Orientation and symmetries of Alexandrov spaces with applications in positive curvature

TL;DR

This paper introduces new tools in Alexandrov geometry, including ramified orientable double covers and a refined Slice Theorem, to classify compact positively curved Alexandrov spaces with high symmetry.

Contribution

The paper develops novel methods for Alexandrov spaces and applies them to classify positively curved spaces with maximal symmetry rank.

Findings

Introduction of ramified orientable double covers

A refined Slice Theorem for compact Lie group actions

Classification results for spaces with maximal symmetry rank

Abstract

We develop two new tools for use in Alexandrov geometry: a theory of ramified orientable double covers and a particularly useful version of the Slice Theorem for actions of compact Lie groups. These tools are applied to the classification of compact, positively curved Alexandrov spaces with maximal symmetry rank.