INTRODUCTION TO ALGEBRAIC TOPOLOGY
MARTIN ˇCADEK
4. Homology of CW-complexes and applications
4.1. First applications of homology. Using homology groups we can easily prove
the following statements:
(1) Sn
is not a retract of Dn+1
.
(2) Every map f : Dn
→ Dn
has a fixed point, i.e. there is x ∈ Dn
such that
f(x) = x.
(3) If ∅ = U ⊆ Rn
and ∅ = V ⊆ Rm
are open homeomorphic sets, then n = m.
Outline of the proof. (1) Suppose that there is a retraction r : Dn+1
→ Sn
. Then we
get the commutative diagram
Z = Hn(Sn
)
id //
i∗ ((QQQQQQQQQQQQ
Hn(Sn
) = Z
Hn(Dn+1
) = 0
r∗
66mmmmmmmmmmmm
which is a contradiction.
(2) Suppose that f : Dn
→ Dn
has no fixed point. Then we can define the map
g : Dn
→ Sn−1
where g(x) is the intersection of the ray from f(x) to x with Sn−1
.
However, this map would be a retraction, a contradiction with (1).
(3) The proof of the last statement follows from the isomorphisms:
Hi(U, U−{x}) ∼= Hi(Rn
, Rn
−{x}) ∼= ˜Hi−1(Rn
−{x}) ∼= ˜Hi−1(Sn−1
) =
Z for i = n,
0 for i = n.
4.2. Degree of a map. Consider a map f : Sn
→ Sn
. In homology f∗ : ˜Hn(Sn
) →
Hn(Sn
) has the form
f∗(x) = ax, a ∈ Z.
The integer a is called the degree of f and denoted by deg f.
The degree has the following properties:
(1) deg id = 1.
(2) If f ∼ g, then deg f = deg g.
(3) If f is not surjective, then deg f = 0.
(4) deg(fg) = deg f · deg g.
(5) Let f : Sn
→ Sn
, f(x0, x1, . . . , xn) = (−x0, x1, . . . , xn). Then deg f = −1.
1
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(6) The antipodal map f : Sn
→ Sn
, f(x) = −x has deg f = (−1)n+1
.
(7) If f : Sn
→ Sn
has no fixed point, then deg f = (−1)n+1
.
Proof. We outline only the proof of (5) and (7). The rest is not difficult and left as an
exercise.
We show (5) by induction on n. The generator of ˜H0(S0
) is 1 − (−1) and f∗ maps
it in (−1) − 1. Hence the degree is −1. Suppose that the statement is true for n. To
prove it for n+1 we use the diagram with rows coming from a suitable Mayer-Vietoris
exact sequence
0 // ˜Hn+1(Sn+1
)
∼= //
f∗

˜Hn(Sn
) //
(f/Sn)∗

0
0 // ˜Hn+1(Sn+1
)
∼= // ˜Hn(Sn
) // 0
If (f/Sn
)∗ is a multiplication by −1, so is f∗.
To prove (7) we show that f is homotopic to the antipodal map through the homo-
topy
H(x, t) =
tf(x) − (1 − t)x
tf(x) − (1 − t)x
.
Corollary. Sn
has a nonzero continuous vector field if and only if n is odd.
Proof. Let Sn
has such a field v(x). We can suppose v(x) = 1. Then the identity is
homotopic to antipodal map through the homotopy
H(x, t) = cos tπ · x + sin tπ · v(x).
Hence according to properties (2) and (6)
(−1)n+1
= deg(− id) = deg(id) = 1.
Consequently, n is odd.
On the contrary, if n = 2k+1, we can define the required vector field by prescription
v(x0, x1, x2, x3, . . . , x2k, x2k+1) = (−x1, x0, −x3, x2, . . . , −x2k+1, x2k).
Exercise. Prove the properties (3), (4) and (6) of the degree.
4.3. Local degree. Consider a map f : Sn
→ Sn
and y ∈ Sn
such that f−1
(y) =
{x1, x2, . . . , xm}. Let Ui be open disjoint neighbourhoods of points xi and V a neighbourhood
of y such that f(Ui) ⊆ V . Then
(f/Ui)∗ : Hn(Ui, Ui − {xi}) ∼= Hn(Sn
, Sn
− {xi}) = Z
−→ Hn(V, V − {y}) ∼= Hn(Sn
, Sn
− {y}) = Z
is a multiplication by an integer which is called a local degree and denoted by deg f|xi.
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Theorem A. Let f : Sn
→ Sn
, y ∈ Sn
and f−1
(y) = {x1, x2, . . . , xm}. Then
deg f =
m
i=1
deg f|xi.
For the proof see [Hatcher], Proposition 2.30, page 136.
The suspension Sf of a map f : X → Y is given by the prescription Sf(x, t) =
(f(x), t).
Theorem B. deg Sf = deg f for any map f : Sn
→ Sn
.
Proof. f induces Cf : CSn
→ CSn
. The long exact sequence for the pair (CSn
, Sn
)
and the fact that SSn
= CSn
/Sn
give rise to the diagram
˜Hn+1(Sn+1
) ∼=
//
Sf∗

˜Hn+1(CSn
, Sn
)
∂∗
∼=
//
Cf∗

˜Hn(Sn
)
f∗

˜Hn+1(Sn+1
) ∼=
// ˜Hn+1(CSn
, Sn
)
∂∗
∼=
// ˜Hn(Sn
)
which implies the statement.
Corollary. For any n ≥ 1 and given k ∈ Z there is a map f : Sn
→ Sn
such that
deg f = k.
Proof. For n = 1 put f(z) = zk
where z ∈ S1
⊂ C. Using the computation based on
local degree as above, we get deg f = k. The previous theorem implies that the degree
of Sn−1
f : Sn
→ Sn
is also k.
4.4. Computations of homology of CW-complexes. If we know a CW-structure
of a space X, we can compute its cohomology relatively easily. Consider the sequence
of Abelian groups and its morphisms
(Hn(Xn
, Xn−1
), dn)
where dn is the composition
Hn(Xn
, Xn−1
)
∂n
−−→ Hn(Xn−1
)
jn−1
−−−→ Hn−1(Xn−1
, Xn−2
).
Theorem. Let X be a CW-complex. (Hn(Xn
, Xn−1
), dn) is a chain complex with
homology
HCW
n (X) ∼= Hn(X).
Proof. First, we show how the groups Hk(Xn
, Xn−1
) look like. Put X−1
= ∅ and
X0
/∅ = X0
{∗}. Then
Hk(Xn
, Xn−1
) = ˜Hk(Xn
/Xn−1
) = ˜Hk( Sn
α) = α Z n = k,
0 n = k.
Now we show that
Hk(Xn
) = 0 for k > n.
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From the long exact sequence of the pair (Xn
, Xn−1
) we get Hk(Xn
) = Hk(Xn−1
). By
induction Hk
(Xn
) = Hk(X−1
) = 0.
Next we prove that
Hk(Xn
) = Hk(X) for k ≤ n − 1.
From the long exact sequence for the pair (Xn+1
, Xn
) we obtain Hk(Xn
) = Hk(Xn+1
).
By induction Hk(Xn
) = Hk(Xn+m
) for every m ≥ 1. Since the image of each singular
chain lies in some Xn+m
we get Hk(Xn
) = Hk(X).
To prove Theorem we will need the following diagram with parts of exact sequences
for the pairs (Xn+1
, Xn
), (Xn
, Xn−1
) and (Xn−1
, Xn−2
).
0
0
&&MMMMMMMMMMMMM Hn(Xn+1
)
OO
Hn(Xn
)
OO
jn
))RRRRRRRRRRRRRR
Hn+1(Xn+1
, Xn
)
∂n+1
OO
dn+1
// Hn(Xn
, Xn−1
)
dn
//
∂n
))SSSSSSSSSSSSSS
Hn−1(Xn−1
, Xn−2
)
Hn−1(Xn−1
)
jn−1
OO
0
OO
From it we get
dndn+1 = jn−1(∂njn)∂n+1 = jn−1(0)∂n+1 = 0.
Further,
Ker dn = Ker ∂n = Im jn
∼= Hn(Xn
)
and
Im dn+1
∼= Im ∂n+1,
since jn−1 and jn are monomorphisms. Finally,
HCW
n (X) =
Ker dn
Im dn+1
∼=
Hn(Xn
)
Im ∂n+1
∼= Hn(Xn+1
) ∼= Hn(X).
Example. Hn(X) = 0 for CW-complexes without cells in dimension n.
Hk(CPn
) =
Z for k ≤ 2n even,
0 in other cases.
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4.5. Computation of dn. Let en
α and en−1
β be cells in dimension n and n − 1 of a
CW-complex X, respectively. Since
Hn(Xn
, Xn−1
) =
α
Z, Hn−1(Xn−1
, Xn−2
) =
β
Z,
they can be considered as generators of these groups. Let ϕα : ∂Dn
α → Xn−1
be the
attaching map for the cell en
α. Then
dn(en
α) =
β
dαβen−1
β
where dαβ is the degree of the following composition
Sn−1
= ∂Dn
α
ϕα
−→ Xn−1
→ Xn−1
/Xn−2
→ Xn
/(Xn−2
∪
γ=β
en−1
γ ) = Sn−1
.
For the proof we refer to [Hatcher], pages 140 and 141.
Exercise. Compute homology groups of various 2-dimensional surfaces (torus, Klein
bottle, projective plane) using their CW-structure with only one cell in dimension 2.
4.6. Homology of real projective spaces. The real projective space RPn
is formed
by cell e0
, e1
, . . . , en
, one in each dimension from 0 to n. The attaching map for the
cell ek+1
is the projection ϕ : Sk
→ RPk
. So we have to compute the degree of the
composition
f : Sk ϕ
−→ RPk
→ RPk
/RPk−1
= Sk
.
Every point in Sk
has two preimages x1, x2. In a neihbourhood Ui of xi f is a
homeomorphism, hence its local degree deg f|xi = ±1. Since f/U2 is the composition
of the antipodal map with f/U1, the local degrees deg f|x1 and deg f|x1 differs by the
multiple of (−1)k+1
. (See the properties (4) and (6) in 4.2.) According to 4.3
deg f = ±1(1 + (−1)k+1
) =
0 for k + 1 odd,
±2 for k + 1 even.
So we have obtained the chain complex for computation of HCW
∗ (RPn
). The result is
Hk(RPn
) =



Z for k = 0 and k = n odd,
Z2 for k odd , 0 < k < n,
0 in other cases.
4.7. Euler characteristic. Let X be a finite CW-complex. The Euler characteristic
of X is the number
χ(X) =
∞
i=0
(−1)k
rank Hk(X).
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Theorem. Let X be a finite CW-complex with ck cells in dimension k. Then
χ(X) =
∞
k=0
(−1)k
ck.
Proof. Realize that ck = rank Hk(Xk
, Xk−1
) = rank Ker dk + rank Im dk+1 and that
rank Hk(X) = rank Ker dk − rank Im dk+1. Hence
χ(X) =
∞
k=0
(−1)k
rank Hk(X) =
∞
k=0
(−1)k
(rank Ker dk − rank Im dk+1)
=
∞
k=0
(−1)k
rank Ker dk +
∞
k=0
(−1)k
rank Im dk =
∞
k=0
(−1)k
ck.
Example. 2-dimensional oriented surface of genus g (the number of handles attached
to the 2-sphere) has the Euler characteristic χ(Mg) = 2 − 2g.
2-dimensional nonorientable surface of genus g (the number of M¨obius bands which
replace discs cut out from the 2-sphere) has the Euler characteristic χ(Ng) = 2 − g.
4.8. Lefschetz Fixed Point Theorem. Let G be a finitely generated Abelian group
and h : G → G a homomorphism. The trace tr h is the trace of the homomorphism
Zn ∼= G/ Torsion G → G/ Torsion G ∼= Zn
induced by h.
Let X be a finite CW-complex. The Lefschetz number of a map f : X → X is
L(f) =
∞
i=0
(−1)i
tr Hif.
Notice that L(idX) = χ(X). Similarly as for the Euler characteristic we can prove
Lemma. Let fn : (Cn, dn) → (Cn, dn) be a chain homomorphism. Then
∞
i=0
(−1)i
tr Hif =
∞
i=0
(−1)i
tr fi
whenever the right hand side is defined.
Theorem (Lefschetz Fixed Point Theorem). If X is a finite simplicial complex or its
retract and f : X → X a map with L(f) = 0, then f has a fixed point.
For the proof see [Hatcher], Chapter 2C. Theorem has many consequences.
Corollary A (Brouwer Fixed Point Theorem). Every continuous map f : Dn
→ Dn
has a fixed point.
Proof. The Lefschetz number of f is 1.
In the same way we can prove
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Corollary B. If n is even, then every continuous map f : RPn
→ RPn
has a fixed
point.
Corollary C. Let M be a smooth compact manifold in Rn
with nonzero vector field.
Then χ(M) = 0.
The converse of this statement is also true.
Outline of the proof. If M has a nonzero vector field, there is a continuous map f :
M → M which is a ”small shift in the direction of the vector field”. Since such a map
has no fixed point, its Lefschetz number has to be zero. Moreover, f is homotopic to
identity and hence
χ(M) = L(idX) = L(f) = 0.
4.9. Homology with coefficients. Let G be an Abelian group. From the singular
chain complex (Cn(X), ∂n) of a space X we make the new chain complex
Cn(X; G) = Cn(X) ⊗ G, ∂G
n = ∂n ⊗ idG .
The homology groups of X with coefficients G are
Hn(X; G) = Hn(C∗(X; G), ∂G
∗ ).
The homology groups defined before are in fact the homology groups with coefficients
Z. The homology groups with coefficients G satisfy all the basic general properties as
the homology groups with integer coefficients with the exception that
Hn(; G) =
0 for n = 0,
G for n = 0.
If the coefficient group G is a field (for instance G = Q or Zp for p a prime), then
homology groups with coefficients G are vector spaces over this field. It often brings
advantages.
The computation of homology with coefficients G can be carried out again using a
CW-complex structure. For instance, we get
Hk(RPn
; Z2) =
Z2 for 0 ≤ k ≤ n,
0 in other cases.
For an application of Z2-coefficients see the proof of the following theorem in [Hatcher],
pages 174–176.
Theorem (Borsuk-Ulam Theorem). Every map f : Sn
→ Sn
satisfying
f(−x) = −f(x)
has an odd degree.
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