On contact invariants in bordered Floer homology

Abstract

In this paper, we define contact invariants in bordered sutured Floer homology. Given a contact 3-manifold with convex boundary, we apply a result of Zarev (arxiv:1010.3496) to derive contact invariants in the bordered sutured modules $\widehat{\mathit{BSA}}$ and $\widehat{\mathit{BSD}}$, as well as in bimodules $\widehat{\mathit{BSAA}},$ $\widehat{\mathit{BSDD}},$ $\widehat{\mathit{BSDA}}$ in the case of two boundary components. In the connected boundary case, our invariants appear to agree with bordered contact invariants defined by Alishahi-F\"oldv\'ari-Hendricks-Licata-Petkova-V\'ertesi (arxiv:2011.08672) whenever the latter are defined, although ours can be defined in broader contexts. We prove that these invariants satisfy a pairing theorem, which is a bordered extension of the Honda-Kazez-Mati\'c gluing map (arxiv:0705.2828) for sutured Floer homology. We also show that there is a correspondence between certain $\mathcal{A}_{\infty}$ operations in bordered type-$A$ modules and bypass attachment maps in sutured Floer homology. As an application, we characterize the Stipsicz-V\'ertesi map from $\mathit{SFH}$ to $\widehat{\mathit{HFK}}$ as an $\mathcal{A}_{\infty}$ action on $\widehat{\mathit{CFA}}$. We also apply the immersed curve interpretation of Hanselman-Rasmussen-Watson (arxiv:1604.03466) to prove results involving contact surgery.