The Hilbert--Smith conjecture for three-manifolds

TL;DR

This paper proves that any faithful action of a locally compact group on a connected three-manifold must be by a Lie group, confirming the Hilbert--Smith conjecture in this setting through topological and group-theoretic analysis.

Contribution

It establishes the Hilbert--Smith conjecture for three-manifolds by showing no faithful actions of p-adic integers exist on such manifolds, using local topological methods.

Findings

No faithful $ ext{Z}_p$ actions on 3-manifolds exist.

Incompressible surfaces fixed by $ ext{Z}_p$ lead to a contradiction.

The approach is local on the manifold.

Abstract

We show that every locally compact group which acts faithfully on a connected three-manifold is a Lie group. By known reductions, it suffices to show that there is no faithful action of $\mathbb Z_p$ (the $p$-adic integers) on a connected three-manifold. If $\mathbb Z_p$ acts faithfully on $M^3$, we find an interesting $\mathbb Z_p$-invariant open set $U\subseteq M$ with $H_2(U)=\mathbb Z$ and analyze the incompressible surfaces in $U$ representing a generator of $H_2(U)$. It turns out that there must be one such incompressible surface, say $F$, whose isotopy class is fixed by $\mathbb Z_p$. An analysis of the resulting homomorphism $\mathbb Z_p\to\operatorname{MCG}(F)$ gives the desired contradiction. The approach is local on $M$.