Simplicial Structures in Topology

224 VI Homotopy Groups
be two homotopies such that F(−,0) = f , G(−,0) =g and, for every t ∈ I, the
equality F(e0,t)=G(e0,t)=λ(t) holds; besides, let f = F(−,1) and g = G(−,1).
We must prove that
φ n νn
λ ([ f ] × [g]) = φ n νn
λ ([ f ]) × φ n λ ([g])
that is to say
with σ( f ∨ g)νn = K(−,1);here
[σ( f ∨ g)νn]=[σ( f ∨ g)νn]
K : S n × I −→ Y
is any homotopy such that K(−,0)=σ( f ∨g)νn and K(e0,t)=λ(t),foreveryt ∈ I.
We choose
K := σ(F ∨ G)(νn × 1I): S n × I −→ Y;
in this case, a simple calculation shows that
(∀t ∧ x ∈ S n ) K(t ∧ x,1)=σ( f ∨ g)νn(t ∧ x)
and so φ n λ is a homomorphism. �
The following result is a direct consequence of the previous theorem.
(VI.3.12) Corollary. For every closed loop at y0, the function
φ n λ : πn(Y,y0) −→ πn(Y,y0)
is an automorphism of πn(Y,y0), which depends only on the class of λ .
We say that the fundamental group π1(Y,y0) acts on πn(Y,y0); as we have already
remarked, when n = 1, this action is achieved by means of inner automorphisms.
Suppose Y to be path-connected; then the set of orbits
πn(Y,y0)/ π1(Y,y0),
induced by the action of π1(Y,y0) on πn(Y,y0), is in relation with the set of free
homotopy classes [S n ,Y ]; actually, we have the following
(VI.3.13) Theorem. Let (Y,y0) be a path-connected based space. Then there exists
a bijection
φ : πn(Y,y0)/ π1(Y,y 0) −→ [S n ,Y ].
We do not prove this theorem here; the reader, who wishes to read a proof of this
result, is asked to seek Corollary 7.1.3 in [26].
Let us suppose Y to be simply connected, thatistosay,π1(Y,y0)=0; any loop
of Y with base at y0 is homotopic to the constant loop cy0 ; thus, by part 2 of
Theorem (VI.3.11), φ n λ is the identity automorphism of πn(Y,y0) and consequently
πn(Y,y0) ≡ [S n ,Y ].