Hilbert's fifth problem

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Hilbert's fifth problem, from the problem-list publicized in 1900 by mathematician 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.

Contents

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

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 look the same everywhere 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 Ck gamut.

Solution

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.

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