Symplectomorphism groups and almost complex structures

TL;DR

This paper explores how the topology of symplectomorphism groups of ruled surfaces varies with the ratio of base to fiber size, revealing stability and change points related to the ratio R.

Contribution

It provides a detailed analysis of the topological changes in symplectomorphism groups as the cohomology class varies, extending previous results for rational ruled surfaces.

Findings

Topological type changes at integer values of R for sphere bases

Stability of the groups as R approaches infinity for general bases

Extension of previous results on rational ruled surfaces

Abstract

This paper studies groups of symplectomorphisms of ruled surfaces for symplectic forms with varying cohomology class. This class is characterized by the ratio R of the size of the base to that of the fiber. By considering appropriate spaces of almost complex structures, we investigate how the topological type of these groups changes as R increases. If the base is a sphere, this changes precisely when R passes an integer, and for general bases it stabilizes as R goes to infinity. Our results extend and make more precise some of the conclusions of Abreu--McDuff concerning the rational homotopy type of these groups for rational ruled surfaces.