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FUKAYA CATEGORIES OF SYMMETRIC PRODUCTS AND BORDERED HEEGAARD-FLOER HOMOLOGY
 

Summary: FUKAYA CATEGORIES OF SYMMETRIC PRODUCTS AND
BORDERED HEEGAARD-FLOER HOMOLOGY
DENIS AUROUX
Abstract. The main goal of this paper is to discuss a symplectic interpretation of
Lipshitz, Ozsv´ath and Thurston's bordered Heegaard-Floer 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 Heegaard-Floer homology [7] extends
Heegaard-Floer homology to an invariant for 3-manifolds with parametrized bound-
ary. Their construction associates to a (marked and parametrized) surface F a certain
algebra A(F), and to a 3-manifold with boundary F a pair of (A-) modules over
A(F), which satisfy a TQFT-like gluing theorem. On the other hand, recent work
of Lekili and Perutz [5] suggests another construction, whereby a 3-manifold with
boundary yields an object in (a variant of) the Fukaya category of the symmetric
product of F.
1.1. Lagrangian correspondences and Heegaard-Floer homology. Given a

  

Source: Auroux, Denis - Department of Mathematics, Massachusetts Institute of Technology (MIT)

 

Collections: Mathematics