Talk:Circle bundle

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Seems to me like we should move this page to principal circle bundle. Shouldn't circle bundle just refer to a fiber bundle where the fibers are circles (a special case of a sphere bundle). In that case, not every circle bundle is principal. For example the Klein bottle can be viewed as a (nontrivial) circle bundle over the circle but the structure group is only Z/2Z. -- Fropuff 03:26, 10 October 2006 (UTC)

Do you know if each circle bundle is principal apart from that example? --kiddo 00:16, 29 October 2006 (UTC)

No, one can construct numerous examples using the associated bundle construction. Let G be a topological group which has a effective action on the circle that is not equivalent to left translations of the circle. Any dihedral group for example. Given a principal G-bundle over any space X one can then construct an associated circle bundle over X which is not a principal circle bundle. The Klein bottle is the simplest example. Here G = Z/2 and the associated principal bundle is the nontrivial double cover of the circle by itself. -- Fropuff 04:41, 31 October 2006 (UTC)