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

98 Chapter 4. Elementary Homotopy Theory can be replaced by the homeomorphic space

F .f 1/ D f.w; v/ 2 F Y FX j w.1/ D f v.1/g:

Then f 2 becomes f 2 W F .f 1/ ! F .f /, .w; v/ 7!.v.1/; w/. The map

j.f / W Y ! F .f 1/; w 7!.w; k /

satisfies f 2 ı j.f / D i.f /.

(4.7.1) Note. The injection j.f / is a homotopy equivalence.

Proof. We construct a homotopy ht of the identity of F .f 1/ which shrinks the path v to its beginning point and drags behind the path w correspondingly. We write ht .w; v/ D .h1t .w; v/; h2t .w; v// 2 F Y FX and define

We treat the next step in a similar manner.

f

The upper triangle is commutative, and (4.7.2) applies to the lower one. The map i.f 1/ is the embedding of the fibre over the base point. Let W Y ! Y , w 7!w be the inverse.

(4.7.2) Note. j.f / ı ı f ' i.f 1/.

Proof. We compose both sides with the h-equivalence r.f / from the proof of

is h-exact.

Proof. From the definitions and standard properties of mapping spaces we haveF .f / X F Y and F .f / X F Y . We use the exponential law for mapping spaces and consider F Y and F Y as subspaces of Y I I . In the first case we have to use all maps which send @I I [ I 0 to the base point, in the base point. Interchanging

is h-exact. We call it the fibre sequence of f . When we apply the functor ŒB; 0 to the fibre sequence we obtain an exact sequence of pointed sets which consists from the fourth place onwards of groups and homomorphisms and from the seventh place onwards of abelian groups ([147]).

Problems

1.Work out the dual of (4.6.5).

2.Describe what happens to the fibre sequence under adjunction. A map

a W T ! F .f / D f.x; w/ 2 X F Y j f .x/ D w.1/g

has two components b W T ! X and ˇ W T ! F Y . Under adjunction, ˇ corresponds to a map B W C T ! Y from the cone over T . The condition f .x/ D w.1/ is equivalent to

The image of is obtained in the following manner: With the quotient map p W C T ! †T we have B D ı p, and b is the constant map c.

3. There exist several relations between fibre and cofibre sequences.

The adjunction .†; / yields in TOP0 the maps W X ! †X (unit of the adjunction) and " W † X ! X (counit of the adjunction). These are natural in the variable X. For each f W X ! Y we also have natural maps

W F .f / ! C.f /; " W †F .f / ! C.f /

and " adjoint to . Verify the following assertions from the definitions.

(1) The next diagram is homotopy commutative

(2) Let i W X ! Z.f / be the inclusion and r W Z.f / ! Y the retraction. A path in Z.f / that starts in and ends in X 0 yields under the projection to C.f / a loop. This gives a map W F .i/ ! C.f /. The commutativity ı F .r/ ' ı holds.

(3) The next diagram is homotopy commutative

†f

†X †Y .

Chapter 5

Cofibrations and Fibrations

This chapter is also devoted to mostly formal homotopy theory. In it we study the homotopy extension and lifting property.

An extension of f W A ! Y along i W A ! X is a map F W X ! Y such that F i D f . If i W A X is an inclusion, then this is an extension in the ordinary sense. Many topological problems can be given the form of an extension problem. It is important to find conditions on i under which the extendibility of f only depends on the homotopy class of f . If this is the case, then f is called a cofibration.

The dual of the extension problem is the lifting problem. Suppose given maps p W E ! B and f W X ! B. A lifting of f along p is a map F W X ! E such

that pF D f . We ask for conditions on p such that the existence of a lifting only depends on the homotopy class of f . If this is the case, then f is called a fibration.

Each map is the composition of a cofibration and a homotopy equivalence and (dually) the composition of a homotopy equivalence and a fibration. The notions are then used to define homotopy fibres (“homotopy kernels”) and homotopy cofibres (“homotopy cokernels”). Axiomatizations of certain parts of homotopy theory (“model categories”) are based on these notions. The notions also have many practical applications, e.g., to showing that maps are homotopy equivalences with additional properties like fibrewise homotopy equivalences.

Another simple typical example: A base point x 2 X is only good for homotopy theory if the inclusion fxg X is a cofibration (or the homotopy invariant weakening, a so-called h-cofibration). This is then used to study the interrelation between pointed and unpointed homotopy constructions, like pointed and unpointed suspensions.

5.1 The Homotopy Extension Property

A map i W A ! X has the homotopy extension property (HEP) for the space Y if for each homotopy h W A I ! Y and each map f W X ! Y with f i.a/ D h.a; 0/

there exists a homotopy H W X I ! Y with H.x; 0/ D f .x/ and H.i.a/; t/ D h.a; t/. We call H an extension of h with initial condition f . The map i W A ! X

102 Chapter 5. Cofibrations and Fibrations

is a cofibration if it has the HEP for all spaces. The data of the HEP are displayed in the next diagram. We set itX W X ! X I , x 7!.x; t/ and e0.w/ D w.0/.

For a cofibration i W A ! X, the extendibility of f only depends on its homotopy class.

From this definition one cannot prove directly that a map is a cofibration, but it suffices to test the HEP for a universal space Y , the mapping cylinder Z.i/ of i. Recall that Z.i/ is defined by a pushout

initial condition b and the homotopy k. The HEP then provides us with a map

i.e., s is an embedding and r a retraction. Let r be a retraction of s. Given f and h, find as above and set H D r. Then H extends h with initial condition f . Altogether we have shown:

(5.1.1) Proposition. The following statements about i W A ! X are equivalent:

(1)i is a cofibration.

(2)i has the HEP for the mapping cylinder Z.i/.

A cofibration i W A ! X is an embedding; and i.A/ is closed in X, if X is a Hausdorff space (Problem 1). Therefore we restrict attention to closed cofibrations

whenever this simplifies the exposition. A pointed space .X; x/ is called wellpointed and the base point nondegenerate if fxg X is a closed cofibration.

(5.1.2) Proposition. If i W A X is a cofibration, then there exists a retraction r W X I ! X 0 [ A I . If A is closed in X and if there exists a retraction r, then i is a cofibration.

Proof. Let Y D X 0 [ A I , f .x/ D .x; 0/, and h.a; t/ D .a; t/. Apply the HEP to obtain a retraction r D H .

If A is closed in X, then g W X 0[A I ! Y , g.x; 0/ D f .x/, .a; t/ D h.a; t/ is continuous. A suitable extension H is given by gr.

(5.1.3) Example. Let r W X I ! X 0 [ A I be a retraction. Set r.x; t/ D

.r1.x; t/; r2.x; t//. Then

H W X I I ! X I; .x; t; s/ 7!.r1.x; t.1 s//; st C .1 s/r2.x; t//

is a homotopy relative to X 0 [ A I of r to the identity, i.e., a deformation retraction.

(5.1.4) Example. The inclusions Sn 1 Dn and @I n I n are cofibrations. A retraction r W Dn ! Sn 1 I [ Dn 0 was constructed in (2.3.5). Þ

It is an interesting fact that one need not assume A to be closed. Strøm [180, Theorem 2] proved that an inclusion A X is a cofibration if and only if the subspace X 0 [ A I is a retract of X I .

If we multiply a retraction by id.Y / we obtain again a retraction. Hence A Y ! X Y is a (closed) cofibration for each Y , if i W A ! X is a (closed) cofibration. Since we have proved (5.1.2) only for closed cofibrations, we mention another special case, to be used in a moment. Let Y be locally compact and i W A ! X a cofibration. Then i id W A Y ! X Y is a cofibration. For a proof use the fact that via adjunction and the exponential law for mapping spaces the HEP of i id for Z corresponds to the HEP of i for ZY .

(5.1.5) Proposition. Let A X and assume that A I X I has the HEP for Y . Given maps ' W A I I ! Y; H W X I ! Y; f " W X I ! Y such that

'.a; s; 0/ D H.a; s/; f ".x; 0/ D H.x; "/; f ".a; t/ D '.a; "; t/

" 2 f0; 1g, a 2 A, x 2 X, s; t 2 I . Then there exists ˆ W X I I ! Y such that

coincide on A .I 0 [ @I I /. Let k W .I I; I 0 [ @I I / ! .I I; I 0/

be a homeomorphism of pairs. Since A I ! X I has the HEP for Y , there exists ‰ W X I I ! Y which extends ' ı .1 k 1/ and ˛ ı .1 k 1/. The

map ˆ D ‰ ı .1 k/ solves the extension problem.

104 Chapter 5. Cofibrations and Fibrations

(5.1.6) Proposition. Let i W A X be a cofibration. Then X @I [ A I X I is a cofibration.

For A D ; we obtain from (5.1.5) that X @I X I is a cofibration, in particular @I I and f0g I are cofibrations. Induction over n shows again that @I n I n is a cofibration.

We list some special cases of (5.1.5) for a cofibration A X.

(5.1.7) Corollary. .1/ Let ˆ W X I ! Y be a homotopy. Suppose ' D ˆjA I is homotopic rel A @I to . Then ˆ is homotopic rel X @I to ‰ W X I ! Y such that ‰jA I D .

.2/ Let ˆ solve the extension problem for .'; f / and ‰ the extension problem for . ; g/. Suppose f ' g rel A and ' ' rel A @I . Then ˆ1 ' ‰1 rel A.

.3/ Let ˆ; ‰ W X I ! Y solve the extension problem for .h; f /. Then there

f

AB

jJ

If j has the HEP for Z, then J has the HEP for Z. If j is a cofibration, then J is a cofibration.

Proof. Suppose h W B I ! Z and ' W Y ! Z are given such that h.b; 0/ D fJ.b/ for b 2 B. We use the fact that the product with I of a pushout is again a pushout. Since j is a cofibration, there exists Kt W X ! Z such that K0 D 'f and Kt j D ht f . By the pushout property, there exists Ht W Y ! Z such that Ht F D Kt and Ht J D ht . The uniqueness property shows H0 D ', since both maps have the same composition with Fj and Jf .

We call J the cofibration induced from j via cobase change along f .

Example. If A X is a cofibration, then fAg X=A is a cofibration. Sn 1 Dn

Our next result, the homotopy theorem for cofibration says, among other things, that homotopic maps induce h-equivalent cofibrations from a given cofibration under a cobase change.

Let j W A ! X be a cofibration and 't W f ' g W A ! B a homotopy. We consider two pushout diagrams.

Since j is a cofibration, there exists a homotopy ˆt W X ! Yf with initial condition ˆ0 D F and ˆt j D jf 't . The pushout property of the Yg -diagram provides us with a unique map D ' such that jg D jf and G D ˆ1. (We use the notation' although the map depends on ˆ1.) Thus ' is a morphism of cofibrationsW jg ! jf between objects in TOPB . Moreover G ' F . We now verify that the homotopy class of is independent of some of the choices involved. Let t be another homotopy from f to g which is homotopic to 't relative to A @I . Let ‰t W X ! Yf be an extension of jf t with initial condition ‰0 D F . LetW A I I ! B be a homotopy rel A @I from ' to . These data give us on X 0 I [ X I @I a map into Yf such that

.x; 0; t/ D F .x/; .x; s; 0/ D ˆ.x; s/; .x; s; 1/ D ‰.x; s/:

By (5.1.5) there exists an extension, still denoted , to X I I such that jf D.j id id/. We multiply the Yg diagram by I and obtain again a pushout. It provides us with a unique homotopy K W Yg I ! Yf such that K ı .G id/ D 1 and K ı .jg id/ D jf ı pr where 1 W X I ! Yf , .x; t/ 7! .x; 1; t/. By construction, K is a homotopy under B from ' to a corresponding map obtained from t and ‰t . We thus have shown that the homotopy class Œ B under B of only depends on the morphism Œ' from f to g in the groupoid ….A; B/. Let us write Œ D ˇŒ' .

We verify that ˇ is a functor ˇ.Œ ~ Œ' / D ˇŒ' ı ˇŒ . Let W g ' h W A !

the pushout data for .j; h/.) Since ' ‰0 D ' G D ˆ1, we can form the product

determined by ' H D ' ‰1 D ' H and ' jh D jf D ' jg D ' jh. Therefore ' represents ˇ.Œ ~ Œ' /.

Let h-COFB denote the full subcategory of h-TOPB with objects the cofibrations under B. Then we have shown above:

106 Chapter 5. Cofibrations and Fibrations

(5.1.9) Theorem. Let j W A ! X be a cofibration. We assign to the object f W A ! B in ….A; B/ the induced cofibration jf W B ! Yf and to the mor-

Since ….A; B/ is a groupoid, Œ ' is always an isomorphism in h-TOPB . We refer to this fact as the homotopy theorem for cofibrations.

(5.1.10) Proposition. In the pushout (5.1.8) let j be a cofibration and f a homotopy equivalence. Then F is a homotopy equivalence.

Proof. With an h-inverse g W B ! A of f we form a pushout

g

BA

Ji

YG Z.

Since gf ' id, there exists, by (5.1.9), an h-equivalence W Z ! X under A such that GF ' id. Hence F has a left h-inverse and G a right h-inverse. Now interchange the roles of F and G.

Problems

1. A cofibration is an embedding. For the proof use that i1 W A ! Z.i/, a 7!.a; 1/ is an embedding. From i1 D rsi1 D ri1X i then conclude that i is an embedding.

Consider a cofibration as an inclusion i W A X. The image of s W Z.i/ ! X I is the subset X 0 [ A I . Since s is an embedding, this subset equals the mapping cylinder, i.e., one can define a continuous map X 0 [ A I by specifying its restrictions to X 0 and A I . (This is always so if A is closed in X, and is a special property of i W A X if i is a cofibration.)

Let X be a Hausdorff space. Then a cofibration i W A ! X is a closed embedding. Let r W X I ! X 0 [ A I be a retraction. Then x 2 A is equivalent to r.x; 1/ D .x; 1/. Hence A is the coincidence set of the maps X ! X I , x 7!.x; 1/, x 7!r.x; 1/ into a

Hausdorff space and therefore closed.

2. If i W K ! L, j W L ! M have the HEP for Y , then j i has the HEP for Y . A homeomorphism is a cofibration. ; X is a cofibration. The sum qij W q Aj ! qXj of cofibrations ij W Aj ! Xj is a cofibration.

3. Let p W P ! Q be an h-equivalence and i W A B a cofibration. Then f W A ! P has an extension to B if and only if pf has an extension to B. Suppose f0; f1 W B ! P agree

on A. If pf0 and pf1 are homotopic rel A so are f0; f1.

4. Compression. Let A X be a cofibration and f W .X; A/ ! .Y; B/ a map which is homotopic as a map of pairs to k W .X; A/ ! .B; B/. Then f is homotopic relative to A to a map g such that g.X/ B.

5. Let A X be a cofibration and A contractible. Then the quotient map X ! X=A is a homotopy equivalence.

j

A C 0A

fF

Since j is a cofibration, so is J . If f is a cofibration, then F is a cofibration. There exists a canonical homeomorphism X [ C 0A=C 0A Š X=A; it is induced by J . Since C 0A is

contractible, we obtain a homotopy equivalence X [ C 0A ! X [ C 0A=C 0A Š X=A.

8. The unpointed suspension †0X of a space X is obtained from X I if we identify each of the sets X 0 and X 1 to a point. If is a basepoint of X, we have the embedding j W I ! †0X, t 7!. ; t/. If f g X is a closed cofibration, then j is a closed (induced) cofibration. The quotient map †0X ! †X is a homotopy equivalence.

5.2 Transport

Let i W K ! A be a cofibration and ' W K I ! X a homotopy. We define a map

'# W Œ.A; i/; .X; '0/ K ! Œ.A; i/; .X; '1/ K ;

called transport along ', as follows: Let f W A ! X with f i D '0 be given. Choose a homotopy ˆt W A ! X with ˆ0 D f and ˆt i D 't . We define '#Œf D Œˆ1 . Then (5.1.5) shows that '# is well defined and only depends on the

The transport functor measures the difference between “homotopic” in TOPK and in TOP. The following is a direct consequence of the definitions.

(5.2.2) Proposition. Let i W K ! A be a cofibration. Let f W .A; i/ ! .X; g/ and f 0 W .A; i/ ! .X; g0/ be morphisms in TOPK . Then Œf D Œf 0 , if and only if there