Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Hilbert--Smith conjecture for three-manifolds   We show that every locally compact group which acts faithfully on a connected
three-manifold is a Lie group. By known reductions, it suffices to show that
there is no faithful action of $\mathbb Z_p$ (the $p$-adic integers) on a
connected three-manifold. If $\mathbb Z_p$ acts faithfully on $M^3$, we find an
interesting $\mathbb Z_p$-invariant open set $U\subseteq M$ with
$H_2(U)=\mathbb Z$ and analyze the incompressible surfaces in $U$ representing
a generator of $H_2(U)$. It turns out that there must be one such
incompressible surface, say $F$, whose isotopy class is fixed by $\mathbb Z_p$.
An analysis of the resulting homomorphism $\mathbb Z_p\to\operatorname{MCG}(F)$
gives the desired contradiction. The approach is local on $M$.

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The categories of this paper are: hep-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: atom-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: q-alg
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: physics
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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin