On boundedness of characteristic class via quasi-morphism

TL;DR

This paper explores the relationship between bounded characteristic classes and quasi-morphisms, applying these concepts to Hamiltonian fibrations and symplectic manifolds to reveal new insights into their structure and bounded cohomology.

Contribution

It characterizes second bounded characteristic classes via quasi-morphisms and applies this to study boundedness of obstruction classes and the non-triviality of bounded cohomology in symplectic geometry.

Findings

Non-existence of foliated structures on some Hamiltonian fibrations.

Non-triviality of the second bounded cohomology group of Hamiltonian diffeomorphisms.

Characterization of bounded characteristic classes via quasi-morphisms.

Abstract

In this paper, we characterize the second bounded characteristic classes of foliated bundles in terms of the non-descendible quasi-morphisms on the universal covering of the structure group. As its application, we study the boundedness of obstruction classes for (contact) Hamiltonian fibrations and show the non-existence of foliated structures on some Hamiltonian fibrations. Moreover, for any closed symplectic manifold, we show the non-triviality of the second bounded cohomology group of the Hamiltonian diffeomorphism group.