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

(1.8.2) Proposition. .1/ Let H be a subgroup of the topological group G. Let the set G=H of cosets gH carry the quotient topology with respect to p W G ! G=H , g 7!gH . Then l W G G=H ! G=H , .x; gH / 7!xgH is a continuous action.

.2/ G=H is separated if and only if H is closed in G. In particular, G is separated if feg is closed.

.3/ Let H be normal in G. Then the factor group G=H with quotient topology is a topological group.

A space G=H with the G-action by left multiplication is called a homogeneous space. The space of left cosets Hg is H nG; it carries a right action.

(1.8.3) Example. Homogeneous spaces are important spaces in geometry. The orthogonal group O.n C 1/ acts on the sphere Sn by matrix multiplication

case we obtain a homeomorphism U.n C 1/=U.n/ Š S2nC1, in the quaternionic case a homeomorphism Sp.n C 1/=Sp.n/ Š S4nC3. Other important homogeneous spaces are the projective spaces, the Grassmann manifolds, and the Stiefel manifolds to be discussed later. Þ

(1.8.4) Proposition. .1/ If x 2 X is closed, then Gx is closed in G.

.2/ If X is a Hausdorff space, then XH is closed.

.3/ Let A be a G-stable subset of the G-space X. Then A=G carries the subspace topology of X=G. In particular XG ! X ! X=G is an embedding.

.4/ Let B X be closed and A X. Then fg 2 G j gA Bg is closed in G.

.5/ Let B X be closed. Then fg 2 G j gB D Bg is closed.

Proof. (1) The isotropy group Gx is the pre-image of x under the continuous map

is open, and we can write p 1.C / D A \ U with an open subset U X. We have A \ U D A \ GU , since A is G-stable. We conclude C D p.p 1C / D A=G\p.GU /. Since GU is open, p.GU / is open, hence C is open in the subspace topology. By continuity of A=G ! X=G, an open subset in the subspace topology

(1)If A and B are compact, then AB is compact.

(2)If A is compact, then the restriction A X ! X of r is proper. If, moreover, B is closed, then AB is closed.

(3)If G is compact, then the orbit map p is proper. Thus X is compact if and only if X=G is compact.

(4)If G is compact and X separated, then X=G is separated.

(5)Let G be compact, A a G-stable closed subset and U a neighbourhood of A in X. Then U contains a G-stable neighbourhood of A.

Proof. (1) A B G X is compact as a product of compact spaces. Hence the continuous image AB of A B under r W G X ! X is compact. (2) The homeomorphism A X ! A X, .s; x/ 7!.s; sx/ transforms r into the projection pr W A X ! X. The projection is proper, since A is compact (see (1.5.1)). Hence the image AB of the closed set A B is closed. (3) Let A X be closed. Then p 1p.A/ D GA is closed, by (2). Hence p.A/ is closed, by definition of the quotient topology. The pre-images of points are orbits; they are compact as continuous images of G. (4) Since p is proper, so is p p. Hence the image of the diagonal under p p is closed. (5) Let U be open. Then p.X X U / is disjoint to p.A/. By (4), X X p 1p.X X U / is open and a G-stable neighbourhood of A contained in U .

The orbit category Or.G/ is the category of homogeneous G-spaces G=H , H closed in G, and G-maps. There exists a G-map G=H ! G=K if and only if H is conjugate to a subgroup of K. If a 1Ha K, then Ra W G=H ! G=K,

gH 7!gaK is a G-map and each G-map G=H ! G=K has this form; moreover Ra D Rb if and only if a 1b 2 K.

An action G V ! V on a real (or complex) vector space V is called a real (or complex) representation of G if the left translations are linear maps. After choice of

a basis, a representation amounts to a continuous homomorphism from G to GLn.R/ or GLn.C/. A homomorphism G ! O.n/ or G ! U.n/ is called an orthogonal or unitary representation. Geometrically, an orthogonal representation is given by an action G V ! V with an invariant scalar product h ; i. The latter means

hgv; gwi D hv; wi for g 2 G and v; w 2 V . In an orthogonal representation, the unit sphere S.V / D fv 2 V j hv; vi D 1g is G-stable.

If E carries a left K-action which commutes with the right G-action (i.e., k.xg/ D

.kx/g), then E G F carries an induced K-action .k; .x; y// 7!.kx; y/. This construction can in particular be applied in the case that E D K, G a subgroup

20 Chapter 1. Topological Spaces

of K and the G- and K-actions on K are given by right and left multiplication. The assignments F 7!K G F and f 7!id G f yield the induction functor indKG W G- TOP ! K- TOP. This functor is left adjoint to the restriction functor resKG W K- TOP ! G- TOP which is obtained by regarding a K-space as a G-space. The natural adjunction

TOPK .indKG X; Y / Š TOPG .X; resKG Y /

sends a G-map f W X ! Y to the K-map .k; x/ 7!kf .x/; in the other direction one restricts a map to X Š G G X K G X. (Here TOPK denotes the set of K-equivariant maps.)

(1.8.6)Theorem. Suppose the Hausdorff group G is locally compact with countable basis. Let X be a locally compact Hausdorff space and G X ! X a transitive action. Then for each x 2 X the map b W G ! X, g 7!gx is open and the induced

such that Bj W g 1, g 2 gj W . Therefore the gj W cover the group.

Let V G be open and g 2 V . There exists a compact neighbourhood W of e such that W D W 1 and gW 2 V . Since G is the union of the gj W and

contains an interior point, and therefore W x contains an interior point wx. Then x

(1.8.7) Corollary. Let the locally compact Hausdorff group G with countable basis act on a locally compact Hausdorff space X. An orbit is locally compact if and only if it is locally closed. An orbit is a homogeneous space with respect to the isotropy group of each of its points if and only if it is locally closed.

Problems

1.Let H be a normal subgroup of G and X a G-space. Restricting the group action to H , we obtain an H -space X. The orbit space H nX carries then an induced G=H -action.

2.Let a pushout in TOP be given with G-spaces A, B, X and G-maps j , f :

f

AB

jJ

Let G be locally compact. Then there exists a unique G action on Y such that F; J become G-maps. The diagram is then a pushout in G-TOP. Hint: (2.4.6)

3. Let Y be a K-space and G a subgroupp of K. Then K G Y ! K=G Y , .k; y/ 7!

.kG; ky/ is a K-homeomorphism. If X is a G-space, then

restricting the given G-action to NH . The action

G=H W H ! G=H; .gH; nH / 7!gnH

is a free right action by G-automorphisms of G=H .

1.9 Projective Spaces. Grassmann Manifolds

Let P .RnC1/ D RP n be the set of one-dimensional subspaces of the vector space RnC1. A one-dimensional subspace of V is spanned by x 2 V X 0. The vectors x and y span the same subspace if and only if x D y for some 2 R D R X 0. We therefore consider P .RnC1/ as the orbit space of the action

R .RnC1 X 0/ ! RnC1 X 0; . ; x/ 7! x:

The quotient map p W RnC1 X 0 ! P .RnC1/ provides P .RnC1/ with the quotient topology. The space RP n is the n-dimensional real projective space. We set p.x0; : : : ; xn/ D Œx0; : : : ; xn and call x0; : : : ; xn the homogeneous coordinates

of the point Œx0; : : : ; xn .

In a similar manner we consider the set P .CnC1/ D CP n of one-dimensional subspaces of CnC1 as the orbit space of the action

C .CnC1 X 0/ ! CnC1 X 0; . ; z/ 7! z:

We have again a quotient map p W CnC1 X 0 ! P .CnC1/. The space CP n is called the n-dimensional complex projective space. (It is 2n-dimensional as a manifold.)

We describe the projective spaces in a different manner as orbit spaces. The

subgroup G D f˙1g R acts on Sn RnC1 by . ; x/ 7! x, called the

antipodal involution. The inclusion i W Sn ! RnC1 induces a continuous bijective map W Sn=G ! .RnC1 X 0/=R . The map j W RnC1 X 0 ! Sn, x 7! kxk 1x

induces an inverse. The quotient Sn=G is compact, since Sn is compact. By (1.4.4),

the quotient is a Hausdorff space. In a similar manner one treats CP n, but now with respect to the action S1 S2nC1 ! S2nC1, . ; z/ 7! z of S1 on the unit sphere S2nC1 CnC1 X 0.

22 Chapter 1. Topological Spaces

Projective spaces are homogeneous spaces. Consider the action of O.n C 1/ on RnC1 by matrix multiplication. If V 2 P .RnC1/ is a one-dimensional space and A 2 O.n C 1/, then AV 2 P .RnC1/. We obtain an induced action

O.n C 1/ P .RnC1/ ! P .RnC1/:

This action is transitive. The isotropy group of Œ1; 0; : : : ; 0 consists of the matrices

We consider these matrices as the subgroup O.1/ O.n/ of O.nC1/. The assignment A 7!Ae1 induces an O.n C 1/-equivariant homeomorphism

b W O.n C 1/=.O.1/ O.n// Š P .RnC1/:

The action of O.n C 1/ on P .RnC1/ is continuous; this follows easily from the continuity of the action O.n C 1/ .RnC1 X 0/ ! RnC1 X 0 and the definition of the quotient topology. Therefore b is a bijective continuous map of a compact space into a Hausdorff space. In a similar manner we obtain a U.n C 1/-equivariant

homeomorphism U.n C 1/=.U.1/ U.n// Š P .CnC1/.

Finally, one can define the quaternionic projective space HP n in a similar manner as a quotient of HnC1 X 0 or as a quotient of S4nC3.

We generalize projective spaces. Let W be an n-dimensional real vector space. We denote by Gk .W / the set of k-dimensional subspaces of W . We define a topology on Gk .W /. Suppose W carries an inner product. Let Vk .W / denote

the set of orthonormal sequences .w1; : : : ; wk / in W considered as a subspace of W k . We call Vk .W / the Stiefel manifold of orthonormal k-frames in W . We

have a projection p W Vk .W / ! Gk .W / which sends .w1; : : : ; wk / to the subspace Œw1; : : : ; wk spanned by this sequence. We give Gk .W / the quotient topology determined by p. The space Gk .W / can be obtained as a homogeneous space. Let W D Rn with standard inner product and standard basis e1; : : : ; en. We have a continuous action of O.n/ on Vk .Rn/ and Gk .Rn/ defined by .A; .v1; : : : ; vk // 7!

.Av1; : : : ; Avk / and such that p becomes O.n/-equivariant. The isotropy groups of .e1; : : : ; ek / and Œe1; : : : ; ek consist of the matrices

respectively. The map A 7!.Ae1; : : : ; Aek / induces equivariant homeomorphisms in the diagram

This shows that Gk .Rn/ is a compact Hausdorff space. It is called the Grassmann manifold of k-dimensional subspaces of Rn.

In a similar manner we can work with the k-dimensional complex subspaces of Cn and obtain an analogous diagram of U.n/-spaces (complex Stiefel and Grassmann manifolds):

Chapter 2

The Fundamental Group

In this chapter we introduce the homotopy notion and the first of a series of algebraic invariants associated to a topological space: the fundamental group.

Almost every topic of algebraic topology uses the homotopy notion. Therefore it is necessary to begin with this notion. A homotopy is a continuous family ht W X ! Y of continuous maps which depends on a real parameter t 2 Œ0; 1 . (One may interpret this as a “time-dependent” process.) The maps f0 and f1 are then called homotopic, and being homotopic is an equivalence relation on the set of continuous maps X ! Y . This equivalence relation leads to a quotient category of the category TOP of topological spaces and continuous maps, the homotopy category h-TOP. The importance of this notion is seen from several facts.

(1)The classical tools of algebraic topology are functors from a category of spaces to an algebraic category, say of abelian groups. These functors turn out to be homotopy invariant, i.e., homotopic maps have the same value under the functor.

(2)One can change maps by homotopies and spaces by homotopy equivalences. This fact allows for a great flexibility. But still global geometric information is retained. Basic principles of topology are deformation and approximation. One idea of deformation is made precise by the notion of homotopy. Continuity is an ungeometric notion. So often one has to deform a continuous map into a map with better properties.

(3)The homotopy notion leads in an almost tautological way to algebraic structures and categorical structures. In this chapter we learn about the simplest example, the fundamental group and the fundamental groupoid.

The passage to the homotopy category is not always a suitable view-point. In general it is better to stay in the category TOP of topological spaces and continuous maps (“space level” as opposed to “homotopy level”). We thus consider homotopy as an additional structure. Then classical concepts can be generalized by using the homotopy notion. For instance one considers diagrams which are only commutative up to homotopy and the homotopies involved will be treated as additional information. One can also define generalized group objects where multiplication is only associative up to homotopy. And so on.

The passage from TOP to h-TOP may be interpreted as a passage from “continuous mathematics” to “discrete mathematics”.

The homotopy notion allows us to apply algebraic concepts to continuous maps. It is not very sensible to talk about the kernel or cokernel of a continuous map.

But we will see later that there exist notions of “homotopy-kernels” (then called homotopy fibres) and “homotopy-cokernels” (then called homotopy cofibres). This is the more modern view-point of a large variety of homotopy constructions. In general terms: The idea is to replace the categorical notions limit and colimit by appropriate homotopy notions.

The prototype of a functor from spaces to groups is the fundamental group functor. Historically it is the first of such functors. It was introduced by Poincaré, in different context and terminology. In general it is difficult to determine the fundamental group of a space. Usually one builds up a space from simpler pieces and then one studies the interrelation between the groups of the pieces. This uses the functorial aspect and asks for formal properties of the functor. We prove the basic theorem of Seifert and van Kampen which roughly says that the functor transforms suitable pushouts of spaces into pushouts of groups. This may not be the type of algebra the reader is used to, and it can in fact be quite complicated. We describe some related algebra (presentation of groups by generators and relations) and discuss a number of geometric results which seem plausible from our intuition but which cannot be proved (in a systematic way) without algebraic topology. The results are of the type that two given spaces are not homeomorphic – and this follows, if their fundamental groups are different. Finally we show that each group can be realized as a fundamental group (this is the origin of the idea to apply topology to group theory).

The study of the fundamental group can be continued with the covering space theory where the fundamental group is exhibited as a symmetry group. This symmetry influences almost every other tool of algebraic topology (although we do not always carry out this influence in this text).

The chapter contains two sections on point-set topology. We discuss standard spaces like spheres, disks, cells, simplices; they will be used in many different contexts. We present the compact-open topology on spaces of continuous maps; they will be used for the dual definition of homotopy as a continuous family of paths, and this duality will henceforth be applied to many homotopy constructions and notions.

2.1 The Notion of Homotopy

A path in a topological space X from x to y is a continuous map u W Œa; b ! X such that u.a/ D x and u.b/ D y. We say that the path connects the points u.a/ and u.b/. We can reparametrize and use the unit interval as a source Œ0; 1 ! X,

t 7!u..1 t/a C tb/. In the general theory we mostly use the unit interval. If u W Œ0; 1 ! X is a path from x to y, then the inverse path u W t 7!w.1 t/ is a path from y to x. If v W Œ0; 1 ! X is another path from y to z, then the product path u v, defined by t 7!u.2t/ for t 1=2 and v.2t 1/ for t 1=2, is a path from x to z. We also have the constant path kx with value x.

A space is connected if it is not the topological sum of two non-empty subspaces.

Thus X is disconnected if and only if X contains a subset X which is open, closed, and different from ; and X. A decomposition of X is a pair U; V of open, non-

empty, disjoint subsets with union X. A space X is disconnected if and only if there exists a continuous surjective map f W X ! f0; 1g; a decomposition is given by U D f 1.0/, V D f 1.1/. The continuous image of a connected space is connected. Recall from calculus: A R is connected if and only if A is an interval. (An interval is a subset which contains with x; y also Œx; y .)

(2.1.2) Proposition. Let .Aj j j 2 J / be a family of connected subsets of X such

The union of the connected sets in X which contain x is thus a closed connected subset. We call it the component X.x/ of x in X. If y 2 X.x/, then X.y/ D X.x/.

A component of X is a maximal connected subset. Any space is the disjoint union of its components. A space is totally disconnected if its components consist of

single points. Since intervals are connected a path connected space is connected. A product …j Xj is connected if each Xj is connected. The component of

.xj / 2 …j Xj is the product of the components of the xj .

(2.1.3) Example. The space

X D Π1; 0 0 [ 0 Π1; 1 [ f.x; sin. x 1/ j 0 < x 1g

is connected but not path connected. The union S of X with f˙1g Œ 2; 0 [ Œ 2; 2 f 2g is called the pseudo-circle. The complement R2 X S has two path

components.

Let X and Y be topological spaces and f; g W X ! Y continuous maps. A homotopy from f to g is a continuous map

H W X Œ0; 1 ! Y; .x; t/ 7!H.x; t/ D Ht .x/

such that f .x/ D H.x; 0/ and g.x/ D H.x; 1/ for x 2 X, i.e., f D H0 and g D H1. We denote this situation by H W f ' g. One can consider a homotopy as

a dynamical process, the parameter t is the time and Ht is a time-dependent family of maps. One also says that f is deformed continuously into g. Another (dual)

view-point of a homotopy is: a parametrized family of paths. We use the letter I for the unit interval Œ0; 1 . If we write a homotopy in the form Ht , we understand that

H W X I ! Y , .x; t/ 7!Ht .x/ is continuous in both variables simultaneously. We call f and g homotopic if there exists a homotopy from f to g. (One can, of