Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and 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.
A modern formulation of the problem (in its simplest interpretation) is as follows:
An equivalent formulation of this problem closer to that of Hilbert, in terms of composition laws, goes as follows:[2]
Let
V ⊆ U be open subsets of Euclidean space, such that there is a continuous function
f : V × V → U satisfying the group axiom of
associativity. Does it follow that
f must be
smooth (
up to continuous reparametrisation)?
In this form the problem was solved by Montgomery–Zippin and Gleason.
A stronger interpretation (viewing G as a transformation group rather than an abstract group) results in the Hilbert–Smith conjecture about group actions on manifolds, which in full generality is still open. It is known classically for actions on 2-dimensional manifolds and has recently been solved for three dimensions by John Pardon.
An important condition in the theory is no small subgroups. A topological group G, or a partial piece of a group like F above, is said to have no small subgroups 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 Zp as additive group does not, because N will contain the subgroups: pk Zp, 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 Zp can act faithfully on a closed manifold. Gleason, Montgomery and Zippin characterized Lie groups amongst locally compact groups, as those having no small subgroups.
According to Morikuni (1961, p. i), "the final answer to Hilbert’s Fifth Problem"; however this is not so clear since there have been other such claims, based on different interpretations of Hilbert's statement of the problem given by various researchers. For a review of such claims (ignoring the contributions of Yamabe) see Rosinger (1998, pp. xiii–xiv and pp. 169–170)
John, von Neumann (1933). "Die Einführung analytischer parameter in topologischen Gruppen". Annals of Mathematics. 34 (1): 170–190. doi:10.2307/1968347. JSTOR 1968347.
- Morikuni, Goto (1961). "Hidehiko Yamabe (1923–1960)". Osaka Mathematical Journal. 13 (1): i–ii. MR 0126362. Zbl 0095.00505.
- Rosinger, Elemér E. (1998). Parametric Lie Group Actions on Global Generalised Solutions of Nonliear PDE. Including a solution to Hilbert's Fifth Problem. Mathematics and Its Applications. Vol. 452. Doerdrecht–Boston–London: Kluwer Academic Publishers. pp. xvii+234. ISBN 0-7923-5232-7. MR 1658516. Zbl 0934.35003.
- Tao, Terence (2014). Hilbert’s fifth problem and related topics. Graduate Studies in Mathematics. American Mathematical Society. pp. xiii+338. ISBN 978-1-4704-1564-8. Zbl 1298.22001.
- Montgomery, Deane; Zippin, Leo (1955). Topological Transformation Groups. Interscience Tracts in Pure and Applied Mathematics. Interscience Publishers. p. 281.
- Yamabe, Hidehiko, On an arcwise connected subgroup of a Lie group, Osaka Mathematical Journal v.2, no. 1 Mar. (1950), 13–14.
- Irving Kaplansky, Lie Algebras and Locally Compact Groups, Chicago Lectures in Mathematics, 1971.
- Benyamini, Yoav; Lindenstrauss, Joram (2000). Geometric nonlinear functional analysis. Colloquium publications. American Mathematical Society.
- Enflo, Per. (1970) Investigations on Hilbert’s fifth problem for non locally compact groups. (Ph.D. thesis of five articles of Enflo from 1969 to 1970)