Simplicial Structures in Topology

VI.3 Homotopy Groups 229
(VI.3.22) Theorem. The function of sets
is a bijection.
We now define the operation
ψ : [(I n ,∂I n ),(Y,y0)] rel∂ I n −→ πn(Y,y0)
[ f ] rel∂ I n ↦→ [ f ]
rel∂ I n
× : [(I n ,∂I n ),(Y,y0)] rel∂I n × [(I n ,∂I n ),(Y,y0)] rel∂I n −→
−→ [(I n ,∂I n ),(Y,y0)] rel∂I n
as follows: let [ f ] rel∂I n and [g] rel∂ I n be any two elements in [(I n ,∂I n ),(Y,y0)] rel∂ I n;
let us suppose that these two classes are represented, respectively, by the functions
f and g; we now consider the function
such that
Note that
Thus, by definition,
f ∗ g: (I n ,∂I n ) → (Y,y0)
�
f (2x1,...,xn−1,xn) 0 ≤ x1 ≤
( f ∗ g)(x1,...,xn)=
1 2
g(2x1 − 1,...,xn−1,xn) 1
2 ≤ x1 ≤ 1.
(∀(x1,...,xn) ∈ ∂I n )(f ∗ g)(x1,...,xn)=y0.
[ f ] rel∂ I n
rel∂I n
× [g] rel∂I n :=[f ∗ g] rel∂I n.
This operation is well defined, in other words, it does not depend on the element
representing the class.
(VI.3.23) Theorem. The set [(In ,∂ In rel∂ In
),(Y,y0)] rel∂ In with the operation × is a
group isomorphic to πn(Y,y0).
Proof. By Theorem (VI.3.22), it is sufficient to prove that the function
keeps the operations. Let
ψ : [(I n ,∂I n ),(Y,y0)] rel∂ I n −→ πn(Y,y0)
f ,g: (I n ,∂I n ) −→ (Y,y0)
be two given functions; let us break them down into functions through S n ,thatis
to say,