Cw Complexes

8/12/2019 Cw Complexes

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CW complexes

Soren Hansen

This note is meant to give a short introduction to CW complexes.

1. Notation and conventionsIn the following a space is a topological space and a map f :XY between topological spaces

XandY is a function which is continuous. IfXis a space, then a subspace ofXis a subsetA Xwith the relative (or induced) topology (induced by the topology in X). The n-dimensional disk(just called the n-disk in the following) is the following subspace ofRn:

Dn ={ x Rn : |x| 1 },

where| |: Rn [0,[ is the standard norm on Rn. Thus the n-disk is the closed n-disk and is aclosed subset ofRn. The open n-disk, denoted int(Dn), is the interior ofDn in Rn. Thus

int(Dn) ={ x Rn : |x|< 1 }.

The boundary ofDn in Rn is the standard (n 1)-sphere

Sn1 ={x Rn : |x|= 1}.

We note that the 0-disk D0 is equal to R0 ={0}by definition. We have int(D0) =D0 ={0}.A topological spaceXis called quasi-compact if every open cover ofXhas a finite subcover, i.e.

whenever {Ui}iIis a family of open subsets ofX s.t. X=iIUi then there exist i1, . . . , in Is.t. X=

nj=1 Uij . A topological space Xis called compact if it is Hausdorff and quasi-compact.

As is standard we will write iff for if and only if.

2. Cell decompositions and CW-complexes

Definition 2.1. Ann-cell is a space homeomorphic to the open n-disk int(Dn). A cell is a spacewhich is an n-cell for some n 0.

Note that int(Dm) and int(Dn) are homeomorphic if and only ifm = n. This e.g. follows bynoting that int(Dn) is homeomorphic to Rn (via the mapx tan(|x|/2)x), and by the fact thatRm is homeomorphic to Rn iffm = n, cf. [Ha, Theorem 2.26 p. 126]. Thus we can talk about the

dimensionof a cell. An n-cell will be said to have dimension n.

Definition 2.2. A cell-decomposition of a space X is a family E={e | I} of subspaces ofXsuch that each e is a cell and

X=Ie

(disjoint union of sets). The n-skeleton ofXis the subspace

Xn =I:dim(e)ne.

Note that ifEis a cell-decomposition of a space X, then the cells ofEcan have many different

dimensions. E.g. one cell-decomposition ofS1 is given by E = {ea, eb}, where ea is an arbitrarypoint p S1 and eb = S

1 \ {p}. Hereea is a 0-cell and eb is a 1-cell. There are no restrictionson the number of cells in a cell-decomposition. Thus we can have uncountable many cells in sucha decomposition. E.g. any space Xhas a cell-decomposition where each point ofX is a 0-cell. Afinite cell-decomposition is a cell decomposition consisting of finitely many cells.

Definition 2.3. A pair (X, E) consisting of a Hausdorff spaceXand a cell-decompositionEofXis called a CW-complex if the following 3 axioms are satisfied:

1


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Axiom 1: (Characteristic Maps) For each n-cell e E there is a map e : Dn Xrestricting to a homeomorphism e|int(Dn): int(D

n) e and taking Sn1 into Xn1.Axiom 2: (Closure Finiteness) For any cell e E the closure e intersects only a finite

number of other cells in E.Axiom 3: (Weak Topology) A subset A X is closed iffA e is closed in X for eache E.

Here eof course is the closure ofe in X. Note that the Axioms 2 and 3 are only needed in caseE is infinite (i.e. they are automatically satisfied ifEis finite). It is not difficult to give examples ofpairs (X, E) withXa Hausdorff space and Ean infinite cell-decomposition ofXsuch that Axiom1 is satisfied and either Axiom 2 or Axiom 3 is satisfied, see e.g. [J, p. 97]. Thus Axiom 2 and 3are independent of each other. Note that the characteristic map for a 0-celle X is simply themap mapping 0 to the one-point space e.

Lemma 2.4. Let (X, E) be a Hausdorff spaceX together with a cell-decompositionE. If (X, E)satisfies Axiom 1 in Definition 2.3 then we have

e= e(Dn)

for any cell e E. In particular e is a compact subspace ofX and the cell boundary e\ e =e(S

n1) lies inXn1.

Proof. For any mapf :Y Z between topological spaces Y and Zand any subset B Y wehave f(B) f(B), see e.g. [D, Theorem III.8.3 pp. 79-80] or [A, Theorem 2.9 p. 33]. Thus

e= e(int(Dn)) e(Dn) e.

But e(Dn) is compact hence closed in X since X is Hausdorff. Thus e(D

n) = e. By Axiom 1we have e(int(D

n)) =e and e(Sn1) e= so e(Sn

1) = e \ e.

Note, ifXis not Hausdorff we still have e(Dn) ebut we dont necessarily have equality. We

have no garantee that e(Dn) is closed in X.

3. SubcomplexesLet (X, E) be a CW-complex, E Ea set of cells in it and

X =eEe.

Lemma 3.1. The following 3 conditions are equivalent:

(a) The pair(X, E) is a CW-complex.(b) The subsetX is closed inX.(c) The closuree X for eache E, wheree is the closure ofe inX.

For a proof, see [J, p. 98].

Definition 3.2. Let (X, E) be a CW-complex and let (X, E) be as above. Then (X, E) is calleda subcomplex (of (X, E)) if the 3 equivalent conditions (a), (b) and (c) in the above lemma aresatisfied.

We have some immediate consequences:

Corollary 3.3. Let(X, E) be a CW-complex. Then

(1) Let {Ai | i I} be any family of subcomplexes of (X, E). Then iIAi and iIAi aresubcomplex of(X, E).

(2) Then-skeletonXn is a subcomplex of(X, E) for eachn 0.


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(3) Let{ei|i I} be an arbitrary family ofn-cells inE. ThenXn1 (iIei) is a subcomplex.

Proof. For (1) note that iIAi is a subcomplex by characterization (c) in Lemma 3.1 andiIAi is a subcomplex by (b) in that lemma. Both (2) and (3) follow by using characterization(c) in Lemma 3.1 together with Lemma 2.4.

Note that (2) is a special case of (3). By the above we have that the n-skeleton Xn of aCW-complex (X, E) is a closed subset ofX.

4. Identification Topology and Quotient Spaces

In the next section we need a general proceedure for constructing new spaces from old spacesby gluing spaces together via maps. Let us in this section describe the general concepts from pointset topology needed.

4.1. Identification Topology. Let Xbe a topological space and let Y be an arbitrary set andlet p : X Y be a surjection. Then we can define a topology in Y by: a subset U Y isopen iffp1(U) is open in X. This topology is the largest topology in Y for which p: XY iscontinuous. We call it the identification topology in Y determined by p, and p: XY is calledan identification map.

IfX and Y are two spaces and p :X Ya surjective map, then p is called an identificationmap if the topology in Yis the identification topology determined by p.

Lemma 4.1. LetXbe a compact space andYa Hausdorff space and letp : X Ybe a surjectivemap. Thenp is an identification map.

Proof. It is enough to prove thatCY is closed iffp1(C) is closed. Since p is continuous weonly have to prove that C is closed in Y ifp1(C) is closed in X. But ifp1(C) is closed in Xthen it is compact, since X is compact. Thus C = p(p1(C)) is compact in the Hausdorff spaceY, hence C is closed in Y.

Lemma 4.2. Letp : X Y be an identification map and letZ be a space. Thenf : Y Z iscontinuous ifff p: X Z is continuous.

Proof left to the reader.

4.2. Quotient spaces. Let X be a set and let be an equivalence relation on X. Let X/be the set of equivalence classes and let : X X/ be the canonical projection, i.e. thefunction mappingx to the equivalence class containing x. Recall here that the equivalence classesare mutually disjoint subsets ofX and that X is the disjoint union of these equivalence classes.Oppositely, if we have given a disjoint family {Ai}iIof subsets ofX coveringX, i.e.X=iIAi,then we can define an equivalence relation on X by x y if and only if i I s.t. x, y Ai.

The equivalence classes for that equivalence relation are nothing but our subsets Ai. Thus anequivalence relation in X is nothing but a partition ofX into subsets.Now, given a space X and an equivalence relation we equip X/ with the identification

topology determined by the canonical projection :XX/. This topology is normally calledthe quotient topology and X/ is called a quotient space ofX(the quotient ofX by).

Ifp: X Y is an identification map, then we can identify Y with a quotient space. Namely,the subsets p1(y), y Y, gives a partition ofX. Thus the equivalence relation in X inducedby this partition is x x iff p(x) = p(x). We thus have a bijection q : X/ Y given by


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q((x)) = p(x), where : X X/ is the canonical projection. By Lemma 4.2 both q andq1 are continuous since p and are both identification maps and q = p and q1 p = are continuous. Thus q : X/ Yis a homeomorphism so from a topological point of view weconsiderY andX/ to be the same space. Thus quotient spaces and identification spaces are oneand the same thing.

There are many standard constructions in algebraic topology using the above ideas. In particularwe mention:

Collapsing a subspace to a point. LetXbe a topological space and letA be some non-emptysubspace. Then we letX/A= X/, where the equivalence classes w.r.t. areAand the singletons{x}, x X\ A.

The Wedge of Spaces. Given two spaces XandYand chosen points x0 Xandy0 Ywe let

X Y = (X Y)/{x0, y0}.

That is, we collapse the subset {x0, y0} to a point. Here X Y is the topological disjoint unionofX and Y (sometimes called the topological sum ofX and Y), see below. We note that theconstruction depends on the pointsx0and y0. However, different choices can lead to homeomorphicresults. Thus e.g. S1 S1 does not depend on the choice of the points x0 and y0.

More generally we can talk about the wedge of a family of spaces {Xi}iI, namely ifxi Xiwe let

iIXi= iIXi/A,

whereA= {xi}iI, and where iIXi is the topological disjoint union of the spaces Xi.

Here and in the following we use the following construction. Let{Xi}iIbe a family of topo-logical spaces. Then the topological disjoint unionX=Xi is the following topological space. Asa set it is equal to iI{i} Xi. We identify{i} Xi with Xi. The topology on X is given by:U is open in X iffU Xi is open in Xi for all i I. Note that iffi :Xi Y, i I is a familyof maps then we get a unique map f =iIfi :XY s.t. f|Xi =fi for each i I. A functionf :X Y is continuous ifff|Xi :Xi Yis continuous for each i I.

Attaching a space to another via map. LetXand Ybe two spaces and let A be a subspaceofXand f :A Y a map. Then

XfY :=X Y/,

whereX Yis given the disjoint union topology, see above, and the equivalence classes w.r.t. are the singletons {p}, p X\ A and p Y \ f(A), and the subsets {y} f1(y), y f(A). Wewill only need this in situations where A is a closed subset ofX.

Lemma 4.3. Let the situation be as above withA a closed subset ofX, let: X Y XfYbe the canonical projection and let i : Y XY and j : X\A XY be the inclusionmaps. Then i : Y XfY is an embedding with image a closed subset ofXfY and j :X\ A XfY is an embedding with image an open subset ofXfY.

Exercise 4.4. Prove the above lemma.

5. CW-complexes considered as spaces obtained by attaching cells to each other

We start by a small

Lemma 5.1. Let(X, E)be a CW-complex and letYbe any space and letf :X Y be a function.Then the following are equivalent.

(1) f :XY is continuous.(2) The restrictionf|e: e Y is continuous for alle E.


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(3) The restrictionf|Xn :Xn Y is continuous for alln 0.

Exercise 5.2. Prove the above lemma.

Proposition 5.3. Let(X, E) be a CW-complex. ThenXn is obtained fromXn1 by attaching ofthen-cells inX.

Proof. LetEn be the n-cells in X and let for each e En, e : Dn Xbe the characteristicmap of e. Since e(S

n1) Xn1 we can consider e = e|Sn1 : Sn1 Xn1. For each

e En, let Dne = {e} Dn and identify this with Dn. (Here e is just some index.) We let

Z = eEnDne , A = eEnD

ne and = eEne : A X

n1. Thus |Dne = e. (HereDne ={e} S

n1=Sn1.) We let

Y =ZXn1

and let n: Z Xn1 Ybe the canonical projection. We have a surjective map

f= (eEne) j : Z Xn1 Xn,

wherej :Xn1 Xn is the inclusion. The proof is finalized by the following exercise.

Exercise 5.4. Let the situation be as in the above proof. Prove the following facts:

(1) There is a unique bijection : Y Xn such that n= f.(2) The bijection is a homeomorphism. (Hint: To prove that 1 is continuous use Lemma 5.1

and prove that1|e= n|eife Xn1 and1|e e= n|Dne :D

ne Y for eache En.

Prove and use that e: Dne e is an identification map.)

(3) The composition n|Xn1 :Xn1 Xn is the inclusion.

We can now formulate an alterantive definition of CW-complexes.

Proposition/Definition 5.5. A CW-complex is a space together with a filtration of subspaces

= X1 X0 X1 . . . Xn . . . X

such that(1) Xn is obtained by attaching ofn-cells to Xn1. That is, we have maps: D

n X

n1,

In, n 0, and homeomorphismsn :Yn Xn such thatn n|Xn1 :Xn1 Xn

is the inclusion, where

Yn =Zn nXn1,

whereZn = InDn andn = In : InD

n X

n1, and where n : Zn

Xn1 Yn is the canonical projection.(2) X=n0Xn.(3) Xcarries the weak topology w.r.t. the family of spaces{Xn}n0. That is, a subsetA X

is closed iffA Xn is closed inXn for alln 0.

Thus by (1) we have (inductively) a unique topology on X

n

for each n 0 starting by notingthat X0 necessarily have the discrete topology (since X0 =I0D0), and by (3) the total space

Xthen has the weak topology w.r.t. these topological spaces.Note that we have not included in the definition above that X is Hausdorff. That follows

automatically as the proof will reveal.

Proof. Exercise. See later version.


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References

[A] M. A. Armstrong, Basic Topology, Undergraduate Texts in Mathematics, SpringerVerlag (1983).[D] J. Dugundji, Topology, Allyn and Bacon, Inc. (1966).[Ha] A. Hatcher, Algebraic Topology, Cambridge University Press (2002).[J] K. Janich, Topology, Undergraduate Texts in Mathematics, SpringerVerlag (1984).

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