Hilbert–Smith conjecture: Difference between revisions
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Because of known structural results on ''G'', it is enough to deal with the case where ''G'' is the additive group ''Z<sub>p</sub>'' of [[p-adic integer]]s, for some [[prime number]] ''p''. An equivalent form of the conjecture is that ''Z<sub>p</sub>'' has no faithful group action on a topological manifold. |
Because of known structural results on ''G'', it is enough to deal with the case where ''G'' is the additive group ''Z<sub>p</sub>'' of [[p-adic integer]]s, for some [[prime number]] ''p''. An equivalent form of the conjecture is that ''Z<sub>p</sub>'' has no faithful group action on a topological manifold. |
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The naming of the conjecture is for [[David Hilbert]], and the American topologist [[Paul A. Smith]]. It is considered by some to be a better formulation of [[Hilbert's fifth problem]], than the characterisation in the category of [[topological group]]s of the [[Lie group]]s often cited as a solution. |
The naming of the conjecture is for [[David Hilbert]], and the American topologist [[Paul A. Smith]].<ref name=Smith1941>{{cite book | last = Smith | first = Paul A. | author-link = Paul A. Smith | chapter = Periodic and nearly periodic transformations | title = Lectures in Topology | editor1-first = R. | editor1-last = Wilder | editor2-first = W | editor2-last = Ayres | publisher = University of Michigan Press | place = Ann Arbor, MI | year = 1941 | pages = 159–190 }}</ref> It is considered by some to be a better formulation of [[Hilbert's fifth problem]], than the characterisation in the category of [[topological group]]s of the [[Lie group]]s often cited as a solution. |
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In 1997, [[Dušan Repovš]] and Evgenij Ščepin proved the Hilbert-Smith conjecture for groups acting by Lipschitz maps on a Riemannian manifold using the covering, fractal and cohomological dimension theory. |
In 1997, [[Dušan Repovš]] and Evgenij Ščepin proved the Hilbert-Smith conjecture for groups acting by Lipschitz maps on a Riemannian manifold using the covering, fractal and cohomological dimension theory. |
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| ⚫ | <ref name=Repovs1997>{{cite journal | first = Dušan | last = Repovš | authorlink = Dušan Repovš |first2 = Evgenij V. | last2 = Ščepin | title = A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps | periodical = [[Mathematische Annalen]] | volume = 308 | issue = 2 |date=June 1997 | pages = 361–364 | doi=10.1007/s002080050080}}</ref> |
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In 1999, [[Gaven Martin]] extended their dimension-theoretic argument to quasiconformal actions on a Riemannian manifold and gave applications concerning unique analytic continuation for Beltrami systems. |
In 1999, [[Gaven Martin]] extended their dimension-theoretic argument to quasiconformal actions on a Riemannian manifold and gave applications concerning unique analytic continuation for Beltrami systems.<ref name=Gaven1999> |
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In 2013, [[John Pardon]] proved the three-dimensional case of the Hilbert–Smith conjecture. |
In 2013, [[John Pardon]] proved the three-dimensional case of the Hilbert–Smith conjecture.<ref name=pardon2013> |
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| ⚫ | {{cite journal | first = John | last = Pardon | authorlink = John Pardon | title = The Hilbert–Smith conjecture for three-manifolds | periodical = [[Journal of the American Mathematical Society]] | volume = 26 | issue = 3 |date= 2013 | pages = 879–899 | doi=10.1090/s0894-0347-2013-00766-3| arxiv = 1112.2324 }}</ref> |
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==References== |
==References== |
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{{Reflist}} |
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*{{Citation | last = Smith | first = Paul A. | author-link = Paul A. Smith | chapter = Periodic and nearly periodic transformations | title = Lectures in Topology | editor1-first = R. | editor1-last = Wilder | editor2-first = W | editor2-last = Ayres | publisher = University of Michigan Press | place = Ann Arbor, MI | year = 1941 | pages = 159–190 }} |
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==Further reading== |
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*{{Citation | last = Chu | first = Hsin | year = 1973 | contribution = On the embedding problem and the Hilbert-Smith conjecture | title = Recent Advances in Topological Dynamics | editor1-last = Beck | editor1-first = Anatole | series = Lecture Notes in Mathematics | volume = 318 | publisher = [[Springer-Verlag]] | pages = 78–85 }} |
*{{Citation | last = Chu | first = Hsin | year = 1973 | contribution = On the embedding problem and the Hilbert-Smith conjecture | title = Recent Advances in Topological Dynamics | editor1-last = Beck | editor1-first = Anatole | series = Lecture Notes in Mathematics | volume = 318 | publisher = [[Springer-Verlag]] | pages = 78–85 }} |
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{{DEFAULTSORT:Hilbert-Smith conjecture}} |
{{DEFAULTSORT:Hilbert-Smith conjecture}} |
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Revision as of 10:14, 3 February 2020
 | This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. (October 2016) |
In mathematics, the Hilbert–Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups G that can act effectively (faithfully) on a (topological) manifold M. Restricting to G which are locally compact and have a continuous, faithful group action on M, it states that G must be a Lie group.
Because of known structural results on G, it is enough to deal with the case where G is the additive group Zp of p-adic integers, for some prime number p. An equivalent form of the conjecture is that Zp has no faithful group action on a topological manifold.
The naming of the conjecture is for David Hilbert, and the American topologist Paul A. Smith.[1] It is considered by some to be a better formulation of Hilbert's fifth problem, than the characterisation in the category of topological groups of the Lie groups often cited as a solution.
In 1997, Dušan Repovš and Evgenij Ščepin proved the Hilbert-Smith conjecture for groups acting by Lipschitz maps on a Riemannian manifold using the covering, fractal and cohomological dimension theory. [2]
In 1999, Gaven Martin extended their dimension-theoretic argument to quasiconformal actions on a Riemannian manifold and gave applications concerning unique analytic continuation for Beltrami systems.[3]
In 2013, John Pardon proved the three-dimensional case of the Hilbert–Smith conjecture.[4]
References
- ^ Smith, Paul A. (1941). "Periodic and nearly periodic transformations". In Wilder, R.; Ayres, W (eds.). Lectures in Topology. Ann Arbor, MI: University of Michigan Press. pp. 159–190.
- ^ Repovš, Dušan; Ščepin, Evgenij V. (June 1997). "A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps". Mathematische Annalen. 308 (2): 361–364. doi:10.1007/s002080050080.
- ^
Martin, Gaven (1999). "The Hilbert-Smith conjecture for quasiconformal actions". Electronic Research Announcements of the American Mathematical Society. 5 (9): 66–70.
{{cite journal}}: Cite has empty unknown parameter:|1=(help) - ^ Pardon, John (2013). "The Hilbert–Smith conjecture for three-manifolds". Journal of the American Mathematical Society. 26 (3): 879–899. arXiv:1112.2324. doi:10.1090/s0894-0347-2013-00766-3.
Further reading
- Chu, Hsin (1973), "On the embedding problem and the Hilbert-Smith conjecture", in Beck, Anatole (ed.), Recent Advances in Topological Dynamics, Lecture Notes in Mathematics, vol. 318, Springer-Verlag, pp. 78–85