Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On three-dimensional Alexandrov spaces   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.

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The categories of this paper are: physics
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The categories of this paper are: stat
8
The categories of this paper are: hep-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: cs
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The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-th
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
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The categories of this paper are: math-ph
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The categories of this paper are: chao-dyn