Semi-global Kuranishi charts and the definition of contact homology

Bao, Erkao; Honda, Ko

doi:10.48550/arxiv.1512.00580

Cited by 10 publications

(11 citation statements)

References 21 publications

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“…For a general Liouville domain M , the SFT definition of SH * (M ) requires a choice of a virtual perturbation scheme making the moduli spaces regular; see [51,66,9,78]. An example of a Liouville domain whose SFT version of symplectic cohomology can be defined using elementary transversality arguments is T * L where L is a smooth manifold admitting a metric of non-positive curvature, for instance, the torus [17].…”

Section: Symplectic Cohomologymentioning

confidence: 99%

String topology with gravitational descendants, and periods of Landau-Ginzburg potentials

Tonkonog

2018

Preprint

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This paper introduces new operations on the string topology of a smooth manifold: gravitational descendants of its cotangent bundle, which are augmentations of the Chas-Sullivan L∞ algebra structure of the loop space. The definition extends to Liouville domains. Descendants of the n-torus are computed.To a monotone Lagrangian torus in a symplectic manifold, one associates a Laurent polynomial called the Landau-Ginzburg potential, by counting holomorphic disks. This paper proves the following mirror symmetry prediction: the constant terms of the powers of an LG potential are equal to descendant Gromov-Witten invariants of the ambient manifold.

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Section: Symplectic Cohomologymentioning

confidence: 99%

String topology with gravitational descendants, and periods of Landau-Ginzburg potentials

Tonkonog

2018

Preprint

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“…In dimension three, in the absence of contractible Reeb orbits, and when paired with the action filtered versions of [HN2, Thm 1.6, 1.9], the definition provided by Bao-Honda in [BaHon1] can be shown to be isomorphic to the cylindrical contact homology. Using virtual techniques, Bao-Honda [BaHon2] give a definition of the full contact homology differential graded algebra for any closed contact manifold in any dimension. The approaches of Pardon and the latter of Bao-Honda make use of Kuranishi structures to construct contact and symplectic invariants and while they hold more generally, they are more difficult to work with in computations and applications.…”

Section: Motivation and Resultsmentioning

confidence: 99%

Automatic transversality in contact homology II: filtrations and computations

Nelson

2019

Proc. London Math. Soc.

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This paper is the sequel to the previous paper [Nelson, Abh. Math. Semin. Univ. Hambg. 85 (2015) , which showed that sufficient regularity exists to define cylindrical contact homology in dimension three for nondegenerate dynamically separated contact forms, a subclass of dynamically convex contact forms. The Reeb orbits of these so-called dynamically separated contact forms satisfy a uniform growth condition on their Conley-Zehnder indices with respect to a free homotopy class. Given a contact form which is dynamically separated up to large action, we demonstrate a filtration by action on the chain complex and show how to obtain the desired cylindrical contact homology by taking direct limits. We give a direct proof of invariance of cylindrical contact homology within the class of dynamically separated contact forms, and elucidate the independence of the filtered cylindrical contact homology with respect to the choice of the dynamically separated contact form and almost complex structure. We also show that these regularity results are compatible with geometric methods of computing cylindrical contact homology of prequantization bundles, proving a conjecture of Eliashberg [Symplectic field theory and its applications, International Congress of Mathematicians I (European Mathematical Society, Zürich, 2007) 217-246] in dimension three.Definition 1.6. Let (M, λ) be a contact 3-manifold with c 1 (ker λ) = 0 such that all the Reeb orbits of R λ are contractible. Then λ is said to be dynamically separated whenever the following conditions hold.(I) If γ is a closed simple Reeb orbit then 3 μ CZ (γ) 5. (II) If γ k is the k-fold cover of a simple orbit γ then μ CZ (γ k ) = μ CZ (γ k−1 ) + 4. * (Y, λ, J).Corollary 1.18. CH EGH * is an invariant of closed contact manifolds (Y, ξ) for which there exists a pair (λ, J) where λ is a nondegenerate hypertight contact form with ker(λ) = ξ, and J is an admissible λ-compatible almost complex structure.Again, we will upgrade these results to hold for dynamically convex contact forms in dimension three in a forthcoming paper by Hutchings and Nelson. In contrast to [30] and a forthcoming paper by Hutchings and Nelson, this paper is concerned with the more restricted class of dynamically separated contact forms which allows us to directly obtain regularity for S 1 -independent pseudoholomorphic cylinders in cobordisms. Filtered cylindrical contact homologyThe action of a Reeb orbit γ is given by A(γ) := γ λ.. Since J is a λ-compatible almost complex structure on the symplectization it follows [40, Lemma 2.18] that the cylindrical contact homology differential(s) decreases the action, for example, ifThus, given any real number L it makes sense to define the filtered cylindrical contact homology, denoted by CH EGH,L * (M, λ, J), to be the homology of the subcomplex C EGH,L * (M, λ, J) of the chain complex spanned by generators of action less than L. The invariance of these filtered cylindrical contact homology groups is more subtle than in the unfiltered case, as they typically depend ...

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“…Q , which is defined by (full) contact homology. For constructions of contact homology, see [2], [15], [16] and references therein. The following argument follows [16].…”

Section: In Case (A) Ie I + (Ymentioning

confidence: 99%

Strong closing property of contact forms and action selecting functors

Irie¹

2022

Preprint

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We introduce a notion of strong closing property of contact forms, inspired by the C ∞ closing lemma for Reeb flows in dimension three. We then prove a sufficient criterion for strong closing property, which is formulated by considering a monoidal functor from a category of manifolds with contact forms to a category of filtered vector spaces. As a potential application of this criterion, we propose a conjecture which says that a standard contact form on the boundary of any symplectic ellipsoid satisfies strong closing property.

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Semi-global Kuranishi charts and the definition of contact homology

Cited by 10 publications

References 21 publications

String topology with gravitational descendants, and periods of Landau-Ginzburg potentials

String topology with gravitational descendants, and periods of Landau-Ginzburg potentials

Automatic transversality in contact homology II: filtrations and computations

Strong closing property of contact forms and action selecting functors

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