  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.
