• DOI:10.1093/IMRN/RNU101
• Corpus ID: 119138702

On three-dimensional Alexandrov spaces

@article{GalazGarca2013OnTA,
  title={On three-dimensional Alexandrov spaces},
  author={Fernando Galaz‐Garc{\'i}a and Luis Guijarro},
  journal={arXiv: Differential Geometry},
  year={2013}
}

• F. Galaz‐García, Luis Guijarro
• Published 15 July 2013
• Mathematics
• arXiv: Differential Geometry

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.