Bordered Floer homology and contact structures

Abstract

We introduce a contact invariant in the bordered sutured Heegaard Floer homology of a three-manifold with boundary. The input for the invariant is a contact manifold (M, ξ, F) whose convex boundary is equipped with a signed singular foliation F closely related to the characteristic foliation. Such a manifold admits a family of foliated open book decompositions classified by a Giroux Correspondence, as described in [LV20]. We use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well-defined classes cD and cA in the corresponding bordered sutured modules.Foliated open books exhibit user-friendly gluing behavior, and we show that the pairing on invariants induced by gluing compatible foliated open books recovers the Heegaard Floer contact invariant for closed contact manifolds. We also consider a natural map associated to forgetting the foliation F in favor of the dividing set, and show that it maps the bordered sutured invariant to the contact invariant of a sutured manifold defined by Honda-Kazez-Matić. CONTENTS1. Introduction 1 2. Preliminaries 4 3. Construction of the contact invariant 18 4. Invariance 23 5. Gluing 29 6. Relationship to the HKM contact element 33 References 36 2010 Mathematics Subject Classification. 57M27.

A friendly introduction to the bordered contact invariant

“…Let (M, ξ, F) be a foliated contact three-manifold. In [1], a sorted foliated open book for (M, ξ, F) was used to construct a Heegaard diagram for an associated bordered sutured manifold (M, , Z), along with a preferred generator of the diagram. The homotopy equivalence class of this generator in the resulting bordered sutured Floer homology is an invariant of the foliated contact three-manifold [1, Theorem 1].…”

“…In fact, the union of the first and last page naturally describes a (cornered) handlebody decomposition for M. Using the data of ({S i }, h, {γ ± i }), we describe a multipointed bordered Heegaard diagram H = ( , α, β, Z) for this handlebody decomposition, along with a preferred generator. We outline the construction below; see [1,Section 3].…”

“…In this paper, we discuss a recent extension of the contact class to three-manifolds with boundary. Namely, in [1], a contact invariant was defined in the bordered sutured Floer homology of a foliated contact three-manifold (M, ξ, F), which is a contact manifold with a certain type of singular foliation on the boundary. We associate to a foliated contact three-manifold a bordered sutured manifold (M, , Z).…”

“…The resulting sutures are particularly simple, so one can think of (M, , Z) as a bordered manifold (M, Z) of a type slightly more general than in [14]. Below, we rephrase the main results of [1], translating from "bordered sutured" to "multipointed" language. Section 4 explores the correspondence between these two viewpoints in more detail.…”

“…This paper offers a hands-on introduction to the bordered contact invariant, favoring geometric intuition over the formal proofs that may be found in [12] and [1]. We assume minimal background in contact geometry, so Section 2 focuses on understanding contact structures via characteristic foliations.…”