Simplicial Structures in Topology

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