On three-dimensional Alexandrov spaces

TL;DR

This paper extends classical results on three-dimensional manifolds to Alexandrov spaces, classifying positively curved spaces, analyzing the Poincaré Conjecture, and proving a geometrization analogue for these singular spaces.

Contribution

It provides a classification of 3D Alexandrov spaces with curvature bounds and extends key topological results like the Poincaré and geometrization conjectures to these singular spaces.

Findings

Closed positively curved Alexandrov spaces with singularities are suspensions of the real projective plane.

The only homotopy sphere among 3D Alexandrov spaces is the 3-sphere.

An analogue of the geometrization conjecture is proved for these spaces.

Abstract

We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspension of the real projective plane; we use this to classify, up to homeomorphism, closed, positively curved Alexandrov spaces of dimension three. Second, we classify closed three-dimensional Alexandrov spaces of nonnegative curvature. Third, we study the well-known Poincar\'e Conjecture in dimension three, in the context of Alexandrov spaces, in the two forms it is usually formulated for manifolds. We first show that the only three-dimensional Alexandrov space that is also a homotopy sphere is the 3-sphere; then we give examples of closed, geometric, simply connected three-dimensional Alexandrov spaces for five of the eight Thurston geometries, proving along the way the impossibility of getting such examples for the Nil, $\widetilde{\mathrm{SL}_2(\mathbb{R})}$ and Sol geometries. We conclude the paper by proving the analogue of the geometrization conjecture for closed three-dimensional Alexandrov spaces.