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 conjecture for closed three-dimensional Alexandrov spaces.

Zugehörige Institution(en) am KIT |
Institut für Algebra und Geometrie (IAG) |

Publikationstyp |
Buchaufsatz |

Publikationsjahr |
2013 |

Sprache |
Englisch |

Identifikator |
KITopen-ID: 1000051159 |

Erschienen in |
arXiv [math.DG] |

Seiten |
arXiv:1307.3929 |

Externe Relationen |
Siehe auch |

KIT – Die Forschungsuniversität in der Helmholtz-Gemeinschaft

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