Editing Hilbert's fifth problem (section)

== Formulation of the problem ==
A modern formulation of the problem (in its simplest interpretation) is as follows:{{sfn|Tao|2014|loc=Theorem 1.1.13}}

{{block indent|Let {{math|''G''}} be a [[topological group]] that is also a [[topological manifold]] (that is, [[locally homeomorphic]] to a [[Euclidean space]]). Does it follow that {{math|''G''}} must be [[Topological group#Homomorphisms|isomorphic]] (as a topological group) to a [[Lie group]]?}}

An equivalent formulation of this problem closer to that of Hilbert, in terms of composition laws, goes as follows:<ref>Hilbert, David. [https://en.wikisource.org/wiki/Mathematical_Problems "5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group"]. ''Mathematical Problems'' – via Wikisource.</ref>

{{block indent|Let {{math|''V'' ⊆ ''U''}} be open subsets of Euclidean space, such that there is a continuous function {{math|''f'' : ''V'' × ''V'' → ''U''}} satisfying the group axiom of [[associativity]]. Does it follow that {{math|''f''}} must be [[smooth function|smooth]] ([[up to]] continuous reparametrisation)?}}

In this form the problem was solved by Montgomery–Zippin and Gleason.

A stronger interpretation (viewing {{math|''G''}} as a [[transformation group]] rather than an abstract group) results in the [[Hilbert–Smith conjecture]] about [[group action]]s on manifolds, which in full generality is still open. It is known classically for actions on 2-dimensional manifolds and has recently been solved for three dimensions by [[John Pardon]].<ref name="Pardon5th">{{cite journal | last=Pardon | first=John | title=The Hilbert–Smith conjecture for three-manifolds | journal=Journal of the American Mathematical Society | volume=26 | issue=3 | date=19 March 2013 | issn=0894-0347 | doi=10.1090/S0894-0347-2013-00766-3 | doi-access=free | pages=879–899 | url=https://www.ams.org/jams/2013-26-03/S0894-0347-2013-00766-3/S0894-0347-2013-00766-3.pdf | access-date=12 February 2025}}</ref>