Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Monodromy, liftings of holomorphic maps, and extensions of holomorphic
  motions   We study monodromy of holomorphic motions and show the equivalence of
triviality of monodromy of holomorphic motions and extensions of holomorphic
motions to continuous motions of the Riemann sphere. We also study liftings of
holomorphic maps into certain Teichm\"uller spaces. We use this "lifting
property" to prove that, under the condition of trivial monodromy, any
holomorphic motion of a closed set in the Riemann sphere, over a hyperbolic
Riemann surface, can be extended to a holomorphic motion of the sphere, over
the same parameter space. We conclude that this extension can be done in a
conformally natural way.

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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: stat
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: acc-phys
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: econ
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: funct-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-ex