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The Hilbert-Smith conjecture for three-manifolds


Author: John Pardon
Journal: J. Amer. Math. Soc. 26 (2013), 879-899
MSC (2010): Primary 57S10, 57M60, 20F34, 57S05, 57N10; Secondary 54H15, 55M35, 57S17
DOI: https://doi.org/10.1090/S0894-0347-2013-00766-3
Published electronically: March 19, 2013
MathSciNet review: 3037790
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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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Additional Information

John Pardon
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Email: pardon@math.stanford.edu

DOI: https://doi.org/10.1090/S0894-0347-2013-00766-3
Received by editor(s): April 10, 2012
Received by editor(s) in revised form: October 27, 2012, and November 25, 2012
Published electronically: March 19, 2013
Additional Notes: The author was partially supported by a National Science Foundation Graduate Research Fellowship under grant number DGE–1147470.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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