Given any topological group G, the topological classiﬁcation of G-principal bundles over a ﬁnite CW-complex X is long-known to be given by the set of free homotopy classes of maps from X to the corresponding classifying space BG. This classical result has been long-used to provide such classiﬁcation in terms of explicit characteristic classes. However, even when X has dimension 2, it seems there is a case in which such explicit classiﬁcation has not been explicitly considered. This is the case where G is a Lie group, whose group of components acts non-trivially on its fundamental group π_1G. In this note we deal with this case by obtaining the classiﬁcation, in terms of characteristic classes, of G-principal bundles over a ﬁnite CW-complex of dimension 2, with G is a Lie group such that π_0G is abelian.
Year of publication: 2020