# Odd sphere bundles, symplectic manifolds, and their intersection theory

## TL;DR

This paper reveals that certain symplectic A∞-algebras introduced by Tsai-Tseng-Yau are topologically equivalent to de Rham algebras on odd sphere bundles, enabling intersection theory and functorial analysis in symplectic topology.

## Contribution

It establishes a topological interpretation of symplectic A∞-algebras as de Rham algebras on sphere bundles, linking algebraic invariants to geometric structures.

## Key findings

- A∞-algebras are equivalent to de Rham algebras on sphere bundles when the symplectic form is integral.
- The symplectic A∞-algebras satisfy the Calabi-Yau property for closed symplectic manifolds.
- These algebras can define an intersection theory for coisotropic and isotropic chains.

## Abstract

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-Yau's 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.