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On Three-Dimensional Alexandrov Spaces. [Preprint]

Galaz-Garcia, F.; Guijarro, L.

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, SL2(R)˜ and Sol geometries. We conclude the paper by proving the analogue of the geometrization conject ... mehr


Zugehörige Institution(en) am KIT Institut für Algebra und Geometrie (IAG)
Publikationstyp Buchaufsatz
Jahr 2013
Sprache Englisch
Identifikator KITopen ID: 1000051159
Erschienen in arXiv [math.DG]
Seiten arXiv:1307.3929
URLs http://arxiv.org/abs/1307.3929
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