Flat structures on surface bundles

TL;DR

This paper explores flat surface bundles with non-trivial normal bundles, computes the Abelianisation of surface diffeomorphism groups with marked points, and extends formulas relating Euler classes and MMM-classes to higher genus surfaces.

Contribution

It introduces new examples of flat surface bundles with closed leaves and non-trivial normal bundles, and generalizes existing formulas to higher genus surfaces.

Findings

Existence of flat surface bundles with non-trivial normal bundles.

Computed Abelianisation of surface diffeomorphism groups with marked points.

Extended formulas for Euler class and MMM-class to higher genus surfaces.

Abstract

We show that there exist flat surface bundles with closed leaves having non-trivial normal bundles. This leads us to compute the Abelianisation of surface diffeomorphism groups with marked points. We also extend a formula of Tsuboi that expresses the Euler class of a flat circle bundle in terms of the Calabi invariant of certain Hamiltonian diffeomorphisms to surfaces of higher genus and derive a similar formula for the first MMM-class of surface bundles with punctured fibre.