Simplicial Structures in Topology

222 VI Homotopy Groups
Let f := F(−,1). Note that the maps f and f are homotopic to each other by means
of a free homotopy (actually, the restriction of F to e0 ×I coincides with the path λ);
moreover, f (e0)=y1. We thus define the relation
with the condition φ λ ([ f ]) = [ f ].
(VI.3.11) Theorem. The relation φ n λ
such that:
φ n λ : πn(Y,y0) −→ πn(Y,y1)
defined by the path λ is a group isomorphism
1. If μ : I → Y is a path from y0 to y1 homotopic rel∂I to the path λ,thenφ n μ = φ n λ .
2. The constant path cy0 at y0 induces the identity isomorphism
φ n cy 0 : πn(Y,y0) → πn(Y,y0).
3. If η : I → Y is a path from y1 to y2 ∈ Y,then
φ n λ ∗η = φ n ηφ n λ .
Proof. We have not yet established whether φ n λ is well defined; this will follow from
the proof of 1. Let
G: | •
σn+1|×I → Y
be a homotopy induced by f and μ, in other words, such that
G(−,0)= f and G(e0,t)=μ(t).
We define g = G(−,1) and note that the composite homotopy
H = F −1 ∗ G: | •
σn+1|×I −→ Y
is a free homotopy from f to g, that may be transformed into a based homotopy, as
follows. From the condition μ ∼ λ rel∂I, we obtain a based homotopy
K : I × I −→ Y
such that K(−,0)=μ −1 ∗ λ and K(−,1)=cy 1 . Note that the restriction H|e0 × I
coincides with the closed path μ −1 ∗ λ. We consider the polyhedra
and
M = | •
σn+1|×I
L = | •
σn+1|×{0}∪e0 × I ∪| •
σn+1|×{1};
since L is a subpolyhedron of M, the Homotopy Extension Property holds for the
pair (M,L) and, therefore, there exists a retraction
r : M × I −→ M ×{0}∪L × I.