Loops on spheres having a compact-free inner mapping group

TL;DR

This paper classifies 1-dimensional loops on spheres and projective spaces with a compact-free inner mapping group, showing they are homeomorphic to a circle and explicitly describing their differentiable structures.

Contribution

It proves that such loops are homeomorphic to a circle and provides an explicit classification of differentiable 1-dimensional compact loops using Fourier series.

Findings

Loops on spheres with a compact-free inner mapping group are homeomorphic to a circle.

Explicit classification of differentiable 1-dimensional compact loops.

Identification of the topological and differentiable structure of these loops.

Abstract

We prove that any topological loop homeomorphic to a sphere or to a real projective space and having a compact-free Lie group as the inner mapping group is homeomorphic to the circle. Moreover, we classify the differentiable $1$-dimensional compact loops explicitly using the theory of Fourier series.