Mathematical Research Letters

Volume 26 (2019)

Number 5

Bordered theory for pillowcase homology

Pages: 1467 – 1516



Artem Kotelskiy (Department of Mathematics, Princeton University, Princeton, N.J., U.S.A.; and Department of Mathematics, Indiana University, Bloomington, In., U.S.A.)


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