2015
DOI: 10.1007/s00029-015-0206-x
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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 cont… Show more

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Cited by 39 publications
(102 citation statements)
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“…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.…”
Section: The Morphism H Encodes Maps Hmentioning
confidence: 93%
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“…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.…”
Section: The Morphism H Encodes Maps Hmentioning
confidence: 93%
“…We use this construction in [2] to define the first invariant of contact manifolds in the instanton Floer setting.…”
Section: Introductionmentioning
confidence: 99%
“…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.…”
Section: Instanton Floer Homologymentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 98%
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