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.
We introduce filtered cohomologies of differential forms on symplectic
manifolds. They generalize and include the cohomologies
discussed in Papers 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.