Symplectomorphism groups and almost complex structures

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.

24 pages, with minor corrections

Cited by in corpus (15)

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• Moduli space of $J$-holomorphic subvarieties
• Relative family Gromov-Witten invariants and symplectomorphisms
• Stability of the symplectomorphism group of rational surfaces
• Higher Maslov Indices
• Continuous families of Hamiltonian torus actions
• From Smooth to Almost Complex
• Maps with symplectic graphs
• Symplectic isotopy on non-minimal ruled surfaces