The Hilbert-Smith conjecture concerns the transformation groups of manifolds, in particular the limitations on which topological groups can act effectively and continuously on a manifold. Restricted to locally compact groups with a continuous, faithful action, the conjecture states that such a group must be a Lie group; an equivalent form states that the additive group of p-adic integers has no faithful group action on a topological manifold. Partial cases have been proved: Dusan Repovs and Evgenij Scepin proved it in 1997 for groups acting by Lipschitz maps on a Riemannian manifold, Gaven Martin extended this to quasiconformal actions in 1999, and John Pardon proved the three-dimensional case in 2013, but the general conjecture remains open. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementThe Hilbert-Smith conjecture states that if a locally compact topological group G acts faithfully and continuously on a connected manifold M, then G must be a Lie group. 1 Progress Toward ResolutionJohn Pardon proved the three-dimensional case of the Hilbert-Smith conjecture in 2013; the general conjecture remains open. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
Sources
1. Hilbert-Smith Conjecture (Wikipedia)
Wikimedia FoundationLead section
In 2013, John Pardon proved the three-dimensional case of the Hilbert-Smith conjecture.
Lead section, second sentence
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.
Results section
In 2013, John Pardon proved the three-dimensional case of the Hilbert-Smith conjecture.
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