Simplicial Structures in Topology

VI.3 Homotopy Groups 227
Since the map 1S n : Sn → S n has obviously degree 1, the degree function that we
have defined is surjective.
It must be proved that the degree function is injective; we do not prove it here,
but the reader can find the proof in the work of H. Whitney [34]. However, Theorem
(VI.3.17) may also be proved in different ways (see, for instance, Sect. V.3 of [26]).
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Lemma (VI.3.18) proves Theorem (VI.3.17).
So far, not all homotopy groups of the spheres are known; yet, we have two
important results due to Jean–Pierre Serre [31] and based on difficult techniques of
homological algebra:
(VI.3.19) Theorem. If n is odd and m �= n, then πm(Sn ,e0) is a finite group.
(VI.3.20) Theorem. If n is even, then
1. πm(Sn ,e0) is finite if m �= n and m �= 2n − 1.
2. π2n−1(Sn ,e0) is the direct sum of an infinite cyclic group and a finite group
(eventually trivial).
VI.3.3 Another Approach to Homotopy Groups
In the literature, the homotopy groups are also described in a different way but
equivalent to ours; sometimes it is convenient to study the homotopy groups under
this other point of view. We begin our work with a lemma that is very important to
the development of our theme.
(VI.3.21) Lemma. Given two maps f ,g ∈ CTop((X,A),(Y,B)) such that f |A =
g|A, let
H : (X × I,A × I) −→ (Y,B)
be a homotopy relative to A from f to g. Then, there exists a based homotopy
between the maps
induced by f and g.
H : X/A × I −→ Y /B
f ,g: X/A −→ Y /B
Proof. The quotient space X/A is given by the following pushout
cA
A
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∗
iA ��
X
qX
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X/A