Simplicial Structures in Topology

IV.1 Cohomology with Coefficients in G 153
we conclude that the sequence
0
��
Hom(Z,G)
��
Hom(Z,G)
�2 ��
Hom(Z2,G)
is exact. However, Hom(Z,G) ∼ = G and since �2 is precisely the multiplication by 2
in G, we conclude from the last exact sequence that Hom(Z2,G) ∼ = ker(2: G → G).
In particular, Hom(Z2,Z)=0.
As in the case for changing the coefficients in homology, where we have applied
the functor −⊗G to a chain complex (C,∂ ) – in particular, the complex (C(K),∂) –
we can apply the functor Hom(−,G) to (C,∂).
We now construct the cohomology (with coefficients in an Abelian group G)ofa
chain complex (C, ∂). Theimageof(C,∂) by the functor Hom(−,G) is the graded
Abelian group
Hom(C,G)={C n (C,G)} := {Hom(Cn,G)}.
We denote the adjoint homomorphism of the boundary homomorphism
with ∂ n−1 ,thatistosay,
∂n : Cn → Cn−1
∂ n−1 := Hom(∂n,G): C n−1 (C,G) → C n (C,G) .
Since ∂n∂n+1 = 0, we immediately conclude that ∂ n ∂ n−1 = 0andsothat
The quotient group
im∂ n−1 := B n (C,G) ⊂ ker∂ n := Z n (C,G) .
H n (C;G)=Z n (C,G)/B n (C,G)
is the nth-cohomology group of the chain complex (C,∂) with coefficients in G.
In particular, if (C,∂) =(C(K),∂ ) is the positive free chain complex associated
with the oriented chain simplex K =(X,Φ), the graded Abelian group H∗ (K;Z)
is the simplicial cohomology of K with integral coefficients. In this case, since the
Abelian group Cn(K) is generated by the oriented n-simplexes σ of K,wedefinethe
homomorphisms
�
1 , τ = σ
cσ : Cn(K) → Z , (∀τ ∈ Φ,dimτ = n) cσ (τ)=
0 , τ �= σ
Consequently,
C n (K)=Hom(Cn(K),Z)
is the free Abelian group generated by the homomorphisms cσ ,whereσ runs over
the set of all oriented n-simplexes of K.