INTRODUCTION TO ALGEBRAIC TOPOLOGY
MARTIN ˇCADEK
5. Singular cohomology
Cohomology forms a dual notion to homology. To every topological space we assign
a graded group H∗
(X) equipped with a ring structure given by a product ∪ : Hi
(X)×
Hj
(X) → Hi+j
(X). In this section we give basic definitions and properties of singular
cohomology groups which are very similar to those of homology groups.
5.1. Cochain complexes. A cochain complex (C, δ) is a sequence of Abelian groups
(or modules over a ring) and their homomorphisms indexed by integers
. . .
δn−2
−−−−→ Cn−1 δn−1
−−−−→ Cn δn
−−→ Cn+1 δn+1
−−−−→ . . .
such that
δn
δn−1
= 0.
δn
is called a coboundary operator. A cochain homomorphism of cochain complexes
(C, δC) and (D, δD) is a sequence of homomorphisms of Abelian groups (or modules
over a ring) fn
: Cn
→ Dn
which commute with the coboundary operators
δn
Dfn = fn+1
δn
C.
5.2. Cohomology of cochain complexes. The n-th cohomology group of a cochain
complex (C, δ) is the group
Hn
(C) =
Ker δn
Im δn−1
.
The elements of Ker δn
= Zn
are called cocycles of dimension n and the elements of
Im δn−1
= Bn
are called coboundaries (of dimension n). If a cochain complex is exact,
then its cohomology groups are trivial.
The component fn
of the cochain homomorphism f : (C, δC) → (D, δD) maps
cocycles into cocycles and coboundaries into coboundaries. It enables us to define
Hn
(f) : Hn
(C) → Hn
(D)
by the prescription Hn
(f)[c] = [fn
(c)] where [c] ∈ Hn
(C) and [fn
(c)] ∈ Hn
(D) are
classes represented by the elements c ∈ Zn
(C) and fn
(c) ∈ Zn
(D), respectively.
5.3. Long exact sequence in cohomology. A sequence of cochain homomorphisms
· · · → A
f
−−→ B
g
−−→ C → . . .
1
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is exact if for every n ∈ Z
· · · → An fn
−−→ Bn
gn
−−→ Cn
→ . . .
is an exact sequence of Abelian groups. Similarly as for homology groups we can prove
Theorem. Let 0 → A
i
−→ B
j
−−→ C → 0 be a short exact sequence of cochain complexes.
Then there is a so called connecting homomorphism δ∗
: Hn
(C) → Hn+1
(A)
such that the sequence
. . .
δ∗
−−→ Hn
(A)
Hn(i)
−−−→ Hn
(B)
Hn(j)
−−−→ Hn
(C)
δ∗
−−→ Hn+1
(A)
Hn+1(i)
−−−−→ . . .
is exact.
5.4. Cochain homotopy. Let f, g : C → D be two cochain homomorphisms. We
say that they are cochain homotopic if there are homomorphisms sn
: Cn
→ Dn−1
such that
δn−1
D sn
+ sn+1
δn
C = fn
− gn
for all n.
The relation to be cochain homotopic is an equivalence. The sequence of maps sn
is
called a cochain homotopy. Similarly as for homology we have
Theorem. If two cochain homomorphism f, g : C → D are cochain homotopic, then
Hn
(f) = Hn
(g).
5.5. Singular cohomology groups of a pair. Consider a pair of topological spaces
(X, A), an inclusion i : A → X and an Abelian group G. Let
C(X, A) = (Cn(X)/Cn(A), ∂n)
be the singular chain complex of the pair (X, A). The singular cochain complex
(C(X, A; G), δ) for the pair (X, A) is defined as
Cn
(X, A; G) = Hom (Cn(X, A), G) ∼= {h ∈ Hom(Cn(X), G); h|Cn(A) = 0}
= Ker i∗
: Hom(Cn(X), G) −→ Hom(Cn(A), G).
and
δn
(h) = h ◦ ∂n+1 for h ∈ Hom(Cn(X, A), G).
The n-th cohomology group of the pair (X, A) with coefficients in the group G is the
n-th cohomology group of this cochain complex
Hn
(X, A; G) = Hn
(C(X, A; G), δ).
We write Hn
(X; G) for Hn
(X, ∅; G). A map f : (X, A) → (Y, B) induces the cochain
homomorphism Cn
(f) : Cn
(Y ; G) → Cn
(X; G) by
Cn
(f)(h) = h ◦ Cn(f)
which restricts to a cochain homomorphism Cn
(Y, B; G) → Cn
(X, A; G) since f(A) ⊆
B. In cohomology it induces the homomorphism
f∗
= Hn
(f) : Hn
(Y, B) → Hn
(X, A).
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Moreover, Hn
(id(X,A)) = idHn(X,A;G) and Hn
(fg) = Hn
(g)Hn
(f). We can conclude
that Hn
is a contravariant functor (cofunctor) from the category Top2
into the category
AG of Abelian groups.
5.6. Long exact sequence for singular cohomology. Consider inclusions of spaces
i : A → X, i : B → Y and maps j : (X, ∅) → (X, A), j : (Y, ∅) → (Y, B) induced
by idX and idY , respectively. Let f : (X, A) → (Y, B) be a map. Then there are
connecting homomorphisms δ∗
X and δ∗
Y such that the following diagram
. . .
δ∗
X
// Hn
(X, A; G)
j∗
// Hn
(X; G)
i∗
// Hn
(A; G)
δ∗
X
// Hn+1
(X, A; G)
j∗
// . . .
. . .
δ∗
Y
// Hn
(X, B; G)
j ∗
//
f∗
OO
Hn
(Y ; G)
i ∗
//
f∗
OO
Hn
(B; G)
δ∗
Y
//
(f/B)∗
OO
Hn+1
(Y, B; G)
j ∗
//
f∗
OO
. . .
commutes and its horizontal sequences are exact.
The proof follows from Theorem 5.3 using the fact that
0 → Cn
(X, A; G)
Cn(j)
−−−→ Cn
(X; G)
Cn(i)
−−−→ Cn
(A; G) → 0
is a short exact sequence of cochain complexes as it follows directly from the definition
of Cn
(X, A; G).
Remark A. Consider the functor I : Top2
→ Top2
which assigns to every pair (X, A)
the pair (A, ∅). The commutativity of the last square in the diagram above means
that δ∗
is a natural transformation of contravariant functors Hn
◦ I and Hn+1
defined
on Top2
.
Remark B. It is useful to realize how δ∗
: Hn
(A; G) → Hn+1
(X, A; G) looks like.
Every element of Hn
(A; G) is represented by a cochain q ∈ Hom(Cn(A); G) with a zero
coboundary δq ∈ Hom(Cn+1(A); G). Extend q to Q ∈ Hom(Cn(X); G) in arbitrary
way. Then δQ ∈ Hom(Cn+1(X), G) restricted to Cn+1(A) is equal to δq = 0. Hence it
lies in Hom(Cn+1(X, A); G) and from the definition in 5.3 we have
δ∗
[q] = [δQ].
5.7. Homotopy invariance. If two maps f, g : (X, A) → (Y, B) are homotopic, then
they induce the same homomorphisms
f∗
= g∗
: Hn
(Y, B; G) → Hn(X, A; G).
Proof. We already know that the homotopy between f and g induces a chain homotopy
s∗ between C∗(f) and C∗(g). Then we can define a cochain homotopy between C∗
(f)
and C∗
(g) as
sn
(h) = h ◦ sn−1 for h ∈ Hom(Cn(Y ); G)
and use Theorem 5.4.
Corollary. If X and Y are homotopy equivalent spaces, then
Hn
(X) ∼= Hn
(Y ).
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5.8. Excision Theorem. Similarly as for singular homology groups there are two
equivalent versions of this theorem.
Theorem A (Excision Theorem, 1st version). Consider spaces C ⊆ A ⊆ X and
suppose that ¯C ⊆ int A. Then the inclusion
i : (X − C, A − C) → (X, A)
induces the isomorphism
i∗
: Hn
(X, A; G)
∼=
−→ Hn
(X − C, A − C; G).
Theorem B (Excision Theorem, 2nd version). Consider two subspaces A and B of a
space X. Suppose that X = int A ∪ int B. Then the inclusion
i : (B, A ∩ B) → (X, A)
induces the isomorphism
i∗
: Hn
(X, A; G)
∼=
−→ Hn
(B, A ∩ B; G).
The proof of Excision Theorem for singular cohomology follows from the proof of
the homology version.
5.9. Cohomology of finite disjoint union. Let X = k
α=1 Xα be a disjoint union.
Then
Hn
(X; G) =
k
α=1
Hn
(Xα).
The statement is not generally true for infinite unions.
5.10. Reduced cohomology groups. For every space X = ∅ we define the augmented
cochain complex ( ˜C∗
(X; G), ˜δ) as follows
˜Cn
(X; G) = Hom( ˜Cn(X); G)
with ˜δn
h = h◦ ˜∂n+1 for h ∈ Hom( ˜Cn(X); G). See 3.14. The reduced cohomology groups
˜Hn(X; G) with coefficients in G are the cohomology groups of the augmented cochain
complex. From the definition it is clear that
˜Hn
(X; G) = Hn
(X; G) for n = 0
and
˜Hn
(∗; G) = 0 for all n.
For pairs of spaces we define ˜Hn
(X, A; G) = Hn
(X, A; G) for all n. Then theorems
on long exact sequence, homotopy invariance and excision hold for reduced cohomology
groups as well.
Considering a space X with base point ∗ and applying the long exact sequence for
the pair (X, ∗), we get that for all n
˜Hn
(X; G) = ˜Hn
(X, ∗; G) = Hn
(X, ∗; G).
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Using this equality and the long exact sequence for unreduced cohomology we get that
H0
(X; G) ∼= H0
(X, ∗; G) ⊕ H0
(∗; G) ∼= ˜H0
(X) ⊕ G.
Analogously as for homology groups we have
Lemma. Let (X, A) be a pair of CW-complexes. Then
˜Hn
(X/A; G) = Hn
(X, A; G)
and we have the long exact sequence
· · · → ˜Hn
(X/A; G) → ˜Hn
(X; G) → ˜Hn
(A; G) → ˜Hn+1
(X/A; G) → . . .
5.11. The long exact sequence of a triple. Consider a triple (X, B, A), A ⊆
B ⊆ X. Denote i : (B, A) → (X, A) and j : (X, A) → (X, B) maps induced by the
inclusion B → X and idX, respectively. Analogously as for homology one can derive
the long exact sequence of the triple (X, B, A)
. . .
δ∗
−−→ Hn
(X, B; G)
j∗
−−→ Hn
(X, A; G)
i∗
−−→ Hn
(B, A; G)
δ∗
−−→ Hn+1
(X, B; G)
j∗
−−→ . . .
5.12. Singular cohomology groups of spheres. Considering the long exact sequence
of the triple (∆n
, δ∆n
, V = δ∆n
− ∆n−1
): we get that
Hi
(∆n
, ∂∆n
; G) =
G for i = n,
0 for i = n.
The pair (Dn
, Sn−1
) is homeomorphic to (∆n
, ∂∆n
). Hence it has the same cohomology
groups. Using the long exact sequence for this pair we obtain
˜Hi
(Sn
; G) = Hi+1
(Dn+1
, Sn
) =
0 for i = n,
G for i = n.
5.13. Mayer-Vietoris exact sequence. Denote inclusions A∩B → A, A∩B → B,
A → X, B → X by iA, iB, jA, jB, respectively. Let C → A, D → B and suppose
that X = int A ∪ int B, Y = int C ∪ int D. Then there is the long exact sequence
. . .
δ∗
−−→ Hn
(X, Y ; G)
(j∗
A,j∗
B)
−−−−−→ Hn
(A, C; G) ⊕ Hn
(B, D; G)
i∗
A−i∗
B
−−−−−→ Hn(A ∩ B, C ∩ D; G)
δ∗
−−→ Hn+1
(X, Y ; G) −→ . . .
Proof. The coverings U = {A, B} and V = {C, D} satisfy conditions of Lemma 3.12.
The sequence of chain complexes
0 −→ Cn(A ∩ B, C ∩ D)
i
−→ Cn(A, C) ⊕ Cn(B; D)
j
−→ CU,V
n (X, Y ) −→ 0
where i(x) = (x, x) and j(x, y) = x − y is exact. Applying Hom(−, G) we get a new
short exact sequence of cochain complexes
0 −→ Cn
U,V(X, Y ; G)
j∗
−→ Cn
(A, C; G) ⊕ Cn
(B, D; G)
i∗
−→ Cn
(A ∩ B, C ∩ D; G) −→ 0
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and it induces a long exact sequence. Using Lemma 3.12 we get that Hn
(CU,V(X, Y ; G)) =
Hn
(X, Y ; G), which completes the proof.
5.14. Computations of cohomology of CW-complexes. If we know a CWstructure
of a space X, we can compute its cohomology in the same way as homology.
Consider the chain complex from Section 4
(Hn(Xn
, Xn−1
), dn).
Theorem. Let X be a CW-complex. The n-th cohomology group of the cochain com-
plex
(Hom(Hn(Xn
, Xn−1
; G), dn
) dn
(h) = h ◦ dn
is isomorphic to the n-th singular cohomology group Hn
(X; G).
Exercise A. After reading the next section try to prove the theorem above using the
results and proofs from Section 4.
Exercise B. Compute singular cohomology of real and complex projective spaces
with coefficients Z and Z2.
CZ.1.07/2.2.00/28.0041
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