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
A Lie group here is understood as a group object in the category of finite-dimensional smooth manifolds locally modeled on . Evidently not all topological groups admit Lie group structures (consider for example the compact group of -adic integers under addition); a necessary condition is that the underlying space of 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 ).
A topological group underlies a (unique) Lie group structure if and only if the underlying space of is a topological manifold.
A deeper structural theorem from which theorem 1 can be deduced is
Let be a locally compact group, and let be an open neighborhood of the identity in . Then there exists an open subgroup of , and a compact normal subgroup of contained in , such that 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.
Suppose is a locally compact group. A necessary and sufficient condition for is that it satisfies the “no small subgroups” property (NSS for short) that there exist a neighborhood the identity so small that it contains no nontrivial subgroups.
First let us prove that Lie groups are NSS. Certainly is NSS, and so is . This means we can find a starlike neighborhood of the origin in , small enough so that the exponential map maps diffeomorphically onto its image , and so that contains no line through the origin. Then contains the image of no 1-parameter subgroup , otherwise we get a line :
whose image is contained in , and this is a contradiction. Since contains no 1-parameter subgroups, it follows easily that contains no non-trivial subgroups.
In the other direction, suppose given such a . It follows from the Gleason-Yamabe theorem that there is an open subgroup of that is a Lie group. (To be completed.)