A Contact Invariant in Sutured Monopole Homology

Abstract

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured monopole Floer homology theory (SHM). Our invariant can be viewed as a generalization of Kronheimer and Mrowka's contact invariant for closed contact 3-manifolds and as the monopole Floer analogue of Honda, Kazez, and Matić's contact invariant in sutured Heegaard Floer homology (SFH). In the process of defining our invariant, we construct maps on SHM associated to contact handle attachments, analogous to those defined by Honda, Kazez, and Matić in SFH. We use these maps to establish a bypass exact triangle in SHM analogous to Honda's in SFH. This paper also provides the topological basis for the construction of similar gluing maps in sutured instanton Floer homology, which are used in Baldwin and Sivek [Selecta Math. (N.S.), 22(2) (2016), 939-978] to define a contact invariant in the instanton Floer setting.

Instanton Floer homology and contact structures

“…1 We also show that this contact invariant behaves naturally with respect to the contact handle attachment maps, per the following (stated later as Theorem 4.8).…”

“…Among these are the following two results (stated later as Theorems 4.10 and 4.12). Interestingly, the proofs of both theorems below are substantially different from those of their counterparts in [1] in the sutured monopole homology setting.For the next theorem, suppose (Y, ξ) is a closed contact 3-manifold and let Y (n) denote the sutured manifold obtained by removing n disjoint Darboux balls for any n ≥ 1.As we shall see, the corollary below (stated later as Corollary 4.13) follows from Theorems 1.4 and 1.2. Note that one needs some kind of naturality for a statement like that in Conjecture 1.6 since "linear independence" has little meaning if elements are only well-defined up to isomorphism.…”

“…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].…”

“…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.…”

“…We define a map H i : SHI(−M, −Γ) → SHI(−M i , −Γ i ) which depends only on the smooth data involved in this handle attachment. Our construction of these maps is almost identical to that of the analogous maps in sutured monopole homology [1]; in particular, these maps are defined in terms of the maps on instanton Floer homology induced by natural cobordisms between closures. Suppose now that (M, Γ, ξ) is a sutured contact manifold.…”