Hilbert's fifth problem

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Hilbert's fifth problem, from the Hilbert problems list promulgated in 1900 by David Hilbert, concerns the characterization of Lie groups.

A formulation that was accepted for a long period was that the question was to characterize Lie groups as the topological groups that were also topological manifolds. In terms closer to those that Hilbert would have used, near the identity element e of the group G in question, we have some open set U in Euclidean space containing e, and on some open subset V of U we have a continuous mapping

F:V × V → U

that satisfies the group axioms where those are defined. This much is a fragment of a typical locally Euclidean topological group. The problem is then to show that F is a smooth function near e (since topological groups are homogeneous spaces, they everywhere look the same as they do near e).

Another way to put this is that the possible differentiability class of F doesn't matter: the group axioms collapse the whole C^k gamut.

The first major result was that of John von Neumann in 1929, for compact groups. 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.

In 1953, Hidehiko Yamabe obtained the final answer to Hilbert’s Fifth Problem :A connected locally compact group G is a projective limit of a sequence of Lie groups; and, if G has no small subgroups, then it is a Lie group.

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 2005.

An important condition in the theory is no small subgroups. G, or a partial piece of a group like F above, is said to satisfy the no small subgroups condition 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_p as additive group does not, because N will contain the subgroups

p^kZ_p

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_p can act faithfully on a closed manifold. Gleason, Montgomery and Zippin characterized Lie groups amongst locally compact groups, as those having no small subgroups.

[edit] References

D. Montgomery and L. Zippin, Topological Transformation Groups
Yamabe, Hidehiko, On an arcwise connected subgroup of a Lie group, Osaka Mathematical Journal v.2, no. 1 Mar. (1950) pp.13-14.
I. Kaplansky, Lie Algebras and Locally Compact Groups, Chicago Lectures in Mathematics, 1971.