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