Math 2.0

Manifolds and cell complexes Covering spaces and transformation groups Conjecture

Hilbert–Smith conjecture

Builds on Locally compact groupLie groupTopological manifoldFaithful action

Statement

Let MM be a connected topological manifold and let GG be a locally compact group. Suppose that GG acts on MM continuously and faithfully. That is, the map G×M→MG \times M \to M is continuous, and only the identity of GG fixes every point of MM. Then GG is a Lie group.

Remarks

The conjecture is a form of Hilbert's fifth problem, which Hilbert stated for transformation groups; it is named after David Hilbert and Paul A. Smith. It is known to be equivalent to the claim that, for every prime pp, the group ℤp\Z_p of pp-adic integers cannot act continuously and faithfully on a connected manifold. The conjecture is proved for manifolds of dimension at most 33; the case of dimension 33 is due to John Pardon. It is also proved for actions by Lipschitz maps and for actions by diffeomorphisms. In general it is open.

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Faithful action, concept, no evidence levelFaithful actionconceptLie group, concept, no evidence levelLie groupconceptLocally compact group, concept, no evidence levelLocally compact groupconceptTopological manifold, concept, no evidence levelTopological manifoldconceptHilbert–Smith conjecture, conjecture, no evidence levelHilbert–Smith conject…conjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Faithful action concept
  2. Lie group concept
  3. Locally compact group concept
  4. Topological manifold concept
  5. Hilbert–Smith conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert%E2%80%93Smith_conjecture
About this page
TagDS2AS9TSZP
URIm2://DS2AS9TSZP
Kindconjecture
WritingW0 stub
MSC57S05, 57S10, 22A05
StatusThis copy of the site has no build data. Run m2 status DS2AS9TSZP.
AboutLocally compact group, Topological manifold
UsesLocally compact group, Lie group, Topological manifold, Faithful action
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