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.
Highlights
Floer-theoretic invariants of contact structures—in particular, those defined by Kronheimer and Mrowka in [26] and by Ozsvath and Szaboin [41]—have led to a number of spectacular results in low-dimensional topology over the last decade or so
We show that our contact invariant is ‘natural’ in the sense that it is preserved by the canonical isomorphisms relating the different sutured monopole homology groups associated to a given sutured contact manifold, something which has not been completely established on the Heegaard Floer side
To define the contact invariant, we introduce the notion of a contact closure of (M, Γ, ξ ), which is a closure D of (M, Γ ) together with a contact structure ξon Y extending ξ and satisfying certain conditions
Summary
Floer-theoretic invariants of contact structures—in particular, those defined by Kronheimer and Mrowka in [26] and by Ozsvath and Szaboin [41]—have led to a number of spectacular results in low-dimensional topology over the last decade or so. We prove that the result of contact (+1)-surgery along such a knot can be viewed naturally as a contact closure of (M , Γ , ξ ), and we define H2 in terms of the map on Floer homology induced by the corresponding 2handle cobordism.
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