Maps between CW complexes

Exercises

1.4 Maps between CW complexes

(Xⁿ× {0})∪((Xⁿ⁻¹∪A)×I) is a strong deformation retract of (Xⁿ∪A)×I.

It follows immediately thatZn−1 is a strong deformation retract of Zn. By 1.3.14,Z₋1 is a strong deformation ofZ, as claimed. The last part is proved

as in the proof of 1.3.12.

Examination of the proof of 1.3.15 shows a little more which can be useful:

Addendum 1.3.16.Let(X, A)be a CW pair and letF : (X×{0})∪(A×I)→ Z be a map. The mapF extends to a mapF˜ :X×I→Z such that, for every cell eαof X−A,F˜1(eα) =F0(eα)∪F˜(^•eα×I).

The definition of the homotopy extension property has nothing to do with CW complexes. Therefore the last part of 1.3.15 is more powerful than it might appear; it says that the existence of a CW pair structure on the pair of spaces (X, A) ensures that (X, A) has the homotopy extension property with respect to every space.

Corollary 1.3.17.If(X, A)is a CW pair andZ is a contractible space, every

map A→Z extends to a map X→Z.

Exercises

1. Prove 1.3.4–1.3.7.

For the next two exercises (X, A) is a CW pair.

2. Prove that if A is a weak deformation retract of X then A is a deformation retract ofX.

3. Prove that ifA is a deformation retract of X thenA is a strong deformation retract ofX.

Hint for 2. and 3.: Apply 1.3.15 appropriately.

1.4 Maps between CW complexes 29 Proposition 1.4.1.Let m > n. Let U be an open subset of Rⁿ such that the spaceclU is compact. Letf : clU →B^mbe such that f(frU)⊂S^m−1. Then f is homotopic, rel fr U, to a map g such that some point of B^m−S^m−1 is not in the image ofg.

Proof. There are many proofs in the literature. The most common involve

“simplicial approximation” off (see, for example, [146, 7.6.15], ). We give a proof here which assumes instead knowledge of multivariable calculus.

Let V =f⁻¹(B^m−S^m−1). Define : V →(0,∞) by (x) = distance¹⁴ fromxto frV. Using partitions of unity (see [85, pp. 41–45] ), there exists a C¹map ˆg:V →B^m−S^m−1such that|f(x)−ˆg(x)|< (x) for allx∈V. Since

|f(x)−g(x)ˆ | →0 asxapproaches fr_RmV, ˆg can be extended continuously to g : clU →B^m by definingg = ˆg onV andg =f on (clU)−V. Thenf is homotopic tog by the homotopy

H_t(x) = (1−t)f(x) +tg(x).

We haveg(clU)∩(B^m−S^m−1) =g(V). We will show thatg(V) is a proper subset ofB^m−S^m−1.

IdentifyRⁿ withRⁿ× {0} ⊂R^m. ThenV ⊂Rⁿ× {0} ⊂R^m andg(V) = gp(V) wherep:R^m→Rⁿ is projection.

Anm-cube CinR^mof side λ >0 is a productC=I₁×. . .×I_mof closed intervals each of lengthλ. The m-dimensional measure ofC isλ^m. A subset X ⊂R^m hasmeasure 0 if for every >0,X can be covered by a countable set ofm-cubes the sum of whose measures is less¹⁵than.

Clearly V ⊂ R^m has m-dimensional measure 0. Since ˆg is C¹, so also is ˆgp |: p⁻¹(V) → B^m −S^m−1. By the Mean Value Theorem, given any compact subsetK⊂p⁻¹(V), there existsµ(K)>0 such that|ˆgp(x)−gp(y)ˆ | ≤ µ(K)|x−y|for all x, y ∈K; we may takeµ(K) = sup{||D(ˆgp)x|| | x∈K}. It follows that ifC⊂Kis anm-cube of sideλ, thengp(C) lies in anm-cube C of side√

m.µ(K).λ. ThusC has measure (√

m.µ(K))^m.(measure ofC).

It is a theorem of elementary topology that every open cover of an open subset of Rⁿ has a countable subcover; see pages 174 and 65 of [51] for a proof. Hence we may write V =

∞ i=1

V_i where each set V_i lies in the interior of a closedm-ballBi⊂p⁻¹(V). Let >0 be given. Each Vi can be covered by countably manym-cubesC_i1, C_i2, . . .such that eachC_ij⊂intB_i, and

j

measure (Cij)< . Hence, by the above discussion,g(Vi) can be covered by countably many cubesC_ij such that

j

measure (C_ij )< (√

m.µ(B_i))^m. So

14All distances in euclidean spaces refer to the usual euclidean metric |x−y| = (P

(x²_i −y_i²))^1/2.

15Of course, this ism-dimensional Lebesgue measure, but we need almost no mea- sure theory!

gp(Vi) has measure 0. But, clearly, the countable union of sets of measure 0 has measure 0, so gp(V)(= g(V)) has measure 0. Clearly, B^m−S^m−1 does

not have measure 0. Sog(V) is a proper subset.

Proposition 1.4.2.Leteⁿ_αbe ann-cell of the CW complexX, and letz∈e^◦n_α. Then Xⁿ−^◦eⁿ_α is a strong deformation retract of Xⁿ− {z}.

Proof. Leth : (Bⁿ, Sⁿ⁻¹) →(eⁿ_α,e^•n_α) be a characteristic map. For any y ∈ Bⁿ−Sⁿ⁻¹, the formula x→x+ (1− |x|)y defines a homeomorphism ofBⁿ fixing each point ofSⁿ⁻¹and sending 0 toy. Hence we may assumeh(0) =z.

LetH : (Bⁿ−{0})×I→Bⁿ−{0}be the homotopy (x, t)→(1−t)x+tx/|x|. Note thatH is a strong deformation retraction ofBⁿ−{0}ontoSⁿ⁻¹. It is an easy exercise to check that a functionHexists making the following diagram commute:

(Bⁿ− {0})×I −−−−→^H Bⁿ− {0}

⏐⏐ ^h|×id

⏐⏐ ^h| (eⁿ_α− {z})×I −−−−→^H eⁿ_α− {z}

i.e., that if (h(x₁), t₁) = (h(x₂), t₂) thenh(H(x₁, t₁)) =h(H(x₂, t₂)). Sinceh is a quotient map, so ish|, and also (h|)×id by 1.3.11. ThusHis continuous and is therefore the required strong deformation retraction.

If X is a CW complex and S ⊂ X, the carrier of S is the intersection of all subcomplexes ofX which contain S. It is a subcomplex ofX, denoted C(S).

Theorem 1.4.3. (Cellular Approximation Theorem) Let f : X → Y be a map between CW complexes and let A be a subcomplex of X such that f |A:A→Y is cellular. Then f is homotopic, relA, to a cellular map.

Proof. We will constructH :X×I→Y such thatH0=f,Ht|A=f |Afor allt, andg:=H1is cellular. ThisH is defined by induction on (Xⁿ∪A)×I.

For each vertexe⁰ ofX⁰−A, there is a unique celldsuch thatf(e⁰)∈d.^◦ Define H | {e⁰} ×I to be any path inC(d) fromf(e⁰) to a vertex of C(d):

call that vertexg(e⁰). (There is such a path becauseC(d) is path connected.) AssumeH already defined on (Xⁿ⁻¹∪A)×Iwith the desired properties.

Extend H to agree with f on Xⁿ× {0}. By 1.3.15, H extends to a map H¯ : (Xⁿ∪A)×I → Y. Let ¯g = ¯H1. Then ¯g | Xⁿ⁻¹ = g | Xⁿ⁻¹ and is cellular, but ¯g might not map n-cells of X −A into Yⁿ. We claim that ¯g is homotopic rel Xⁿ⁻¹∪A to a cellular map. For eachn-cell eⁿ of X, the carrier of ¯g(eⁿ),C(¯g(eⁿ)), is a finite subcomplex ofY, by 1.2.13. Letd^mbe a top-dimensional cell ofC(¯g(eⁿ)). Writeq :eⁿ →C(¯g(eⁿ)) for the restriction of ¯g. If m ≤ n, the next step can be omitted, so suppose m > n. Since q(^•eⁿ)⊂Yⁿ⁻¹, q⁻¹(d−d)^• ⊂eⁿ−^•eⁿ. We want a homotopy of q, rele^•n, to

1.5 Neighborhoods and complements 31 a map which misses a point of d^◦m, for then, by 1.4.2, there is a homotopy of q, rel^•eⁿ, to a map ofeⁿ into C(¯g(eⁿ))−d, and after finitely many such^◦ homotopies, rel^•eⁿ, we will obtain a map of eⁿ intoYⁿ.

If q(eⁿ)∩d^◦ = ∅, there is nothing to do. Otherwise, let y ∈ q(eⁿ)∩d.^◦ q⁻¹(y) is a compact set lying in^◦eⁿ. LetDbe a neighborhood ofy ind^◦which is homeomorphic to a closed ball, and let U = q⁻¹(intD). Then cl U is a compact subset ofe^◦n and q|: clU →D. By 1.4.1, the desired homotopy of q, releⁿ−U (hence rel^•eⁿ), exists.

Doing this on each n-cell of X, we get the desired homotopy of ¯g, and, hence, in the obvious way, the desiredH : (Xⁿ∪A)×I→Y. Addendum 1.4.4.The homotopy H : X×I →Y obtained in the proof of 1.4.3 has the property that for each cell eof X,H(e×I)⊂C(f(e)).

Proof. Assume the induction hypothesisH(e^m×I)⊂C(f(e^m)) for allm < n.

It is clearly true when n = 1. Let e^•n ⊂ ^◦e^m_α¹

1 ∪. . .∪e^◦m_α^r

r. Then ¯g(eⁿ) ⊂ H¯(eⁿ ×I) = H(eⁿ×0)∪H(e^•n×I) ⊂ f(eⁿ)∪

r i=1

C(f(e^m_αⁱ

i)) ⊂ C(f(eⁿ)).

(The last inclusion holds because, if ^◦e^m∩^•eⁿ = ∅, then C(e^m) ⊂ C(eⁿ).) The other half of the homotopy ofeⁿ takes place in C(¯g(eⁿ))⊂C(f(eⁿ)). So

H(eⁿ×I)⊂C(f(eⁿ)).

Corollary 1.4.5.LetX be a CW complex;X is path connected iffX¹ is path connected.

Proof. As in the proof of 1.4.3, there is a path from any point ofX to a point ofX¹, soX is path connected ifX¹ is path connected. For the converse, let ω:I→X be a path withω(0), ω(1)∈X¹. Without loss of generality assume ω(0), ω(1) ∈ X⁰. Apply 1.4.3 to produce a cellular path, necessarily in X¹,

fromω(0) toω(1).