The Hilbert–Smith conjecture for three-manifolds

Pardon, John

doi:10.1090/S0894-0347-2013-00766-3

The Hilbert–Smith conjecture for three-manifolds
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by John Pardon

J. Amer. Math. Soc. 26 (2013), 879-899

DOI: https://doi.org/10.1090/S0894-0347-2013-00766-3

Published electronically: March 19, 2013

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Abstract:

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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