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

118 Chapter 5. Cofibrations and Fibrations

W C I I ! B relative to C @I . We obtain in a similar manner a map from a lifting ‰t of t pf . Claim: The maps ' and are homotopic over C . In order to verify this, we lift the homotopy ı.pf id id/ W Yf I I ! B to a homotopy with initial data .y; s; 0/ D ˆ.y; s/, .y; s; 1/ D ‰.y; s/, and .y; 0; t/ D F .t/ by an application of (5.1.5). The homotopy H W .y; t/ 7! .y; 1; t/ yields, by the pullback property of the right square, a homotopy K W Yf I ! Yg such that GK D H and pg K D pr ıpf . By construction, K is a homotopy over C from f to g . The reader should now verify the functoriality Œ ' D Œ Œ ' .

Let h-FIBC be the full subcategory of h-TOPC with objects the fibrations over C .

(5.5.9) Proposition. Let p W X ! B be a fibration. We assign to f W C ! B the induced fibration pf W Yf ! C and to the morphism Œ' W f ! g in ….C; B/ the morphism Œ ' . This yields a functor ….C; B/ ! h-FIBC .

Since ….C; B/ is a groupoid, Œ ' is always an isomorphism in h-TOPB . This fact we call the homotopy theorem for fibrations.

As a special case of (5.5.9) we obtain the fibre transport. It generalizes the fibre transport in coverings. Let p W E ! B be a fibration and w W I ! B a path from b to c. We obtain a homotopy equivalence TpŒw W Fb ! Fc which only depends on the homotopy class Œw of w, and TpŒu v D TpŒv TpŒu . This yields a functor Tp W ….B/ ! h-TOP. In particular the fibres over points in the same path component of B are h-equivalent.

(5.5.10) Proposition. In the pullback (5.5.7) let p be a fibration and f a homotopy equivalence. Then F is a homotopy equivalence.

Proof. The proof is based on (5.5.9) and follows the pattern of (5.1.10).

(5.5.11) Remark. The notion of fibration and cofibration are not homotopy invari-

ant. The projection I 0 [ 0 I ! I , .x; t/ 7!x is not a fibration, but the map is over I h-equivalent to id. One definition of an h-fibration p W E ! B is

that homotopies X I ! B which are constant on X Œ0; " ; " > 0 can be lifted

with a given initial condition; a similar definition for homotopy extensions gives the notion on an h-cofibration. In [46] you can find details about these notions.

Problems

1.A composition of fibrations is a fibration. A product of fibrations is a fibration. ; ! B is a fibration.

2.Suppose p W E ! B has the HLP for Y I n. Then each homotopy h W Y I n I ! B

has a lifting to E with initial condition given on Y .I n 0 [ @I n I /.

3. Let p W E ! B I be a fibration and p0 W E0 ! B its restriction to B 0 D B. Then there exists a fibrewise h-equivalence from p0 id.I / to p which is over B 0 the inclusion

E0 ! E.

4.Go through the proof of (5.5.9) and verify a relative version. Let .C; D/ be a closed cofibration. Consider only maps C ! B with a fixed restriction d W D ! B and homotopies

relative to D. Let pD W YD ! B be the pullback of p along d . Then the maps pf have the form .pf ; pD / W .Yf ; YD / ! .C; D/. By (5.5.6), .Yf ; YD / is a closed cofibration, and by (5.5.3) the homotopies ˆt can be chosen constant on YD . The maps ' W Yf ! Yg are then the identity on YD . The homotopy class of ' is unique as a map over C and under YD .

5.Let .B; C / be a closed deformation retract with retraction r W B ! C . Let p W X ! B be a fibration and pC W XC ! C its restriction to C . Then there exists a retraction R W X ! XC such that pC R D rp.

5.6 Transport

We construct a dual transport functor. Let p W E ! B be a fibration, ' W Y I ! B a homotopy and ˆ W Y I ! E a lifting along p with initial condition f . We define

'# W Œ.Y; '0/; .E; p/ B ! Œ.Y; '1/; .E; p/ B ; Œf 7!Œˆ1 :

One shows that this map is well defined and depends only on the homotopy class

(5.6.1) Proposition. The assignments f 7!Œf; p B and Œ' 7!'# are a functor, called transport functor, from ….Y; B/ into the category of sets. Since ….Y; B/ is a groupoid, '# is always bijective.

(5.6.3) Theorem. Let f W X ! Y be an h-equivalence and p W E ! B be a fibration. Then f W Œv; p B ! Œvf; p B is bijective for each v W Y ! B.

Proof. The proof is based on (5.6.1) and (5.6.2) and formally similar to the proof of (5.2.4).

(5.6.4) Theorem. Let p W X ! B and q W Y ! B be fibrations. Let h W X ! Y be an h-equivalence and a map over B. Then f is an h-equivalence over B.

Proof. The proof is based on (5.6.3) and formally similar to the proof of (5.2.5).

(5.6.5) Corollary. Let q W Y ! C be a fibration and a homotopy equivalence.

Chapter 6

Homotopy Groups

The first fundamental theorem of algebraic topology is the Brouwer–Hopf degree theorem. It says that the homotopy set ŒSn; Sn has for n 1 a homotopically defined ring structure. The ring is isomorphic to Z, the identity map corresponds to 1 2 Z and the constant map to 0 2 Z. The integer associated to a map f W Sn ! Sn is called the degree of f . We have proved this already for n D 1. Also in the general case “degree n” roughly means that f winds Sn n-times around itself. In order to give precision to this statement, one has to count the number of pre-images of a “regular” value with signs. This is related to a geometric interpretation of the degree in terms of differential topology.

Our homotopical proof of the degree theorem is embedded into a more general investigation of homotopy groups. It will be a simple formal consequence of the so-called excision theorem of Blakers and Massey. The elegant elementary proof of this theorem is due to Dieter Puppe. It uses only elementary concepts of homotopy theory, it does not even use the group structure. (The excision isomorphism is the basic property of the homology groups introduced later where it holds without any restrictions on the dimensions.)

Another consequence of the excision theorem is the famous suspension theorem of Freudenthal. There is a simple geometric construction (the suspension) which leads from ŒSm; Sn to ŒSmC1; SnC1 . Freudenthal’s theorem says that this process after a while is “stable”, i.e., induces a bijection of homotopy sets. This is the origin of the so-called stable homotopy theory – a theory which has developed into a highly technical mathematical field of independent interest and where homotopy theory has better formal and algebraic properties. (Homology theory belongs to stable homotopy.)

The degree theorem contains the weaker statement that the identity of Sn is not null homotopic. It has the following interpretation: If you extend the inclusion Sn 1 Rn to a continuous map f W Dn ! Rn, then there exists a point x with f .x/ D 0. For n D 1 this is the intermediate value theorem of calculus; the higher dimensional analogue has other interesting consequences which we discuss under the heading of the Brouwer fixed point theorem.

This chapter contains the fundamental non-formal results of homotopy theory. Based on these results, one can develop algebraic topology from the view-point of homotopy theory. The chapter is essentially independent of the three previous chapters. But in the last section we refer to the definition of a cofibration and a suspension.

n.X; / D Œ.I n; @I n/; .X; f g/ Š ŒI n=@I n; X 0

with the group structure defined below. For n D 1 it is the fundamental group. The definition of the set n.X; / also makes sense for n D 0, and it can be identified with the set 0.X/ of path components of X with Πas a base point. The

As in the case of the fundamental group one shows that this composition law is a group structure. The next result is a consequence of (4.3.1); a direct verification along the same lines is easy. See also (2.7.3) and the isomorphism (2) below.

A group structure Ci , 1 i n is defined again by the formula (1) above. There is no group structure in the case n D 0.

We now consider n as a functor. Composition with f W .X; A; / ! .Y; B; / induces f W n.X; A; / ! n.Y; B; /; this is a homomorphism for n 2. Sim-

morphism j W n.X; / ! n.X; A; / is obtained by interpreting the first group as n.X; f g; / and then using the map induced by the inclusion .X; f g; /

.X; A; /. Maps which are pointed homotopic induce the same homomorphisms. The group n.X; A; / is commutative for n 3.

We rewrite the homotopy groups in terms of mapping spaces. This is not strictly necessary for the following investigations but sometimes technically convenient.

the constant map is the base point. The space .X/ D .X/ is the loop space of X.

is exact. The maps i and j are induced by the inclusions.

Proof. We prove the exactness for the portion involving 0 and 1 in an elementary manner. Exactness at 0.A; / and the relations @j D 0 and j i D 0 are left to the reader.

Let w W I ! X represent an element in 1.X; A; / with @Œw D 0. This means: There exists a path u W I ! A with u.0/ D w.1/ and u.1/ D . The product w u is then a loop in X. The homotopy H which is defined by

wu

shows j Œw u D Œw . Thus we have shown exactness at 1.X; A; /.

124 Chapter 6. Homotopy Groups

Given a loop w W I ! X. Let H W .I; @I; 0/ I ! .X; A; / be a homotopy from w to a constant path. Then u W s 7!H.1; s/ is a loop in A. We restrict H to the boundary of the square and compose it with a linear homotopy to prove

with base point . ; k/, k W I ! f g the constant path. This space is homeomorphic

to

F .X; A/ D fw 2 XI j w.0/ D ; w.1/ 2 Ag;

i. / becomes the inclusion .X/ F .X; A/, and 1 the evaluation F .X; A/ ! A,

By standard properties of adjunction we see that the assignment Œf 7!Œfx is a well-defined bijection

.4/ nC1.X; A; / Š n.F .X; A/; /;

and in fact a homomorphism with respect to the composition laws Ci for 1 i n. These considerations also make sense for n D 0. In the case that A D f g, the space F .X; A/ is the loop space .X/.

The exact sequence is also obtained from the fibre sequence of . Under the identifications (4) the boundary operator is transformed into

1 W ŒI n=@I n; F .X; A/ 0 ! ŒI n=@I n; A 0; and nC1.X; / ! nC1.X; A; / is transformed into

i. / W ŒI n=@I n; .X/ 0 ! ŒI n=@I n; F .X; A; / 0:

Now apply B D I n=@I n to the fibre sequence (4.7.4) of W A X to see the exactness of a typical portion of the homotopy sequence. Þ

(6.1.4) Remark. In the sequel it will be useful to have different interpretations for elements in homotopy groups. (See also the discussion in Section 2.3.) We set S.n/ D I n=@I n and D.n C 1/ D CS.n/, the pointed cone on S.n/. We have homeomorphisms

S.n/ ! @I nC1=J n; D.n C 1/ ! I nC1=J n;

the first one x 7!.x; 1/, the second one the identity on representatives in I n I ; moreover we have the embedding S.n/ ! D.nC1/, x 7!.x; 1/ which we consider as an inclusion. These homeomorphisms allow us to write

n.X; / Š ŒI n=@I n; X 0 D ŒS.n/; X 0;

nC1.X; A; / Š Œ.I nC1=J n; @I nC1=J n/; .X; A/ 0 Š Œ.D.nC1/; S.n//; .X; A/ 0;

.DnC1; Sn/ ! .X; A/ and elements in n.X; / by pointed maps Sn ! X. In comparing these different models for the homotopy groups it is important to remember the homeomorphism between the standard objects (disks and spheres), since there are two homotopy classes of homeomorphisms. Þ

Problems

1. n.A; A; a/ D 0. Given f W .I n; @I n; J n 1/ ! .A; A; a/. Then a null homotopy is ft .x1; : : : ; xn/ D f .x1; x2; : : : ; .1 t/xn/.

isomorphism colimi n.Xi ; / Š n.X; /.

3. Let .X; A; B; b/ be a pointed triple. Define the boundary operator @ W n.X; A; b/ !n 1.A; b/ ! n 1.A; B; b/ as the composition of the previously defined operator with the map induced by the inclusion. Show that the sequence

126 Chapter 6. Homotopy Groups

5. Let f W .X; x/ ! .Y; y/ be a pointed map. One can embed the induced morphism f W n.X; x/ ! n.Y; y/ into an exact sequence which generalizes the case of an inclusion f . Let Z.f / be the pointed mapping cylinder of f and f D pi W X ! Z.f / ! Y the standard factorization into an inclusion and a homotopy equivalence, as explained in (5.3.1).

One can define the groups n.Z.f /; X; / without using the mapping cylinder. Consider commutative diagrams with pointed maps ' and ˆ.

We consider .'; ˆ/ W j ! f as a morphism in the category of pointed arrows. Letn.f / denote the set of homotopy classes of such morphisms. For f W X Y we obtain the previously defined n.Y; X; /. The projection p W i ! f induces an isomorphismn.Z.f /; X; / D n.i/ ! n.f /. One can also use the fibre sequence of f .

6.2 The Role of the Base Point

We have to discuss the role of the base point. This uses the transport along paths.

and problems in that section. In order to be independent of that section, we also repeat a proof in the present context.

(6.2.1) Proposition. The assignment ŒV0 7!ŒV1 is a well-defined map

v# W n.X; v.0// ! n.X; v.1//

which only depends on the morphism Œv in the fundamental groupoid ….X/. The relation .v w/# D w# ıv# holds, and thus we obtain a transport functor from ….X/ which assigns to x0 2 X the group n.X; x0/ and to a path v the morphism v#.

.f; v/O and Wt a homotopy which extends .g; w/O . These data combine to a map on

T D I n 0 I [ @I n I I [ I n I @I I nC2 as follows: On I n 0 I

we use ', on @I n I I we use O , on I n I 0 we use V , and on I n I 1 we use W . If we interchange the last two coordinates then T is transformed into J nC1. Therefore our map has an extension to I nC2, and its restriction to I n 1 I is a homotopy from V1 to W1. This shows the independence of the representatives f and v. The other properties are clear from the construction.

There is a similar transport functor in the relative case. We start with a function f W .I n; @I n; J n 1/ ! .X; A; a0/ and a path v W I ! A from a0 to a1. We consider

(6.2.2) Proposition. The assignment ŒV0 7!ŒV1 is a well-defined map

v# W n.X; A; a0/ ! n.X; A; a1/

which only depends on the morphism Œv in the fundamental groupoid ….A/. For n 2 the map v# is a homomorphism. As above we have a transport functor from ….A/.

Since v# is always bijective, homotopy groups associated to base points in the same path component are isomorphic.

We list some naturality properties of the transport functors. As a special case of the functor property we obtain right actions of the fundamental groups:

We also have an action of 1.A; a/ on n.X; a/ via the natural homomorphism; more generally, we can make the n.X; a/ into a functor on ….A/ by viewing a path in A as a path in X. From the constructions we see:

(6.2.3) Proposition. The morphisms in the exact homotopy sequence of the pair

.X; A/ are natural transformations of transport functors on ….A/. In particular, they are 1.A; /-equivariant with respect to the actions above.

Continuous maps f W .X; A/ ! .Y; B/ are compatible with the transport functors

f .w#.˛// D .f w/#.f .˛//:

Let ft W .X; A/ ! .Y; B/ be a homotopy and set w W t 7!f .a; t/. Then the diagram

is commutative. As in the proof of (2.5.5) one uses this fact to show: