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

Part of: Fiber spaces and bundles Operations and obstructions

Published online by Cambridge University Press:  06 January 2022

André G. Oliveira Open the ORCID record for André G. Oliveira [Opens in a new window]
André G. Oliveira*
Affiliation:
Centro de Matemática da Universidade do Porto, CMUP Faculdade de Ciências, Universidade do Porto Rua do Campo Alegre 687, 4169-007 Porto, Portugalwww.fc.up.pt email: [email protected]://sites.google.com/view/aoliveira On leave from: Departamento de Matemática, Universidade de Trás-os-Montes e Alto Douro, UTAD Quinta dos Prados, 5000-911 Vila Real, Portugalwww.utad.pt email: [email protected]
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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, 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 nontrivially on its fundamental group $\pi_1G$ . Here, we deal with this case and obtain 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 $\pi_0G$ is abelian.

Keywords

Principal bundlescharacteristic classesCW-complexes

MSC classification

Primary: 55R15: Classification 55S35: Obstruction theory
Secondary: 55R10: Fiber bundles 55S45: Postnikov systems, $k$-invariants
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 3 , September 2022 , pp. 675 - 690
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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