Simplicial Structures in Topology

III.6 Homology of the Product of Two Polyhedra 147
There are natural transformations
η : C× −→ C ⊗C
τ : C ⊗C −→ C×
which extend the identity homomorphism 1Z : Z → Z. Besides, ητ and τη are
homotopic to the natural transformations given by the identities.
III.6.2 Homology of the Product of Two Polyhedra
Let two polyhedra |K|,|L|∈Csim be given. Our aim here is to compute the homology
groups of the polyhedron |K|×|L| in terms of those of |K| and |L|.
By the Eilenberg–Zilber Theorem (III.6.3), the chain complexes C(K) ⊗ C(L)
and C(K × L) are chain equivalent; therefore, we have a graded group isomorphism
H∗(|K|×|L|;Z) ∼ = H∗(C(K) ⊗C(L)).
Following the steps taken in Sect. II.5 when studying the relationship between
H∗(K;G) and H∗(K,Z), we interpret the graded Abelian groups Z(K)={Zn(K)|n ≥
0} and B(K)={Bn(K)|n ≥ 0} as chain complexes with trivial boundary operator 0;
we then construct the chain complexes
1. (Z(K) ⊗C(L),0 ⊗ d L )
2. (C(K) ⊗C(L),∂ K ⊗ ∂ L )
3. ( � B(K) ⊗C(L),0 ⊗ ∂ L ),where � B(K) n = Bn−1(K)
since the chain complex sequence
Z(K) ��
i ��
C(K)
d K
��
��
B(K) �
is exact and short, with an argument similar to that used in Lemma (II.5.1), we
conclude that
(Z(K) ⊗C(L),0 ⊗ ∂ L ) ��
��
(C(K) ⊗C(L),d K ⊗ ∂ L )
��
��
( B(C) � L
⊗C(L),0 ⊗ ∂ )
is a short exact sequence.
By the Long Exact Sequence Theorem (II.3.1), we obtain the exact sequence
...→ Hn(Z(K) ⊗C(L)) i∗
→ Hn(C(K) ⊗C(L)) ∂∗
→ Hn( � B(K) ⊗C(L))
λn
→ Hn−1(Z(K) ⊗C(L)) → ... .