homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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 except
for $n = 0$ in which case $\pi_k(S^k) = \mathbb{Z}$;
$k = 2m$ and $n = 2m -1$ in which case
for $F_m$ a 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 September 5, 2020 at 14:39:21. See the history of this page for a list of all contributions to it.