The Hilbert-Smith Conjecture

For every $n\ge 1$, every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected Hausdorff second-countable topological $n$-manifold without boundary is a Lie group. The core theorem is that every continuous $\mathbb{Z}_p$-action on such a manifold has nonzero kernel, by a contradiction between $p^k$ equal integral classes summing to a fixed class and a fixed denominator bound in a Witt-group lattice. Noncompact, nonorientable and nontriangulable manifolds and arbitrary stabilizers are allowed. Manifolds with boundary are not treated.

For every $n\ge 1$, every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected Hausdorff second-countable topological $n$-manifold without boundary is a Lie group. The core theorem is that every continuous $\mathbb{Z}_p$-action on such a manifold has nonzero kernel, by a contradiction between $p^k$ equal integral classes summing to a fixed class and a fixed denominator bound in a Witt-group lattice. Noncompact, nonorientable and nontriangulable manifolds and arbitrary stabilizers are allowed. Manifolds with boundary are not treated.