{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T21:52:03Z","timestamp":1747173123019,"version":"3.40.5"},"reference-count":22,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2022,1,6]],"date-time":"2022-01-06T00:00:00Z","timestamp":1641427200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Glasgow Math. J."],"published-print":{"date-parts":[[2022,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Given any topological group <jats:italic>G<\/jats:italic>, the topological classification of principal <jats:italic>G<\/jats:italic>-bundles over a finite CW-complex <jats:italic>X<\/jats:italic> is long known to be given by the set of free homotopy classes of maps from <jats:italic>X<\/jats:italic> to the corresponding classifying space <jats:italic>BG<\/jats:italic>. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when <jats:italic>X<\/jats:italic> has dimension 2, there is a case in which such explicit classification has not been explicitly considered. This is the case where <jats:italic>G<\/jats:italic> is a Lie group, whose group of components acts nontrivially on its fundamental group <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089521000434_inline1.png\"\/><jats:tex-math>\n$\\pi_1G$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Here, we deal with this case and obtain the classification, in terms of characteristic classes, of principal <jats:italic>G<\/jats:italic>-bundles over a finite CW-complex of dimension 2, with <jats:italic>G<\/jats:italic> is a Lie group such that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089521000434_inline2.png\"\/><jats:tex-math>\n$\\pi_0G$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is abelian.<\/jats:p>","DOI":"10.1017\/s0017089521000434","type":"journal-article","created":{"date-parts":[[2022,1,6]],"date-time":"2022-01-06T15:25:17Z","timestamp":1641482717000},"page":"675-690","source":"Crossref","is-referenced-by-count":0,"title":["Principal bundles on two-dimensional CW-complexes with disconnected structure group"],"prefix":"10.1017","volume":"64","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4850-6222","authenticated-orcid":false,"given":"Andr\u00e9 G.","family":"Oliveira","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2022,1,6]]},"reference":[{"key":"S0017089521000434_ref10","volume-title":"Princeton Mathematical Series","volume":"38","author":"Lawson","year":"1989"},{"key":"S0017089521000434_ref2","doi-asserted-by":"publisher","DOI":"10.4310\/MRL.2010.v17.n5.a4"},{"key":"S0017089521000434_ref11","first-page":"155","article-title":"Classifying Spaces and Fibrations","volume":"1","author":"May","year":"1975","journal-title":"Mem. 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