Monopoles and contact structures

Kronheimer, P. B.; Mrowka, Tomasz

doi:10.1007/s002220050183

Cited by 133 publications

(272 citation statements)

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“…There is a geometric interpretation of Spin c structures in four dimensions, analogous to Turaev's interpretation of Spin c structures in three-dimensions; compare [19] and [13].…”

Section: Spin C Structuresmentioning

confidence: 99%

Holomorphic disks and topological invariants for closed three-manifolds

2004

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“…There is a geometric interpretation of Spin c structures in four dimensions, analogous to Turaev's interpretation of Spin c structures in three-dimensions; compare [19] and [13].…”

Section: Spin C Structuresmentioning

confidence: 99%

Holomorphic disks and topological invariants for closed three-manifolds

2004

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“…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 .…”

Section: Introductionmentioning

confidence: 99%

Instanton Floer homology and contact structures

Baldwin

Sivek

2015

Sel. Math. New Ser.

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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...

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“…An example is M (− The second problem is distinguishing the tight contact structures. For Stein fillable contact structures this is made easy by a result from Seiberg-Witten theory due to Lisca and Matić [31] and Kronheimer and Mrowka [28], which gives necessary conditions for holomorphically fillable contact structures to be isotopic. Another way to distinguish contact structures is through their homotopy classification as 2-planes fields given by Gompf [19] in terms of algebraic topological invariants.…”

Section: What We Can Do and What We Can't Domentioning

confidence: 99%