On isomorphism classes of locally unitary groups

Abstract

Since the solution of the celebrated Hilbert’s Fifth Problem, one knows that a locally euclidean topological group is a Lie group. In accordance with Hilbert’s original idea of getting rid of the differentiability hypothesis in Lie theory, one can look for a homotopy analogue of this result. Unfortunately the most straightforward idea is false: Hilton, Roitberg and Stasheff have exhibited a topological group having the homotopy type of a compact manifold and which is not homotopy equivalent to any Lie group (see [Sta69]). Nevertheless, the problem of finding a homotopy characterisation of Lie groups remains open and has been intensively investigated for the last three decades. Obviously, the first step of such a programme is a better understanding of the homotopy properties of Lie groups. One has to restrict this study to the class of compact connected Lie groups: the Iwasawa decomposition tells us that any connected Lie group has the homotopy type of a compact one.