Simplicial Structures in Topology

II.5 Homology with Coefficients 89
Cn(K) being the free Abelian groups of formal linear combinations with coefficients
in Z of the n-simplexes of K. We now wish to generalize our homology with coefficients
in the Abelian group Z to a homology with coefficients drawn from any
Abelian group G.
We begin by reviewing the construction of the tensor product of two Abelian
groups A, B: by definition, A ⊗ B is the Abelian group generated by the set of
elements
{a ⊗ b|a ∈ A , b ∈ B}
where (∀a,a ′ ∈ A , b,b ′ ∈ B)
1. (a + a ′ ) ⊗ b = a ⊗ b + a ′ ⊗ b ,
2. a ⊗ (b + b ′ )=a ⊗ b + a ⊗ b ′ .
We notice that the function
A ⊗ Z → A , a ⊗ n ↦→ na
is a group isomorphism, that is to say, A⊗Z ∼ = A (similarly, Z⊗A ∼ = A). The reader
may easily prove that
(A ⊕ B) ⊗C ∼ = (A ⊗C) ⊕ (B ⊗C)
for any three Abelian groups A, B,andC. Finally, given two group homomorphisms
φ : A → A ′ and ψ : B → B ′ , the function φ ⊗ ψ : A ⊗ B → A ′ ⊗ B ′ defined by φ ⊗
ψ(a ⊗ b)=φ(a) ⊗ ψ(b) is a homomorphism of Abelian groups.
In this way, by fixing an Abelian group G we are able to construct a covariant
functor
−⊗G: Ab → Ab
that transforms a group A into A ⊗ G and a morphism φ : A → B into the morphism
φ ⊗ 1G.
We extend this functor to chain complexes. We transform a given chain complex
(C,∂) ∈ C in (C ⊗ G,∂ ⊗ 1G), by setting
(C ⊗ G)n := Cn ⊗ G
for every n ∈ Z, and by defining the homomorphisms
Since
(∂ ⊗ 1G)n := ∂n ⊗ 1G : Cn ⊗ G → Cn−1 ⊗ G .
(∂ ⊗ 1G)n−1(∂ ⊗ 1G)n =(∂n−1 ⊗ 1G)(∂n ⊗ 1G)=∂n−1∂n ⊗ 1G = 0 ,
we conclude that (C ⊗ G,∂ ⊗ 1G) is a chain complex whose homology groups are
the homology groups of (C,∂ ) with coefficients in G. Thenth-homology group of
(C,∂) with coefficients in G is defined by the quotient group