International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 21, Pages 3487-3495
Flat circle bundles, pullbacks, and the circle made
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 John Oprea and Daniel Tanré. 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.