Toric orbit spaces which are manifolds

TL;DR

This paper characterizes when the orbit space of a smooth manifold under a compact torus action is a topological manifold, extending previous results and connecting to combinatorial and topological theories.

Contribution

Provides a new proof for characterizing torus actions with manifold orbit spaces, applicable to manifolds with boundary, and explores interdisciplinary connections.

Findings

Characterization of torus actions with manifold orbit spaces

Extension of previous results to manifolds with boundary

Connections to combinatorial and topological theories

Abstract

We characterize the actions of compact tori on smooth manifolds for which the orbit space is a topological manifold (either closed or with boundary). For closed manifolds the result was originally proved by Styrt in 2009. We give a new proof for closed manifolds which is also applicable to manifolds with boundary. In our arguments we use the result of Provan and Billera who characterized matroid complexes which are pseudomanifolds. We study the combinatorial structure of torus actions whose orbit spaces are manifolds. In two appendix sections we give an overview of two theories related to our work. The first one is the combinatorial theory of Leontief substitution systems from mathematical economics. The second one is the topological Kaluza--Klein model of Dirac's monopole studied by Atiyah. The aim of these sections is to draw some bridges between disciplines and motivate further studies in toric topology.