Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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154 IV Cohomology<br />
(IV.1.2) Remark. It is easily proved that, for every cn+1 ∈ Cn+1(K) and for every<br />
c n ∈ C n (K),<br />
(∂ n (c n ))(cn+1)=c n ∂n+1(cn+1) ;<br />
<strong>in</strong> particular, if c n = ∑<br />
σ∈Φ<br />
dimσ=n<br />
nσ cσ ,wehave<br />
∂ n (c n )= ∑<br />
σ∈Φ<br />
dimσ=n<br />
nσ ∂ n (cσ )<br />
and therefore, for every oriented (n + 1)-simplex τ of K,<br />
We now prove that<br />
∂ n (c n )(τ)= ∑<br />
σ∈Φ<br />
dimσ=n<br />
nσ cσ(∂n+1(τ)) .<br />
H ∗ (−;Z): Csim −→ Ab Z<br />
is a contravariant functor. Let K, L be two simplicial complexes and f : K → L be<br />
a simplicial function. We know that, for each n ≥ 0, f def<strong>in</strong>es a homomorphism<br />
Cn( f ): Cn(K) → Cn(L); therefore, for every n ≥ 0, we have a homomorphism of<br />
Abelian groups<br />
C n ( f ): C n (L) −→ C n (K)<br />
such that C n ( f )(c n )=c n Cn( f ) for every c n ∈ C n (L). It is easily proved that, for<br />
every n ≥ 0,<br />
C n+1 ( f )∂ n L = ∂ n K Cn ( f ) ;<br />
consequently, C n ( f ) <strong>in</strong>duces a homomorphism of Abelian groups<br />
H n ( f ): H n (L;Z) → H n (K;Z).<br />
Here, the situation is completely analogous to the one for homology.<br />
As <strong>in</strong> the case of simplicial homology, we def<strong>in</strong>e the cohomology H ∗ (K,L;Z)<br />
of a pair of simplicial complexes (K,L): for that, it is enough to apply the functor<br />
Hom(−,Z) to the cha<strong>in</strong> complex<br />
(C(K,L),∂ K,L ) := {Cn(K)/Cn(L),∂ K,L<br />
n } .<br />
What is more, (K,L) produces a long exact sequence <strong>in</strong> simplicial cohomology,<br />
that is to say, the follow<strong>in</strong>g Cohomology Long Exact Sequence Theorem<br />
holds:<br />
(IV.1.3) Theorem. Let (K,L) be a pair of simplicial complexes. For every n > 0,<br />
there exists a homomorphism<br />
˜λ n : H n (L;Z) → H n+1 (K,L;Z)