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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. 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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  • Hilbert–Smith conjecture
  • 希尔伯特-史密斯猜想
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  • 數學上的希爾伯特-史密斯猜想,是關於流形的變換群,特別是忠實地作用在一個拓撲流形上的拓撲群的限制。這猜想說若一個的拓撲群G有一個連續且忠實的群作用在拓撲流形M上,則G必定是一個李群。 基於G的結構的已知結果,僅需證明當G是p進數Zp的加法群時(p是素數),G無忠實的群作用在拓撲流形上。 這個猜想以大衛·希爾伯特和美國拓撲學家命名。有些人認為這個猜想是對希爾伯特第五問題更好的表述。 這猜想的一般情形現在仍未解決。2013年,證明了這猜想對三維流形的情形成立。
  • 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. 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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  • 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. 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. 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 2013, John Pardon proved the three-dimensional case of the Hilbert–Smith conjecture.
  • 數學上的希爾伯特-史密斯猜想,是關於流形的變換群,特別是忠實地作用在一個拓撲流形上的拓撲群的限制。這猜想說若一個的拓撲群G有一個連續且忠實的群作用在拓撲流形M上,則G必定是一個李群。 基於G的結構的已知結果,僅需證明當G是p進數Zp的加法群時(p是素數),G無忠實的群作用在拓撲流形上。 這個猜想以大衛·希爾伯特和美國拓撲學家命名。有些人認為這個猜想是對希爾伯特第五問題更好的表述。 這猜想的一般情形現在仍未解決。2013年,證明了這猜想對三維流形的情形成立。
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