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 homo-topical obstruction is described which detects when an action is Hamiltonian. This new entity, the AA-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.

Homotopy theory and circle actions on symplectic manifolds

“…More interesting applications may be found in [LO2]. For example, we mention the following result which is a mixture of the ideas presented above and work of Gottlieb [G1].…”

“…But then we have a free (so without fixed points) circle action with ι * (ν S 1 ) = α = 0 since H 1 (M ) = π/ [π, π]. This contradicts the fact due to Ono [On1] (and generalized in 78 J. OPREA [LO2]) that, in the presence of the Lefschetz type condition, a symplectic circle action S 1 × M → M on a closed symplectic manifold M has fixed points if and only if the orbit map (restricting the action to a fixed m 0 ∈ M ) ι : S 1 → M is trivial on homology. Hence, the non-abelian hypothesis is incorrect and M is a torus.…”

“…Frankel [Fr] proved that if M is Kähler, then an action is Hamiltonian if and only if there are fixed points. This has been generalized to the case of Lefschetz type by Ono [On1] and to the case of c-symplectic manifolds of Lefschetz type in [LO2]. For the notion of Lefschetz type, see §3 below.…”

“…Theorem 3.7 ( [BG], [Mc3], [LO1], [LO2]). A symplectic nilmanifold M of Lefschetz type is diffeomorphic to a torus.…”

Dolbeault homotopy theory and compact nilmanifolds

“…In fact, since the minimal model (Λ * g * , d) is formal if and only if g is Abelian [16], a well known result of Deligne, Griffiths, Morgan and Sullivan [12] implies that Γ\G has no Kähler structure unless it is a torus (see also [6,18,19]). However, many compact nilmanifolds have symplectic structures [7,28].…”