Hilbert–Smith conjecture
Conjecture in topology
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In mathematics, the Hilbert–Smith conjecture is concerned with the possible transformation groups of manifolds. The conjecture says that if a locally compact topological group G act faithfully and continuous on a connected manifold M, then G must be a Lie group.
The naming of the conjecture is for David Hilbert, and Paul A. Smith.[1] It is considered by some to be a better formulation of Hilbert's fifth problem, than the characterisation of topological groups being also a topological manifolds as Lie groups often cited as a solution.
Results
Because of known structural results on G, it is enough to deal with the case where G is the additive group of p-adic integers, for some prime number p. An equivalent form of the conjecture is that has no faithful group action on a topological manifold.
In 1997, Dušan Repovš and Evgenij Ščepin proved the Hilbert–Smith conjecture for groups acting by Lipschitz maps on a Riemannian manifold using covering, fractal, and cohomological dimension theory.[2]
In 1999, Gaven Martin extended their dimension-theoretic argument to quasiconformal actions on a Riemannian manifold and gave applications concerning unique analytic continuation for Beltrami systems.[3]
In 2013, John Pardon proved the three-dimensional case of the Hilbert–Smith conjecture.[4]