Hilbert–Smith conjecture explained

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 groups G which are locally compact and have a continuous, faithful group action on M, the conjecture 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

\Z _{p} of p-adic integers, for some prime number p. An equivalent form of the conjecture is that \Z _{p} 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 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]

Further reading

• Web site: Tao . Terence . Terence Tao. 2011 . The Hilbert-Smith conjecture . .

Notes and References

1. Book: Smith, Paul A. . Paul A. Smith . Periodic and nearly periodic transformations . Lectures in Topology . R. . Wilder . W . Ayres . . Ann Arbor, MI . 1941 . 159–190 .
2. Dušan . Repovš . Dušan Repovš . Evgenij V. . Ščepin . A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps . . 308 . 2 . June 1997 . 361–364 . 10.1007/s002080050080.
3. Gaven . Martin . The Hilbert-Smith conjecture for quasiconformal actions . Electronic Research Announcements of the American Mathematical Society. 5 . 9 . 1999. 66–70 .
4. John . Pardon . John Pardon . The Hilbert–Smith conjecture for three-manifolds . . 26 . 3 . 2013 . 879–899 . 10.1090/s0894-0347-2013-00766-3. 1112.2324 . 96422853 .