Homotopy

Exercises

1.3 Homotopy

A)→Z and g:Bⁿ(A)→Z be maps such thatfn−1◦hα|Sⁿ_α⁻¹=g|S_αⁿ⁻¹. Then there is a unique mapfn : (Xⁿ∪A)→Zsuch thatfn|Xⁿ⁻¹∪A=fn−1

andfn◦hα=g|B_αⁿ.

Proof. By 1.2.7,Xⁿ∪Ais obtained fromXⁿ⁻¹∪Aby attachingn-cells. The result follows from the properties of the quotient topology stated in Sect. 1.1.

Remark 1.2.24.A Hausdorff spaceZ is paracompact if every open coverU of Z has a locally finite open refinementV; this means:V is a locally finite open cover ofZand every element ofV is a subset of some element ofU. It is a fact that every CW complex is paracompact; for a proof see [105]. This arises, for example, in the proof that a fiber bundle whose base space is a CW complex has the homotopy lifting property.

Historical Note:CW complexes were introduced by J.H.C. Whitehead in [154].

In his exposition primacy was given to the “open cells”e^◦α rather than the “closed cells” eα, presumably because each ^◦eα is homeomorphic to an open ball while eα

need not be homeomorphic to a closed ball. The shift of primacy to “closed cells”

has become standard.

Exercises

1. IfX⁰=∅thenX =∅; why?

2. Show that a compact subset of a CW complex lies in a finite subcomplex.

3. In Example 1.2.10(a) and (b) it is asserted that some familiar spaces have partic- ular CW complex structures. Prove that these spaces (endowed with the topol- ogy inherited from the Euclidean spaces in which they live) are homeomorphic to the indicated CW complexes.

1.3 Homotopy

Let X and Y be spaces and let f0, f1 : X → Y be maps. One says that f0

is homotopic to f1, denoted f0 f1, if there exists a map F : X×I → Y such that for allx∈X,F(x,0) =f0(x) and F(x,1) =f1(x). The mapF is called a homotopy fromf0 tof1. One often writes Ft:X →Y for the map x→F(x, t); thenF0=f0andF1=f1. IfF is a homotopy fromf0tof1, one writesF :f0f1.

Proposition 1.3.1.Homotopy is an equivalence relation on the set of maps fromX toY.

Proof. Given f : X → Y, define F : f f byF(x, t) = f(x) for all t ∈ I;

F = f ◦ (projection: X ×I → X), the composition of two maps. So F is a map. Thus reflexivity. GivenF :f g, define F :g f byF(x, t) =

F(x,1−t). The functionq:I→I,t→1−t, is a map.F=F◦(id×q), soFis a map. Thus symmetry. GivenF:f gandG:gh, defineH :X×I→Y by H(x, t) =F(x,2t) when 0 ≤t ≤ ¹2 and byH(x, t) = G(x,2t−1) when

1

2 ≤ t ≤ 1. Let q1 : [0,¹₂] → I be the map t → 2t. Let q2 : [¹₂,1] → I be the mapt→2t−1. OnX×[0,¹₂],H agrees with the mapF ◦(id×q1); on X×[¹₂,1],H agrees with the mapG◦(id×q2). SinceX×[0,¹₂] andX×[¹₂,1]

are closed in X×I, and the restriction ofH to each is a map, H is a map.

Thus transitivity.

A map f : X → Y is a homotopy equivalence if there exists a map g : Y → X such that g ◦f id : X → X and f ◦ g id : Y → Y. Such a map g is called ahomotopy inverse of f. The spaces X and Y have the same homotopy type if there exists a homotopy equivalence from X to Y. If g is a homotopy inverse of the homotopy equivalence f, then g is also a homotopy equivalence. A map homotopic to a homotopy equivalence is a homotopy equivalence. Any two homotopy inverses of f are homotopic, and any map homotopic to a homotopy inverse of f is a homotopy inverse of f. The map id_X is a homotopy equivalence. A homeomorphism is a homotopy equivalence.

Definitions parallel to those of the last paragraph hold for maps of pairs.

A map f : (X, A)→ (Y, B) is ahomotopy equivalence if there exists a map g : (Y, B)→(X, A) such that g◦f id : (X, A)→(X, A) and f◦g id : (Y, B)→(Y, B). Such a mapg is called a homotopy inverse off. The pairs (X, A) and (Y, B) have the same homotopy type if there exists a homotopy equivalence from (X, A) to (Y, B), etc.

Let (X, A) and (Y, B) be pairs of spaces, let f0, f1 : (X, A) →(Y, B) be maps of pairs, and letX⊂X. One says thatf0ishomotopic tof1relative to X, denotedf0f1relX, if there exists a mapF : (X×I, A×I)→(Y, B) such that for all x ∈ X, F(x,0) = f0(x) and F(x,1) = f1(x), and for all x ∈ X and t ∈ I, F(x, t) = f0(x). Then F is a homotopy relative to X fromf0tof1. By a proof similar to that of 1.3.1 we have:

Proposition 1.3.2.Homotopy relative to X is an equivalence relation on

the set of maps from (X, A)to(Y, B).

It is customary not to distinguish between the spaceXand the pair (X,∅), so that homotopy between maps of spaces is considered to be a special case of homotopy between maps of pairs.

Proposition 1.3.3.Let f₀, f₁ : (X, A) → (Y, B) be homotopic rel X, let g0, g1: (Y, B)→(Z, C)be homotopic relY, wheref0(X)⊂Y. Then g0◦f0

andg1◦f1 are homotopic relX.

Proof. Let F : f₀ f₁ and G : g₀ g₁ be homotopies which behave as required on A, B, C and X. Let p: X×I → I be projection. Let (F, p) : X×I → Y ×I denote the function (x, t) →(F(x, t), p(x, t)) = (F(x, t), t).

(F, p) is a map. The required homotopy isG◦(F, p) :X×I→Z.

1.3 Homotopy 25 The spaceX iscontractible if it has the same homotopy type as the one- point space. Equivalently,X is contractible if idX is homotopic to some con- stant map fromX toX.

Proofs of 1.3.4–1.3.7 below are left as exercises:

Proposition 1.3.4.Any two maps from a space to a contractible space are

homotopic.

Proposition 1.3.5.Any two maps from a contractible space to a path con-

nected space are homotopic.

Proposition 1.3.6.The product of two contractible spaces is contractible.

Example 1.3.7.The spacesR,R+, andIare contractible. Hence also (by 1.3.6) Rⁿ,Rⁿ+ and Bⁿ are contractible. Ifp∈Sⁿ, thenSⁿ− {p} is homeomorphic toRⁿ; henceSⁿ− {p} is contractible.

The subspaceA⊂X is aretract ofX if there is a mapr:X →A such thatA→ⁱ X−→^r Ais the identity. Such a mapr is a retraction of X toA.

The subspace A ⊂ X is a strong deformation retract of X if there is a retraction r : X → A and a homotopy F : X ×I → X relative to A such that F0 = idX and F1 = i◦r. Such a homotopy F is a strong deformation retraction of X to A. A pair (A, A) is a strong deformation retract of the pair (X, X) if (A, A)⊂(X, X) and the retractionrand the homotopyF in the previous sentence are maps of pairs.¹³

Proposition 1.3.8.If A is a strong deformation retract of X then A →X is a homotopy equivalence.

Proof. In the notation above,ris a homotopy inverse for inclusion.

For building homotopies involving CW complexes the following example is fundamental.

Lemma 1.3.9.The subspace(Bⁿ× {0})∪(Sⁿ⁻¹×I)is a strong deformation retract ofBⁿ×I.

Proof. One standard proof involves “radial projection” of Bⁿ×I onto the subspace(Bⁿ× {0})∪(Sⁿ⁻¹×I) from the “light source” (0, . . . ,0,2)∈Rⁿ⁺¹. We give a variation which avoids complicated formulas.

Let f : Bⁿ → I be the map f(x) = 0 if |x| ≤ ¹₂ and f(x) = 2|x| −1 if

1

2 ≤ |x| ≤1. LetY ={(x, t)∈Bⁿ×R|f(x)≤t≤1}. LetY₀={(x, t)∈Y |

13IfF is not required to be relA whiler remains a retraction, one says thatAis adeformation retract ofX. If this is further weakened by only requiringr◦ito be homotopic to idAthenAis aweak deformation retract ofX. Note thatAis a weak deformation retract ofX iffi:A →Xis a homotopy equivalence. WhenA is a subcomplex of the CW complexX the three notions of deformation retract coincide; see Exercises 2 and 3 below.

t=f(x)}. There is a strong deformation retractionF : Y ×I→ Y of Y to Y0, namely F(x, t, s) = (x,(1−s)t+sf(x)). And there is a homeomorphism h: (Bⁿ×I,(Bⁿ×{0})∪(Sⁿ⁻¹×I))→(Y, Y0), namelyh(x, t) = (¹₂(t+ 1)x, t).

We are preparing for the important Theorem 1.3.15. The building blocks of the proof are Lemma 1.3.9 and the following (compare 1.2.23):

Proposition 1.3.10.Let (X, A) be a CW pair and let the n-cells of X which are not in A be indexed by A. Let {hα : Bⁿ_α → X | α ∈ A}

be a set of characteristic maps for those n-cells. Let Z be a space and let Fn−1 : (Xⁿ⁻¹∪A)×I → Z and g : Bⁿ(A)×I → Z be maps such that Fn−1◦(hα×id) | S_αⁿ⁻¹×I = g | S_αⁿ⁻¹×I. Then there is a unique map Fn : (Xⁿ ∪A)×I → Z such that Fn | (Xⁿ⁻¹ ∪ A)×I = Fn−1 and Fn◦(hα×id) =g|Bⁿ_α×I.

The proof of 1.3.10 requires a non-obvious lemma from general topology;

see [51, p. 262]:

Lemma 1.3.11.LetZ andW be Hausdorff spaces. Ifq:Z →W is a quotient map thenq×id :Z×I→W×I is a quotient map.

Proof (Proof of 1.3.10). By 1.2.7, Xⁿ ∪A is obtained from Xⁿ⁻¹∪A by attaching n-cells. Thus there is a quotient mapp: (Xⁿ⁻¹∪A)

Bⁿ(A)→ Xⁿ∪Awhich agrees with inclusion onXⁿ⁻¹∪Aand withhαonB_αⁿ⊂Bⁿ(A).

By 1.3.11,p × id : ((Xⁿ⁻¹ ∪ A) × I)

(Bⁿ(A) × I)→(Xⁿ∪A)×I is a quotient map agreeing with inclusion on the first summand and withhα×id on theB_αⁿ×Ipart of the second summand, for eachα∈ A. The desired result therefore follows from the properties of the quotient topology stated in Sect.

1.1.

Proposition 1.3.12.IfY is obtained fromAby attachingn-cells, then(Y× {0})∪(A×I) is a strong deformation retract of Y ×I. Hence any map (Y × {0})∪(A×I)→Z extends to a map Y ×I→Z.

Proof. By 1.3.9, there is a strong deformation retractionF : Bⁿ×I×I → Bⁿ×IofBⁿ×Ito (Bⁿ×{0})∪(Sⁿ⁻¹×I). LetAindex then-cells. Consider the diagram

(A

Bⁿ(A))×I×I

p×id×id

f

((Q

QQ QQ QQ QQ QQ Q

Y ×I×I

f˜

//

_ _ _ _

_ Y ×I

Here, p : A

Bⁿ(A) → Y is a quotient map, and f agrees with A×I×I ^projection // A×I ^inclusion //Y ×I on A × I × I and with Bⁿ_α×I×I ^F //Bⁿ_α×I ^(p|)×id //Y ×I on B_αⁿ ×I×I for each α ∈ A.

1.3 Homotopy 27 By 1.3.11,p×id×id is a quotient map. By the properties of quotient maps stated in Sect. 1.1, the map ˜f exists and is the desired strong deformation retraction. For the last part, any mapg: (Y× {0})∪(A×I)→Z is extended

byg◦f˜1to a well-defined mapY ×I→Z.

The last sentence of 1.3.12 illustrates the following basic property. A pair of spaces (Y, A) has thehomotopy extension property with respect to a space Z if every map (Y× {0})∪(A×I)→Z extends to a mapY×I→Z. In the presence of this property, a map onAcan be extended toX if it is homotopic to a map which can be extended toX.

By 1.2.7 we have:

Proposition 1.3.13.IfAis a subcomplex ofX, thenXⁿ∪Ais a subcomplex ofX. MoreoverXⁿ∪Ais obtained from Xⁿ⁻¹∪Aby attaching n-cells.

Proposition 1.3.14.Let Z =

n≥1

Zn be a CW complex, where each Zn is a subcomplex ofZ. For all n >1, assumeZn−1 is a strong deformation retract ofZn (in particular Zn−1⊂Zn). Then Z1 is a strong deformation retract of Z.

Proof. Letn≥2. LetF⁽ⁿ⁾:Zn×[¹_n,_n−¹₁]→Zn satisfyF_t⁽ⁿ⁾(x) =xift= _n¹ or ifx∈ Zn−1, and F⁽ⁿ⁾₁

n−1(Zn) =Zn−1. Let ˆF⁽ⁿ⁾₁

n−1 :Zn → Zn−1 be the map induced byF⁽ⁿ⁾₁

n−1. DefineG⁽ⁿ⁾:Z_n×I→Z_nto agree with projection (x, t)→ xon Zn×[0,_n¹], with F⁽ⁿ⁾ on Zn×[_n¹,_n−¹₁], with F⁽ⁿ⁻¹⁾◦Fˆ⁽ⁿ⁾₁

n−1 ×id on Z_n×[_n−¹₁,_n−¹₂], withF⁽ⁿ⁻²⁾◦Fˆ⁽ⁿ₁⁻¹⁾

n−2 ×id

◦Fˆ⁽ⁿ⁾₁

n−1×id

onZ_n×[_n¹₋₂,_n¹₋₃], . . ., and withF⁽²⁾◦Fˆ⁽³⁾₁

2 ×id

◦. . .◦Fˆ⁽ⁿ⁾₁

n−1×id

onZ_n×[¹₂,1]. ThenG⁽ⁿ⁾is a strong deformation retraction ofZn toZ1, andG⁽ⁿ⁾agrees withG⁽ⁿ⁻¹⁾onZn−1×I.

DefineG:Z×I→Z to agree withG⁽ⁿ⁾ onZn×I for alln≥2. By 1.2.12, Gis continuous. Clearly,Gis the required strong deformation retraction ofZ

toZ1.

Now we are ready for the Homotopy Extension Theorem for CW com- plexes:

Theorem 1.3.15. (Homotopy Extension Theorem) If (X, A) is a CW pair, then(X× {0})∪(A×I)is a strong deformation retract ofX×I. Hence (X, A)has the homotopy extension property with respect to every space.

Proof. Let Z_n = (X × {0})∪((Xⁿ ∪A)×I) and let Z = X ×I. Then Z₋₁= (X× {0})∪(A×I), andZ=

n≥−1

Z_n. By 1.3.13 and 1.2.19, eachZ_nis a subcomplex ofZ, and ofZ_n+1. By 1.3.13,Xⁿ∪Ais obtained fromXⁿ⁻¹∪A by attachingn-cells. Hence, by 1.3.12, ((Xⁿ∪A)× {0})∪((Xⁿ⁻¹∪A)×I) =

(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.