Simplicial Structures in Topology

II.3 Homological Algebra 65
II.3 Introduction to Homological Algebra
In the previous section, we have seen that we can associate a graded Abelian
group C(K) ={Cn(K)} with any simplicial complex K and a homomorphism
∂n : Cn(K) → Cn−1(K), such that ∂n−1∂n = 0, to each integer n; these homomorphisms
define a graded Abelian group H∗(K;Z) ={Hn(K;Z)}. All this can be
viewed in the framework of a more general and more useful context.
A chain complex (C,∂) is a graded Abelian group C = {Cn} together with
an endomorphism ∂ = {∂n} of degree −1, called boundary homomorphism3 ∂ =
{∂n : Cn → Cn−1}, such that ∂ 2 = 0; this means that, for every n ∈ Z, ∂n∂n+1 = 0.
Hence
Bn = im∂n+1 ⊂ Zn = ker∂n
and so we can define the graded Abelian group
H∗(C)={Hn(C)=Zn/Bn | n ∈ Z};
this is the homology of C.
A chain homomorphism between two chain complexes (C,∂) and (C ′ ,∂ ′ ) is a
graded group homomorphism f = { fn : Cn → C ′ n} of degree 0 commuting with the
boundary homomorphism, that is to say, for every n ∈ Z, fn−1∂n = ∂ ′ n fn.
Chain complexes and chain homomorphisms form a category C, thecategory of
chain complexes.
It is costumary to visualize chain complexes as diagrams
···
��
Cn+1
∂n+1 ��
Cn
and their morphisms as commutative diagrams
···
···
��
Cn+1
��
��
′
C n+1
fn+1
∂n+1 ��
Cn
��
∂ ′ n+1 ��
′
C n
fn
∂n ��
Cn−1
∂n ��
Cn−1
��
∂ ′ n ��
′
C n−1
fn−1
��
···
��
···
��
···
The previous definitions are clearly inspired by what we did to define the homology
groups of a simplicial complex; indeed, we emphasize the fact that, for every
simplicial complex X, the graded Abelian group {Cn(K)|n ∈ Z} together with its
boundary homomorphism ∂ K = {∂ K n |n ∈ Z} is a chain complex (C(K),∂ K ). The
chain complex C(K) is said to be positive because its terms of negative index are 0.
In particular, for every simplicial function f : K → M, the homomorphism
is a chain homomorphism.
C( f ): C(K) → C(M)
3 In some textbooks, it is called differential operator.