INTRODUCTION TO ALGEBRAIC TOPOLOGY
MARTIN ˇCADEK
12. Homotopy and CW-complexes
This section demonstrates the importance of CW-complexes in homotopy theory.
The main results derived here are Whitehead theorem and theorems on approximation
of maps by cellular maps and spaces by CW-complexes.
12.1. n-connectivity. A space X is n-connected if πi(X, x0) = 0 for all 0 ≤ i ≤ n
and some base point x0 ∈ X (and consequently, for all base points).
A pair (X, A) is called n-connected if each component of path connectivity of X
contains a point from A and πi(X, A, x0) = 0 for all x0 ∈ A and all 1 ≤ i ≤ n
We say that a map f : X → Y is an n-equivalence if f∗ : πi(X, x0) → πi(Y, f(x0)) is
an isomorphism for all x0 ∈ X if 0 ≤ i < n and an epimorphism for all x0 if i = n.
Exercise. Prove that a pair (X, A) is n-connected if and only if the inclusion i : A →
X is an n-equivalence.
12.2. Compression lemma is an important technical tool in what follows.
Lemma A (Compression lemma). Let (X, A) be a pair of CW-complexes and (Y, B)
a pair with B = ∅. Suppose that πn(Y, B, y0) = 0 for all y0 ∈ B whenever there is a
cell in X − A of dimension n. Then every f : (X, A) → (Y, B) is homotopic rel A
with a map g : X → B.
A

f/A
// B

X
f
//
g
∼
>>~
~
~
~
Y
If n = 0, the condition π0(Y, B, y0) = 0 means that (Y, B) is 0-connected.
Proof. By induction we will define maps fn : X → Y such that fn(Xn
∪ A) ⊆ B,
and fn is homotopic to fn−1 rel A ∪ Xn−1
. Put f−1 = f. Suppose that we have
fn−1 and there is a cell en
in X − A. Let ϕ : Dn
→ X be its characteristic map.
Then fn−1ϕ : (Dn
, ∂Dn
) → (Y, B) represents zero element in πn(Y, B). According to
Proposition 10.2 it means that fn−1ϕ : (Dn
, ∂Dn
) → (Y, B) is homotopic rel ∂Dn
to
a map hn : (Dn
, ∂Dn
) → (B, B). Doing it for all cells of dimension n in X − A we
obtain a map gn : Xn
∪A → B homotopic rel A∪Xn−1
with fn−1 restricted to Xn
∪A.
Using the homotopy extension property of the pair (X, Xn
∪ A) we can conclude that
gn can be extended to a map fn : X → Y which is homotopic rel A ∪ Xn−1
to fn−1.
1
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Now for x ∈ Xn
define g(x) = fn(x) = gn(x). By the same trick as in the proof of
Theorem 2.7 we can construct a homotopy rel A between f and g.
The proof of the following extension lemma is similar but easier and hence left to
the reader.
Lemma B (Extension lemma). Consider a pair (X, A) of CW-complexes and a map
f : A → Y . If Y is path connected and πn−1(Y, y0) = 0 whenever there is a cell in
X − A of dimension n, then f can be extended to a map X → Y .
12.3. Whitehead Theorem. The compression lemma has two important conse-
quences.
Corollary. Let h : Z → Y be an n-equivalence and let X be a finite dimensional
CW-complex. Then the induced map h∗ : [X, Z] → [X, Y ] is
(1) a surjection if dim X ≤ n,
(2) a bijection if dim X ≤ n − 1.
Proof. First, we will suppose that h : Z → Y is an inclusion and apply the compression
lemma. Put B = Z, A = ∅ and consider a map f : X → Y . If dim X ≤ n then all
the assumptions of the compression lemma are satisfied. Consequently, there is a map
g : X → Z such that hg ∼ f. Hence h∗ : [X, Z] → [X, Y ] is surjection.
Let dim X ≤ n − 1 and let g1, g2 : X → Z be two maps such that hg1 ∼ hg2 via a
homotopy F : X ×I → Y . Then we can apply the compression lemma in the situation
of the diagram
X × {0, 1}
g1∪g2
//

Z
h

X × I F
//
H
::t
t
t
t
t
Y
to get a homotopy H : X × I → Z between g1 and g2.
If h is not an inclusion, we use the mapping cylinder Mh. (See 1.5 for the definition
and basic properties.) Let f : X → Y be a map. Apply the result of the previous part
of the proof to the inclusion iZ : Z → Mh and to the map iY f : X → Y → Mh to get
g : X → Z such that iZg ∼ iY f.
Z
h
~~||||||||
iZ

h
BBBBBBBB
X
g
77nnnnnnnn
f
// Y iY
// Mh p
// Y
Since the right triangle in the diagram commutes and the middle one commutes up to
homotopy and piY = idY, we get
hg = piZg ∼ piY f = f.
The statement (2) can be proved in a similar way.
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A map f : X → Y is called a weak homotopy equivalence if f∗ : πn(X, x0) →
πn(Y, f(x0)) is an isomorphism for all n and all base points x0.
Theorem (Whitehead Theorem). If a map h : Z → Y between two CW-complexes is
a weak homotopy equivalence, then h is a homotopy equivalence.
Moreover, if Z is a subcomplex of Y and h is an inclusion, then Z is even deformation
retract of Y .
Proof. Let h be an inclusion. We apply the compression lemma in the following situ-
ation:
Z
idZ
//
h

Z
h

Y
idY
//
g
??~
~
~
~
Y
Then gh ∼ idY rel Z and consequently hg = idZ. So Z is a deformation retract of Y .
The proof in a general case again uses mapping cylinder Mh.
12.4. Simplicial approximation lemma. The following rather technical statement
will play an important role in proofs of approximation theorems in this section and in
the proof of homotopy excision theorem in the next section. Under convex polyhedron
we mean an intersection of finite number of halfspaces in Rn
with nonempty interior.
Lemma (Simplicial approximation lemma). Consider a map f : In
→ Z. Let Z be a
space obtained from a space W by attaching a cell ek
. Then f is rel f−1
(W) homotopic
to f1 for which there is a simplex ∆k
⊂ ek
with f−1
1 (∆k
) a union (possibly empty) of
finitely many convex polyhedra such that f1 is the restriction of a linear surjection
Rn
→ Rk
on each of them.
The proof is elementary but rather technical and we omit it. See [Hatcher], Lemma
4.10, pages 350–351.
12.5. Cellular approximation. We recall that a map g : X → Y between two
CW-complexes is called cellular, if g(Xn
) ⊆ Y n
for all n.
Theorem (Cellular approximation theorem). If f : X → Y is a map between CWcomplexes,
then it is homotopic to a cellular map. If f is already cellular on a subcomplex
A, then f is homotopic to a cellular map rel A.
Corollary A. πk(Sn
) = 0 for k < n.
Corollary B. Let (X, A) be a pair of CW-complexes such that X − A contains only
cells of dimension greater then n. Then (X, A) is n-connected.
Proof of the cellular approximation theorem. By induction we will construct maps fn :
X → Y such that f−1 = f, fn is cellular on Xn
and fn ∼ fn−1 rel Xn−1
∪ A. Then we
can define g(x) = fn(x) for x ∈ Xn
and by the same trick as in the proof of Theorem
2.7 we can construct homotopy rel A between f and g.
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Suppose we have already fn−1 and there is a cell en
such that fn−1(en
) does not lie in
Y n
. Then f(en
) meets a cell ek
in Y of dimension k > n. According to the simplicial
approximation lemma fn−1 restricted to en is homotopic rel ∂en
to h : en → Y with
the property that there is a simplex ∆k
⊂ ek
and h(en
) ⊂ Y − ∆k
. (Since n < k,
there is no linear surjection Rn
→ Rk
.) ∂ek
is a deformation retract of ek − ∆k
and
that is why h is homotopic rel ∂en
to a map g : en → Y − ek
. Since f(en
) meets only
a finite number of cells, repeating the previous step we get a map fn defined on en
such that fn(en
) ⊆ Y n
and homotopic rel ∂en
to fn−1/ en. In the same way we can
define fn on A ∪ Xn
homotopic to fn−1/A ∪ Xn
rel A ∪ Xn−1
. Then using homotopy
extension property for the pair (X, A ∪ Xn
) we obtain fn : X → Y homotopic to fn−1
rel A ∪ Xn−1
.
12.6. Approximation by CW-complexes. Consider a pair (X, A) where A is a
CW-complex. An n-connected CW model for (X, A) is an n-connected pair of CWcomplexes
(Z, A) together with a map f : Z → X such that f/A = idA and f∗ :
πi(Z, z0) → πi(X, f(z0)) is an isomorphism for i > n and a monomorphism for i = n
and all base points z0 ∈ Z.
If we take A a set containing one point from every path component of X, then
0-connected CW model gives a CW-complex Z and a map Z → X which is a weak
homotopy equivalence.
Theorem A (CW approximation theorem). For every n ≥ 0 and for every pair
(X, A) where A is a CW-complex there exists n-connected CW-model (Z, A) with the
additional property that Z can be obtained from A by attaching cells of dimensions
greater than n.
Proof. We proceed by induction constructing Zn = A ⊂ Zn+1 ⊂ Zn+2 ⊂ . . . with Zk
obtained from Zk−1 by attaching cells of dimension k, and a map f : Zk → X such
that f/A = idA and f∗ : πi(Zk) → πi(X) is a monomorhism for n ≤ i < k and an
epimorphism for n < i ≤ k. For simplicity we will consider X and A path connected
with a fixed base point x0 ∈ A.
Suppose we have already f : Zk → X. Let ϕα : Sk
→ Zk be maps representing
generators in the kernel of f∗ : πk(Zk) → πk(X). Put
Yk+1 = Zk ∪ϕα
α
Dk+1
α .
Since the map f : Zk → X restricted to the boundaries of new cells is trivial, it can
be extended to a map f : Yk+1 → X.
By the cellular approximation theorem πi(Yk+1) = πi(Zk) for all i ≤ k − 1. Hence
the new f∗ has the same properties as the old f∗ on homotopy groups πi with i ≤
k − 1. Since the composion πk(Zk) → πk(Yk+1) → πk(X) is surjective according to the
induction assumptions, the homomorphism f∗ : πk(Yk+1) → πk(X) has to be surjective
as well.
Now we prove that it is injective. Let [ϕ] ∈ πk(Yk+1) and let fϕ ∼ 0. By cellular
approximation ϕ : Sk
→ Yk+1 is homotopic to ϕ : Sk
→ Y k
k+1 = Zk ⊆ Yk+1 and
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[fϕ] = 0 in πk(X). Hence [ϕ] ∈ Ker f∗ is a sum of [ϕα], and consequenly, it is zero in
πk(Yk+1).
Next, let maps ψα : Sk+1
α → X represent generators of πk+1(X). Put
Zk+1 = Yk+1 ∨
α
Sk+1
α
and define f = ψα on new (k + 1)-cells. It is clear that f∗ : πk+1(Zk+1) → πk+1(X)
is a surjection. Using cellular approximation it can be shown that πi(Zk+1, Yk+1) =
0 for i ≤ k. From the long exact sequence of the pair (Zk+1, Yk+1) we get that
πi(Yk+1) = πi(Zk+1) for i ≤ k − 1. Consequently, f∗ : πi(Zk+1) → πi(X) is an
isomorphism for n < i ≤ k − 1 and a monomorphism for i = n. The same long exact
sequence implies that πk(Yk+1) → πk(Zk+1) is surjective. We have already proved that
f∗ : πk(Yk+1) → πk(X) is an isomorphism. From the diagram
πk(Yk+1)
iso &&MMMMMMMMMM
epi
// πk(Zk+1)
f∗

πk(X)
we can see that f∗ : πk(Zk+1) → πk(X) is also an isomorphism.
Corollary. If (X, A) is an n-connected pair of CW-complexes, then there is a pair
(Z, A) homotopy equivalent to (X, A) rel A such that the cells in Z −A have dimension
greater than n.
Proof. Let f : (Z, A) → (X, A) be an n-connected model for (X, A) obtained by
attaching cells of dimension > n to A. Then f∗ : πj(Z) → πj(X) is a monomorphism
for j = n and an isomorphism for j > n. We will show that f∗ is an isomorphism also
for j ≤ n. Consider the diagram:
A
iZ

iX
@@@@@@@@
Z
f
// X
The inclusions iX and iZ are n-equivalences. Consequently, f∗iZ∗ = iX∗ : πj(A) →
πj(X) is an epimorphism for j = n. Hence so is f∗. Next, iX∗ and iZ∗ are isomorphisms
for j < n, hence so is f∗.
Finally, according to Whitehead Theorem, the weak homotopy equivalence f between
two CW-complexes is a homotopy equivalence.
Theorem B. Let f : (Z, A) → (X, A) and f : (Z , A ) → (X , Z ) be two n-connected
CW-models. Given a map g : (X, A) → (X , A ) there is a map h : (Z, A) → (Z , A )
such that the following diagram commutes up to homotopy rel A:
Z
f
//
h

X
g

Z
f
// X
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The map h is unique up to homotopy rel A.
Proof. By the previous corollary we can suppose that Z −A has only cells of dimension
≥ n + 1. We can define h/A as g/A.
A
h/A
//

Z
f

Z
gf
// X
Replace X by the mapping cylinder Mf which is homotopy equivalent to X . Since
f : Z → X is an n-connected model, from the long exact sequence of the pair
(Mf , Z ) we get that πi(Mf , Z ) = 0 for i ≥ n + 1. According to compression lemma
12.2 there exists h : Z → Z such that the diagram
A
h/A
//

Z

Z //
h
>>|
|
|
|
|
Mf
commutes up to homotopy rel A. This h has required properties. The proof that it is
unique up to homotopy follows the same lines.
CZ.1.07/2.2.00/28.0041
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