The Hilbert-Smith conjecture asserts that a locally compact topological group admitting a faithful continuous group action on a connected space that is a topological manifold is a Lie group. Faithfulness means that only the identity element acts trivially on the entire topological manifold; individual points may have nontrivial stabilizers. Pardon (2013) proved the three-dimensional case.
Hilbert-Smith Conjecture
See also
Group Action, Lie Group, Topological Group, Topological ManifoldExplore with Wolfram|Alpha
References
Pardon, J. "The Hilbert-Smith Conjecture for Three-Manifolds." J. Amer. Math. Soc. 26, 879-899, 2013. https://doi.org/10.1090/S0894-0347-2013-00766-3.Cite this as:
Weisstein, Eric W. "Hilbert-Smith Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Hilbert-SmithConjecture.html