- References
{{no footnotes|date=October 2016}}In mathematics, the Hilbert–Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups G that can act effectively (faithfully) on a (topological) manifold M. Restricting to G which are locally compact and have a continuous, faithful group action on M, it states that G must be a Lie group. Because of known structural results on G, it is enough to deal with the case where G is the additive group Zp of p-adic integers, for some prime number p. An equivalent form of the conjecture is that Zp has no faithful group action on a topological manifold. The naming of the conjecture is for David Hilbert, and the American topologist Paul A. Smith. It is considered by some to be a better formulation of Hilbert's fifth problem, than the characterisation in the category of topological groups of the Lie groups often cited as a solution. In 1999 Gaven Martin proved the Hilbert-Smith conjecture for groups acting quasiconformally on a Riemannian manifold and gave applications concerning unique analytic continuation for Beltrami systems. In 2013, John Pardon proved the three-dimensional case of the Hilbert–Smith conjecture. References- {{Citation | last = Smith | first = Paul A. | author-link = Paul A. Smith | chapter = Periodic and nearly periodic transformations | title = Lectures in Topology | editor1-first = R. | editor1-last = Wilder | editor2-first = W | editor2-last = Ayres | publisher = University of Michigan Press | place = Ann Arbor, MI | year = 1941 | pages = 159–190 }}
- {{Citation | last = Chu | first = Hsin | year = 1973 | contribution = On the embedding problem and the Hilbert-Smith conjecture | title = Recent Advances in Topological Dynamics | editor1-last = Beck | editor1-first = Anatole | series = Lecture Notes in Mathematics | volume = 318 | publisher = Springer-Verlag | pages = 78–85 }}
- {{Citation | first = Dušan | last = Repovš | first2 = Evgenij V. | last2 = Ščepin | title = A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps | periodical = Mathematische Annalen | volume = 308 | issue = 2 |date=June 1997 | pages = 361–364 | doi=10.1007/s002080050080}}
- {{Citation | first = Gaven | last = Martin |title = The Hilbert-Smith conjecture for quasiconformal actions | periodical = Electronic Research Announcements of the American Mathematical Society| volume = 5 | issue = 9 |date= 1999| pages = 66-70 |}}
- {{Citation | first = John | last = Pardon | authorlink = John Pardon | title = The Hilbert–Smith conjecture for three-manifolds | periodical = Journal of the American Mathematical Society | volume = 26 | issue = 3 |date= 2013 | pages = 879–899 | doi=10.1090/s0894-0347-2013-00766-3| arxiv = 1112.2324 }}
{{DEFAULTSORT:Hilbert-Smith conjecture}} 4 : Topological groups|Group actions (mathematics)|Conjectures|Structures on manifolds |