Cohomologically Symplectic Spaces: Toral Actions and the Gottlieb Group

Abstract

Aspects of symplectic geometry are explored from a homotopical viewpoint. In particular, the question of whether or not a given toral action is Hamiltonian is shown to be independent of geometry. Rather, a new homotopical obstruction is described which detects when an action is Hamiltonian. This new entity, the ${\lambda _{\hat \alpha }}$-invariant, allows many results of symplectic geometry to be generalized to manifolds which are only cohomologically symplectic in the sense that there is a degree $2$ cohomology class which cups to a top class. Furthermore, new results in symplectic geometry also arise from this homotopical approach.