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Register a new userOdd sphere bundles, symplectic manifolds, and their intersection theory
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Recently, Tsai-Tseng-Yau constructed new invariants of symplectic manifolds: a sequence of Aoo-algebras built of differential forms on the symplectic manifold. We show that these symplectic Aoo-algebras have a simple topological interpretation. Namely, when the cohomology class of the symplectic form is integral, these Aoo-algebras are equivalent to the standard de Rham differential graded algebra on certain odd-dimensional sphere bundles over the symplectic manifold. From this equivalence, we deduce for a closed symplectic manifold that Tsai-Tseng-Yaus symplectic Aoo-algebras satisfy the Calabi-Yau property, and importantly, that they can be used to define an intersection theory for coisotropic/isotropic chains. We further demonstrate that these symplectic Aoo-algebras satisfy several functorial properties and lay the groundwork for addressing Weinstein functoriality and invariance in the smooth category.
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We show that the exterior derivative operator on a symplectic manifold has a natural decomposition into two linear differential operators, analogous to the Dolbeault operators in complex geometry. These operators map primitive forms into primitive forms and therefore lead directly to the construction of primitive cohomologies on symplectic manifolds. Using these operators, we introduce new primitive cohomologies that are analogous to the Dolbeault cohomology in the complex theory. Interestingly, the finiteness of these primitive cohomologies follows directly from an elliptic complex. We calculate the known primitive cohomologies on a nilmanifold and show that their dimensions can vary depending on the class of the symplectic form.
We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered cohomologies give a two-sided resolution of Lefschetz maps, and thereby, they are directly related to the kernels and cokernels of the Lefschetz maps. We also introduce a novel, non-associative product operation on differential forms for symplectic manifolds. This product generates an A-infinity algebra structure on forms that underlies the filtered cohomologies and gives them a ring structure. As an application, we demonstrate how the ring structure of the filtered cohomologies can distinguish different symplectic four-manifolds in the context of a circle times a fibered three-manifold.
We classify symplectically foliated fillings of certain contact foliated manifolds. We show that up to symplectic deformation, the unique minimal symplectically foliated filling of the foliated sphere cotangent bundle of the Reeb foliation in the 3-sphere is the associated disk cotangent bundle. En route to the proof, we study another foliated manifold, namely the product of a circle and an annulus with almost horizontal foliation. In this case, the foliated unit cotangent bundle does not have a unique minimal symplectic filling. We classify the foliated fillings of this manifold up to symplectic deformation equivalence using combinatorial invariants of the filling.
The paper is devoted to study of Massey products in symplectic manifolds. Theory of generalized and classical Massey products and a general construction of symplectic manifolds with nontrivial Massey products of arbitrary large order are exposed. The construction uses the symplectic blow-up and is based on the author results, which describe conditions under which the blow-up of a symplectic manifold X along its submanifold Y inherits nontrivial Massey products from X ot Y. This gives a general construction of nonformal symplectic manifolds.
We introduce a class of Weinstein domains which are sublevel sets of flexible Weinstein manifolds but are not themselves flexible. These manifolds exhibit rather subtle behavior with respect to both holomorphic curve invariants and symplectic flexibility. We construct a large class of examples and prove that every flexible Weinstein manifold can be Weinstein homotoped to have a nonflexible sublevel set.
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