Naturality in sutured monopole and instanton homology

Baldwin, John A.; Sivek, Steven

doi:10.4310/jdg/1432842360

Cited by 31 publications

(170 citation statements)

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“…Then H i (θ(M, Γ, ξ)) = θ(M i , Γ i , ξ i ). 2 The invariant θ shares several important features with Honda, Kazez, and Matić's invariant and with our contact invariant in sutured monopole homology (besides the one above). Among these are the following two results (stated later as Theorems 4.10 and 4.12).…”

Section: H : Shi(−h(s)) → Shi(−m −γ)mentioning

confidence: 98%

Section: Instanton Floer Homology and Contact Structuresmentioning

confidence: 99%

“…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. This is precisely the sort of consideration that motivated our work in [2].As we explain in Section 5, a positive answer to Conjecture 1.6 would imply the following link between the fundamental group of a contact 3-manifold and properties of its Stein fillings. Conjecture 1.7.…”

mentioning

confidence: 99%

“…A positive answer to this conjecture would allow us assign well-defined maps to cobordisms between sutured manifolds in the instanton Floer setting-in the language of [2], to extend SHI to a functor from CobSut to C-PSys-following Juhasz's strategy [9], as explained in the analogous context of sutured monopole in [1,Subsection 1.3]. And this, in turn, would allow us to define invariants of bordered 3-manifolds using instanton Floer homology, following a strategy of Zarev [20]; again, see [1,Subsection 1.3] for the analogous discussion in the sutured monopole Floer setting.…”

Section: H : Shi(−h(s)) → Shi(−m −γ)mentioning

confidence: 99%

“…In [2], we introduced a refinement of their construction which assigns to (M, Γ) a projectively transitive system of C-modules, denoted by SHI(M, Γ). This system records the collection of C-modules-all isomorphic to SHI(M, Γ)-associated to different closures of (M, Γ) together with canonical isomorphisms relating these modules, where these isomorphisms are well-defined up to multiplication in C × .…”

Section: Introductionmentioning

confidence: 99%

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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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Section: H : Shi(−h(s)) → Shi(−m −γ)mentioning

confidence: 98%

Section: Instanton Floer Homology and Contact Structuresmentioning

confidence: 99%

mentioning

confidence: 99%

Section: H : Shi(−h(s)) → Shi(−m −γ)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Instanton Floer homology and contact structures

Baldwin

Sivek

2015

Sel. Math. New Ser.

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Itsy Bitsy Topological Field Theory

Mathews

2013

Ann. Henri Poincaré

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We extend the topological field theory ("itsy bitsy topological field theory"') of our previous work from mod-2 to twisted coefficients. This topological field theory is derived from sutured Floer homology but described purely in terms of surfaces with signed points on their boundary (occupied surfaces) and curves on those surfaces respecting signs (sutures). It has informationtheoretic ("itsy") and quantum-field-theoretic ("bitsy") aspects. In the process we extend some results of sutured Floer homology, consider associated ribbon graph structures, and construct explicit admissible Heegaard decompositions.

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Decomposing sutured monopole and instanton Floer homologies

Ghosh

2023

Sel. Math. New Ser.

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Naturality in sutured monopole and instanton homology

Cited by 31 publications

References 19 publications

Instanton Floer homology and contact structures

Instanton Floer homology and contact structures

Itsy Bitsy Topological Field Theory

Decomposing sutured monopole and instanton Floer homologies

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