Hilbert's fifth problem

Hilbert's fifth problem, from the Hilbert problems list promulgated in 1900 by David Hilbert, concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathematics; its importance there and in theoretical physics (for example quark theory) grew steadily in the twentieth century. In rough terms, Lie group theory is the common ground of group theory and the theory of topological manifolds. The question Hilbert asked was an acute one of making this precise: is there any difference if a restriction to smooth manifolds is imposed?

The expected answer was in the negative (the classical groups, the most central examples in Lie group theory, are smooth manifolds). This was eventually confirmed in the early 1950s. Since the precise notion of "manifold" was not available to Hilbert, there is room for some debate about the formulation of the problem in contemporary mathematical language.

Classic formulation

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.

olution

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.

Alternate formulation

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.

No small subgroups

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"^"k""Z"_"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.

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.
*Irving Kaplansky, "Lie Algebras and Locally Compact Groups", Chicago Lectures in Mathematics, 1971.