Hilbert's 5th problem
Assuming a small amount of background, the authors give full proofs of the results by Gleason, Mongomery-Zippin, and Yamabe that characterize Lie groups from among locally compact topological groups. Assuming that \(G\) is a locally compact topological group the following are equivalent: (1) \(G\) is a Lie group. (2) \(G\) has no small subgroups. (3) \(G\) is locally euclidean. (4) \(G\) is locally connected and has no small connected subgroups. (5) \(G\) is locally connected and bounded in dimension. Every compact group \(G\) has an open subgroup that can be approximated by Lie groups, that is, every neighbourhood of the identity of \(G\) contains a compact normal subgroup \(N\) of \(G\) such that \(G/N\) is a Lie group. The paper also gives an account of the work of Hirschfeld, who used nonstandard analysis to simplify some tricky parts of the work by Gleason and Montgomery, and further simplifies them.
scientific article; zbMATH DE number 6545824
scientific article