Simplicial Structures in Topology

III.1 The Category of Polyhedra 107
A ×{0} 1A × i0 ��
A × I
��
��
��
��
��
��
i × 1 {0} i × 1 {0}
1A × i0
X ×{0}
G
�X ���
���
�H
���
f
��
Z
The function F = Hr: X × I → Z is the extension of the homotopy that we were
seeking. �
(III.1.7) Theorem. Let L be a subcomplex of the simplicial complex K. Then, the
pair of polyhedra (|K|,|L|) has the homotopy extension property.
Proof. We start the proof by assuming that |K| and |L| are the geometric realizations
of the simplicial complexes K =(X,Φ) and L =(Y,Ψ), respectively. By
Lemma (III.1.6), we only need to prove that there exists a retraction of |K|×I onto
�|K|. Let σ = {x0,...,xn} be an n-simplex of K and let |σ| be the geometric realization
of (σ,℘(σ) � /0). Letb(σ) be the barycenter of |σ|. Foreveryp ∈|σ| and
for every real number t ∈ I, let the point tb(σ)+(1 − t)p ∈|σ| be denoted by [p,t]
(we recall that |σ| is a convex space – see Theorem (II.2.9)). We now consider the
function
Hσ : |σ|×I × I −→ |σ|×I
defined by the equations:
Hσ([p,t],s,u)=
�
([p,(1 − u)t + u(2t−s)
2−s ],(1 − u)s) , s ≤ 2t
([p,(1 − u)t],(1 − u)s + u(s−2t)
1−t , 2t ≤ s .
To understand geometrically how we have come to this function, suppose that |σ|
is a 2-simplex that we have placed in the plane (x,y) of R3 with its barycenter
b(σ) at the origin (0,0,0); we then project |σ|×I on |σ|×{0}∪| •
σ|×I from the
point (0,0,2) (here | •
σ| is the boundary of |σ|). Note that, in this way, we obtain
a retraction of |σ|×I onto |σ|×{0}∪| •
σ|×I. In the general case, when σ is an
n-simplex, the function Hσ (−,1) is a retraction. 1
For each integer n ≥−1, we define
Mn = |K|×{0}∪|K n ∪ L|×I
where K n is the simplicial subcomplex determined by all simplexes of K with
dimension ≤ n, and such that K −1 = /0; then,
M−1 = � |K| = |K|×{0}∪|L|×I .
1 | • σ| is the geometric realization of the simplicial complex • σ = {σ ′ ⊂ σ|dimσ ′ ≤ n − 1} .