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