nLab
Serre finiteness theorem

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

In stable 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 except

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

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

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

   for $F_m$ a 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)