Simplicial Structures in Topology

162 IV Cohomology
(IV.2.2) Corollary. The graded Abelian group Z ∗ (K;Z) is a graded ring with identity;
the graded Abelian group B ∗ (K;Z) is one of its (bilateral) ideals.
Proof. It is enough to prove that the cup product has the following properties:
∪: Z p (K;Z) × Z q (K;Z) −→ Z p+q (K;Z) ,
∪: Z p (K;Z) × B q (K;Z) −→ B p+q (K;Z) ,
∪: B p (K;Z) × Z q (K;Z) −→ B p+q (K;Z) .
We prove only the last one. Let c p = ∂ p−1 (c p−1 ) ∈ B p (K;Z) and c q ∈ Z q (K;Z) be
given arbitrarily. Then,
The quotient ring
c p ∪ c q = ∂ p+q−1 (c p−1 ∪ c q ) . �
H ∗ (K;Z)=Z ∗ (K;Z)/B ∗ (K;Z)
is called cohomology ring of the (finite and oriented) simplicial complex K with
coefficients in Z. This ring is skew-commutative; in fact, we have the following
result:
(IV.2.3) Theorem. For every x ∈ H p (K;Z) and y ∈ H q (K;Z),
Proof. Let us suppose that
x ∪ y =(−1) pq y ∪ x .
x = c p + B p (K;Z) and y = c q + B q (K;Z) ;
we wish to prove that, for every generator {x0,...,xp+q} of Cp+q(K),
c p ∪ c q ({x0,...,xp+q})=(−1) pq c q ∪ c p ({x0,...,xp+q}) .
Since we are working with oriented simplexes, according to our rules we have, for
every ℓ-simplex {x0,x1,...,xℓ},
and so
{x0,x1,...,xℓ} =(−1) 1 2 ℓ(ℓ−1) {xℓ,xℓ−1,...,x0}
c p ∪ c q ({x0,x1,...,xp+q} =
= c p ∪ c q ((−1) 1 2 (p+q)(p+q−1) ({xp+q,...,x0})
=(−1) 1 2 (p+q)(p+q−1) c p ({xp+q,...,xp}) × c q ({xp,...,x0})
=(−1) 1 2 (p+q)(p+q−1) c q ({xp,...,x0}) × c p ({xp+q,...,xp})