This presentation will give an overview of the history leading to the present status of the Hilbert-Smith conjecture.

Hilbert’s problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics.  The classical formulation of Hilbert’s fifth problem asks whether topological groups that have the topological structure of a manifold, are necessarily Lie groups. This is indeed, the case, thanks to following theorem of Gleason and Montgomery-Zippin:

Theorem 1 (Hilbert’s fifth problem): Let Γ be a topological group which is locally Euclidean. Then Γ is isomorphic to a Lie group.

The Hilbert-Smith conjecture is a generalization of Hilbert’s fifth problem, in which a space acted on by the group has the manifold structure, rather than the group itself:

Conjecture 2 (Hilbert-Smith conjecture): Let Γ be a locally compact topological group which acts continuously and faithfully (or effectively) on a connected finite-dimensional manifold M. Then Γ is isomorphic to a Lie group.

There are several special cases for which the conjecture has been proved. For manifolds of dimension 3 or less it is known the Conjecture is true.

Next month we will discuss current evidence for and against the conjecture holding for higher dimensional manifolds. Of course dimension 4 likely holds key information.