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

We describe low dimensional homology groups of ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{+}^{\delta }{S}^1$ in terms of Haefliger’s classifying space $B{\bar{\mathrm{\Gamma }}}_1$ 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 ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{+}^{\omega ,\delta }{S}^1$ should contain two kinds of many torsion classes which vanish in ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{+}^{\delta }{S}^1$. This is an informal note on our discussions about the above question.