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

TL;DR

This paper explores the homology of diffeomorphism groups of the circle, examines the vanishing of the rational Euler class in real analytic flat $S^1$-bundles, and discusses torsion classes in these groups.

Contribution

It relates low dimensional homology of diffeomorphism groups to Haefliger's classifying space and investigates the behavior of the Euler class in real analytic flat bundles.

Findings

Homology groups of $ ext{Diff}^ullet_+S^1$ described via Haefliger's space.

If the rational Euler class vanishes, certain torsion classes must appear in the homology of real analytic diffeomorphisms.

Discussion on the potential existence of torsion classes in $ ext{Diff}_+^{ ext{ω},ullet} S^1$.

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.