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

Since g f D .gf / D Œ'.v id/ # id , we conclude from (5.2.1) and (5.2.3) that g f is bijective, hence g is surjective. The bijectivity of f g shows that g is also injective. Therefore g is bijective and hence f is bijective too.

(5.2.5) Proposition. Let i W K ! X and j W K ! Y be cofibrations and f W X ! Y an h-equivalence such that f i D j . Then f is an h-equivalence under K.

Proof. By (5.2.4), we have a bijective map

f W Œ.Y; j /; .X; i/ K ! Œ.Y; j /; .Y; j / K :

Hence there exists Œg with f Œg K D Œfg K D Œid K . Since f is an h-equivalence, so is g. Since also g is bijective, g has a homotopy right inverse under K. Hence g and f are h-equivalences under K.

(5.2.6) Proposition. Let i W A ! X be a cofibration and an h-equivalence. Then i is a deformation retract.

Proof. The map i is a morphism from id.A/ to i. By (5.2.5), i is an h-equivalence under A. This means: There exists a homotopy X I ! X rel A from the identity

with a cofibration i and h-equivalences and 0. Given v W A ! X and ' W v ' u. Then there exists v0 W A0 ! X0 and '0 W 0v0 ' u0 such that v0i D f v and

't0 i D g't .

Proof. We have bijective maps (note 0f v D g v ' gu D u0i)

.g'/# ı 0 W Œ.A0; i/; .X0; f v/ A ! Œ.A0; i/; .Y 0; 0f v/ A ! Œ.A0; i/; .Y 0; u0i/ A:

Let v0 W A0 ! X0 be chosen such that .g'/# 0 Œv0 A D Œu0 A. This means: v0i D f v; and 0v0 has a transport along g' to a map which is homotopic under A to u0.

This yields a homotopy '00 W 0v0 ' u0 such that '00.i.a/; t/ D g'.a; min.2t; 1//. The homotopy ' k W .a; t/ 7!'.a; min.2t; 1// is homotopic rel A @I to '. We now use (5.1.6) in order to change this '00 into another homotopy '0 with the desired properties.

If we apply (5.2.7) in the case that u and u0 are the identity we obtain the next result (in different notation). It generalizes (5.2.5).

(5.2.8) Proposition. Given a commutative diagram

with cofibrations i, j and h-equivalences f and F . Given g W B ! A and ' W gf ' id, there exists G W Y ! X and ˆ W GF ' id such that Gj D ig and ˆt i D i't . In particular: .F; f / is an h-equivalence of pairs, and there exists a homotopy

is given with cofibration aj , bj and h-equivalences fj . Let X be the colimit of the aj and Y the colimit of the bj . Then the map f W X ! Y induced by the fj is a homotopy equivalence.

Proof. We choose inductively h-equivalences Fn W Yn ! Xn such that anFn 1 D Fnbn and homotopies 'n W Xn I ! Xn from Fnfn to id.Xn/ such that an'n 1 D 'n.an id/. This is possible by (5.2.7). The Fn and 'n induce F W Y ! X and ' W X I ! X; Ff ' id. Hence F is a left homotopy inverse of f .

Problems

1. Let i W K ! A and j W K ! B be cofibrations. Let ˛ W .B; j / ! .A; i/ be a morphism under K, W X ! Y a continuous map, and ' W K I ! X a homotopy. Then

110 Chapter 5. Cofibrations and Fibrations

commutes; here Œ˛; K Œf D Œ f ˛ .

2. Apply the transport functor to pointed homotopy sets. Assume that the inclusion f g A is a cofibration. For each path w W I ! X we have the transport

w# W ŒA; .X; w.0// 0 ! ŒA; .X; w.1// 0:

As a special case we obtain a right action of the fundamental group (transport along loops)

ŒA; X 0 1.X; / ! ŒA; X 0; .x; ˛/ 7!x ˛ D ˛#.x/:

Let v W ŒA; X 0 ! ŒA; X denote the forgetful map which disregards the base point. The map v induces an injective map from the orbits of the 1-action into ŒA; X . This map is

bijective, if X is path connected.

A space is said to be A-simple if for each path w the transport w# only depends on the

endpoints of w; equivalently, if for each x 2 X the fundamental group 1.X; x/ acts trivially on ŒA; .X; x/ 0. If A D Sn, then we say n-simple instead of A-simple. We call X simple if

it is A-simple for each well-pointed A.

3.The action on ŒI=@I; X 0 D 1.X/ is given by conjugation. Hence this action is trivial if and only if the fundamental group is abelian.

4.Let ŒA; X 0 carry a composition law induced by a comultiplication on A. Then w# is a

homomorphism. In particular 1.X/ acts by homomorphisms. (Thus, if the composition law on ŒA; X 0 is an abelian group, then this action makes this group into a right module over the integral group ring Z 1.X/.)

which can be obtained as follows. Extend the homotopy I ! A _ S.1/, t 7!t 2 S.1/ to a homotopy ' W A I ! A _ S.1/ with the initial condition A A _ S.1/ and set D '1. Express in terms of and the comultiplication of S.1/ the fact that the induced map is a group action ( is a coaction up to homotopy).

6. Let .X; e/ be a path connected monoid in h-TOP0. Then the 1.X; e/-action on ŒA; X 0 is trivial.

5.3 Replacing a Map by a Cofibration

We recall from Section 4.1 the construction of the mapping cylinder. Let f W X ! Y be a map. We construct the mapping cylinder Z D Z.f / of f via the pushout

Since h i0; i1 i is a closed cofibration, the maps h s; j i, s and j are closed cofibrations. We also have the projection q W Z.f / ! Y , .x; t/ 7!f .x/, y 7!y. In the case that f W X Y , let p W Y ! Y =X be the quotient map. We also have the quotient

map P W Z.f / ! C.f / D Z.f /=j.X/ onto the mapping cone C.f /. (Now we consider the unpointed situation. The “direction” of the unit interval is different from the one in the previous chapter.) We display the data in the next diagram. The map r is induced by q.

(5.3.1) Proposition. The following assertions hold:

(1)j and s are cofibrations.

(2)sq is homotopic to the identity relative to Y . Hence s is a deformation retraction with h-inverse q.

(3)If f is a cofibration, then q is a homotopy equivalence under X and r the induced homotopy equivalence.

(4)c.f / W Y ! C.f / is a cofibration.

Proof. (1) was already shown.

(2)The homotopy contracts the cylinder X I to X 0 and leaves Y fixed, ht .x; c/ D .x; tc C 1 t/, ht .y/ D y.

(3)is a consequence of (5.2.5).

(4)c.f / is induced from the cofibration i0 W X ! X I=X 1 via cobase

homotopy equivalence q. Factorizations of this type are unique in the following sense. Suppose f D q0j 0 W X ! Z0 ! Y is another such factorization. Then iq0 W Z0 ! Z satisfies iq0j 0 ' i. Since j 0 is a cofibration, we can change iq0 ' k such that kj 0 D j . Since iq0 is an h-equivalence, the map k is an h-equivalence under X, by (5.2.5). Also qk ' q0. This expresses a uniqueness of the factorization. If f D qj W X ! Z ! Y is a factorization into a cofibration j and a homotopy equivalence q, then Z=j.X/ is called the (homotopical) cofibre of f . The uniqueness of the factorization implies uniqueness up to homotopy equivalence of the cofibre. If f W X Y is already a cofibration, then Y ! Y =X is the projection onto the cofibre; in this case q W Z ! Y is an h-equivalence under X.

The factorization of a map into a cofibration and a homotopy equivalence is a useful technical tool. The proof of the next proposition is a good example.

112 Chapter 5. Cofibrations and Fibrations

(5.3.2) Proposition. Let a pushout diagram

f

AB

jJ

with a cofibration j be given. Then the diagram is a homotopy pushout.

Proof. Let qi W A ! Z.j / ! X be the factorization of j . Since q is a homotopy equivalence under A, it induces a homotopy equivalence

q [A id W Z.f / [A B D Z.f; j / ! X [A B D Y

be given. Suppose the inner and the outer square are homotopy cocartesian. If ˛, ˇ, are homotopy equivalences, then ı is a homotopy equivalence.

Proof. From the data of the diagram we obtain a commutative diagram

where ' and '0 are the canonical maps. By hypothesis, ' and '0 are homotopy equivalences. By (4.2.1) the map Z.ˇ; ˛; / is a homotopy equivalence.

(5.3.4) Proposition. Given a commutative diagram as in the previous proposition. Assume that the squares are pushout diagrams. Then ı is induced by ˛, ˇ, . Suppose that ˛, ˇ, are homotopy equivalences and that one of the maps k, l and one of the maps k0, l0 is a cofibration. Then ı is a homotopy equivalence.

Proof. From (5.3.2) we see that the squares are homotopy cocartesian. Thus we can apply (5.3.3).

Problems

1. A map f W X ! Y has a left homotopy inverse if and only if j W X ! Z.f / has a retraction r W Z.f / ! X. The map f is a homotopy equivalence if and only if j is a deformation retract.

2. In the case of a pointed map f W .X; / ! .Y; / one has analogous factorizations into a cofibration and a homotopy equivalence. One replaces the mapping cylinder Z.f / with the pointed mapping cylinder Z0.f / defined by the pushout

pointed cofibrations. We have a diagram as for (5.3.1) with pointed homotopy equivalences s; q and C 0.f / D Z0.f /=j.X/ the pointed mapping cone, the pointed cofibre of f .

3. h i0; i1 i W X _ X ! XI is an embedding.

4. Let f W X ! Y and g W Y ! Z be pointed maps. We have canonical maps ˛ W C.f / ! C.gf / and ˇ W C.gf / ! C.g/; ˛ is the identity on the cone and maps Y by g, and ˇ is the identity on Z and maps the cone by f id. Show that ˇ is the pointed homotopy cofibre of ˛.

5.4 Characterization of Cofibrations

We look for conditions on A X which imply that this inclusion is a cofibration. We begin by reformulating the existence of a retraction (5.1.2).

(5.4.1) Proposition. There exists a retraction r W X I ! A I [ X 0 if and only if the following holds: There exists a map u W X ! Œ0; 1Œ and a homotopy

'W X I ! X such that:

(1)A u 1.0/

(2)'.x; 0/ D x for x 2 X

(3)'.a; t/ D a for .a; t/ 2 A I

(4)'.x; t/ 2 A for t > u.x/.

Proof. Suppose we are given a retraction r. We set '.x; t/ D pr1 ı r.x; t/ and u.x/ D maxft pr2 ı r.x; t/ j t 2 I g. For (4) note the following implications: t > u.x/, pr2 r.x; t/ > 0, r.x; t/ 2 A I , '.x; t/ 2 A. The other properties

remarkable property of being the zero-set of a continuous real-valued function. Þ

(5.4.3) Lemma. Let u W X ! I and A D u 1.0/. Let ˆ W f ' g W X ! Z rel A.

We call .X; A/ a neighbourhood deformation retract (NDR ), if there exist a homotopy W X I ! X and a function v W X ! I such that:

(1)A D v 1.0/

(2).x; 0/ D x for x 2 X

(3).a; t/ D a for .a; t/ 2 A I

(4).x; 1/ 2 A for 1 > v.x/.

The pair . ; v/ is said to be an NDR-presentation of .X; A/.

(5.4.4) Proposition. .X; A/ is a closed cofibration if and only if it is an NDR.

Proof. If A X is a closed cofibration, then an NDR-presentation is obtained

(5.4.5) Theorem (Union Theorem). Let A X, B X, and A \ B X be closed cofibrations. Then A [ B X is a cofibration.

Proof ([112]). Let ' W .A [ B/ I ! Z be a homotopy and f W X ! Z an initial condition. There exist extensions ˆA W X I ! Z of 'jA I and ˆB W B I ! Z of 'jB I with initial condition f . The homotopies ˆA and ˆB coincide on

.A \ B/ I . Therefore there exists ‰ W ˆA ' ˆB rel .A \ B/ I [ X 0. Let p W X I ! X I= be the quotient map which identifies each interval

fcg I , c 2 A \ B to a point. Let T W I I ! I I switch the factors. Then ‰ ı .id T factors over p id and yields W .X I= / I ! Z.

Let u W X ! I and v W X ! I be functions such that A D u 1.0/ and B D v 1.0/. Define j W X ! X I= by j.x/ D .x; u.x/=.u.x/ C v.x/// for x … A \ B and by j.x/ D .x; 0/ D .x; t/ for x 2 A \ B. Using the compactness

(5.4.6) Theorem (Product Theorem). Let A X and B Y be closed cofibrations. Then the inclusion X B [ A Y X Y is a cofibration.

Proof. A X X Y , X B X Y , and A B D .A Y / \ .X B/

X B X Y are cofibrations. Now apply (5.4.5).

Therefore s is a section of r. Conversely, given data .a; h/ for a homotopy lifting problem. They combine to a map W X ! W .p/. The composition H D s with a section s is a solution of the lifting problem. Therefore we have shown:

(5.5.1) Proposition. The following statements about p W E ! B are equivalent:

(5.5.2) Proposition. Let p W E ! B have the HLP for X. Let i W A X be a closed cofibration and an h-equivalence. Let f W X ! B be given and a W A ! E a lifting of f over A, i.e., pa D f i. Then there exists a lifting F of f which extends a.

Proof. By (5.2.6) and (5.4.2) we know: There exists u W X ! I and ' W X I ! X rel A such that A D u 1.0/, '1 D id.X/, '0.X/ A. Set r W X ! A, x 7!'0.x/.

Define a new homotopy ˆ W X I ! X by ˆ.x; t/ D '.x; tu.x/ 1/ for t < u.x/ and ˆ.x; t/ D '.x; 1/ D x for t u.x/. We have seen in (5.4.3) that ˆ is continuous. Apply the HLP to h D f ˆ with initial condition b D ar W X ! E. The verification

h.x; 0/ D f ˆ.x; 0/ D f '.x; 0/ D f r.x/ D par.x/ D pb.x/

shows that b is indeed an initial condition. Let H W X I ! E solve the lifting problem for h; b. Then one verifies that F W X ! E, x 7!H.x; u.x// has the desired properties.

(5.5.3) Corollary. Let p W E ! B have the HLP for X I and let i W A X be a closed cofibration. Then each homotopy h W X I ! B with initial condition given on A I [ X 0 has a lifting H W X I ! E with this initial condition.

Proof. This is a consequence of (5.1.3) and (5.5.2).

(5.5.4) Proposition. Let i W A B be a (closed) cofibration of locally compact spaces. The restriction from B to A yields a fibration p W ZB ! ZA.

Let p W X ! B be a fibration. Then pZ W XZ ! BZ is a fibration for locally compact Z.

Proof. Use adjunction and the fact that X A ! X B is a cofibration for each X.

(5.5.5) Proposition. Let p W E ! B be a fibration. Then r W EI ! W .p/, v 7!

.v.0/; pv/ is a fibration.

Proof. A homotopy lifting problem for X and r is transformed via adjunction into a lifting problem for p and X I with initial condition given on the subspace

X .I 0 [ 0 I /.

(5.5.6) Proposition. Let p W E ! B be a fibration, B0 B and E0 D p 1.B0/. If B0 B is a closed cofibration, then E0 E is a closed cofibration.

Proof. Let u W B ! I and h W B I ! B be an NDR-presentation of B0 B. Let H W X I ! X solve the homotopy lifting problem for h.p id/ with initial condition id.X/. Define K W X ! X by K.x; t/ D H.x; min.t; up.x///. Then

.K; up/ is an NDR-presentation for X0 X.

The proof of the next formal proposition is again left to the reader.

(5.5.7) Proposition. Let a pullback in TOP be given.

YF X

qp

If q has the HLP for Z, then so also has p. If p is a fibration, then q is a fibration.

We call q the fibration induced from the fibration p via base change along f . In the case that f W C B the restriction p W p 1.C / ! C can be taken as the induced fibration.

The homotopy theorem for fibrations says, among other things, that homotopic maps induce h-equivalent fibrations.

Let p W X ! B be a fibration and ' W f ' g W C ! B a homotopy. We consider

There exists a homotopy ˆt W Yf ! X such that ˆ0 D F and pˆt D 't pf . The