Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Heegaard genus of amalgamated 3-manifolds   Let M and M' be simple 3-manifolds, each with connected boundary of genus at
least two. Suppose that M and M' are glued via a homeomorphism between their
boundaries. Then we show that, provided the gluing homeomorphism is
`sufficiently complicated', the Heegaard genus of the amalgamated manifold is
completely determined by the Heegaard genus of M and M' and the genus of their
common boundary. Here, a homeomorphism is `sufficiently complicated' if it is
the composition of a homeomorphism from the boundary of M to some surface S,
followed by a sufficiently high power of a pseudo-Anosov on S, followed by a
homeomorphism to the boundary of M'. The proof uses the hyperbolic geometry of
the amalgamated manifold, generalised Heegaard splittings and minimal surfaces.

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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: quant-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-th
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The categories of this paper are: q-alg
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: physics
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: cs
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The categories of this paper are: gr-qc