Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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114 III Homology of Polyhedra<br />
that is to say, {g(σ 0 ),...,g(σ n )}⊂s( f (p)). S<strong>in</strong>ce<br />
|g|(p)=<br />
n<br />
∑<br />
i=0<br />
αig(σ i ) ,<br />
we may conclude that |g|(p) ∈|s( fF(p))|. �<br />
(III.2.5) Theorem. The geometric realization of a simplicial approximation<br />
g: K (r) → Lofamap f: |K|→|L| is homotopic to f F.<br />
Proof. By def<strong>in</strong>ition, given any p ∈|K (r) |, the segment with ends fF(p) and |g|(p)<br />
is entirely conta<strong>in</strong>ed <strong>in</strong> the convex space |s( fF(p))| ⊂|L| (see Theorem (II.2.9)).<br />
The map<br />
H : |K (r) |×I →|L| , H(p,t)=tfF(p)+(1 − t)|g|(p)<br />
is a homotopy between fF and |g|. �<br />
(III.2.6) Theorem. Let g: K (r) → L and g ′ : K (s) → L be two simplicial approximations<br />
of a map f : |K| →|L|. Suppose that s < r and π (r,s) : K (r) → K (s) is the<br />
composition of projections. Then the simplicial functions<br />
are contiguous.<br />
g: K (r) → L and g ′ π (r,s) : K (r) → L<br />
Proof. We have to prove that, for every simplex σ ∈ Φ (r) , there is a simplex τ ∈ Ψ<br />
such that g(σ) ⊂ τ and g ′ π (r,s) (σ) ⊂ τ.<br />
For every σ ∈ Φ (r) , let us def<strong>in</strong>e τ = s( f (b(σ))) where b(σ) is the barycenter of<br />
σ. S<strong>in</strong>ceg is a simplicial approximation of f ,<br />
|g|(b(σ)) ∈|s( f (b(σ)))| ;<br />
but g(σ) is the smallest simplex of L whose geometric realization conta<strong>in</strong>s the po<strong>in</strong>t<br />
|g|(b(σ)) and so, g(σ) ⊂ τ.<br />
On the other hand, let P(σ) be the smallest simplex of K (s) such that |σ| ⊂<br />
|P(σ)|. Clearly,<br />
|π (r,s) (σ)|⊂|P(σ)|<br />
and<br />
|g ′ (π (r,s) (σ))|⊂|g ′ (P(σ))| .<br />
From b(σ) ∈|σ|⊂|P(σ)|, it follows that<br />
|g ′ |(b(σ)) ∈|g ′ (P(σ))|<br />
and because |g ′ |(b(σ)) ∈|s( f (b(σ)))|, it also follows that<br />
|g ′ (P(σ))|⊂|s( f (b(σ)))|.