Simplicial Structures in Topology

158 IV Cohomology
When we apply this result to these exact sequences, we get the Universal Coefficients
Theorem in Cohomology:
(IV.1.7) Theorem. The cohomology of a free positive complex (C,∂) with coefficients
in an Abelian group G is determined by the following short exact sequences:
Ext(Hn−1(C),G) ��
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H n (C;G)
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Hom(Hn(C),G) .
We note that for an oriented simplicial complex K, we apply the functor
Hom(−,G) to the positive free chain complex (C(K),∂) to obtain, for every n ≥ 0,
the following short exact sequences:
Ext(Hn−1(K;Z),G) ��
��
H n (K;G)
IV.1.1 Cohomology of Polyhedra
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Hom(Hn(K;Z),G) .
We now wish to study the cohomology (with coefficients in Z, to make it simple) as
a contravariant functor from the category P of polyhedra to the category Ab Z ,
H ∗ (−;Z): P −→ Ab Z .
The definition of the functor on the objects is obvious:
(∀|K|∈P) H ∗ (|K|;Z)=H ∗ (K;Z).
As for the morphisms, let f : |K| →|L| be a continuous function; by the Simplicial
Approximation Theorem, there exists a simplicial function g: K (r) → L that
approximates f simplicially. It produces a homomorphism
H n (g;Z): H n (L;Z) → H n (K (r) ;Z)
for every n ∈ Z. All results in Sect. III.2, needed to prove that a map f : |K|→|L|
defines a homomorphism H∗( f ;Z) between the homology of the polyhedron |K|
and the homology of |L|, hold in cohomology; in particular, we point out that if a
chain complex C is acyclic, also the chain complex Hom(C,Z) is acyclic (use the
Universal Coefficients Theorem in Cohomology). In addition, if we return to the
proof of Theorem (III.2.2), we see that, for any projection π r : K (r) → K, wemay
find a homomorphism of chain complexes
ℵ r : C(K) → C(K (r) )
such that ℵ r C(π r ) is chain homotopic to 1 C(K (r), andC(π r )ℵ r is chain homotopic
to 1 C(K). This means that, homologically speaking, the homomorphisms induced by
ℵ r and C(π r ) are isomorphisms and the inverse of each other. We now consider the
cochain complexes