On contact invariants in bordered Floer homology

TL;DR

This paper introduces contact invariants within bordered sutured Floer homology, extending their applicability and establishing a pairing theorem, with implications for contact surgery and connections to existing invariants.

Contribution

It defines new contact invariants in bordered sutured Floer homology, broadening the contexts where these invariants can be applied and connecting them to existing theories.

Findings

Contact invariants satisfy a pairing theorem extending Honda-Kazez-Matić gluing map.

Established correspondence between $ ext{A}_ ext{infty}$ operations and bypass maps.

Characterized the Stipsicz-Vértési map as an $ ext{A}_ ext{infty}$ action.

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.