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
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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 $\pi_{n+k}(S^k)$ 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)
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
An entirely different proof, using only elementary concepts, is given in
Last revised on February 3, 2019 at 15:38:12. See the history of this page for a list of all contributions to it.