Cohomogeneity One Alexandrov Spaces

TL;DR

This paper classifies closed cohomogeneity one Alexandrov spaces in dimensions 3 and 4, providing a structure theorem and identifying new non-manifold examples such as the spherical suspension of RP^2.

Contribution

It offers a structure theorem for these spaces and classifies them in low dimensions, including non-manifold examples not present in lower-dimensional cases.

Findings

Classification of 3- and 4-dimensional cohomogeneity one Alexandrov spaces

Identification of the spherical suspension of RP^2 as a non-manifold example

Complete classification of spaces with isometric T^{n-1} actions

Abstract

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary, we obtain the classification of closed, $n$-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric $T^{n-1}$ action. In contrast to the 1- and 2-dimensional cases, where it is known that an Alexandrov space is a topological manifold, in dimension 3 the classification contains, in addition to the known cohomogeneity one manifolds, the spherical suspension of $RP^2$, which is not a manifold.