Math 2.0

Topological groups, Lie groups Topological groups: basic notions Problem

Hilbert's fifth problem

Builds on Topological groupLie groupTopological manifold

Problem

In its standard modern reading, the problem asks the following. Let GG be a topological group whose underlying space is a topological manifold. Is GG necessarily a Lie group? That is, does GG admit a smooth structure, compatible with its topology, for which multiplication and inversion are smooth?

Remarks

The answer is yes. It was obtained in 1952 by Andrew Gleason and by Deane Montgomery and Leo Zippin. Hidehiko Yamabe later extended their methods to all locally compact groups, giving the Gleason–Yamabe theorem. A key step is the notion of a group with no small subgroups: a locally compact group with this property is a Lie group. Hilbert posed the problem in his 1900 list of problems, in terms of continuous transformation groups and without assuming differentiability. The version for groups acting on manifolds, rather than groups that are manifolds, is the Hilbert–Smith conjecture, which is still open.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Lie group, concept, no evidence levelLie groupconceptTopological group, concept, no evidence levelTopological groupconceptTopological manifold, concept, no evidence levelTopological manifoldconceptHilbert's fifth problem, problem, no evidence levelHilbert's fifth probl…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Lie group concept
  2. Topological group concept
  3. Topological manifold concept
  4. Hilbert's fifth problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert%27s_fifth_problem
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TagNR70GHZ37D
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AboutLie group, Topological group
UsesTopological group, Lie group, Topological manifold
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Changed2 November 2026

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