7!.x; y; z gQfQ.x//. Then we use the homo-

topies of pairs .x; y fQ.x/; z gQ..1 t/yCtfQ.x/// and .x; y tfQ.x/; z gQfQ.x//.

Þ

Problems

1.Verify in detail that the commutativity of the diagram in (7.4.4) implies that D is a functor.

2.Use the homotopy invariance of the duality functor and generalize (7.1.3) as follows. Suppose X Rn and Y Rm are closed subsets which are properly homotopy equivalent. Let n m.

(1)If Rn X X 6D ;, then Rm X Y 6D ;:

(2)Let Rn 6D X. For each choice of a base point Rm X Y has the same stable homotopy type as †n m.Rn X X/.

(3)If Rn X X is empty and Rm X Y is non-empty, then Rm X Y has the stable homotopy type of Sm n 1.

(4)If n D m then the complements of X and Y have the same number of path components.

3.Let X Rn and Y Rm be closed subsets and f W X ! Y a proper map. Consider the closed subspace W Rm R Rn of points

Then W is homeomorphic to the mapping cylinder Z.f / of f .

4. Let X Rn and f W X ! Rn Rm the standard embedding x 7!.x; 0/. Then Df is represented by the homotopy equivalence

C.RnjX Rmj0/ ˛ C.RnjX/ ^ C m ' C.RnjX/ ^ Sm:

(Direct proof or an application of (7.4.3).)

5. Let X Rn be compact. Suppose kxk r > 0 for x 2 X. Then the constant function '.t/ D r is a scaling function for each f W X ! Y . Show that the map

m n n m n m

C.R jY / ^ C ! C ^ C.R jY / ! C.R jX/ ^ C

which is obtained from the definition in (7.4.6) is homotopic to the map C.D#f / of the general definition.

7.5 Duality

We have associated to a proper map between closed subsets of Euclidean spaces a dual morphism in the stable category ST. If X Rn then the stable homotopy type of Rn X X or C.RnjX/ is to be considered as a dual object of X. There is a

categorical notion of duality in tensor categories.

Let A and B be pointed spaces. An n-duality between .A; B/ consists of an evaluation

" W B ^ A ! Sn

and a coevaluation

W Sn ! A ^ B

such that the following holds:

(1) The composition

.1 ^ "/. ^ 1/ W Sn ^ A ! A ^ B ^ A ! A ^ Sn

is homotopic to the interchange map .

(2) The composition

." ^ 1/.1 ^ / W B ^ Sn ! B ^ A ^ B ! Sn ^ B

is homotopic to . 1/n .

We now construct an n-duality for .B; A/ D .C.RnjK/; KC/ where K Rn is a suitable space. In the general definition of an n-duality above we now replace Sn by C n. The evaluation is defined to be

n n C.d / n

" W C.R jK/ ^ C.K; ;/ ! C.R jK KjK/ ! C.R j0/ where d is the difference map

d W .Rn K; .Rn X K/ K/ ! .Rn; Rn X 0/; .x; k/ 7!x k as a map of pairs. This definition works for arbitrary K Rn.

176 Chapter 7. Stable Homotopy. Duality

Let K Rn be compact and D Rn a large disk which contains K. Let V be an open neighbourhood of K. Consider the following diagram

with the diagonal W x 7!.x; x/. We apply the mapping cone functor and (7.2.1), (7.2.7). The maps i and j induce h-equivalences. We obtain

V W C n ! V C ^ C.RnjK/:

We want to replace V by K in order to obtain the desired map. This can be done if we assume that there exists a retraction r W V ! K of K V . Then we can compose with rC W V C ! KC and obtain a coevaluation

W C n ! C.K; ;/ ^ C.RnjK/:

We call a closed subspace K Rn a Euclidean neighbourhood retract (D ENR) if there exists a retraction r W V ! K from a suitable neighbourhood V of K in Rn. We mention here that this is a property of K that does not depend on the particular embedding into a Euclidean space; see (18.4.1).

The basic duality properties of " and are:

(7.5.1) Proposition. The maps " and are an n-duality for the pair .KC; C.RnjK//.

Proof. For the proof of the first assertion we consider the diagram

with ˛.x; y/ D .y; x/, ˇ.x; y/ D .y; x y/, and D .1 d /. 1/ W .x; y/ 7!

.x; x y/. The homotopy .x; y; t/ 7!.tx C .1 t/y; x y/ shows that the right square is h-commutative and the homotopy .x; y; t/ 7!.y; x ty/ shows that the triangle is h-commutative. The axiom (1) of an n-duality now follows if we write out the morphisms according to their definition and use the result just proved.

For the proof of the axiom (2) we start with the diagram

with ˛.x; y/ D . y; x/, ˇ.x; y/ D .x y; x/, and .x; y/ D .x y; y/. The homotopy .x; y; t/ 7!.x ty .1 t/r.y/; y/ shows that the bottom triangle is h- commutative; the homotopy .x; y; t/ 7!.x y; .1 t/x Cty/ shows ' ˇ.1 j /;

Given a natural duality for objects via evaluations and coevaluations one can define the dual of an induced map. We verify that we recover in the case of compact ENR the morphisms constructed in the previous section. The following three proposition verify that the duality maps have the properties predicted by the categorical duality theory.

(7.5.2) Proposition. Let X Rn be compact and a retract of a neighbourhood V . The following diagram is homotopy commutative

Proof. We reduce the problem to maps of pairs. We use the simplified definition (7.4.6) of the duality map. First we have the basic reduction

RmjY Rnj0 RmjY RnjD ! RmjY RnjX RmjY V jX:

Then the remaining composition RmjY V jX ! Rmj0 RnjX which involves , C.f /, " is the assignment .y; x/ 7!.y f r.x/; x/. The other map is .y; x/ 7!

.y fQ.x/; x/. Now we observe that we can arrange that fQjV D f r (by possibly passing to a smaller neighbourhood, see Problem 1).

Dual maps are adjoint with respect to evaluation and coevaluation. This is the content of (7.5.3) and (7.5.4).

(7.5.3) Proposition. The following diagram is homotopy commutative