nLab
Hilbert's fifth problem

Context

Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Statement

Hilbert’s fifth problem, from his famous list of problems in his address to the International Congress of Mathematicians in 1900, is conventionally understood as broadly asking

Which topological groups admit Lie group structures?

A Lie group here is understood as a group object in the category of finite-dimensional smooth manifolds locally modeled on n\mathbb{R}^n. Evidently not all topological groups GG admit Lie group structures (consider for example the compact group of pp-adic integers under addition); a necessary condition is that the underlying space of GG be a topological manifold (or equivalently, using the group structure, that there is a neighborhood of the identity that is homeomorphic to a Euclidean space n\mathbb{R}^n).

Results

Theorem 1 (Gleason; Montgomery-Zippin)

A topological group GG underlies a (unique) Lie group structure if and only if the underlying space of GG is a topological manifold.

A deeper structural theorem from which theorem 1 can be deduced is

Theorem 2 (Gleason-Yamabe)

Let GG be a locally compact group, and let UU be an open neighborhood of the identity in GG. Then there exists an open subgroup VV of GG, and a compact normal subgroup HH of VV contained in UU, such that V/HV/H is isomorphic to a Lie group.

An exposition of this incredible theorem was given in a series of blog posts by Terry Tao; see the reference below for a suitable entry point.

Here are some sample theorems which follow from the Gleason-Yamabe theorem.

Theorem

Suppose GG is a locally compact group. A necessary and sufficient condition for GG is that it satisfies the “no small subgroups” property (NSS for short) that there exist a neighborhood the identity UU so small that it contains no nontrivial subgroups.

Proof

First let us prove that Lie groups GG are NSS. Certainly \mathbb{R} is NSS, and so is n\mathbb{R}^n. This means we can find a starlike neighborhood WW of the origin in Lie(G)Lie(G), small enough so that the exponential map exp:Lie(G)G\exp: Lie(G) \to G maps diffeomorphically onto its image U=exp(W)U = \exp(W), and so that WW contains no line through the origin. Then UU contains the image of no 1-parameter subgroup ϕ:G\phi \colon \mathbb{R} \to G, otherwise we get a line LL:

L=(ϕUexp 1Lie(G))L = (\mathbb{R} \stackrel{\phi}{\to} U \stackrel{\exp^{-1}}{\to} Lie(G))

whose image is contained in WW, and this is a contradiction. Since UU contains no 1-parameter subgroups, it follows easily that UU contains no non-trivial subgroups.

In the other direction, suppose given such a UU. It follows from the Gleason-Yamabe theorem that there is an open subgroup VV of GG that is a Lie group. (To be completed.)

References

  • Terence Tao, Hilbert’s fifth problem and Gleason metrics. What’s new (weblog), June 17, 2011 (link).
Revised on May 10, 2013 18:11:36 by Urs Schreiber (82.169.65.155)