Torus actions on Alexandrov 4-spaces

TL;DR

This paper classifies four-dimensional Alexandrov spaces with torus symmetries, extending known results for manifolds and showing they are homeomorphic to certain orbifolds with isometric torus actions.

Contribution

It generalizes the equivariant classification of 4-manifolds with torus actions to Alexandrov spaces and relates them to Riemannian orbifolds.

Findings

Alexandrov spaces with torus actions are homeomorphic to 4D Riemannian orbifolds

Provides an equivariant classification extending Orlik and Raymond's work

Partial homeomorphism classification achieved

Abstract

We obtain an equivariant classification for orientable, closed, four-dimensional Alexandrov spaces admitting an isometric torus action. This generalizes the equivariant classification of Orlik and Raymond of closed four-dimensional manifolds with torus actions. Moreover, we show that such Alexandrov spaces are equivariantly homeomorphic to $4$-dimensional Riemannian orbifolds with isometric $T^2$-actions. We also obtain a partial homeomorphism classification.