The Künneth theorem for the Fukaya algebra of a product of Lagrangians

Abstract

Given a compact Lagrangian submanifold [Formula: see text] of a symplectic manifold [Formula: see text], Fukaya, Oh, Ohta and Ono construct a filtered [Formula: see text]-algebra [Formula: see text], on the cohomology of [Formula: see text], which we call the Fukaya algebra of [Formula: see text]. In this paper, we describe the Fukaya algebra of a product of two Lagrangians submanifolds [Formula: see text]. Namely, we show that [Formula: see text] is quasi-isomorphic to [Formula: see text], where [Formula: see text] is the tensor product of filtered [Formula: see text]-algebras defined in [L. Amorim, Tensor product of filtered [Formula: see text]-algebras, J. Pure Appl. Algebra 220(12) (2016) 3984–4016]. As a corollary of this quasi-isomorphism, we obtain a description of the bounding cochains on [Formula: see text] and of the Floer cohomology of [Formula: see text].