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

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 $L_1\times L_2$.

v2: one mistake fixed; to appear in International Journal of Mathematics

References in corpus (3)

• Technical details on Kuranishi structure and virtual fundamental chain
• Kuranishi structure, Pseudo-holomorphic curve, and Virtual fundamental chain: Part 1
• Tensor product of filtered $A_{\infty }$-algebras

Cited by in corpus (2)

• Tensor product of filtered $A_{\infty }$-algebras
• $SO(3)$-Floer homology of 3-manifolds with boundary 1