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 $\pi_{n+k}(S^k)$, for $k \geq 1$, are finite groups (in fact finite abelian groups), hence pure torsion, except
for $n = 0$ in which case $\pi_k(S^k) = \mathbb{Z}$ is the integers;
$k = 2m$ and $n = 2m -1$ in which case
for $F_m$ some finite group.
(Serre 53, see Ravenel 86, Chapter I, Lemma 1.1.8)
The non-torsion element in $\pi_{4k-1}(S^{2k})$ 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
Last revised on December 8, 2021 at 09:22:46. See the history of this page for a list of all contributions to it.