Hilbert's fifth problem: Review (Q1848082)

The author gives an overview of results concerning Hilbert's fifth problem which can be stated as follows: Let \(G\) be a locally Euclidean topological group, let \(M\) be a locally Euclidean topological space, i.e., \(M\) is a topological manifold, and let a continuous action \(\Phi:G\times M\to M\) of \(G\) on \(M\) be given. Is it then always possible to choose the local coordinates in \(G\) and \(M\) in such a way that the action \(\Phi\) becomes real analytic? In other words, is it possible to give the topological manifolds \(G\) and \(M\) real analytic structures such that \(\Phi\) becomes real analytic?