Simplicial Structures in Topology

III.6 Homology of the Product of Two Polyhedra 143
Proof. In regard to the first part of the statement, we need only to note that
Hi(σ j × σn− j,Z) ∼ = Hi(|σ j|×|σn−j|,Z) ∼ = Hi(|σ n |,Z)
�
∼=
Z
0
if i = 0
ifi > 0.
For the proof of the second part, we recall Lemma (II.3.8), which tells us that
I. There exists a function
such that εη = 1,
II. There exists a homotopy
such that
η : Z → C0(σn)
s: C(σn) → C(σn)
1. ∂1s0 = 1 − ηε
2. ∂n+1sn + sn−1∂n = 1foreveryn ≥ 1
We define the morphism
on generators by the formula
S : C(σn) ⊗C(σn) → C(σn) ⊗C(σn)
S(x ⊗ y)=s(x) ⊗ y + ηε(x) ⊗ s(y)
provided that ηε(x)=0if|x| �= 0. Since ε∂1 = 0, we conclude that
d ⊗ S(x ⊗ y)=d ⊗ [s(x) ⊗ y)+ηε(x) ⊗ s(y)]
On the other hand,
and then,
while
= d(s(x)) ⊗ y +(−1) |x|+1 s(x) ⊗ d(y)+ηε(x) ⊗ d(s(y)).
S(d ⊗ (x ⊗ y)) = S(d(x) ⊗ y +(−1) |x| x ⊗ d(y))
= s(d(x)) ⊗ y +(−1) |x| s(x) ⊗ d(y)+ηε(x) ⊗ s(d(y))
(∀x ⊗ y ∈ (C(σn) ⊗C(σn))i with i > 0) (d ⊗ S + Sd ⊗ )(x ⊗ y)=x ⊗ y
(∀x ⊗ y ∈ (C(σn) ⊗C(σn))0) d ⊗ S(x ⊗ y)=(1 − ηε⊗ ηε)(x ⊗ y).
We conclude that properties II.1 and 2 of Lemma (II.3.8) hold for η ⊗ η and for S.
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