Simplicial Structures in Topology

166 IV Cohomology
It is easily proved that the only 1-cocycles are ξ 1 1 = {0,4}∗ + {4,5} ∗ + {5,0} ∗
and ξ 1 2 = {0,6}∗ + {6,7} ∗ + {7,0} ∗ . These cocycles are indipendent and their cohomology
classes generate H 1 (S 2 ∨ (S 1 ∨ S 1 );Z); also easily proved is the fact that
ξ 1 1 ∪ ξ 1 2 = 0. The cohomology rings of T 2 and S 2 ∨ (S 1 ∨ S 1 ) are therefore different;
consequently, the polyhedra T 2 and S 2 ∨ (S 1 ∨ S 1 ) cannot be of the same homotopy
type.
IV.3 The Cap Product
In this section, we study a product between homology and cohomology classes.
More precisely, let K be a simplicial complex that we once again assume to be finite
and oriented. Consider the chain and the cochain groups associated with K
Cn(K;Z) and C n (K;Z)=Hom(Cn(K),Z) ,
where n is an integer such that 0 ≤ n ≤ dimK. Foreveryd ∈ C p (K;Z) and for every
(p + q)-simplex {x0,...,xp,...,xp+q} of K (that is to say, a generator of Cp+q(K)),
we define
d ∩{x0,...,xp,...,xp+q} := d({x0,...,xp}){xp,...,xp+q}∈Cq(K);
we linearly extend the definition d ∩c to every (p +q)-chain c ∈ Cp+q(K;Z);inthis
way we obtain a bilinear relation
∩: C p (K;Z) ×Cp+q(K;Z) −→ Cq(K;Z)
called cap product. Note that the cap product is not defined if q < 0orq > dimK − p.
In particular, if p = q, foreveryd ∈ C p (K;Z) and every p-simplex {x0,...,xp,},
we have
d ∩{x0,...,xp} = d({x0,...,xp}){xp}∈C0(K;Z).
Let ε : C0(K;Z) → Z be the augmentation homomorphism of the positive chain
complex C(K,Z); by definition,
and by linearity
ε(d ∩{x0,...,xp})=d({x0,...,xp})
ε(c ∩ d)=d(c)
for every c ∈ Cp(K;Z) and any d ∈ Cp (K;Z). The next two results establish a
relation between cap and cup products.
(IV.3.1) Theorem. For every c ∈ Cp+q+r(K;Z), d∈ Cp (K;Z), and e ∈ Cq (K;Z),
the equality
d ∩ (e ∩ c)=(d∪ e) ∩ c
holds.