This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (October 2016) (Learn how and when to remove this template message)
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.
- Smith, Paul A. (1941), "Periodic and nearly periodic transformations", in Wilder, R.; Ayres, W (eds.), Lectures in Topology, Ann Arbor, MI: University of Michigan Press, pp. 159–190
- Chu, Hsin (1973), "On the embedding problem and the Hilbert-Smith conjecture", in Beck, Anatole (ed.), Recent Advances in Topological Dynamics, Lecture Notes in Mathematics, 318, Springer-Verlag, pp. 78–85
- Repovš, Dušan; Ščepin, Evgenij V. (June 1997), "A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps", Mathematische Annalen, 308 (2), pp. 361–364, doi:10.1007/s002080050080
- Martin, Gaven (1999), "The Hilbert-Smith conjecture for quasiconformal actions", Electronic Research Announcements of the American Mathematical Society, 5 (9), pp. 66–70
- Pardon, John (2013), "The Hilbert–Smith conjecture for three-manifolds", Journal of the American Mathematical Society, 26 (3), pp. 879–899, arXiv:1112.2324, doi:10.1090/s0894-0347-2013-00766-3