Simplicial Structures in Topology

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