Hilbert–Smith conjecture

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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.

A proof of the conjecture was announced in 2002 by Louis McAuley, but it has not been accepted by the mathematical community. The conjecture is still widely considered to be open.

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.