Simplicial Structures in Topology

94 II Simplicial Complexes
and consider the following free chain complexes with augmentation to the Abelian
group H (viewed as a Z-module): 5
1. (C,∂), with C1 = B, C0 = Z, ∂1 = i, ε = q, Ci = 0foralli�= 0,1and∂i = 0for
all i ≥ 2;
2. (C ′ ,∂ ′ ), with C ′ 1 = R, C′ 0 = F, ∂ ′ 1 = j, ε′ = q ′ , C ′ i = 0foralli�= 0,1and∂ ′
i = 0
for all i ≥ 2.
These chain complexes are free and acyclic; by Theorem (II.3.6), we obtain chain
morphisms f : C → C ′ and g: C ′ → C whose composites fg and gf are chain homotopic
to the respective identities. The tensor product with G is a functor that
preserves compositions of morphisms; therefore, their tensor products by G produce
the chain morphisms
and besides,
f ⊗ 1G : C ⊗ G → C ′ ⊗ G
g ⊗ 1G : C ′ ⊗ G → C ⊗ G
( f ⊗ 1G)(g ⊗ 1G) and (g ⊗ 1G)( f ⊗ 1G)
are still chain homotopic to their respective identities. This means that the induced
morphisms in homology
��
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H1( f ⊗ 1G)
ker(i ⊗ 1G) ker( j ⊗ 1G)
H1(g ⊗ 1G)
are the inverse of each other and likewise for
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H0( f ⊗ 1G)
coker(i ⊗ 1G) coker( j ⊗ 1G)
H0(g ⊗ 1G)
This implies that neither ker(i ⊗ 1G) nor coker(i ⊗ 1G) depends on the chosen presentation
of H.
Hence, by following the argument in Lemma (II.5.3),
coker(i ⊗ 1G) ∼ = H ⊗ G
regardless of which free presentation of H we take.
5 Chain complexes can be constructed over Λ-modules, with Λ a commutative ring with unit
element.