Simplicial Structures in Topology

IV.1 Cohomology with Coefficients in G 155
for which the following sequence of cohomology groups
...→ H n (K;Z) Hn (i)
−→ H n (L;Z) ˜ λ n
−→ H n+1 (K,L;Z) q∗ (n+1)
−→ H n+1 (K;Z) → ...
is exact.
The proof of this theorem follows the steps taken in proving the corresponding
theorem in homology Theorem (II.3.1); we only wish to point out that the main
result needed for this proof is the following lemma:
(IV.1.4) Lemma. If
A ��
f
��
B
is a short exact sequence of free groups and G is an Abelian group, then also the
sequence
Hom(C,G) ��
�g
g
��
Hom(B,G)
��
��
C
�f
��
��
Hom(A,G)
is exact.
The reader may take the proof of Lemma (II.5.1) as a basis for proving this one;
actually, the only tensor product property used in Lemma (II.5.1) is that −⊗G is
a functor which takes sums of morphisms into sums of morphisms; this very same
property holds also for the (contravariant) functor Hom(−,G).
The cohomology determined by an oriented chain simplex K with coefficients in
an Abelian group G is defined in the same manner as the one with coefficients in
Z, except that we apply to the chain complex (C(K),∂) the contravariant functor
Hom(−,G) instead of Hom(−,Z).
We wish to determine the cohomology with coefficients in G of a simplicial complex,
based on its homology with coefficients in Z and the Abelian group G. We
seek to use the same previous ideas to our benefit, bearing in mind that the functor
A ↦→ A ⊗ G is covariant, whereas A ↦→ Hom(A,G) is contravariant. We begin by reviewing
what has been done so far in general terms. A cochain complex (or cocomplex)
is a graded Abelian group {Cn } together with an endomorphism of degree +1,
called coboundary homomorphism ∂ ∗ = {∂ n : Cn → Cn+1 ,},forwhich∂n+1∂n = 0.
A cochain homomorphism between two cochain complexes (C,∂ ∗ ) and (C ′ ,∂ ′∗ ) is
a homomorphism between the graded Abelian groups C and C ′ that commutes with
the coboundary homomorphisms.
Also the other concepts defined for chain complexes have their dual correspondents:
the elements of Cn are called n-cochains; the elements of
are n-cocycles, and those of
Z n (C) := ker(∂ n : C n → C n+1 )
B n (C) := im(∂ n−1 : C n−1 → C n )
are n-coboundaries; finally, the quotient H n (C) := Z n (C)/B n (C) is the cohomology
group of the cocomplex (C,∂ ∗ ) in dimension n.