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International Journal of Mathematics and Mathematical Sciences
Volume 2005, Issue 21, Pages 3487-3495

Flat circle bundles, pullbacks, and the circle made discrete

1Department of Mathematics, Cleveland State University, Cleveland 44115-2214, OH, USA
2Département de Mathématiques, Université de Lille 1, Villeneuve d'Ascq Cedex 59655, France

Received 15 December 2004; Revised 5 June 2005

Copyright © 2005 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We use basic homotopical methods applied to Lie groups made discrete to prove that the real Euler class of a circle bundle vanishes if and only if the bundle is flat.