Principal bundles on 2-dimensional CW-complexes with disconnected structure group

Abstract

Given any topological group $G$, the topological classification of principal $G$-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 principal $G$-bundles over a finite CW-complex of dimension $2$, with $G$ is a Lie group such that $π_0G$ is abelian.