Hilbert–Smith conjecture

The Hilbert–Smith conjecture states that every locally compact group acting effectively by homeomorphisms on a connected manifold is a Lie group.

Connect

Locally compact group: A topological group in which every point has a compact neighborhood. The conjecture asks which locally compact groups can act effectively on manifolds.
David Hilbert: A German mathematician whose 1900 list of problems shaped twentieth-century mathematics. His fifth problem provided the historical source of the conjecture's name and motivation.
Topological group: A group equipped with a topology for which multiplication and inversion are continuous. Locally compact groups in the conjecture are topological groups with an additional compactness condition.
p-adic group action: An action of a group built from p-adic numbers or integers on a topological space. Proving that p-adic groups cannot act effectively on connected manifolds would settle the conjecture.
Hilbert's fifth problem: David Hilbert's problem asking whether locally Euclidean topological groups are Lie groups. A positive answer would extend Lie-group rigidity from groups with manifold topology to effective manifold actions.
Effective group action: A group action in which only the identity element acts as the identity transformation everywhere. Effectiveness rules out hiding nontrivial group elements in the action's kernel.
John von Neumann: A mathematician who proved a major case of Hilbert's fifth problem for compact groups. His compact-group result helped establish the structural background from which the conjecture emerged.
Manifold: A space that locally resembles Euclidean space. The conjecture concerns group actions on connected manifolds, including topological manifolds.
Smith theory: The study of fixed-point sets of periodic transformations, especially actions of finite p-groups. Its fixed-point methods are among the tools used to investigate possible p-adic actions.
Transformation group theory: The study of groups acting continuously or smoothly on topological spaces and manifolds. The conjecture is a foundational structural question for this field.