 
Summary: FUKAYA CATEGORIES OF SYMMETRIC PRODUCTS AND
BORDERED HEEGAARDFLOER HOMOLOGY
DENIS AUROUX
Abstract. The main goal of this paper is to discuss a symplectic interpretation of
Lipshitz, Ozsv´ath and Thurston's bordered HeegaardFloer homology [7] in terms
of Fukaya categories of symmetric products and Lagrangian correspondences. More
specifically, we give a description of the algebra A(F) which appears in the work of
Lipshitz, Ozsv´ath and Thurston in terms of (partially wrapped) Floer homology for
product Lagrangians in the symmetric product, and outline how bordered Heegaard
Floer homology itself can conjecturally be understood in this language.
1. Introduction
Lipshitz, Ozsv´ath and Thurston's bordered HeegaardFloer homology [7] extends
HeegaardFloer homology to an invariant for 3manifolds with parametrized bound
ary. Their construction associates to a (marked and parametrized) surface F a certain
algebra A(F), and to a 3manifold with boundary F a pair of (A) modules over
A(F), which satisfy a TQFTlike gluing theorem. On the other hand, recent work
of Lekili and Perutz [5] suggests another construction, whereby a 3manifold with
boundary yields an object in (a variant of) the Fukaya category of the symmetric
product of F.
1.1. Lagrangian correspondences and HeegaardFloer homology. Given a
