Simplicial Structures in Topology

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