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

Preprint

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.

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Year of publication: 2020

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