4:00 pm Thursday, April 14, 2016

John Pardon (Stanford)

Abstract: Hilbert's Fifth Problem asks whether every topological group
which is a manifold is in fact a (smooth!) Lie group; this was solved
in the affirmative by Gleason and Montgomery--Zippin. A stronger
conjecture is that a locally compact topological group which acts
faithfully on a manifold must be a Lie group. This is the
Hilbert--Smith Conjecture, which in full generality is still wide
open. It is known, however (as a corollary to the work of Gleason and
Montgomery--Zippin) that it suffices to rule out the case of the
additive group of $p$-adic integers acting faithfully on a manifold.
I will present a solution in dimension three. The proof uses tools
from low-dimensional topology, for example incompressible surfaces,
minimal surfaces, and a property of the mapping class group.