Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Topology and geometry of flagness and beltness of simple orbifolds   We consider a class of right-angled Coxeter orbifolds, named as simple
orbifolds, which are a generalization of simple polytopes. Similarly to
manifolds over simple polytopes, the topology and geometry of manifolds over
simple orbifolds are closely related to the combinatorics and orbifold
structure of simple orbifolds. We generalize the notions of flag and belt in
the setting of simple polytopes into the setting of simple orbifolds.
  To describe the topology and geometry of a simple orbifold in terms of its
combinatorics, we focus on {\em simple handlebodies} (that is, simple orbifolds
which can be obtained from simple polytopes by gluing some disjoint specific
codimension-one faces). We prove the following two main results in terms of
combinatorics, which can be understood as "Combinatorial Sphere Theorem" and
"Combinatorial Flat Torus Theorem" on simple handlebodies:
  (A) A simple handlebody is orbifold-aspherical if and only if it is flag.
  (B) There exists a rank-two free abelian subgroup in $\pi_1^{orb}(Q)$ of an
orbifold-aspherical simple handlebody $Q$ if and only if it contains an
$\square$-belt.
  Furthermore, based on such two results and some results of geometry, it is
shown that the existence of some curvatures on a certain manifold cover
(manifold double) over a simple handlebody $Q$ can be characterized in terms of
the combinatorics of $Q$. In 3-dimensional case, together with the theory of
hyperbolic 3-manifolds, we can induce a pure combinatorial equivalent
description for a simple $3$-handlebody to admit a right-angled hyperbolic
structure, which is a natural generalization of Pogorelov Theorem.

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The categories of this paper are: adap-org
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The categories of this paper are: hep-lat
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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: q-bio
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The categories of this paper are: acc-phys
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: atom-ph