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