Let S be a surface with a metric of constant curvature and finite area. Is there a classification of the circle bundles over S?
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1$\begingroup$ What do you mean by "classification"? $\endgroup$– Igor RivinCommented May 10, 2014 at 0:58
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1$\begingroup$ And what do you mean by "circle bundle"? The true bundles are classified by $H^2$, with metric and area completely irrelevant. $\endgroup$– Alex DegtyarevCommented May 10, 2014 at 5:57
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$\begingroup$ I mean a bundle whose the fibers are circles. By classification i mean one way of distinguish the total spaces. $\endgroup$– Vanderson LimaCommented May 10, 2014 at 18:14
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2 Answers
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answered May 10, 2014 at 0:59
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Flat orientable circle bundles (topological, not necessarily linear) are classified by homeomorphisms of the fundamental group to $homeo^+(S^1)$, and these are in a weak sense (up to topological semiconjugacy) classified by their bounded Euler class as an element in $H^2_b(S,Z)$, the second bounded cohomology with integer coefficients. This group fits into a sequence $Hom(\pi_1S,R/Z)\to H^2_b(S,Z)\to H_2^b(S,R)$ and $H_b^2(S,R)$ is infinite-dimensional by Brooks-Series.
See Ghys' Survey Paper "Groups acting on the circle".
