On the moduli space of flat symplectic surface bundles

TL;DR

This paper establishes homological stability for symplectomorphisms of surfaces, constructs an isomorphism with infinite loop space homology, and applies these results to characteristic classes of surface bundles.

Contribution

It introduces a homotopy theoretic approach to study the stable homology of symplectomorphism groups and surface bundle characteristic classes.

Findings

Proves homological stability for symplectomorphisms of surfaces.

Constructs an isomorphism with infinite loop space homology.

Provides a homotopy theoretic proof of the Kotschick-Morita theorem.

Abstract

In this paper, we prove homological stability of symplectomorphisms and extended hamiltonians of surfaces made discrete. We construct an isomorphism from the stable homology group of symplectomorphisms and extended Hamiltonians of surfaces to the homology of certain infinite loop spaces. We use these infinite loop spaces to study characteristic classes of surface bundles whose holonomy groups are area preserving, in particular we give a homotopy theoretic proof of the Kotschick-Morita theorem.