Simplicial Structures in Topology

II.5 Homology with Coefficients 91
and
(r ⊗ 1G)( f ⊗ 1G)=1A⊗G (g ⊗ 1G)(s ⊗ 1G)=1C⊗G
( f ⊗ 1G)(r ⊗ 1G)+(s ⊗ 1G)(g ⊗ 1G)=1B⊗G.
The first of these relations tells us that ( f ⊗1G) is injective; the second, that (g⊗1G)
is surjective, and the third, that im( f ⊗ 1G) =ker(g ⊗ 1G) because, if we take x ∈
B ⊗ G such that (g ⊗ 1G)(x)=0, then
x =(f ⊗ 1G)(r ⊗ 1G)(x)+(s ⊗ 1G)(g ⊗ 1G)(x)
=(f ⊗ 1G)((r ⊗ 1G)(x)) ∈ im( f ⊗ 1G).
Returning to our free chain complex (C,∂), we notice that for each integer n,the
sequence
Zn(C) ⊗ G ��
��
Cn ⊗ G ∂n ⊗ 1G�� ��
Bn−1(C) ⊗ G
is short exact. In addition, we observe that the graded Abelian groups Z(C) =
{Zn(C)|n ∈ Z} and B(C)={Bn(C)|n ∈ Z} may be viewed as chain complexes with
trivial boundary operator 0; we then construct the chain complexes
1. (Z(C) ⊗ G,0 ⊗ 1G);
2. (C ⊗ G,∂ ⊗ 1G);
3. ( � B(C) ⊗ G,0 ⊗ 1G), where � B(C) n = Bn−1(C)
and observe that in view of the preceding short exact sequence of Abelian groups,
we have a short exact sequence of chain complexes
(Z(C) ⊗ G,0 ⊗ 1G) ��
��
(C ⊗ G,∂ ⊗ 1G)
��
��
( B(C) � ⊗ G,0 ⊗ 1G).
By the Long Exact Sequence Theorem (II.3.1), we obtain the long exact sequence
of homology groups
···
Hn( � B(C) ⊗ G)
��
Hn(Z(C) ⊗ G)
��
Hn(C ⊗ G)
��
Hn−1(Z(C) ⊗ G)
��
···
in other words, by considering the format of the boundary operators, we have the
following exact sequence of Abelian groups:
Hn(C;G)
···
��
Bn(C) ⊗ G in ⊗ 1G ��
Zn(C) ⊗ G
hn ��
Bn−1(C) ⊗ G in−1 ⊗ 1G ��
Zn−1(C) ⊗ G
��
jn ��
��
···
Note that in is the inclusion of Bn(C) in Zn(C) and jn is the induced homomorphism
by the inclusion of Zn(C) in Cn; the reader is also asked to notice that the connecting
homomorphism λn+1 in Theorem (II.3.1) coincides with in ⊗ 1G.
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