homotopy theory, (∞,1)-category theory, homotopy type theory
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There are several theorems by Serre which deserve to be called “finiteness theorems”.
On the homotopy groups of spheres
(Serre finiteness theorem)
The homotopy groups of spheres , for , are finite groups except
for in which case ;
and in which case
for a finite group.
(Serre 53, see Ravenel 86, Chapter I, Lemma 1.1.8)
The non-torsion element in may be seen explicitly in the minimal Sullivan model for the rational 2k-sphere, see there for more.
The original proof is due to
The statement is reviewed in
An entirely different proof, using only elementary concepts, is given in
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