A-infinity algebras, strand algebras, and contact categories

TL;DR

This paper explores the A-infinity algebra structures on contact category algebras derived from bordered sutured Floer theory, providing explicit constructions and analyzing their properties and higher-order operations.

Contribution

It introduces explicit constructions of A-infinity structures on contact category algebras and investigates their properties and conditions for operation vanishing.

Findings

Explicit A-infinity structures constructed

Conditions for vanishing and nonvanishing of operations established

Enhanced understanding of tensor products of strand diagrams

Abstract

In previous work we showed that the contact category algebra of a quadrangulated surface is isomorphic to the homology of a strand algebra from bordered sutured Floer theory. Being isomorphic to the homology of a differential graded algebra, this contact category algebra has an A-infinity structure, allowing us to combine contact structures not just by gluing, but also by higher-order operations. In this paper we investigate such A-infinity structures and higher order operations on contact structures. We give explicit constructions of such A-infinity structures, and establish some of their properties, including conditions for the vanishing and nonvanishing of A-infinity operations. Along the way we develop several related notions, including a detailed consideration of tensor products of strand diagrams.