nLab
Hilbert's fifth problem

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 ${ℝ}^n$. Evidently not all topological groups $G$ admit Lie group structures (consider for example the compact group of $p$-adic integers under addition); a necessary condition is that the underlying space of $G$ 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 ${ℝ}^n$).

Results

Theorem 1 (Gleason; Montgomery-Zippin)

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

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

Theorem 2 (Gleason-Yamabe)

Let $G$ be a locally compact group, and let $U$ be an open neighborhood of the identity in $G$. Then there exists an open subgroup $V$ of $G$, and a compact normal subgroup $H$ of $V$ contained in $U$, such that $V/H$ 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 $G$ is a locally compact group. A necessary and sufficient condition for $G$ is that it satisfies the “no small subgroups” property (NSS for short) that there exist a neighborhood the identity $U$ so small that it contains no nontrivial subgroups.

Proof

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

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

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

In the other direction, suppose given such a $U$. It follows from the Gleason-Yamabe theorem that there is an open subgroup $V$ of $G$ 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).