Bordered theory for pillowcase homology

TL;DR

This paper develops an algebraic framework for pillowcase homology by associating modules to immersed curves in the pillowcase, enabling algebraic computation of Lagrangian Floer homology.

Contribution

It introduces an algebraic model for pillowcase homology using A-infinity modules, extending the geometric construction to an algebraic setting.

Findings

Constructed algebra A associated with the pillowcase.

Defined A-infinity modules M(L) for immersed curves L.

Proved isomorphism between Floer homology and algebraic pairing of modules.

Abstract

We construct an algebraic version of Lagrangian Floer homology for immersed curves inside the pillowcase. We first associate to the pillowcase an algebra A. Then to an immersed curve L inside the pillowcase we associate an A infinity module M(L) over A. Then we prove that Lagrangian Floer homology HF(L,L') is isomorphic to a suitable algebraic pairing of modules M(L) and M(L'). This extends the pillowcase homology construction - given a 2-stranded tangle inside a 3-ball, if one obtains an immersed unobstructed Lagrangian inside the pillowcase, one can further associate an A infinity module to that Lagrangian.