Cambridge Journal of Mathematics

Volume 6 (2018)

Number 3

Odd sphere bundles, symplectic manifolds, and their intersection theory

Pages: 213 – 266



Hiro Lee Tanaka (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Li-Sheng Tseng (Department of Mathematics, University of California at Irvine)


We introduce and investigate a new perspective which relates invariants of a symplectic manifold to topological invariants of certain odd-dimensional sphere bundles over the symplectic manifold. Specifically, we show that the novel symplectic $A_{\infty}$-algebras of differential forms recently found by Tsai–Tseng–Yau are in fact equivalent to the standard de Rham differential graded algebra of the odd sphere bundles when the cohomology class of the symplectic form is integral. As applications of this equivalence, we deduce for a closed symplectic manifold that Tsai–Tseng–Yau’s symplectic $A_{\infty}$-algebras satisfy the Calabi–Yau property and argue that they can be used to define an intersection theory for cycles built from co-isotropic/isotropic submanifolds. We further demonstrate that these symplectic $A_{\infty}$-algebras satisfy several functorial properties and lay the groundwork for addressing Weinstein functoriality.

Received 30 November 2017

Published 28 August 2018