Homotopy theory and circle actions on symplectic manifolds

Abstract

Introduction. Traditionally, soft techniques of algebraic topology have found much use in the world of hard geometry. In recent years, in particular, the subject of symplectic topology (see [McS]) has arisen from important and interesting connections between symplectic geometry and algebraic topology. In this paper, we will consider one application of homotopical methods to symplectic geometry. Namely, we shall discuss some aspects of the homotopy theory of circle actions on symplectic manifolds. Because this paper is meant to be accessible to both geometers and topologists, we shall try to review relevant ideas in homotopy theory and symplectic geometry as we go along. We also present some new results (e.g. see Theorem 2.12 and §5) which extend the methods reviewed earlier. This paper then serves two roles: as an exposition and survey of the homotopical approach to symplectic circle actions and as a first step to extending the approach beyond the symplectic world.