2Singular (co)homology
III Algebraic Topology
2.1 Chain complexes
This course is called algebraic topology. We’ve already talked about some
topology, so let’s do some algebra. We will just write down a bunch of definitions,
which we will get to use in the next chapter to define something useful.
Definition
(Chain complex)
.
A chain complex is a sequence of abelian groups
and homomorphisms
· · · C
3
C
2
C
1
C
0
0
d
3
d
2
d
1
d
0
such that
d
i
◦ d
i+1
= 0
for all i.
Very related to the notion of a chain complex is a cochain complex, which is
the same thing with the maps the other way.
Definition
(Cochain complex)
.
A cochain complex is a sequence of abelian
groups and homomorphisms
0 C
0
C
1
C
2
C
3
· · ·
d
0
d
1
d
2
such that
d
i+1
◦ d
i
= 0
for all i.
Note that each of the maps d is indexed by the number of its domain.
Definition (Differentials). The maps d
i
and d
i
are known as differentials.
Eventually, we will get lazy and just write all the differentials as d.
Given a chain complex, the only thing we know is that the composition of
any two maps is zero. In other words, we have
im d
i+1
⊆ ker d
i
. We can then
ask how good this containment is. Is it that
im d
i+1
=
ker d
i
, or perhaps that
ker d
i
is huge but
im d
i+1
is trivial? The homology or cohomology measures
what happens.
Definition (Homology). The homology of a chain complex C
·
is
H
i
(C
·
) =
ker(d
i
: C
i
→ C
i−1
)
im(d
i+1
: C
i+1
→ C
i
)
.
An element of H
i
(C
·
) is known as a homology class.
Dually, we have
Definition (Cohomology). The cohomology of a cochain complex C
·
is
H
i
(C
·
) =
ker(d
i
: C
i
→ C
i+1
)
im(d
i−1
: C
i−1
→ C
i
)
.
An element of H
i
(C
·
) is known as a cohomology class.
More names:
Definition
(Cycles and cocycles)
.
The elements of
ker d
i
are the cycles, and
the elements of ker d
i
are the cocycles.
Definition
(Boundaries and coboundaries)
.
The elements of
im d
i
are the
boundaries, and the elements of im d
i
are the coboundaries.
As always, we want to talk about maps between chain complexes. These are
known as chain maps.
Definition
(Chain map)
.
If (
C
·
, d
C
·
) and (
D
·
, d
D
·
) are chain complexes, then a
chain map
C
·
→ D
·
is a collection of homomorphisms
f
n
:
C
n
→ D
n
such that
d
D
n
◦ f
n
=
f
n−1
◦ d
C
n
. In other words, the following diagram has to commute for
all n:
C
n
D
n
C
n−1
D
n−1
f
n
d
C
n
d
D
n
f
n−1
There is an obvious analogous definition for cochain maps between cochain
complexes.
Lemma.
If
f
·
:
C
·
→ D
·
is a chain map, then
f
∗
:
H
n
(
C
·
)
→ H
n
(
D
·
) given
by [
x
]
7→
[
f
n
(
x
)] is a well-defined homomorphism, where
x ∈ C
n
is any element
representing the homology class [x] ∈ H
n
(C
·
).
Proof.
Before we check if it is well-defined, we first need to check if it is defined
at all! In other words, we need to check if
f
n
(
x
) is a cycle. Suppose
x ∈ C
n
is a
cycle, i.e. d
C
n
(x) = 0. Then we have
d
D
n
(f
n
(x)) = f
n−1
(d
C
n
(x)) = f
n−1
(0) = 0.
So f
n
(x) is a cycle, and it does represent a homology class.
To check this is well-defined, if [
x
] = [
y
]
∈ H
n
(
C
·
), then
x − y
=
d
C
n+1
(
z
) for
some
z ∈ C
n+1
. So
f
n
(
x
)
− f
n
(
y
) =
f
n
(
d
C
n+1
(
z
)) =
d
D
n+1
(
f
n+1
(
z
)) is a boundary.
So we have [f
n
(x)] = [f
n
(y)] ∈ H
n
(D
·
).