continious cohomology of groups of volume-preserving and symplectic diffepmorphisms, measurable transfer and higher asymtotic cycles

TL;DR

This paper develops new cohomology classes called Borel-Bott classes for volume-preserving and symplectomorphism groups, revealing their role in the symplectic topology and the cohomology of the mapping class group.

Contribution

It introduces Borel-Bott classes as real counterparts to Z/ classes and demonstrates their significance in the cohomology of symplectomorphism and mapping class groups.

Findings

Borel-Bott classes constructed for symplectic and volume-preserving diffeomorphisms.

First class's restriction generates the second bounded cohomology of the mapping class group.

New insights into the cohomological structure of symplectomorphism groups.

Abstract

I construct the real counterparts (which I call Borel-Bott classes) of the R/Z classes constructed in "Characteristic classes in symplectic topology", to appear, in the cohomology of volume-preserving and symplectomorhisms of a compact (symplectic) manifold.I show that, for the symplectic action of the mapping class group in the moduli space of stable vector bundles over a Riemann surface, the restriction of the first constructed class from the symplectomorphism group gives a generator for the second (bounded) cohomology of the mapping class group.