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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Subscript ModifyingAbove alpha With caret"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>α</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\lambda _{\hat \alpha }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant, allows many results of symplectic geometry to be generalized to manifolds which are only <italic>cohomologically symplectic</italic> in the sense that there is a degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> cohomology class which cups to a top class. Furthermore, new results in symplectic geometry also arise from this homotopical approach.