INTRODUCTION TO ALGEBRAIC TOPOLOGY
MARTIN ČADEK
2. CW-COMPLEXES
2.1. Constructive definition of CW-complexes. CW-complexes are all the spaces which can be obtained by the following construction:
(1) We start with a discrete space X°. Single points of X° are called O-dimensional cells.
(2) Suppose that we have already constructed Xn_1. For every element a of an index set Jn take a map fa : S^1 = dD™ —> Xn~x and put
Xn = |J (X-1 U/a Dl) .
a
Interiors of discs D™ are called n-dimensional cells and denoted by e™.
(3) We can stop our construction for some n and put X = Xn or we can proceed with n to infinity and put
00
X = (J Xn.
n=0
In the latter case X is equipped with inductive topology which means that A C X is closed (open) iff A H Xn is closed (open) in Xn for every n.
Example A. The sphere Sn is a CW-complex with one cell e° in dimension 0, one cell en in dimension n and the constant attaching map / : Sn~1 —> e°.
Example B. The real projective space IRPn is the space of 1-dimensional linear sub-spaces in IRn+1. It is homeomorhic to
Sn/(v ~ -v) = Dn/(w ~ -w),    for w G dDn = Sn-\
However, Sn-1/{w ~ -w) = KF1-1. So RFn arises from IRPn_1 by attaching one n-dimensional cell using the projection / : S^1 KF1-1. Hence MPn is a CW-complex with one cell in every dimension from 0 ton.
We define IRP°° = U^°=1 ^Pn. It is again a CW-complex.
Example C. The complex projective space CPn is the space of complex 1-dimensional linear subspaces in Cn+1. It is homeomorhic to
S2n+1/(v ~ Xv) ^ {(w, y/1 - \w\2) e Cn+1; \\w\\ < l}/((w,0) ~ X(w,0), \\w\\ = 1) = D2n/(w ~\w; we dD2n)
for all A G C, |A| = 1. However, dD2n/(w ~ Aw) = CPn_1. So CPn arises from CP™"1 by attaching one 2n-dimensional cell using the projection / : S2n~l = dD2n —> CPn_1. Hence CPn is a CW-complex with one cell in every even dimension from 0 to 2n.
1
2
Define CP00 = \J™=1 CPn. It is again a CW-complex.
2.2. Another definition of CW-complexes. Sometimes it is advantageous to be able to describe CW-complexes by their properties. We carry it out in this paragraph. Then we show that the both definitions of CW-complexes are equivalent.
Definition. A cell complex is a Hausdorff topological space A such that
(1) A as a set is a disjoint union of cells ea
X=\Jea.
aeJ
(2) For every cell ea there is a number, called dimension.
Xn=    |J ea
dim ea<n
is the n-skeleton of A.
(3) Cells of dimension 0 are points.  For every cell of dimension > 1 there is a characteristic map
Va : (Dn, S'^1) -> (A, A™"1)
which is a homeomorphism of intDn onto ea.
The cell subcomplex Y of a cell complex A is a union Y = [jaeK ea , K C J, which is a cell complex with the same characterictic maps as the complex A. A CW-complex is a cell complex satisfying the following conditions:
(C) Closure finite property. The closure of every cell belongs to a finite subcomplex,
i. e. subcomplex consisting only from a finite number of cells. (W) Weak topology property. F is closed in A if and only if F H ea is closed for every a.
Example. Examples of cell complexes which are not CW-complexes:
(1) S2 where every point is 0-cell. It does not satisfy property (W).
(2) D3 with cells e3 = int_B3, e° = {x} for all x G S2. It does not satisfy (C).
(3) A = {1/n; n > 1} U {0} C R. It does not satisfy (W).
(4) A = \J™=1{x e R2; \\x-(1/n, 0)|| = 1/n} C R2. If it were a CW-complex, the set {(1/n, 0) G R2; n > 1} would be closed in A, and consequently in IR2.
2.3. Equivalence of definitions.
Proposition. The definitions 2.1 and 2.2 of CW-complexes are equivalent.
Proof. We will show that a space A constructed according to 2.1 satisfies definition 2.2. The proof in the opposite direction is left as an exercise to the reader.
The cells of dimension 0 are points of A0. The cells of dimension n are interiors of discs D™ attached to An_1 with charakteristic maps
: (Z^Sr1) ^ (A-1 U^I&X**-1)
3
induced by identity on D™. So X is a cell complex. From the construction 2.1 it follows that X satisfies property (W). It remains to prove property (C). We will carry it out by induction.
Let n = 0. Then e° = eQa.
Let (C) holds for all cells of dimension < n — 1. e™ is a compact set (since it is an image of -D"). Its boundary <9e™ is compact in Xn~x. Consider the set of indices
K = {f3e J; &£ne^0}.
If we show that K is finite, from the inductive assumption we get that e™ lies in a finite subcomplex which is a union of finite subcomplexes for eg, (3 G K.
Choosing one point from every intersection <9e™ H ep, (3 G K we form a set A. A is closed since any intersection with a cell is empty or a onepoint set. Simultaneously, it is open, since every its element a forms an open subset (for A — {a} is closed). So A is a discrete subset in the compact set <9e™, consequently, it is finite. □
2.4. Compact sets in CW complexes.
Lemma. Let X be a CW-complex. Then any compact set A C X lies in a finite subcomplex, particularly, there is n such that A C Xn.
Proof. Consider the set of indices
K = {f3e J; Ane/3^0}.
Similarly as in 2.3 we will show that K is a finite set. Then A C IJ/3eA"^/3 an<^ every eg lies in a finite subcomplexes. Hence A itself is a subset of a finite subcomplex. □
2.5. Cellular maps. Let X and Y be CW-complexes. A map / : X —> Y is called a cellular map if f{Xn) C Fn for all n. In Section 5 we will prove that every map g : X —> Y is homotopic to a cellular map / : X —> Y. If moreover, g restricted to a subcomplex A C X is already cellular, / can be chosen in such a way that / = g on A.
2.6. Spaces homotopy equivalent to CW-complexes. One can show that every open subset of IRn is a CW-complex. In [Hatcher], Theorem A. 11, it is proved that every retract of a CW-complex is homotopy equivalent to a CW-complex. These two facts imply that every compact manifold with or without boundary is homotopy equivalent to a CW-complex. (See [Hatcher], Corollary A. 12.)
2.7. CW complexes and HEP. The most important result of this section is the following theorem:
Theorem. Let A be a subcomplex of a CW-complex X. Then the pair (X, A) has the homotopy extension property.
4
Proof. According to the last theorem in Section 1 it is sufficient to prove that X x {0} U A x I is a retract of X x J. We will prove that it is even a deformation retract. There is a retraction rn : Dn x J —y Dn x {0} U S"*-1 x J. (See Section 1.) Then hn : _Dn x I x J —>• Dn x J defined by
is a deformation retraction, i.e. a homotopy between id and rn.
Put = A, Yn = Xn U A. Using hn we can define a deformation retraction Fn : Yn x J x J -> Fn x J for the retract Fn x {0} U F^1 x J of Yn x J. Now define the deformation retraction H : X x I x I —)■ X x I for the retract X x {0} U A x I succesively on the subspaces X x {0} x I UYn x I x I with values in X x {0} U Yn x I. For n = 0 put
Suppose that we have already defined H on X x {0} UYn 1 x I. On X x {0} U Fn x I we put
#(rr, s, t) = (x, s)      for (x, s) G X x {0} or t G [0,1 /2n+1],
H(x,s,t) = Hn(x,s,2n+1(t- l/2n+1))   forrr GFn and t G [l/2n+1, l/2n],
H(x,s,t) = H(H(x,s,l/2n),t)    forrr GFn and t G [1/2", 1].
H : Xxlxl —> Xxl is continuous since so are its restrictions on Xx{0}xIUYnxIxI and the space X x I x I is a direct limit of the subspaces X x {0} x I U Fn x I x I.
hn(x, s, t) = (1 —        s) + trn(x, s
H(x, s, t) = (x, s)      for (x, s) G X x {0} or t G [0,1 /2], H(x,s,t) = H0(x,s,2(t - 1/2))   for x G F° and t G [1/2,1].
\
\ X x {0} U72 x I
s
s
X x I
X x {0} U A x I
FIGURE 2.1. Image of H depending on t
□
2.8. First criterion for homotopy equivalence.
Proposition. Suppose that a pair (X, A) has the homotopy extension property and that A is contractible (in A). Then the canonical projection q : X —> X/A is a homotopy equivalence.
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Proof. Since A is contractible, there is a homotopy h : A x I ^ A between icU and constant map. This homotopy together with idx : X —> X can be extended to a homotopy / : X x I —>• X. Since f(A,t) C A for all t £ I, there is a homotopy / : X/A x/-> XA4 such that the diagram
X x I —--^X
g g
XIA x I-*■ XIA
I
commutes. Define g : X/A —> X by pQa?]) = f(x, 1). Then idx ~ g ° q via the homotopy / and idx/A ^ Q ° 9 via the homotopy /. Hence X is homotopy equivalent to X/A. □
Exercise A. Using the previous criterion show that S2/S° ~ S2 V S1.
Exercise B. Using the previous criterion show that the suspension and the reduced suspension of a CW-complex are homotopy equivalent.
2.9. Second criterion for homotopy equivalence.
Proposition. Let (X, A) be a pair of CW-complexes and let Y be a space. Suppose that f,g : A —^ Y are homotopic maps. Then lUjľ and X Ug Y are homotopy equivalent.
Proof. Let F : Ax I —> Y be & homotopy between / and g. We will show that XUfY and X UgY are both deformation retracts of {X x I) U f Y. Consequently, they have to be homotopy equivalent.
We construct a deformation retraction in two steps.
(1) (X x {0}) U/ Y is a deformation retract of (X x {0} U A x I) UF Y.
(2) (IxjojuAx/) UFľisa deformation retract of (X x I) U f Y.
□
Exercise. Let (X, A) be a pair of CW-complexes. Suppose that A is a contractible in X, i. e. there is a homotopy F : A —> X between idx and const. Using the first criterion show that X/A = IU CA/CA ~IU C A. Using the second criterion prove that X U CA ~IV SA. Then
X/A -XV S A.
Apply it to compute Sn/S\ i < n.
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