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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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The Hilbert–Smith conjecture for three-manifolds
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by John Pardon PDF
J. Amer. Math. Soc. 26 (2013), 879-899 Request permission

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
  • MR Author ID: 857067
  • Email: pardon@math.stanford.edu
  • 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.
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 3037790