Simplicial Structures in Topology

IV.2 The Cohomology Ring 163
(for Z is commutative)
=(−1) 1 2 (p+q)(p+q−1) (−1) 1 2 q(q−1) (−1) 1 2 p(p−1)
= c q ({x0,...,xq}) × c p ({xq,...,xp+q})
=(−1) pq c q ({x0,...,xq}) × c p ({xq,...,xp+q}) .�
Hence, H ∗ (K;Z) is a graded skew-commutative ring with identity. We now prove
that H ∗ (−;Z) is a contravariant functor from the category of simplicial complexes
Csim to the category of graded commutative rings with identity.
(IV.2.4) Lemma. For any c p ∈ C p (L;Z),c q ∈ C q (L;Z), and any simplicial function
f : K → L, the equality
holds.
C p+q ( f )(c p ∪ c q )=C p ( f )(c p ) ∪C q ( f )(c q )
Proof. For each {x0,...,xp,...,xp+q}∈Cp+q(K),
C p+q ( f )(c p ∪ c q )({x0,...,xp,...,xp+q})
=(c p ∪ c q )Cp+q( f )({x0,...,xp,...,xp+q})
�
cp ({ f (x0),..., f (xp)}) × c
=
q ({ f (xp),..., f (xp+q)}) (∀i�= j) f (xi) �= f (x j)
0 otherwise.
On the other hand,
(C p ( f )(c p ) ∪C q ( f )(c q ))({x0,...,xp,...,xp+q})
=(c p Cp( f ) ∪ c q Cq( f ))({x0,...,xp,...,xp+q})
�
cp ({ f (x0),..., f (xp)}) × c
=
q ({ f (xp),..., f (xp+q)}) (∀i�= j) f (xi) �= f (x j)
0 otherwise.
�
Consequently, C ∗ ( f ): C ∗ (L;Z) → C ∗ (K;Z) preserves cup products; since the
homomorphisms C p ( f ) commute with the appropriate coboundary operators, the
simplicial function induces a ring homomorphism H ∗ ( f ): H ∗ (L;Z) → H ∗ (K;Z).
Regarding the cohomology of polyhedra, given any polyhedron |K|, wemay
define in H ∗ (|K|;Z) a structure of ring with identity simply because H ∗ (|K|;Z)=
H ∗ (K;Z). We must verify that H ∗ (−;Z) is a contravariant functor from the category
of polyhedra P to the category of graded commutative rings with identity.
In fact, let f : |K| →|L| be a continuous function and g: K (r) → L a simplicial
approximation of f . We know that the homomorphism H ∗ (π r ;Z) induced by the
projection π r : K (r) → K is an isomorphism; in addition,
H ∗ ( f ;Z)=H ∗ (π r ;Z) −1 H ∗ (g;Z): H ∗ (|L|;Z) → H ∗ (|K|;Z).