On ThreeDimensional Alexandrov Spaces. [Preprint]
GalazGarcia, F.; Guijarro, L.
Abstract:
We study threedimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on threedimensional manifolds: First, we show that a closed threedimensional 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 threedimensional Alexandrov spaces of nonnegative curvature. Third, we study the wellknown 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 threedimensional Alexandrov space that is also a homotopy sphere is the 3sphere; then we give examples of closed, geometric, simply connected threedimensional 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 conjecture for closed threedimensional Alexandrov spaces.
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 