The Hilbert–Smith conjecture for three-manifolds

DOI10.1090/S0894-0347-2013-00766-3zbMATH Open1273.57024arXiv1112.2324WikidataQ29041534 ScholiaQ29041534MaRDI QIDQ4924070

Author name not available (Why is that?)

Publication date: 30 May 2013

Published in: (Search for Journal in Brave)

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

m a t h b b Z_{p}

(the

p

-adic integers) on a connected three-manifold. If

m a t h b b Z_{p}

acts faithfully on

M^{3}

, we find an interesting

m a t h b b Z_{p}

-invariant open set

U s u b s e t e q M

with

H_{2} (U) = m a t h b b 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

m a t h b b Z_{p}

. An analysis of the resulting homomorphism

m a t h b b Z_{p} o o p e r a t o r n a m e M C G (F)

gives the desired contradiction. The approach is local on

M

.

Full work available at URL: https://arxiv.org/abs/1112.2324

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