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

198 Chapter 8. Cell Complexes

continuous. We leave it as an exercise to show that this map is actually a homeomorphism. The numbers .˛.e/ j e 2 E/ are the barycentric coordinates of ˛.

We define a further topology on jKj. For s 2 S let .s/ be the standard simplex f.te/ 2 jKj j te D 0 for e … sg. Then jKj is the union of the .s/, and

we write jKjc for jKj with the quotient topology defined by the canonical map

`

s2S .s/ ! jKj. The identity jKjc ! jKjp is continuous but not, in general, a homeomorphism. The next proposition will be proved in the more general context of simplicial diagrams.

(8.1.4) Proposition. jKjc ! jKjp is a homotopy equivalence.

In the sequel we write jKj D jKjc and call this space the geometric realization of K. We define jsj jKj as jsj D f˛ 2 jKj j ˛.e/ 6D0 ) e 2 sg and call this set a closed simplex of jKj. For each simplex s of K the open simplex hsi jKj is the subspace hsi D f˛ 2 jKj j ˛.e/ 6D0 , e 2 sg. The complement jsj n hsi D @jsj is the combinatorial boundary of jsj; it is the geometric realization

of the subcomplex which consists of the proper faces of s. The set jKj is the disjoint

A homeomorphism t W jKj ! X is called a triangulation of X. The triangulation of surfaces was proved by Radó [161], the triangulation of 3-dimensional manifolds by Moise (see [141] for references and proofs; [197, 7.5.1]). Differentiable manifolds can be triangulated, and the triangulation can be chosen in such a way that it is on each simplex a smooth embedding ([193]; [143]).

Since jKjm is separated and id W jKj ! jKjm continuous, jKj is separated. For

finite K the identity jKj ! jKjm is a homeomorphism.

For each vertex e 2 E the set St.e/ D f˛ 2 jKj j ˛.e/ 6D0g is called the star of e. Since ˛ 7!˛.e/ is continuous, the set St.e/ is open in jKjd and therefore

f W jKj ! Rn; ˛ 7!Pe2E ˛.e/xe:

If f is an embedding, we call the image of f a simplicial polyhedron in Rn of type K, and f .jKj/ is a realization of K as a polyhedron in Rn.

Standard tools for the application of simplicial complexes in algebraic topology are subdivision and simplicial approximation [67, p. 124].

Problems

1. id W jKjm ! jKjp is a homeomorphism.

2. Let K D .N0; S/ be the simplicial complex where S consists of all finite subsets of N0. The canonical map jKjc ! jKjp is not a homeomorphism.

3. Let L be a subcomplex of K. We can identify jLj with a subset of jKj, and jLj carries then

hold.

4.Let K be a simplicial complex. Then the following assertions are equivalent: (1) K is

locally finite. (2) jKj is locally compact. (3) The identity jKj ! jKjd is a homeomorphism.

(4) jKj is metrizable. (5) Each point of jKj has a countable neighbourhood basis. (See [44, p. 65].)

5.Let K be a countable, locally finite simplicial complex of dimension at most n. Then K has a realization as a polyhedron in R2nC1. (See [44, p. 66].)

8.2 Whitehead Complexes

We use the standard subsets of Euclidean spaces Sn 1; Dn; En D Dn n Sn 1,

.n 1/. We set S 1 D ; and let D0 be a point, hence E0 D D0. A k-dimen- sional cell (a k-cell) in a space X is a subset e which is, in its subspace topology,

homeomorphic to Ek . A point is always a 0-cell.

A Whitehead complex is a space X together with a decomposition into cells

.e j 2 ƒ/ such that:

A subset A of a Whitehead complex is a subcomplex if it is a union of cells and the closure of each cell in A is contained in A. We will see that a subcomplex together with its cells is itself a Whitehead complex. From the definition of a subcomplex

200 Chapter 8. Cell Complexes

we see that intersections and unions of subcomplexes are again subcomplexes. Therefore there exists a smallest subcomplex X.L/ which contains a given set L.

The decomposition of a Hausdorff space into its points always satisfies (W1)– (W3). We see that (W4) is an important condition. Condition (W3) is also called (C), for closure finite. Condition (W4) is called (W), for weak topology. This is the origin for the name CW-complex. In the next section we consider these complexes from a different view-point and introduce the notion of a CW-complex.

(8.2.1) Lemma. Let ˆ W Dn ! X be a continuous map into a Hausdorff space. Let e D ˆ.En/. Then ˆ.Dn/ D ex. In particular ex is compact.

Proof. ˆ.Dn/ is a compact subset of a Hausdorff space and therefore closed. This

(8.2.2) Example. Suppose X has a cell decomposition into a finite number of cells such that properties (W1) and (W2) hold. Then X is a finite union of closures ex of cells and therefore compact by (8.2.1). Properties (W3) and (W4) are satisfied and X is a Whitehead complex. Þ

(8.2.3) Examples. The sphere Sn has the structure of a Whitehead complex with a single 0-cell and a single n-cell. The map

p

Sn 1 D Sn 1 0. A characteristic map is Dn ! D˙n , x 7!.x; ˙ 1 kxk2/.Þ

(8.2.4) Proposition. Let X be a Whitehead complex.

(1)A compact set K in X meets only a finite number of cells.

(2)A subcomplex which consists of a finite number of cells is compact and closed in X.

(3)X.e/ D X.e/x is for each cell e a finite subcomplex.

(4)A compact subset of a Whitehead complex is contained in a finite subcomplex.

(5)X carries the colimit topology with respect to the finite subcomplexes.

(6)A subcomplex A is closed in X.

Proof. (1) Let E be the set of cells which meet K. For each e 2 E we choose a point xe 2 K \ e and set Z D fxe j e 2 Eg. Let Y Z be any subset. For each cell f of X the closure fx is contained in the union of a finite number of cells. Thus

Y \ fx is a finite set, hence closed in fx since fx is a Hausdorff space. The condition (W4) now says that Y is closed in X and hence in Z. This tells us that Z carries the discrete topology and is closed in X. A discrete closed set in a compact space

by definition of a subcomplex. By (8.2.1), A A is compact and closed.

D x

(3)Induction over dim.e/. If e is a 0-cell, then e is a point and closed, hence a subcomplex and X.e/ D X.e/x D e. Suppose X.f / is finite for each cell f with

dim.f / < n. Let e be an n-cell with characteristic map ˆ.

The set ˆ.Sn 1/ is contained in the union of cells of dimension at most n 1,

hence is contained in ex X e.

Then ex X e D ˆ.Sn 1/ is compact, hence contained in a finite number of cells e1; : : : ; ek , by (1), which are contained in Xn 1, by (W2). By induction hypothesis, the set C D e [ X.e1/ [ [ X.ek / is a finite subcomplex which contains e and hence X.e/. Therefore X.e/ is finite. Since X.e/ is closed, by (2), we have ex X.e/ and X.e/x X.e/.

(4)This is a consequence of (1) and (3).

(5)We show: A X is closed if and only if for each finite subcomplex Y the intersection A \ Y is closed in Y .

Suppose the condition is satisfied, and let f be an arbitrary cell. Then A\X.f / is closed in X.f /, hence, by (2) and (3), closed in X; therefore A \ fx D A \ X.f / \ fx is closed in X and in fx, hence closed in X by condition (W4).

(6)If Y is a finite subcomplex, then A \ Y is a finite subcomplex, hence closed.

By (5), A is closed.

(8.2.5) Proposition. A subcomplex Y of a Whitehead complex X is a Whitehead complex.

Proof. Let e be a cell in Y and ˆ W Dn ! X a characteristic map. Then ˆ.Dn/ D ex Y , since Y is closed. Hence ˆ can be taken as a characteristic map for Y .

It remains to verify condition (W4). Let L Y and suppose L \ ex is closed in ex for each cell e in Y . We have to show that L is closed in Y . We show that L is closed in X. Let f be a cell of X. By (W3), fx is contained in a finite union e1 [ [ ek of cells. Let e1; : : : ; ej be those which are contained in Y . Then

fx \ Y e1 [ [ ej ex1 [ [ exj Y

202 Chapter 8. Cell Complexes

(8.2.6) Proposition. Let X be a Whitehead complex. Then:

(1)X carries the colimit topology with respect to the family .Xn j n 2 N0/.

(2)Let .e j 2 ƒ.n// be the family of n-cells of X with characteristic maps ˆ W Dn ! Xn and restrictions ' W Sn 1 ! Xn 1. Then

Proof. (1) Suppose A \ Xn is closed in Xn for each n. Then for each n-cell e of X the set A \ ex D A \ ex \ Xn is closed in ex. By (W4), A is closed in X.

(2) The diagram is a pushout of sets. Give Xn the pushout topology and denote this space by Z. By construction, the identity W Z ! Xn is continuous. We show that is also closed. Let V Z be closed. By definition of the pushout topology this means:

(i) V \ Xn 1 is closed in Xn 1.

(ii) ˆ 1.V / \ Dn is closed in Dn, hence also compact.

We conclude that ˆ.ˆ 1.V / \ Dn/ D V \ ˆ.Dn/ D V \ ex is closed in ex , being a continuous image of a compact space in a Hausdorff space. From (i) and (ii) we

inclusions of the finite pointed subcomplexes F X induce a canonical map colimF k .F; / ! k .X; /. This map is an isomorphism.

Recall from Section 7.9 the notion of a k-space and the k-space k.X/ obtained from a space X.

(8.2.8) Proposition. Let X have a cell decomposition such that (W1)–(W3) hold and such that each compact set is contained in a finite number of cells. Then k.X/ is a Whitehead complex with respect to the given cell decomposition and the same characteristic maps. Moreover, X is a Whitehead complex if and only if k.X/ D X.

Proof. Let ˆ W Dn ! X be a characteristic map for the cell e. Since ex is compact it has the same topology in k.X/. Hence ˆ W Dn ! k.X/ is continuous. Since ˆ is a quotient map and ˆ 1.e/ D En, we see that e has the same topology in k.X/ and X. Thus e is a cell in k.X/ with characteristic map ˆ.

Let A \ ex be closed in ex for each cell e. Let K k.X/ be compact. By hypothesis, K is contained in a finite number of cells, say K e1 [ [ ek . Then A \ K D ..A \ ex1/ [ [ .A \ exk // \ K is closed in K. Hence A is k-closed.

characteristic map for e f . For this purpose use a homeomorphism

.DmCn; SmCn 1/ ! .Dm Dn; Dm Sn 1 [ Sm 1 Dn/:

With this cell structure, X Y satisfies conditions (W1)–(W3) in the definition of a Whitehead complex. In general, property (W4) may not hold. In this case one re-topologizes X Y such that the compact subsets do not change. The space X k Y D k.X Y / is then a Whitehead complex (see (8.2.8)).

Problems

1. R carries the structure of a Whitehead complex with 0-cells fng, n 2 Z and 1-cellsn; n C 1Œ, n 2 Z. There is an analogous Whitehead complex structure W .ı/ on Rn with 0-cells the set of points ı.k1; : : : ; kn/, kj 2 Z, ı > 0 fixed and the associated ı-cubes. Thus, given a compact set K Rn and a neighbourhood U of K, there exists another neighbourhood L of K contained in U such that L is a subcomplex of the complex W .ı/. In this sense, compact subsets can be approximated by finite complexes.

2. The geometric realization of a simplicial complex is a Whitehead complex.

8.3 CW-Complexes

We now use (8.2.6) as a starting point for another definition of a cell complex. Let

.X; A/ be a pair of spaces. We say, X is obtained from A by attaching an n-cell, if there exists a pushout

'

Sn 1 A

\\

Dn ˆ X.

Then A is closed in X and X n A is homeomorphic to En via ˆ. We call X n A an n-cell in X, ' its attaching map and ˆ its characteristic map.

(8.3.1) Proposition. Let a commutative diagram with closed embeddings j; J be given:

f

AY

jJ

Suppose F induces a bijection X X A ! Z X Y . Then the diagram is a pushout, provided that (1) F .X/ Z is closed; (2) F W X ! F .X/ is a quotient map. Condition .2/ holds if X is compact and Z Hausdorff.

204 Chapter 8. Cell Complexes

Proof. Let g W X ! U and h W Y ! U be given such that gj D hf . The diagram is a set-theoretical pushout. Therefore there exists a unique set map ' W Z ! U with 'F D g, 'J D h. Since J is a closed embedding, 'jJ.Y / is continuous. Since F is a quotient map, 'jF .X/ is continuous. Thus ' is continuous, since F .X/ and J.Y / are closed sets which cover Z.

(8.3.2) Note. Let X be a Hausdorff space and A a closed subset. Suppose there exists a continuous map ˆ W Dn ! X which induces a homeomorphism ˆ W En ! X n A. Then X is obtained from A by attaching an n-cell.

(8.3.3) Example. The projective space RP n is obtained from RP n 1 by attaching an n-cell. The projective space CP n is obtained from CP n 1 by attaching a 2n- cell.

We recall that CP n 1 is obtained from S2n 1 by the equivalence relation

We can also attach several n-cells simultaneously. We say X is obtained from A by attaching n-cells if there exists a pushout

The index j just enumerates different copies of the same space. Again, A is then

of dimension, the integer n is determined by X n A.) We allow J D ;; in that case A D X. We write ˆj D ˆjDjn and 'j D 'jSn 1 and call ˆj the characteristic map of the n-cell ˆ.Ejn/ and 'j its attaching map.

8.3. CW-Complexes 205

Let us give another interpretation: X D X.'/ is the double mapping cylinder

is replaced by a homotopic map , then X.'/ and X. / are h-equivalent under A. Let f W A ! Y be a given map. Assume that X is obtained from A by attaching

In view of this note we call the homotopy classes Œf 'j the obstructions to

extending f .

Let A be a subspace of X. A CW-decomposition of .X; A/ consists of a sequence of subspaces A D X 1 X0 X1 X such that:

(1)X D [n 0Xn.

(2)For each n 0, the space Xn is obtained from Xn 1 by attaching n-cells.

(3)X carries the colimit topology with respect to the family .Xn/.

Xk is a subspace of the colimit X of a sequence Xj Xj C1 . If the inclusions are closed, then Xk is closed in X. This is an immediate consequence of the definition of the colimit topology.

A pair .X; A/ together with a CW-decomposition .Xn j n 1/ is called a relative CW-complex. In the case A D ; we call X a CW-complex. The space Xn is the n-skeleton of .X; A/ and .Xn j n 1/ is the skeleton filtration. The cells of Xn n Xn 1 are the n-cells of .X; A/. We say, .X; A/ is finite (countable etc.) if

X n A consists of a finite (countable etc.) number of cells. If X D Xn, X ¤ Xn 1 we denote by n D dim.X; A/ the cellular dimension of .X; A/. If A D ;, then A is suppressed in the notation. We call X a CW-space if there exists some cellular

decomposition X0 X1 of X.

Let X be a Whitehead complex. From (8.2.6) we obtain a CW-decomposition of X. The converse also holds: From a CW-decomposition we obtain a decomposition into cells and characteristic maps; it remains to verify that X is a Hausdorff space and carries the colimit topology with respect to the closures of cells (see (8.3.8)).

In the context of CW-complexes .X; A/, the symbol Xn usually denotes the n-skeleton and not the n-fold Cartesian product.

(8.3.5) Note. If .X; A/ is a relative CW-complex, then also .X; Xn/ and .Xn; A/ are relative CW-complexes, with the obvious skeleton-filtration inherited from

.Xn j n 1/.

(8.3.6) Example. From (8.3.3) we obtain cellular decompositions of CP n and RP n. The union of the sequence RP n RnC1 defines the infinite projective

In favorable cases, the filtration ..X Y /n j n 1/ is a CW-decomposition of the pair

.X Y; A B/.

Let Y be locally compact. Then .X Y /n is obtained from .X Y /n 1 by attaching n-cells.

3.Let .X; A/ be a relative CW-complex and let C A. Then .X=C; A=C / is a relative CW-complex with CW-decomposition .Xn=C /. Moreover, X=A is a CW-complex.

4.Let A X be a subcomplex. Then X=A is a CW-complex.

5.Let A and B be subcomplexes of X. Then A=.A \ B/ is a subcomplex of X=B.

6.Let A be a subcomplex of B and Y another CW-complex. Then A ^k Y is a subcomplex

of B ^k Y .

7. Let A be a CW-complex. Suppose X is obtained from A by attaching n-cells via attaching maps ' W Sjn 1 ! An 1. Then X is a CW-complex with CW-decomposition Xj D Aj

suspension †X are CW-complexes. (In statements of this type the reader is asked to find a canonical cell decomposition induced from the initial data.)

10. Let .Xj j j 2 J / be a family of pointed CW-complexes with base point a 0-cell. Then

W

j 2J Xj has the structure of a CW-complex such that the summands are subcomplexes. 11. Let p W E ! B be a Serre fibration and .X; A/ a CW-pair. Then each homotopy h W X I ! B has a lifting along p with given initial condition on X 0 [ A I .

12.Suppose X is obtained from A by attaching n-cells. Let p W E ! X be a covering and E0 D p 1.A/. Then E is obtained from E0 by attaching n-cells.

13.Let X be a CW-complex with n-skeleton Xn and p W E ! X a covering. Then E is a CW-complex with n-skeleton En D p 1.Bn/ such that p maps the cells of E homeomor-

phically to cells of X. An automorphism of p maps cells of E homeomorphically to cells.

14.Each neighbourhood U of a point x of a CW-complex contains a neighbourhood V which is pointed contractible to x. A connected CW-complex has a universal covering. The universal covering has a cell decomposition such that its automorphism group permutes the cells freely.

15.Let X and Y be countable CW-complexes. Then X Y is a CW-complex in the product topology.

8.4 Weak Homotopy Equivalences

We now study the notion of an n-connected map and of a weak homotopy equivalence in the context of CW-complexes.

(8.4.1) Proposition. Let .Y; B/ be n-connected. Then a map f W .X; A/ ! .Y; B/ from a relative CW-complex .X; A/ of dimension dim.X; A/ n is homotopic