Odd Sphere Bundles, Symplectic Manifolds, and Their Intersection Theory

Hiro Lee Tanaka Harvard University Li-Sheng Tseng University of California, Irvine

Symplectic Geometry mathscidoc:1702.34002

Recently, Tsai-Tseng-Yau constructed new invariants of symplectic manifolds: a sequence of A-infinity algebras built of differential forms on the symplectic manifold. We show that these symplectic A-infinity algebras have a simple topological interpretation. Namely, when the cohomology class of the symplectic form is integral, these A-infinity 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-Yau's symplectic A-infinity 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 A-infinity algebras satisfy several functorial properties and lay the groundwork for addressing Weinstein functoriality and invariance in the smooth category.
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@inproceedings{hiroodd,
  title={Odd Sphere Bundles, Symplectic Manifolds, and Their Intersection Theory},
  author={Hiro Lee Tanaka, and Li-Sheng Tseng},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170219072026626115472},
}
Hiro Lee Tanaka, and Li-Sheng Tseng. Odd Sphere Bundles, Symplectic Manifolds, and Their Intersection Theory. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170219072026626115472.
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