Simplicial Structures in Topology

III.2 Homology of Polyhedra 115
From this, we conclude that
|g ′ (π (r,s) (σ))|⊂|s( f (b(σ)))|
and, consequently, g ′ (π (r,s) (σ)) ⊂ s( f (b(σ))). �
(III.2.7) Corollary. Amap f: |K|→|L| defines a unique homomorphism of graded
groups
H∗( f ,Z): H∗(|K|,Z) → H∗(|L|,Z) .
Proof. For any simplicial approximation g: K (r) → L of f ,letusset
H∗( f ,Z)=H∗(g,Z)(H∗(π r )) −1
(recall that (H∗(π r )) −1 = H∗(ℵ r ) ).
Suppose g ′ : K (s) → L to be another simplicial approximation of f , with s < r.
The contiguity of the simplicial functions g and g ′ π (r,s) (which follows from the
previous theorem) ensures that
H∗(g ′ π (r,s) )=H∗(g)
(cf. Corollary (III.2.1)). But π r = π s π (r,s) and therefore
H∗(g ′ ,Z)(H∗(π s ,Z)) −1 = H∗(g,Z)(H∗(π r ,Z)) −1 .
Corollary (III.2.7) completes the definition of functor H∗(−,Z) on morphisms;
we repeat it here: for every n ≥ 0 and any map f : |K|→|L|,
Hn( f ,Z) := Hn(g,Z)(Hn(π r ,Z)) −1 : Hn(|K|,Z) → Hn(|L|,Z)
where g: K (r) → L is any simplicial approximation of f .
We now prove that the homomorphism H∗( f ,Z),definedbyamap f : |K|→|L|,
is a homotopy invariant, meaning that the next result is true.
(III.2.8) Theorem. Let f ,g: |K|→|L| be homotopic maps. Then
H∗( f ,Z)=H∗(g,Z) .
Proof. Let H : |K|×I →|L| be the homotopy that links f to g; suppose that Hi0 = f
and Hi1 = g,wherei0 and i1 are the maps
iα : |K|→|K|×I , p ↦→ (p,α) , α = 0,1 .
We represent the intervall I =[0,1] as the geometric realization of the complex
I =(Y,ϒ ) , Y = {0,1} , ϒ = {{0},{1},{0,1}} .
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