nLab 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 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), for k1k \geq 1, are finite groups (in fact finite abelian groups), hence pure torsion, except

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

  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 some finite group.

(Serre 53, see Ravenel 86, Chapter I, Lemma 1.1.8)


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

The statement is reviewed in

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 (doi:10.1023/A:1015370800473)

Last revised on December 8, 2021 at 09:22:46. See the history of this page for a list of all contributions to it.