Simplicial Structures in Topology

VI.3 Homotopy Groups 223
We now construct the following maps:
and
F : (| •
σn+1|×{0}) × I → Y
G: (| •
σn+1|×{1}) × I → Y
such that, for every x ∈| •
σn+1| and every t ∈ I,
Note that
Let
F(x,0,t)= f (x) and G(x,1,t)=g(x).
F(e0,0,t)=G(e0,1,t)=y1.
H : M ×{0}∪L × I −→ Y
be the map defined by the union of the maps H ∪(F ∪K ∪G) (note that the restriction
of H to L ×{0} coincides with f ∪ μ −1 ∗ λ ∪ g); we now consider the homotopy
θ : M × I
r ��
M ×{0}∪L × I
H ��
Y
in other words, the function defined by the commutative diagram
The homotopy
L ×{0}
��
M ×{0}
��
L × I
��
��
H
M × �I
���
���
θ
��
H ���
����
��
Y
•
�H = θ(−,−,1): | σn+1|×I → Y
is a free homotopy from f to g. Hence, [ f ]=[g] ∈ πn(Y,y1) and thus, by setting
μ = λ , we see that φ n λ ([ f ]) does not depend on the choices of f , the representative of
the class [ f ], or the homotopy F with the required properties. We have then proved
that φ n λ is a (well defined) function that satisfies property 1 stated in the theorem.
We leave the proof of properties 2 and 3 to the reader (anyway, the results follow
easily from the definitions). A consequence of properties 1, 2, and 3 is that φ n λ is
injective and surjective.
We now prove that φ n λ is a group homomorphism. Let [ f ],[g] ∈ πn(Y,y0) be given
arbitrarily and let
F,G: S n × I → Y