Quotients of Spheres By Linear Actions of Tori

TL;DR

This paper studies quotients of spheres under linear torus actions, linking their topological properties to matroid theory and Tutte polynomials, and characterizes when these quotients are manifolds or spheres.

Contribution

It introduces a method to analyze the topology of quotient spaces using matroids and Tutte polynomials, providing new insights into their homology and homotopy types.

Findings

Computed the integral homology groups of quotient spaces.

Determined the homotopy type and homology of singular quotient spaces.

Characterized conditions for the quotient to be a manifold or sphere.

Abstract

We consider quotients of spheres by linear actions of real tori. To each quotient we associate a matroid built out of a diagonalization of the torus action. We find the integral homology groups of the resulting quotient spaces in terms of the Tutte polynomial of the matroid. We also find the homotopy type and homology of the singular space of such an action. Lastly, we consider the circumstances under which the orbit space is a manifold or, more specifically, a (homology) sphere.