Serre finiteness theorem



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

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see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


Stable Homotopy theory


There are several theorems by Serre which deserve to be called “finiteness theorems”.

In stable homotopy theory

On the homotopy groups of spheres


(Serre finiteness theorem)

The homotopy groups of spheres π n+k(S k)\pi_{n+k}(S^k) are finite groups except

  1. for n=0n = 0 in which case π k(S k)=\pi_k(S^k) = \mathbb{Z};

  2. k=2mk = 2m and n=2m1n = 2m -1 in which case

    π 4m1(S 2m)F m \pi_{4m - 1}(S^{2m}) \simeq \mathbb{Z} \oplus F_m

    for F mF_m a finite group.

(Serre 53)


The non-torsion element in π 4k1(S 2k)\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

  • Jean-Pierre Serre, Groupes d’homotopie et classes de groupes abelien, Ann. of Math. 58 (1953), 258–294.

An entirely different proof, using only elementary concepts, is given in

  • S. S. Podkorytov, An Alternative Proof of a Weak Form of Serre’s Theorem, Journal of Mathematical Sciences July 2002, Volume 110, Issue 4, pp 2875–2881

Last revised on February 3, 2019 at 15:38:12. See the history of this page for a list of all contributions to it.