Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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168 IV Cohomology<br />
∩: B p (K;Z) × Zp+q(B;Z) −→ Bq(K;Z) ,<br />
∩: Z p (K;Z) × Bp+q(B;Z) −→ Bq(K;Z) .<br />
Proof. The first assertion follows directly from the theorem.<br />
Given d = ∂ p−1 (d ′ ) ∈ B p (K;Z) and c ∈ Zp+q(K;Z),<br />
and, therefore,<br />
∂q+1(d ′ ∩ c)=(−1) p (d ′ ∩ ∂p+q(c) − ∂ p−1 (d ′ ) ∩ c)=(−1) p (d ∩ c)<br />
d ∩ c =(−1) p ∂q+1(d ′ ∩ c) ∈ Bq(K;Z) .<br />
The last case is proved <strong>in</strong> a similar way. �<br />
By what we have showed, the cap product of p-cocha<strong>in</strong>s and (p + q)-cha<strong>in</strong>s<br />
becomes a bil<strong>in</strong>ear relation <strong>in</strong> (co)homology<br />
∩: H p (K;Z) × Hp+q(K;Z) −→ Hq(K;Z)<br />
which is also called cap product. The remarks on cohomology of polyhedra (see<br />
Sect. IV.1.1) allow us to def<strong>in</strong>e<br />
∩: H p (|K|;Z) × Hp+q(|K|;Z) −→ Hq(|K|;Z)<br />
for any polyhedron |K|∈P.<br />
We now consider the morphisms. Let a simplicial function f : K → L,anelement<br />
d of C p (L), andc ∈ Cp+q(K) be given. Then,<br />
(recall that<br />
Cp+q( f )(c) ∈ Cp+q(L) and C p ( f )(d) ∈ C p (K)<br />
C p ( f )=Hom( f ;Z): Hom(Cp(L);Z)=C p (L) → Hom(Cp(K);Z) ).<br />
Therefore, d ∩Cp+q( f )(c) ∈ Cq(L) and C p ( f )(d) ∩ c ∈ Cq(K).<br />
(IV.3.5) Theorem. Given a simplicial function f : K → L, d ∈ C p (L), and c ∈<br />
Cp+q(K),<br />
d ∩Cp+q( f )(c)=Cq( f )(C p ( f )(d) ∩ c) ,<br />
C q ( f ))(d ∩Cp+q( f )(c)) = C p ( f )(d) ∩ c .<br />
Proof. We only prove the first equality. Let us suppose that<br />
c = {x0,...,xp,...,xp+q}<br />
and that for any i �= j between 0 and p + q, f (xi) �= f (x j); it follows that<br />
Cp+q( f )(c)={ f (x0),..., f (xp),..., f (xp+q)} .