nLab
Hilbert's fifth problem

Context

Lie theory

∞-Lie theory (higher geometry)

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)