Instanton Floer homology and contact structures

Abstract

Abstract. We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured instanton Floer homology theory. To the best of our knowledge, this is the first invariant of contact manifolds-with or without boundary-defined in the instanton Floer setting. We prove that our invariant vanishes for overtwisted contact structures and is nonzero for contact manifolds with boundary which embed into Stein fillable contact manifolds. Moreover, we propose a strategy by which our contact invariant might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings. Our construction is inspired by a reformulation of a similar invariant in the monopole Floer setting defined by the authors in [1]. IntroductionFloer-theoretic invariants of contact manifolds have been responsible for many important results in low-dimensional topology. Notable examples include the invariants of closed contact 3-manifolds defined by Kronheimer and Mrowka [10] and by Ozsváth and Szabó [18] in monopole and Heegaard Floer homology, respectively. Also important is the work in [8], where Honda, Kazez, and Matić extend Ozsváth and Szabó's construction, using sutured Heegaard Floer homology to define an invariant of sutured contact manifolds, which are triples of the form (M, Γ, ξ) where (M, ξ) is a contact 3-manifold with convex boundary and Γ ⊂ ∂M is a multicurve dividing the characteristic foliation of ξ on ∂M . Recently, we defined an analogous invariant of sutured contact manifolds in Kronheimer and Mrowka's sutured monopole Floer homology theory [1].The goal of this paper is to define an invariant of sutured contact manifolds in Kronheimer and Mrowka's sutured instanton Floer homology (SHI). To the best of our knowledge, this is the first invariant of contact manifolds-with or without boundary-defined in the instanton Floer setting. Like the Heegaard Floer invariants but in contrast with the monopole invariants, our instanton Floer contact invariant is defined using the full relative Giroux correspondence. Its construction is inspired by a reformulation of the monopole Floer invariant in [1] which was used there to prove that the monopole invariant is well-defined.A unique feature of the instanton Floer viewpoint is the central role played by the fundamental group. Along these lines, we conjecture a means by which our contact invariant in SHI might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings, a relationship which has been largely unexplored to this point. Below, we sketch the construction of our contact invariant, describe some of its most important properties, state some conjectures, and discuss plans for future work which include using the constructions in this paper to define invariants of bordered manifolds in the instanton Floer setting.1.1. A contact invariant in SHI. Suppose (M, Γ) is a balanced sutured manifold. Roughly speaking, a closure of (M, Γ) is formed by gluing on some auxiliary piece a...

A Contact Invariant in Sutured Monopole Homology

“…As mentioned previously, the ideas in this paper are used to define similar contact handle attachment maps in the instanton Floer setting in [2]. These maps give rise to an analogous bypass exact triangle in that setting.…”

“…We use this construction in [2] to define the first invariant of contact manifolds in the instanton Floer setting.…”

Stein fillings and SU(2) representations

“…The discussion below is adapted from [BS16] The product manifold S × [−1, 1] admits an [−1, 1]-invariant contact structure in which each S × {t} is a convex surface with collared Legendrian boundary and dividing set consisting of one boundary-parallel arc on each component of ∂S, oriented in the same direction as ∂S. Upon rounding corners, we obtain a product sutured contact manifold in the terminology of [BS16], denoted by H(S), which is topologically a handlebody with boundary the double of S and dividing set ∂S ⊂ ∂H(S). For notational convenience, we will often simply equate H(S) and S × [−1, 1], as in the definition below.…”

“…Of course, Theorem 1.5 also follows from the Poincaré Conjecture together with Eliashberg's foundational result [Eli90] that S 3 bounds a unique Stein domain, of the form (B 4 , J), but our proof is unique in that it does not make any use of Ricci flow or holomorphic curves. Theorem 1.1 and most of the other results in this article follow from a new theorem on the relationship between Stein fillings and the rank of sutured instanton homology, proved using the contact invariants we defined in [BS16]. We describe this relationship below.…”

“…Suppose (Y, ξ) is a closed contact 3-manifold with base point p. Removing a Darboux ball around p, we obtain a balanced sutured manifold Y (p) = (Y B 3 , S 1 ). In [BS16], we defined an element Θ(ξ) ∈ SHI(−Y (p)) which is an invariant of ξ| Y B 3 up to isotopy rel boundary. Here, SHI refers to our natural refinement of Kronheimer-Mrowka's sutured instanton homology [KM10,BS15].…”