Editing Hilbert's fifth problem

{{Short description|Problem in Lie group theory}}
'''Hilbert's fifth problem''' is the fifth mathematical problem from the [[Hilbert problems|problem list]] publicized in 1900 by mathematician [[David Hilbert]], and concerns the characterization of [[Lie group]]s.

The theory of Lie groups describes [[continuous symmetry]] in mathematics; its importance there and in [[theoretical physics]] (for example [[quark theory]]) grew steadily in the twentieth century. In rough terms, Lie group theory is the common ground of [[group theory]] and the theory of [[topological manifold]]s. The question Hilbert asked was an acute one of making this precise: is there any difference if a restriction to [[smooth manifold]]s is imposed?

The expected answer was in the negative (the [[classical group]]s, the most central examples in Lie group theory, are smooth manifolds). This was eventually confirmed in the early 1950s. Since the precise notion of "manifold" was not available to Hilbert, there is room for some debate about the formulation of the problem in contemporary mathematical language.

== 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>

== Solution ==
The first major result was that of [[John von Neumann]] in 1933,<ref>{{cite journal|last=John|first=von Neumann|title=Die Einführung analytischer parameter in topologischen Gruppen|journal=Annals of Mathematics |year=1933 |volume=34 |pages=170–190 |doi=10.2307/1968347|issue=1|jstor=1968347}}</ref> giving an affirmative answer for [[compact group]]s. The [[locally compact abelian group]] case was solved in 1934 by [[Lev Pontryagin]]. The final resolution, at least in the interpretation of what Hilbert meant given above, came with the work of [[Andrew Gleason]], [[Deane Montgomery]] and [[Leo Zippin]] in the 1950s.

In 1953, [[Hidehiko Yamabe]] obtained further results about topological groups that may not be manifolds:{{efn|According to {{harvtxt|Morikuni|1961|p=i}}, "the final answer to Hilbert’s Fifth Problem"; however this is not so clear since there have been other such claims, based on different interpretations of Hilbert's statement of the problem given by various researchers. For a review of such claims (ignoring the contributions of Yamabe) see {{harvtxt|Rosinger|1998|pp=xiii–xiv and pp. 169–170}}}}

{{block indent|Every locally compact [[connected space|connected]] group is the [[projective limit]] of a sequence of Lie groups. Further, it is a Lie  group if it has no small subgroups.}}

It follows that every locally compact group contains an open subgroup that is a projective limit of Lie groups, by [[Totally disconnected group|van Dantzig's theorem]] (this last statement is called the Gleason–Yamabe Theorem in {{harvtxt|Tao|2014|loc=Theorem 1.1.17}}).

== No small subgroups ==
An important condition in the theory is '''[[no small subgroup]]s'''.  A topological group {{math|''G''}}, or a partial piece of a group like {{math|''F''}} above, is said to have ''no small subgroups'' if there is a neighbourhood {{math|''N''}} of {{math|''e''}} containing no subgroup bigger than {{math|{{mset|''e''}}}}. For example, the [[circle group]] satisfies the condition, while the [[p-adic integers|{{math|''p''}}-adic integers]] {{math|'''Z'''<sub>''p''</sub>}} as [[Abelian group|additive group]] does not, because {{math|''N''}} will contain the subgroups: {{math|''p<sup>k</sup>''&thinsp;'''Z'''<sub>''p''</sub>}}, for all large integers {{math|''k''}}. This gives an idea of what the difficulty is like in the problem. In the Hilbert–Smith conjecture case it is a matter of a known reduction to whether {{math|'''Z'''<sub>''p''</sub>}} can act faithfully on a [[closed manifold]]. Gleason, Montgomery and Zippin characterized Lie groups amongst [[locally compact group]]s, as those having no small subgroups.

== Infinite dimensions ==
Researchers have also considered Hilbert's fifth problem without supposing [[Lie groups#Infinite dimensional Lie groups|finite dimensionality]]. This was the subject of [[Per Enflo]]'s doctoral thesis; his work is discussed in {{harvtxt|Benyamini|Lindenstrauss|2000|loc=Chapter 17}}.

== See also ==
* [[Totally disconnected group]]

== Notes ==
{{notelist}}
{{reflist}}

== References ==
{{refbegin}}
* {{cite journal
  | last = Morikuni
  | first = Goto
  | title = Hidehiko Yamabe (1923–1960)
  | journal = Osaka Mathematical Journal
  | volume = 13
  | issue = 1
  | pages = i–ii
  | year = 1961
  | url = http://projecteuclid.org/euclid.ojm/1200690171
  | mr = 0126362
  | zbl = 0095.00505
}}
* {{cite book
  | last = Rosinger
  | first = Elemér E.
  | title = Parametric Lie Group Actions on Global Generalised Solutions of Nonliear PDE. Including a solution to Hilbert's Fifth Problem
  | place = Doerdrecht–Boston–London
  | publisher = [[Kluwer Academic Publishers]]
  | series = Mathematics and Its Applications
  | volume = 452
  | year = 1998
  | pages = xvii+234
  | url = https://books.google.com/books?id=xqaRjdjkxvUC
  | isbn = 0-7923-5232-7
  | mr = 1658516
  | zbl = 0934.35003
}}
* {{cite book | last=Tao|first= Terence |authorlink = Terence Tao|title=Hilbert's fifth problem and related topics | zbl=1298.22001 |series=Graduate Studies in Mathematics |number=153 | publisher=American Mathematical Society | isbn=978-1-4704-1564-8 | pages=xiii+338 | year=2014}} 
* {{cite book| last1=Montgomery| first1=Deane |first2= Leo|last2= Zippin |title=Topological Transformation Groups | series=Interscience Tracts in Pure and Applied Mathematics | number=1 | publisher=Interscience Publishers | pages=281 | year=1955}}
* Yamabe, Hidehiko, ''On an arcwise connected subgroup of a Lie group'', Osaka Mathematical Journal v.2, no. 1 Mar. (1950), &nbsp;13–14.
* [[Irving Kaplansky]], ''Lie Algebras and Locally Compact Groups'', Chicago Lectures in Mathematics, 1971.
* {{cite book | last1=Benyamini |first1= Yoav |first2=Joram |last2=Lindenstrauss| author-link2=Joram Lindenstrauss |title=Geometric nonlinear functional analysis | series= Colloquium publications |number= 48 |publisher= American Mathematical Society | year=2000}}
* [[Per Enflo|Enflo, Per]]. (1970) [[Per Enflo#Hilbert's fifth problem and embeddings|Investigations on Hilbert’s fifth problem for non locally compact groups]]. (Ph.D. thesis of five articles of [[Per Enflo|Enflo]] from 1969 to 1970)
** Enflo, Per; 1969a: Topological groups in which multiplication on one side is differentiable or linear. ''[[Math. Scand.]],'' 24, &nbsp;195–197.
<!-- **Enflo, Per; 1969: On the non-existence of uniform homeomorphisms between Lp-spaces. ''[[Arkiv för Matematik|Ark. Mat.]]'' '''8''', 103–105. -->
** {{cite journal
  |doi=10.1007/BF02589549
  |author=Per Enflo
  |title=On the nonexistence of uniform homeomorphisms between L<sub>p</sub> spaces
  |journal=[[Arkiv för Matematik|Ark. Mat.]]
  |volume=8
  |year=1969
  |pages=103–105
  |issue=2
  |doi-access=free
}}
** Enflo, Per; 1969b: On a problem of Smirnov. ''Ark. Mat.'' '''8''', &nbsp;107–109.
** {{cite journal|doi=10.1007/BF02771560|doi-access=|title=Uniform structures and square roots in topological groups|journal=[[Israel Journal of Mathematics]]|volume=8|issue=3|pages=230–252|year=1970|last1=Enflo|first1=P.|s2cid=189773170}}
** {{cite journal|doi=10.1007/BF02771561|doi-access=|title=Uniform structures and square roots in topological groups|journal=[[Israel Journal of Mathematics]]|volume=8|issue=3|pages=253–272|year=1970|last1=Enflo|first1=P.|s2cid=121193430}}
<!-- *** Enflo, Per. 1976. Uniform homeomorphisms between Banach spaces. ''Séminaire Maurey-Schwartz (1975–1976), Espaces, {{math|''L''<sup>''p''</sup>}}, applications radonifiantes et géométrie des espaces de Banach'', Exp. No. 18, 7 pp.&nbsp;Centre Math., École Polytech., Palaiseau. ''MR''0477709 (57 #17222) -->
{{refend}}

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