Contents Online

# Mathematical Research Letters

## Volume 26 (2019)

### Number 5

### Bordered theory for pillowcase homology

Pages: 1467 – 1516

DOI: https://dx.doi.org/10.4310/MRL.2019.v26.n5.a11

#### Author

#### Abstract

We construct an algebraic version of Lagrangian Floer homology for immersed curves inside the pillowcase. We first associate to the pillowcase an algebra $\mathcal{A}$. Then to an immersed curve $L$ inside the pillowcase we associate an $A_{\infty}$ module $M(L)$ over $\mathcal{A}$. Then we prove that Lagrangian Floer homology $HF(L, L^{\prime})$ is isomorphic to a suitable algebraic pairing of modules $M(L)$ and $M(L^{\prime})$. 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 $A_{\infty}$ module to that Lagrangian.

Received 15 November 2017

Accepted 2 July 2019

Published 27 November 2019