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

28 Chapter 2. The Fundamental Group

course, define homotopies with Œ0; 1 X. While this does not affect the theory, it does make a difference when orientations play a role.)

The homotopy relation ' is an equivalence relation on the set of continuous maps X ! Y . Given H W f ' g, the inverse homotopy H W .x; t/ 7!H.x; 1 t/ shows g ' f . Let K W f ' g and L W g ' h be given. The product homotopy

KL is defined by

and shows f ' h. The constant homotopy H.x; t/ D f .x/ shows f ' f .

The equivalence class of f is denoted Œf and called the homotopy class of f .

We denote by ŒX; Y the set of homotopy classes Œf of maps f W X ! Y . A homotopy Ht W X ! Y is said to be relative to A X if the restriction Ht jA does

not depend on t (is constant on A). We use the notation H W f ' g (rel A) in this case.

The homotopy relation is compatible with the composition of maps: Let H W f ' g W X ! Y and G W k ' l W Y ! Z be given; then

.x; t/ 7!G.H.x; t/; t/ D Gt Ht .x/

is a homotopy from kf to lg. We see that topological spaces and homotopy classes of maps form a quotient category of TOP, the homotopy category h-TOP, when

composition of homotopy classes is induced by composition of representing maps. If f W X ! Y represents an isomorphism in h-TOP, then f is called a homotopy equivalence or h-equivalence. In explicit terms this means: f W X ! Y is a homotopy equivalence if there exists g W Y ! X, a homotopy inverse of f , such that gf and fg are both homotopic to the identity. Spaces X and Y are homotopy equivalent or of the same homotopy type if there exists a homotopy equivalence X ! Y . A space is contractible if it is homotopy equivalent to a point. A map f W X ! Y is null homotopic if it is homotopic to a constant map; a null homotopy

of f is a homotopy between f and a constant map. A null homotopy of the identity id.X/ is a contraction of the space X.

2.1.4 Categories of homotopies. We generalize (2.1.1) and define a category W .X; Y /. The objects are the continuous maps f W X ! Y . A morphism from f to g is a homotopy H W X Œ0; a ! Y with H0 D f and Ha D g. Composition is defined as in (2.1.1). Þ

As in any category we also have the Hom-functors in h-TOP. Given f W X ! Y , we use the notation

f W ŒZ; X ! ŒZ; Y ; g 7!fg; f W ŒY; Z ! ŒX; Z ; h 7!hf

30 Chapter 2. The Fundamental Group

Proof. The product p id W X I ! Y I is an identification, since I is compact. The composition H ı .p id/ is continuous and therefore H is continuous.

(2.1.7) Proposition. Let Ht W X ! X be a homotopy of the identity H0 D id.X/ such that the subspace ; 6DA X is always mapped into itself, Ht .A/ A. Suppose H1 is constant on A. Then the projection p W X ! X=A (A identified to a point) is an h-equivalence.

Proof. Since H1.A/ is a point, there exists a map q W X=A ! X such that qp D H1. By assumption, this composition is homotopic to the identity. The map p ı Ht W X ! X=A factorizes over p and yields Kt W X=A ! X=A such that Kt p D pHt . By (2.1.6), Kt is a homotopy, K0 D id and K1 D pq.

Problems

1.Suppose there exists a homeomorphism R ! X Y . Then X or Y is a point.

2.Let f W X ! Y be surjective. If X is (path) connected, then Y is (path) connected.

3.Let C be a countable subset of Rn, n 2. Show that Rn X C is path connected.

4.The unitary group U.n/ and the general linear group GLn.C/ are path connected. The orthogonal group O.n/ and the general linear group GLn.R/ have two path components; one of them consists of matrices with positive determinant.

5.Let U Rn be open. The path components of U are open and coincide with the components. The set of path components is finite or countably infinite. An open subset of R is a disjoint union of open intervals.

6.List theorems of point-set topology which show that the product homotopy and the inverse homotopy are continuous. Do the same for the linear homotopy in 2.1.5.

7.A space X is contractible if and only if the identity id.X/ is null homotopic.

8.gf is null homotopic, if f or g is null homotopic.

9.Let A be contractible. Then any two maps X ! A are homotopic.

10.The inclusions O.n/ GLn.R/ and U.n/ GLn.C/ are homotopy equivalences. Let P .n/ denote the space of positive definite real .n; n/-matrices. Then O.n/ P .n/ ! GLn.R/, .X; P / 7!XP is a homeomorphism; P .n/ is star-like with respect to the unit matrix.

11.There exist contractible and non-contractible spaces consisting of two points.

2.2 Further Homotopy Notions

The homotopy notion can be adapted to a variety of other contexts and categories: Consider homotopies which preserve some additional structure of a category. We describe some examples from which the general idea emerges. This section only contains terminology.

The construction of group structures on homotopy sets uses the category of pointed spaces, as we will see shortly. We call a pair .X; x0/ consisting of a space

X and a base point x0 2 X a pointed space. A pointed map f W .X; x0/ ! .Y; y0/

is a continuous map f W X ! Y which sends the base point to the base point. A homotopy H W X I ! Y is pointed if Ht is pointed for each t 2 I . We de-

note by ŒX; Y 0 the set of pointed homotopy classes (fixed base points assumed) or by Œ.X; x0/; .Y; y0/ . We obtain related notions: pointed homotopy equivalence, pointed contractible, pointed null homotopy. We denote the category of pointed

spaces and pointed maps by TOP0, and by h-TOP0 the associated homotopy category. Often a base point will just be denoted by . Also a set with a single element will be denoted by its element. The choice of a base point is an additional structure. There is a functor ˛ from TOP to TOP0 which sends a space X to XC D X C f g, i.e., to X with an additional base point added (topological sum), with the obvious extension to pointed maps. This functor is compatible with homotopies. We also have the forgetful functor ˇ from TOP0 to TOP. They are adjoint TOP0.˛.X/; Y / Š TOP.X; ˇY /, and similarly for the homotopy categories.

The category TOP0 has sums and products. Suppose .Xj ; xj / is a family of

where all base points are identified to a single new base point. We have canonical pointed maps ik W Xk ! j Xj which arise from the canonical inclusions Xk !

The sum and the product in TOP0 also represent the sum and the product in

h-TOP0 (use (2.1.6)).

Let .A; a/ and .B; b/ be pointed spaces. Their smash product is

A ^ B D A B=A b [ a B D A B=A _ B:

(This is not a categorical product. It is rather analogous to the tensor product.) The smash product is a functor in two variables and also compatible with homotopies: Given f W A ! C; g W B ! D we have the induced map

f ^ g W A ^ B ! C ^ D; .a; b/ 7!.f .a/; g.b//;

and homotopies ft ; gt induce a homotopy ft ^gt . Unfortunately, there are point-set

topological problems with the associativity of the smash product (see Problem 14). A pair .X; A/ of topological spaces consists of a space X and a subspace A.

A morphism f W .X; A/ ! .Y; B/ between pairs is a map f W X ! Y such that f .A/ B. In this way we obtain the category of pairs TOP.2/. A homotopy H in this category is assumed to have each Ht a morphism of pairs. We write Œ.X; A/; .Y; B/ for the associated homotopy sets and h-TOP.2/ for the homotopy category.

If .X; A/ is a pair, we usually consider the quotient space X=A as a pointed space (A identified to a point) with base point fAg. If A D ;, then X=A D XC is X with a separate base point.

32 Chapter 2. The Fundamental Group

(2.2.1) Note. A continuous map f W .X; A/ ! .Y; / into a pointed space induces

.X; A/ .Y; B/ D .X Y; X B [ A Y /;

although this is not a categorical product. With this notation .I m; @I m/ .I n; @I n/ D

.I mCn; @I mCn/. In a similar manner we treat other configurations, e.g., triples

.X; A; B/ of spaces A B X and the category TOP.3/ of triples.

Let K and B be fixed spaces. The category TOPK of spaces under K has as

objects the maps i W K ! X. A morphism from i W K ! X to j W K ! Y is a map f W X ! Y such that f i D j . The category TOPB of spaces over B has as

objects the maps p W X ! B. A morphism from p W X ! B to q W Y ! B is a map f W X ! Y such that qf D p. If B is a point, then TOPB can be identified

with TOP, since each space has a unique map to a point. If K D f g is a point, then TOPK is the same as TOP0. If p W X ! B is given, then p 1.b/ is called the fibre of p over b; in this context, B is the base space and X the total space of p. A map in TOPB will also be called fibrewise or fibre preserving.

Categories like TOPK or TOPB have an associated notion of homotopy. A homotopy Ht is in TOPK if each Ht is a morphism in this category. A similar definition is used for TOPB . A homotopy in TOPB will also be called fibrewise or fibre preserving. Again, being homotopic is an equivalence relation in these categories. We denote by ŒX; Y K the set of homotopy classes in TOPK , and by ŒX; Y B the set of homotopy classes in TOPB . The homotopy categories are h-TOPK and h-TOPB . Note that a homotopy equivalence in TOPB , i.e., a fibrewise homotopy equivalence, from p W X ! B to q W Y ! B induces for each b 2 B a homotopy equivalence p 1.b/ ! q 1.b/ between the fibres over b, so this is a continuous family of ordinary homotopy equivalences, parametrized by B ([96], [97], [38], [128]).

A morphism r from i W K ! X to id W K ! K in TOPK is a map r W X ! K

ht W X ! X relative to K such that h0 D id and h1 D ir. In this case we call K a deformation retract of X. The inclusion Sn RnC1 X 0 is a deformation retract.

A morphism s from id W B ! B to p W E ! B in TOPB is a map s W B ! E such that ps D id.B/. It is called a section of p. If p W E ! B is homotopy equivalent in TOPB to id.B/ we call p shrinkable. All fibres of a shrinkable map

are contractible.

Problems

2.The canonical maps ik W Xk ! j Xj are embeddings.

3.The comb space X is defined as B Œ0; 1 [ Œ0; 1 f1g with B D fn 1 j n 2 Ng [ f0g.

Then X is contractible but not pointed contractible with respect to .0; 0/. Let Y D X be another comb space. Then X [ Y Š X _ Y is not contractible. Since .X [ Y /=Y is homeomorphic to X, we see that it does not suffice in (2.1.7) to assume that A is contractible.

These counterexamples indicate the need for base points with additional (local) properties.

4.There exists a contractible subspace X R2 which is not pointed contractible to any of its points.

5.Let the homotopy Ht in (2.1.7) be pointed with respect to some base point a 2 A. Show that p W X ! X=A is a pointed h-equivalence. Is .X; A/ h-equivalent to .X; fag/?

9.Remove a point from the torus S1 S1 and show that the result is h-equivalent to S1 _S1. Is there an analogous result when you remove a point from Sm Sn?

10.Construct an inclusion A X which is a retract and a homotopy equivalence but not a deformation retract.

11.Construct a map p W E ! B such that all fibres p 1.b/ are contractible but which does not have a section. Construct an h-equivalence p W E ! B which has a section but which is not shrinkable.

12.What is the sum of two objects in TOPK ? What is the product of two objects in TOPB ?

13.A pullback of a shrinkable map is shrinkable. A pushout of a deformation retract is a deformation retract.

14.Let Y; Z be compact or X; Z be locally compact. Then the canonical bijection (the

identity) .X ^ Y / ^ Z ! X ^ .Y ^ Z/ is a homeomorphism. (In the category of compactly generated spaces (with its associated product and smash product!) the map is always

W

omorphism if the index set is finite. Show that in this case both spaces are quotients of

.qj Aj / B Š qj .Aj B/.

16.Let A be a compact subset of X and p W X ! X=A be the quotient map. Then for each space Y the product p id.Y / is a quotient map. If X is a Hausdorff space, then p is proper and p id closed.

17.The canonical map X I ! X I=@I ! X ^ I=@I is a quotient map which induces a homeomorphism †X D X I=.X @I [ f g I / Š X ^ I=@I .

18.There is a canonical bijective continuous map .X Y /=.X B [A Y / ! X=A^Y =B

34 Chapter 2. The Fundamental Group

(the identity on representatives). It is a homeomorphism if X Y ! X=A ^ Y =B is a quotient map, e.g., if X and Y =B are locally compact (or in the category of compactly generated spaces).

2.3 Standard Spaces

Standard spaces are Euclidean spaces, disks, cells, spheres, cubes and simplices. We collect notation and elementary results about such spaces. The material will be used almost everywhere in this book. We begin with a list of spaces. The Euclidean norm is kxk.

result of this type is:

(2.3.1) Proposition. Let K Rn be a compact convex subset with non-empty interior Kı. Then there exists a homeomorphism of pairs .Dn; Sn 1/ ! .K; @K/ which sends 0 2 Dn to a pre-assigned x 2 Kı.

Proof. Let K Rn be closed and compact and 0 2 Kı. Verify that a ray from 0 intersects the boundary @K of K in Rn in exactly one point. The map f W @K ! Sn 1, x 7!x=kxk is a homeomorphism. The continuous map ' W Sn 1 Œ0; 1 ! K,

This proposition can be used to deduce a homeomorphism .Dn; Sn 1/ Š

.I n; @I n/. The simplex n is a compact convex subset with interior points in the

hyperplane fx 2 RnC1 j Pi xi D 1g. From this fact we deduce a homeomorphism

.Dn; @Dn/ Š . n; @ n/.

The sphere Sn, as a homeomorphism type, will appear in many different shapes.

identity is the linear homotopy .x; t/ 7!tx C .1 t/iN.x/. Moreover N ı i D id. We see that i is a deformation retract.

Under suitable circumstances each map in a small neighbourhood of f is already homotopic to f . For a general theorem to this effect see (15.8.3). Here we only give a simple, but typical, example. Let f; g W X ! Sn be maps such that kf .x/ g.x/k < 2. Then they are homotopic by a linear homotopy when viewed as maps into RnC1 X f0g. We compose with N and see that f ' g.

If f W Sm ! Sn is a continuous map, then there exists (by the theorem of Stone–Weierstrass, say) a C 1-map g W Sm ! Sn such that kf .x/ g.x/k < 2. This indicates another use of homotopies: Improve maps up to homotopy. If one uses some analysis, namely (the easy part of) the theorem of Sard about the density of regular values, one sees that for m < n a C 1-map Sm ! Sn is not surjective and hence null homotopic. (Later we prove this fact by other methods.) There exist surjective continuous maps S1 ! S2 (Peano curves); this ungeometric behaviour of continuous maps is the source for many of the technical difficulties in topology.Þ

(2.3.4) Proposition. The map p W Sn 1 I ! Dn, .x; t/ 7!.1 t/x is a quotient map. Given F W Dn ! X, the composition Fp W Sn 1 I ! X is a null homotopy of f D F jSn 1. Each null homotopy of a map f W Sn 1 ! X arises from a unique F .

Proof. Since a null homotopy H of f W Sn 1 ! X sends Sn 1 1 to a point, it factors through the quotient map q W Sn 1 I ! Sn 1 I=Sn 1 1. Thus null

phism. If V and W are finite-dimensional real vector spaces, we have a canonical

phism W S.1/ ! S.1/ which transports t 7!1 t into the antipodal map x 7! x on R. We obtain an induced homeomorphism

n W S.n/ D S.1/ ^ ^ S.1/ ! S.1/ ^ ^ S.1/ D S.n/

of the n-fold smash products.

(2.3.5) Example. A retraction r W Dn ! Sn 1 I [ Dn 0 is r.x; t/ D

.2˛.x; t/ 1 x; ˛.x; t/ 2C t/ with ˛.x; t/ D max.2kxk; 2 t/. (See Figure 2.1, a

central projection from the point .0; 2/.) Given a map f W I n ! X and a homotopy h W @I n I ! X with h0 D f j@I n combine to a map g W I n 0 [ @I n I ! X. We compose with a retraction and obtain a homotopy H W I n ! X which extends

h and begins at H0 D f . This homotopy extension property is later studied more

Figure 2.1. A retraction.

(2.3.6) Example. The assignment H W .x; t/ 7!.˛.x; t/ 1.1 C t/ x; 2 ˛.x; t// with the function ˛.x; t/ D max.2kxk; 2 t/ yields a homeomorphism of pairs

.Dn; Sn 1/ .I; 0/ Š Dn .I; 0/, see Figure 2.2.

Similarly for .I n; @I n/ in place of .Dn; Sn 1/, since these two pairs are homeomorphic. Þ

Figure 2.2. A relative homeomorphism.

Problems

1.Construct a homeomorphism .Dm; Sm 1/ .Dn; Sn 1/ Š .DmCn; SmCn 1/.

2.RnC1 X fxg ! Sn, z 7!.z x/=kz xk is an h-equivalence.

3.Let DCn D f.x0; : : : ; xn/ 2 Sn j xn 0g. Show that the quotient map Sn ! Sn=DCn

is an h-equivalence.

4.Let f1; : : : ; fk W Cn ! C be linearly independent linear forms (k n). Then the complement Cn X Sj fj 1.0/ is homotopy equivalent to the product of k factors S1.

5.Sn ! f.x; y/ 2 Sn Sn j x 6Dyg, x 7!.x; x/ is an h-equivalence.

6.Let f; g W X ! Sn be maps such that always f .x/ 6D g.x/. Then f ' g.

7.Let A En be star-shaped with respect to 0. Show that Sn 1 Rn XA is a deformation

retract.

8. The projection p W TS n D f.x; v/ 2 Sn RnC1 j x ? vg ! Sn, .x; v/ 7!x is called the tangent bundle of Sn. Show that p admits a fibrewise homeomorphism with pr W Sn Sn X D ! Sn, .x; y/ 7!x (with D the diagonal).

2.4 Mapping Spaces and Homotopy

It is customary to endow sets of continuous maps with a topology. In this section we review from point-set topology the compact-open topology. It enables us to consider a homotopy H W X I ! Y as a family of paths in Y , parametrized by X. This dual aspect of the homotopy notion will be used quite often. It can be formalized; but we use it more like a heuristic principle to dualize various constructions and notions in homotopy theory (Eckmann–Hilton duality).

We denote by Y X or F .X; Y / the set of continuous maps X ! Y . For K X and U Y we set W .K; U / D ff 2 Y X j f .K/ U g. The compact-open topology (CO-topology) on Y X is the topology which has as a subbasis the sets