Hilbert–Smith conjecture

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 Z_p of p-adic integers, for some prime number p. An equivalent form of the conjecture is that Z_p 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 2013, John Pardon proved the three-dimensional case of the Hilbert–Smith conjecture.

References

Smith, Paul A. (1941), "Periodic and nearly periodic transformations", in Wilder, R.; Ayres, W, 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, 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
Pardon, John (2013), "The Hilbert–Smith conjecture for three-manifolds", Journal of the American Mathematical Society, 26 (3), pp. 879–899, doi:10.1090/s0894-0347-2013-00766-3

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