On three-dimensional Alexandrov 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.