Some Techniques in Homotopy Theory
4.1 Altering a CW complex within its homotopy type
4
simultaneous attaching map for then-cells, then there is a homeomorphism Kn→Kn−1∪f
α∈A
Bαn
which matches cells bijectively.
Another example is the mapping cylinder of a cellular map f : X →Y. This is the adjunction complexY ∪f0(X×I) wheref0:X× {0} →Y is the map (x,0)→f(x). We will denote the mapping cylinder off byM(f). The map X → X×I sending xto (x,1) induces an embedding i: X → M(f);
the identity map of Y induces an embedding j : Y → M(f). Both i and j take each cell of the domain homeomorphically onto a cell of M(f); one frequently suppresses i and j, identifying X with i(X) and Y with j(Y), writingX ⊂ M(f) and Y ⊂ M(f). The map X ×I → Y, taking (x, t) to f(x), and the identity map onY, together induce thecollapse r:M(f)→Y.
The following diagram commutes:
X i //
f??????
?? M(f)
||zzzzzzrzz
Y
Proposition 4.1.2.The map r is a homotopy inverse for j, so r is a ho- motopy equivalence. Indeed there is a strong deformation retraction D : M(f)×I→M(f) ofM(f)ontoY such that D1=r.
Proof. The required D is induced by projection: Y ×I → Y and the map X×I×I→X×I, (x, t, s)→(x, t(1−s)).
The proof of 4.1.2 gives a “canonical” strong deformation retraction of M(f) ontoY. Thus the same proof gives:
Proposition 4.1.3.Let X =A∪B andX =A∪B, where A and B are subcomplexes ofX, while A andB are subcomplexes ofX. Let f :X →X be a cellular map such thatf(A)⊂A andf(B)⊂B. Then there is a strong deformation retraction ofM(f)ontoX which restricts to strong deformation retractions ofM(f |A)onto A,M(f |B)ontoB, andM(f |A∩B)onto
A∩B.
Returning to the general adjunction complex Y ∪fX, where (X, A) is a CW pair, Y is a CW complex, and f : A → Y is a cellular map, we now show that the homotopy type ofY∪fX only depends on the homotopy types of (X, A) and Y, and the homotopy class off. We need some preliminaries (4.1.4 and 4.1.5) which have independent interest.
Let n≥0 be an integer and let (X, A) be a pair of CW complexes with X non-empty. The pair (X, A) is n-connected if for each 0 ≤ k ≤ n every map (Bk, Sk−1)→(X, A) is homotopic relSk−1to a map whose image lies in A. Thus (X,∅) is never n-connected, and (X, A) is 0-connected iff each path component ofX has non-empty intersection withA. For anyx∈X, (X,{x}) is 1-connected iffX is simply connected.
4.1 Altering a CW complex within its homotopy type 103 Proposition 4.1.4. (Whitehead Theorem)1Let(X, A)be a CW pair. The following are equivalent:
(i)A is a strong deformation retract of X;
(ii)A →i X is a homotopy equivalence;
(iii) (X, A)isn-connected for alln.
Proof. (i)⇒(ii) is clear; see 1.3.8.
For (ii)⇒ (iii), letr:X →A be a homotopy inverse fori. Since r|A is homotopic to idA, 1.3.15 implies thatris homotopic to a mapr:X→Asuch thatr |A= idA. So we may assumer|A= idA. Letf : (Bk, Sk−1)→(X, A) be a map wherek≤n, and let H :X×I →X be a homotopy from idX to i◦r. Define
F : (Bk×I× {0})∪(Bk× {0,1} ×I)∪(Sk−1×I×I)→X by
F(x, t, s) =
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
H(f(x), t) onBk×I× {0} f(x) onBk× {0} ×I H(irf(x),1−s) onBk× {1} ×I H(f(x),(1−s)t) onSk−1×I×I.
By 1.3.15,F extends to a map ˆF :Bk×I×I→X. LetG:Bk×I→X be the mapG(x, t) = ˆF(x, t,1). ThenG(x,0) =f(x), andG(x,1) =H(irf(x),0) = irf(x)∈ A. Ifx∈ Sk−1, G(x, t) =H(f(x),0) = f(x)∈A. It is easy to see that ((Bk× {1})∪(Sk−1×I), Sk−1× {0}) is homeomorphic to (Bk, Sk−1) (exercise). It follows thatf is homotopic relSk−1 to a map whose image lies inA (exercise).
For (iii) ⇒ (i), let f0 = idX. The pair (X, A) is 0-connected, so f0 |: X0∪A →X is homotopic, relA, to a map intoA. By 1.3.15,f0is homotopic relA to a map f1 :X → X such that f1(X0∪A)⊂A. The pair (X, A) is 1-connected, so by 1.3.10, f1| : X1∪ A → X is homotopic, rel X0∪A, to a map into A. Again by 1.3.15, f1 is homotopic, rel X0∪A, to a map f2 : X →X such thatf2(X1∪A)⊂A. Proceeding thus by induction, and observing that the homotopyfnfn+1is relXn−1∪A, we get a well defined limit mapf :X →X such thatf |A= id andf(X)⊂A. Combining all the homotopies, we can get a homotopy idX f. The details of this last step are
left as an exercise.
The last two propositions lead us to an important theorem on piecing together homotopy equivalences:
Theorem 4.1.5.Let X = A∪B and X = A ∪B, where A and B are subcomplexes ofX, while A andB are subcomplexes of X. Letf :X→X
1 This is the version we will need most often. The full Whitehead Theorem is stated in Exercise 1 of Sect. 4.4.
be a cellular map such that f(A) ⊂ A and f(B) ⊂ B. If f| : A → A, f| : B → B, and f| : A∩B → A ∩B are all homotopy equivalences, then f :X →X is a homotopy equivalence. Moreover, there is a homotopy inverse g :X →X for f, taking A toA, B to B, and A∩B to A∩B, and homotopies g◦f idX and f ◦g idX which restrict to homotopies g◦f |idA,f ◦g |idA, g◦f |idB,f ◦g |idB,g◦f |idA∩B, and f◦g|idA∩B.
Remark 4.1.6.Even the special case in whichB⊂AandB ⊂Ais of interest.
It says that iff : (A, B)→(A, B) is a cellular map of pairs such that the induced mapsA → A and B → B are homotopy equivalences then f is a homotopy equivalence of pairs. Note, in particular, what this says when B andB are single points.
Proof (of 4.1.5). By 4.1.2 f is a homotopy equivalence iff i : X → M(f) is a homotopy equivalence. Write f1 = f |: A → A, f2 = f |: B → B, f0 = f |: A∩B → A ∩B. For k = 0,1,2,fk is a homotopy equivalence.
So, by 4.1.2, A → M(f1),B → M(f2) andA∩B → M(f0) are homotopy equivalences. By 4.1.4, the pairs (M(f1), A), (M(f2), B) and (M(f0), A∩B) aren-connected for alln. The proof of (iii)⇒(i) in 4.1.4 shows that we can construct a strong deformation retraction of M(f0) toA∩B which extends to strong deformation retractions ofM(f1) toAand ofM(f2) toB. SoX is a strong deformation retract ofM(f). For the second part, combine this with
4.1.3.
Now we are ready for the main theorems, 4.1.7 and 4.1.8. Suppose (X, A) and (X, A) are CW pairs, Y and Y are CW complexes, f : A → Y and f : A → Y are cellular maps, g : (X, A) → (X, A) is a map of pairs, g:X →X,g|:A→A andk:Y →Y are homotopy equivalences, and the following diagram commutes:
X oo ? _
g
A f //
g|
Y
k
Xoo ? _A f
//Y
Theorem 4.1.7.The induced map G: Y ∪fX → Y∪fX is a homotopy equivalence.
Proof. Consider the commutative diagram
M(f)∪X −−−−→h Y ∪fX
p
⏐⏐
⏐⏐G M(f)∪X −−−−→h Y∪fX
4.1 Altering a CW complex within its homotopy type 105 Here and in what follows, we writeM(f)∪X forM(f)∪iX wherei:A→ M(f) is the canonical inclusion – i.e., we literally apply the convention of writingA⊂M(f);pis induced by (g|)×id :A×I→A×I,k:Y →Y and g:X →X;his induced by the collapser:M(f)→Y and idX; similarly for h. By 4.1.5,pis a homotopy equivalence.
To see that h is a homotopy equivalence, consider the commutative dia- gram
M(f)∪X h //
mMMMMMMM&&
MM
MM Y ∪fX
M(q|X)
collapse
99r
rr rr rr rr r See Fig. 4.1. Here, q : Y
X →Y ∪f X is the defining quotient map. The inclusionm:M(f)∪X →M(q|X) sendsX to the copy ofX inM(q|X);
M(f)⊂M(q|X) becauseq|A= inclusion◦f. By 4.1.2, we need only show thatmis a homotopy equivalence. By 1.3.15, (X× {1})∪(A×I) is a strong deformation retract ofX×I. Any strong deformation retraction ofX×I to (X× {1})∪(A×I) induces a strong deformation retraction ofM(q|X) to
M(f)∪X.
A M(f)
M(f)
M(q X)| X A
Y
Y
m
h
collapse
X Y
U
U
f
V
V V
V V
V M(f) X
Y X
Fig. 4.1.
Theorem 4.1.8.If the hypotheses of 4.1.7 are weakened fromk◦f =f◦g| tok◦f f◦g|, it is still the case thatY ∪fX andY∪fX have the same homotopy type.
Proof. Consider the diagram X oo ? _
g
A i //
g|
M(f)
˜k
r
""
DD DD DD DD
Y
||yyyyyykyy
X oo ? _A f
//Y
Hereris the collapse and ˜k=k◦r. The right square homotopy commutes. By 1.3.15, ˜kis homotopic to a map ¯ksuch that ¯k◦i=f◦g|. By 4.1.7,M(f)∪X is homotopy equivalent to Y∪f X. By the proof of 4.1.7, M(f)∪X is
homotopy equivalent toY ∪fX.
Here is a useful application of 4.1.7:
Corollary 4.1.9.Let X be a CW complex andA a contractible subcomplex.
The quotientq:X→X/A is a homotopy equivalence.
Proof. Apply 4.1.7 withX =X, A=A,g= id,Y =A,Y={q(A)}. Theorems 4.1.7 and 4.1.8 are powerful technical tools. For example, 4.1.8 implies that ifY is obtained fromAby attachingn-cells, then the homotopy type of Y only depends on the homotopy classes of the attaching maps. We now discuss an application of this to Tietze transformations.
For each presentationP =W |R, ρof a groupGa procedure was given in Example 1.2.17 for building a presentation complexXP, having just one vertex v, such thatπ1(XP, v) ∼=G(by 3.1.8). The 1-cells and 2-cells of XP
are in bijective correspondence with the setsW andR, respectively, in such a way that if P ⇒P :=W |R, ρis a Tietze transformation of Type I, thenXP is, in a natural way, a subcomplex ofXP, such that all cells ofXP
which are not cells ofXP are 2-cells.
Proposition 4.1.10.LetPbe obtained fromP by a Tietze transformation of Type I. Then there is a homotopy equivalencehmaking the following diagram commute up to homotopy:
XP //
inclusion
=
==
==
==
==
= XP∨
α∈R−R
Sα2
XP h
99s
ss ss ss ss s
Proof. The attaching maps for the 2-cells of XP which are not in XP are
homotopic inXP to constant maps. Apply 4.1.8.
4.1 Altering a CW complex within its homotopy type 107 Similarly, ifP ⇒P=W|R, ρis a Tietze transformation of Type II, thenXP is a subcomplex ofXP and we have:
Proposition 4.1.11.The mapXP →XP is a homotopy equivalence.
Proof. (XP)1=XP1∨
α
Sα1
, i.e. the wedge ofXP1 and a bouquet of circles.
For eachα, the 2-celle2αofXPwhich is not inXP has a characteristic map of the formfα:I2→XP wherefα(I1× {−1})⊂XP,fα(±1, t) =fα(±1,−1) for all t ∈I1, and fα | I1× {1} is a characteristic map for Sα1. The strong deformation retraction ofI2 ontoI1× {−1}, (s, t, u)→(s,(1−u)(1 +t)−1) for 0≤u≤1, induces a strong deformation retraction of XP1 ∪Sα1∪e2αonto XP1. This can be done simultaneously for all α, giving a strong deformation
retraction ofXP ontoXP.
Proposition 4.1.12.Fori= 1and2, letPi:=Wi|Ri, ρibe presentations of the group G. There are homotopy equivalent CW complexes YP1 and YP2
obtained from XP1 andXP2 by attaching 3-cells. Moreover, if P1 andP2 are finite presentations, YP1 and YP2 can be obtained by attaching finitely many 3-cells.
Proof. If in the proof of 4.1.10 we attach a 3-cell toXP∨
α∈R−R
Sα2
, for eachα, by a homeomorphismS2 →S2α we obtain a 3-dimensional complex homotopy equivalent to XP. By 4.1.8, we can attach 3-cells to XP itself to get a 3-dimensional complex homotopy equivalent toXP whose 2-skeleton is XP.
Applying this remark in the context of 3.1.15 gives us:
P1
Type II+3PType I+3P ksType IPType IIks P2.
In terms of associated CW complexes, this gives:
XP1Type II+3XP
Type I+3XP //XP∪
{e3α|α∈ A}:=Z1 XP2
Type II+3XP
Type I+3XP //XP ∪
{e3β|β ∈ B}:=Z2
Indeed, the proof of 3.1.15 shows thatAandBare in bijective correspondence with the setsW1
R2 and W2
R1 respectively. The spaces XP1, XP and Z1 are homotopy equivalent. The spaces XP2, XP and Z2 are homotopy equivalent. By 4.1.8, we can attach 3-cells toXP1 and to XP2 to getYP1 :=
XP1 ∪
{e˜3β | β ∈ B} and YP2 := XP2 ∪
{˜e3α | α ∈ A}, both homotopy equivalent to XP ∪
{e3γ |γ ∈ A
B}. The last sentence of the Proposition
is clear.
Remark 4.1.13.There are examples in [52] of finite CW complexesX and Y each having one vertex, such thatX∨S2is homotopy equivalent toY∨S2∨S2, while there is no CW complexZsuch thatXis homotopy equivalent toZ∨S2. This is related to the existence of finitely generated projectiveZG-modules which are stably free but not free. (See also [7].) In the proof of 4.1.12,XP has the homotopy type of bothXP∨
α∈A
Sα2
andXP∨
⎛
⎝
β∈B
Sβ2
⎞
⎠. Dunwoody’s examples show that one cannot always “cancel” copies ofS2.
Theorem 4.1.14.LetX be a path connected CW complex whose fundamental groupGis finitely generated. Then:
(i)X is homotopy equivalent to a CW complex having finite 1-skeleton.
(ii)If Gis finitely presented, there is a CW complex X, obtained fromX by attaching 3-cells, which is homotopy equivalent to a CW complex having finite 2-skeleton.
(iii)If G is finitely presented, X is homotopy equivalent to a CW complex Z whose 2-skeleton is the wedge of a finite CW complex and a bouquet of 2-spheres.
Proof. By 3.1.13, 3.1.12 and 4.1.9, we may assume thatXhas only one vertex.
WriteXP1 =X2, and letP2 be a presentation ofGwhich is finite or finitely generated as appropriate. WritePi=Wi|Ri. Using 4.1.8 as in the proof of 4.1.12, we getX∪
{3-cells} homotopy equivalent to a CW complexY such thatY2=XP2. Similarly, we getX homotopy equivalent to a CW complexZ such thatZ2=XP2∪
β∈B
Sβ2where|B|=|W2|+|R1|. We claimZ1is finite; a sketch of the argument follows. Usingfor “is homotopy equivalent to”, and using the notation of the proof of 4.1.12, we get:
XP XP∨
⎛
⎝
β∈B
Sβ2
⎞
⎠XP2∨
⎛
⎝
β∈B
Sβ2
⎞
⎠.
XP1 XP∪
{e3α|α∈ A} XP2∨
⎛
⎝
β∈B
Sβ2
⎞
⎠∪
{e˜3α|α∈ A}. Hence, by 4.1.8,
X XP2∨
⎛
⎝
β∈B
Sβ2
⎞
⎠∪
{˜e3α|α∈ A} ∪
{cells of dimension ≥3}.
The “cells of dimension≥3” (other than the ˜e3α’s) are in bijective correspon-
dence with those ofX.
4.1 Altering a CW complex within its homotopy type 109 Example 4.1.15.The “finitely generated” and “finitely presented” parts of 4.1.14 would be more similar if we could say, in the finitely presented case, that Xis homotopy equivalent to a CW complex having finite 2-skeleton. However, this is false. For example, letX be an infinite bouquet of 2-spheres. We have π1(X, v) trivial by 3.1.11; however,H2(X;Z2) is an infinite-dimensionalZ2- module (vector space), since every 2-chain is a cycle and none is a boundary.
IfY is a finite CW complex,H2(Y;Z2) is a finite-dimensionalZ2-vector space sinceC2(Y;Z2) is finitely generated, hence also Z2(Y;Z2). Hence, by 2.7.7, X is not homotopy equivalent to a CW complex with finite 2-skeleton.
The proof of 4.1.14 also proves the following, which will be useful.
Addendum 4.1.16.Let X be a path connected CW complex having mk k- cells for eachk≥0, where0≤mk ≤ ∞. LetP:=W |R, ρbe a presentation of the fundamental group of X. Then X is homotopy equivalent to a CW complexZ with the properties:(i)Z2=XP∨(bouquet of m2+|W|2-spheres);
(ii)Z has m3+m1−m0+ 1 +|R| 3-cells; and (iii)Z has mk k-cells for all k≥4.
Proof. LetT be a maximal tree inX. ThenT hasm0 vertices and (m0−1) 1-cells. So X :=X/T has one vertex, (m1−m0+ 1) 1-cells, andmk k-cells fork≥2. By the proof of 4.1.14,X is homotopy equivalent to a complexZ of the form
Z =XP∨
⎛
⎝
β∈B
Sβ2
⎞
⎠∪
{˜e3α|α∈ A} ∪
{cells of dimension ≥3}
where |B|=|W|+m2, |A| =m1−m0+ 1 +|R|, and, for k≥3,Z hasmk k-cells. By 4.1.9,X is homotopy equivalent toZ.