nLab Serre finiteness theorem

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

In homotopy theory

On the homotopy groups of spheres:

Theorem

(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

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

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

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

   for $F_m$ some finite group.

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

Remark

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.

Related concepts

• Nishida nilpotence theorem

References

The original proof is due to

• Jean-Pierre Serre, Groupes d’homotopie et classes de groupes abelien, Ann. of Math. 58 (1953), 258–294 (jstor:1969789)

The statement is reviewed in

• Douglas Ravenel, Chapter I, Theorem 1.1.8 in: Complex cobordism and stable homotopy groups of spheres. Academic Press Orland (1986) reprinted as: AMS Chelsea Publishing, Volume 347 (2004) (ISBN:978-0-8218-2967-7, webpage, pdf)

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)