The K\"unneth theorem for the Fukaya algebra of a product of Lagrangians

TL;DR

This paper proves a K"unneth-type theorem for the Fukaya algebra of a product of Lagrangian submanifolds, showing it is quasi-isomorphic to the tensor product of their individual Fukaya algebras, and describes related cohomological properties.

Contribution

It establishes a K"unneth theorem for Fukaya algebras of product Lagrangians, linking their algebraic structures explicitly.

Findings

Fukaya algebra of a product Lagrangian is quasi-isomorphic to the tensor product of individual Fukaya algebras.

Provides a description of bounding cochains on the product Lagrangian.

Characterizes Floer cohomology of the product in terms of factors.

Abstract

Given a compact Lagrangian submanifold $L$ of a symplectic manifold $(M,\omega)$, Fukaya, Oh, Ohta and Ono construct a filtered $A_\infty$-algebra $\mathcal{F}(L)$, on the cohomology of $L$, which we call the Fukaya algebra of $L$. In this paper we describe the Fukaya algebra of a product of two Lagrangians submanifolds $L_1\times L_2$. Namely, we show that $\mathcal{F}(L_1\times L_2)$ is quasi-isomorphic to $\mathcal{F}(L_1)\otimes_\infty \mathcal{F}(L_2)$, where $\otimes_\infty$ is the tensor product of filtered $A_\infty$-algebras defined in arXiv:1404.7184. As a corollary of this quasi-isomorphism we obtain a description of the bounding cochains on $\mathcal{F}(L_1\times L_2)$ and of the Floer cohomology of $L_1\times L_2$.