# nLab Serre finiteness theorem

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

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.

###### 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.

## References

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.