On actions of tori and quaternionic tori on products of spheres

TL;DR

This paper investigates how tori and quaternionic tori act on products of spheres, showing that certain orbit spaces are homeomorphic to spheres and generalizing previous results to broader classes of topological groups.

Contribution

It proves that specific torus actions produce orbit spaces homeomorphic to spheres and extends these results to arbitrary compact topological groups.

Findings

Orbit space of a specific torus action is homeomorphic to a sphere

Generalization to actions of arbitrary compact topological groups

Extension of previous results on torus actions of complexity one

Abstract

In this paper we study the actions of tori (standard compact tori, as well as their quaternionic analogues) on products of spheres. It is proved that the orbit space of a specific action of a torus on a product of spheres is homeomorphic to a sphere. A similar statement for a real torus $\mathbb{Z}_2^n$ was proved by the second author in 2019. We also provide a statement about arbitrary compact topological groups, generalizing the mentioned results, as well as the results of the first author about the actions of a compact torus of complexity one.