Bordered Floer homology for sutured manifolds

TL;DR

This paper introduces bordered sutured manifolds and extends bordered Floer homology to these objects, creating a framework that unifies sutured and bordered 3-manifold invariants and offers new computational tools.

Contribution

It defines bordered sutured manifolds and develops a functorial extension of bordered Floer homology to these objects, unifying sutured and bordered invariants.

Findings

Defines a sutured cobordism category with manifolds with corners.

Extends bordered Floer homology to bordered sutured manifolds.

Provides a new proof of Juhasz's surface decomposition formula.

Abstract

We define a sutured cobordism category of surfaces with boundary and 3-manifolds with corners. In this category a sutured 3-manifold is regarded as a morphism from the empty surface to itself. In the process we define a new class of geometric objects, called bordered sutured manifolds, that generalize both sutured 3-manifolds and bordered 3-manifolds. We extend the definition of bordered Floer homology to these objects, giving a functor from a decorated version of the sutured category to A-infinity algebras, and A-infinity bimodules. As an application we give a way to recover the sutured homology SFH(Y,Gamma) of a sutured manifold from either of the bordered invariants CFA(Y) and CFD(Y) of its underlying manifold Y. A further application is a new proof of the surface decomposition formula of Juhasz.