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