Explicit Koszul-dualizing bimodules in bordered Heegaard Floer homology

Abstract

We give a combinatorial proof of the quasi-invertibility of $\widehat{\mathit{CFDD}}\left({\mathbb{I}}_{\mathcal{Z}}\right)$ in bordered Heegaard Floer homology, which implies a Koszul self-duality on the dg-algebra $\mathcal{A}\left(\mathcal{Z}\right)$, for each pointed matched circle $\mathcal{Z}$. We do this by giving an explicit description of a rank 1 model for $\widehat{\mathit{CFAA}}\left({\mathbb{I}}_{\mathcal{Z}}\right)$, the quasi-inverse of $\widehat{\mathit{CFDD}}\left({\mathbb{I}}_{\mathcal{Z}}\right)$. To obtain this description we apply homological perturbation theory to a larger, previously known model of $\widehat{\mathit{CFAA}}\left({\mathbb{I}}_{\mathcal{Z}}\right)$.