Which spheres admit a topological group structure?

Abstract

It is well known since the 1940’s that Sº, S1 and S3 are the only spheres admitting a topological group structure. In this short note we provide an easy and direct proof (without using Lie group theory nor dimension theory) of the fact that S2n does not admit such a structure for any n > 0. The proof is based upon the notion of group actions on a topological space; loosely speaking what makes possible this argument is that there are more self-homeomorphisms of a topological group than of an even sphere.