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| '''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?
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| 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.
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| ==Classic formulation==
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| A formulation that was accepted for a long period was that the question was to characterize Lie groups as the [[topological group]]s that were also [[topological manifold]]s. In terms closer to those that Hilbert would have used, near the [[identity element]] ''e'' of the group ''G'' in question, there is an [[open set]] ''U'' in [[Euclidean space]] containing ''e'', and on some open subset ''V'' of ''U'' there is a [[continuous mapping]]
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| :<math>F : V \times V \rightarrow U </math>
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| that satisfies the [[group axioms]] where those are defined. This much is a fragment of a typical [[manifold|locally Euclidean topological group]]. The problem is then to show that ''F'' is a [[smooth function]] near ''e'' (since topological groups are [[homogeneous space]]s, they look the same everywhere as they do near ''e'').
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| Another way to put this is that the possible [[differentiability class]] of ''F'' does not matter: the group axioms collapse the whole ''C''<sup>''k''</sup> gamut.
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| ==Solution==
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| 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|ref=harv}}</ref> for [[compact group]]s. The [[locally compact abelian group]] case was solved in 1934 by [[Lev Pontryagin]]. The final resolution, at least in this interpretation of what Hilbert meant, came with the work of [[Andrew Gleason]], [[Deane Montgomery]] and [[Leo Zippin]] in the 1950s.
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| In 1953, [[Hidehiko Yamabe]] obtained the final answer<ref>According to {{harvtxt|Morikuni|1961|p=i}}.</ref> to Hilbert’s Fifth Problem: a connected locally compact group {{math|''G''}} is a [[projective limit]] of a sequence of Lie groups, and if {{math|''G''}} "has no small subgroups" (a condition defined below), then ''G'' is a Lie group. However, the question is still debated since in the literature there have been other such claims, largely based on different interpretations of Hilbert's statement of the problem given by various researchers.<ref>For a review of such claims (however completely ignoring the contributions of Yamabe) and for a new one, see {{harvtxt|Rosinger|1998|pp=xiii–xiv and pp. 169–170}}.</ref>
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| More generally, every locally compact, almost connected group is the projective limit of a Lie group. If we consider a general locally compact group {{math|''G''}} and the connected component of the identity {{math|''G''<sub>0</sub>}}, we have a group extension
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| : <math>G_0\rightarrow G\rightarrow G/G_0.\,</math>
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| As a totally disconnected group {{math|''G''}}/{{math|''G''<sub>0</sub>}} has an open compact subgroup, and the pullback {{math|''G''′}} of such an open compact subgroup is an open, almost connected subgroup of {{math|''G''}}. In this way, we have a smooth structure on {{math|''G''}}, since it is homeomorphic to {{math|''G''′ × ''G''′ / ''G''<sub>0</sub>}}, where {{math|''G''′ / ''G''<sub>0</sub>}} is a discrete set.
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| ==Alternate formulation==
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| Another view is that ''G'' ought to be treated as a [[transformation group]], rather than abstractly. This leads to the formulation of the [[Hilbert–Smith conjecture]], unresolved {{As of|2009|lc=on}}.
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| ==No small subgroups==
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| An important condition in the theory is '''[[no small subgroup]]s'''. A topological group ''G'', or a partial piece of a group like ''F'' above, is said to have ''no small subgroups'' if there is a neighbourhood ''N'' of ''e'' containing no subgroup bigger than {''e''}. For example the [[circle group]] satisfies the condition, while the [[p-adic integers]] ''Z''<sub>''p''</sub> as [[Abelian group|additive group]] does not, because ''N'' will contain the subgroups
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| :<math>p^k Z_p</math>
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| for all large integers ''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 ''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.
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| ==Infinite dimensions==
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| Researchers have also considered Hilbert's fifth problem without supposing [[Lie groups#Infinite dimensional Lie groups|finite dimensionality]]. The last chapter of Benyamini and [[Joram Lindenstrauss|Lindenstrauss]] discuss the thesis of [[Per Enflo#Hilbert's fifth problem and embeddings|Per Enflo]], on Hilbert's fifth problem without [[compactness]].
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| ==Notes==
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| {{Reflist|30em}}
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| ==See also==
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| * [[Hans Rådström]]
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| ==References==
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| *{{Cite journal
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| | last = Morikuni
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| | first = Goto
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| | author-link =
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| | title = Hidehiko Yamabe (1923–1960)
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| | journal = Osaka Mathematical Journal
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| | volume = 13
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| | issue = 1
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| | pages = i–ii
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| | year = 1961
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| | url = http://projecteuclid.org/euclid.ojm/1200690171
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| | mr = 0126362
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| | zbl = 0095.00505
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| | ref = harv
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| | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}
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| }}. Available from [[Project Euclid]].
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| *{{Cite book
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| | last = Rosinger
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| | first = Elemér E.
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| | author-link =
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| | title = Parametric Lie Group Actions on Global Generalised Solutions of Nonliear PDE. Including a solution to Hilbert's Fifth Problem
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| | place = Doerdrecht–Boston–London
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| | publisher = [[Kluwer Academic Publishers]]
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| | series = Mathematics and Its Applications
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| | volume = 452
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| | year = 1998
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| | pages = xvii+234
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| | url = http://books.google.com/books?id=xqaRjdjkxvUC&printsec=frontcover#v=onepage&q&f=true
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| | id =
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| | isbn = 0-7923-5232-7
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| | mr = 1658516
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| | zbl = 0934.35003
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| | ref = harv
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| | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}
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| }}.
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| *D. Montgomery and L. Zippin, ''Topological Transformation Groups''
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| * Yamabe, Hidehiko, ''On an arcwise connected subgroup of a Lie group'', Osaka Mathematical Journal v.2, no. 1 Mar. (1950), 13–14.
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| *[[Irving Kaplansky]], ''Lie Algebras and Locally Compact Groups'', Chicago Lectures in Mathematics, 1971.
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| * Benyamini, Yoav and [[Joram Lindenstrauss|Lindenstrauss, Joram]], ''Geometric nonlinear functional analysis'' Colloquium publications, 48. American Mathematical Society.
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| *[[Per Enflo|Enflo, Per]]. (1970) [[Per Enflo#Hilbert.27s_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)
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| **Enflo, Per; 1969a: Topological groups in which multiplication on one side is differentiable or linear. ''Math. Scand.,'' 24, 195–197.
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| <!-- **Enflo, Per; 1969: On the non-existence of uniform homeomorphisms between Lp-spaces.'' Ark. Mat.'' '''8''', 103–105. -->
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| ** {{cite journal
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| |doi=10.1007/BF02589549
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| |author=Per Enflo
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| |title=On the nonexistence of uniform homeomorphisms between L<sub>p</sub> spaces
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| |journal=Ark. Mat.
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| |volume=8
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| |year=1969
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| |pages=103–105
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| |issue=2
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| |ref=harv}}
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| **Enflo, Per; 1969b: On a problem of Smirnov. ''Ark. Math''. '''8''', 107–109.
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| **Enflo, Per; 1970a: Uniform structures and square roots in topological groups I. ''Israel J. Math.'' '''8''', 230–252.
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| **Enflo, Per; 1970b: Uniform structures and square roots in topological groups II. ''Israel J. Math.'' '''8''', 2530–272.
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| <!-- ***Enflo, Per. 1976. Uniform homeomorphisms between Banach spaces. ''Séminaire Maurey-Schwartz (1975—1976), Espaces, <math> L^p </math>, applications radonifiantes et géométrie des espaces de Banach'', Exp. No. 18, 7 pp. Centre Math., École Polytech., Palaiseau. ''MR''0477709 (57 #17222) -->
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| {{Hilbert's problems}}
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| [[Category:Hilbert's problems|#05]]
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| [[Category:Lie groups]]
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| [[Category:Differential structures]]
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