Hilbert–Smith conjecture

The Hilbert–Smith conjecture asserts that a locally compact Hausdorff group acting faithfully and continuously on a connected finite-dimensional topological manifold must be a Lie group. Descended from Hilbert's fifth problem, the question crystallized in the transformation-group theory developed around 1940, with the treatise of Montgomery and Zippin as the classical reference [MontgomeryZippin1955TransformationGroups].

By a standard structure-theoretic reduction, the whole conjecture is equivalent to one crisp statement: the additive group $\mathbb Z_p$ of $p$-adic integers admits no faithful continuous action on such a manifold. The conjecture is known in dimensions up to three — the three-dimensional case was proved by Pardon [Pardon2013HilbertSmith] — and it holds under additional regularity hypotheses on the action. Recent work has also pursued variants in other geometric categories, including a symplectic form of the conjecture [ShelukhinSun2024SymplecticHilbertSmith].

A resolution requires ruling out faithful $\mathbb Z_p$-actions in full generality — or producing one, which would yield a non-Lie locally compact transformation group and refute the conjecture. For arbitrary continuous actions on manifolds of dimension at least four, the problem remains open.