Remarks on flat $S^1$-bundles, $C^{\infty }$ vs $C^{\omega }$

Abstract

We describe low dimensional homology groups of ${\mathrm{Diff}}_{+}^{\delta }{S}^1$ in terms of Haefliger's classifying space $B{\overline{\Gamma }}_1$ by applying a theorem of Thurston. Then we consider the question whether some power of the rational Euler class vanishes for real analytic flat $S^1$-bundles. We show that if it occurs, then the homology group of ${\mathrm{Diff}}_{+}^{\omega ,\delta }{S}^1$ should contain two kinds of many torsion classes which vanish in ${\mathrm{Diff}}_{+}^{\delta }{S}^1$. This is an informal note on our discussions about the above question.

21 pages