Contact homology of Hamiltonian mapping tori

Fabert, Oliver

doi:10.4171/cmh/193

Cited by 10 publications

(67 citation statements)

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“…The algebraic invariants are then defined by replacing the original compactified moduli space by the compactified zero set of the perturbed CauchyRiemann operator. Note that this can be achieved by either thinking about the specialities of the problem and then using special perturbations as in [8] or by building a general framework allowing for arbitrary compact perturbations. The observation that one is only interested in the zero set of the perturbed Cauchy-Riemann operator led to the (relative) virtual moduli cycle techniques in symplectic Floer homology and Gromov-Witten theory for general symplectic manifolds, see [10,[17][18][19], where the construction of the relative virtual moduli cycles in symplectic field theory is sketched in [2].…”

Section: Coherent Compact Perturbationsmentioning

confidence: 99%

“…Following [3,8,21] there exists a Banach space bundle E p,d over a Banach manifold of maps B p,d in which the Cauchy-Riemann operator∂ J extends to a smooth section. In our special case it follows that the fibre is given by…”

Section: Linearized Operatormentioning

confidence: 99%

“…In particular it should hold for the abstract perturbations constructed using the polyfold theory of [15] as well as the domain-dependent Hamiltonian perturbations used in [9]. Since the analytical foundations of symplectic field theory are not yet established, we cannot make the above statement rigorous in full detail.…”

Section: Action Filtration On Rational Symplectic Field Theorymentioning

confidence: 99%

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Obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory

Fabert

2010

Math. Z.

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Branched covers of orbit cylinders are the basic examples of holomorphic curves studied in symplectic field theory. Since all curves with Fredholm index one can never be regular for any choice of cylindrical almost complex structure, we generalize the obstruction bundle technique of Taubes for determining multiple cover contributions from Gromov-Witten theory to the case of moduli spaces with boundary. Our result proves that the differential in rational symplectic field theory and contact homology is strictly decreasing with respect to the natural action filtration.

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Section: Coherent Compact Perturbationsmentioning

confidence: 99%

Section: Linearized Operatormentioning

confidence: 99%

Section: Action Filtration On Rational Symplectic Field Theorymentioning

confidence: 99%

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Obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory

Fabert

2010

Math. Z.

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“…Cylindrical contact homology Symplectic field theory and contact homology invariants (see Eliashberg, Givental and Hofer [10]) exist for mapping tori due to the existence of a Hamiltonian structure (see Bourgeois et al [3], Cieliebak and Mohnke [5] and Fabert [11]). In this setting, the "cylindrical mapping torus contact homology" splits as a direct sum…”

Section: Directions For Further Researchmentioning

confidence: 99%

Symplectic Floer homology of area-preserving surface diffeomorphisms

Cotton-Clay

2009

Geom. Topol.

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The symplectic Floer homology HF_*(f) of a symplectomorphism f:S->S encodes data about the fixed points of f using counts of holomorphic cylinders in R x M_f, where M_f is the mapping torus of f. We give an algorithm to compute HF_*(f) for f a surface symplectomorphism in a pseudo-Anosov or reducible mapping class, completing the computation of Seidel's HF_*(h) for h any orientation-preserving mapping class.Comment: 57 pages, 4 figures. Revision for publication, with various minor corrections. Adds results on the module structure and invariance thereo

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“…As we explain below, a rigorous proof of our conjecture would require a combination and extension of the work in [20] and [16]. We however emphasize that the necessary transversality results for all other occuring moduli spaces are established in the appendix of the first paper [15], building on [13]. This paper is dedicated to the memory of my friend Alex Koenen who died in an hiking accident shortly before the first version of this paper was finished.…”

mentioning

confidence: 96%

Higher algebraic structures in Hamiltonian Floer theory

Fabert

2019

Advances in Geometry

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This is the second of two papers devoted to showing how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures on symplectic cohomology. Using the SFT of Hamiltonian mapping tori we show how to define the analogue of rational Gromov-Witten theory for open symplectic manifolds. More precisely, we show that their symplectic cohomology can be equipped with the structure of a so-called cohomology F-manifold. After discussing applications to the quantum cohomology ring of the quintic, we outline why the the classical (closedstring) mirror symmetry conjecture for open Calabi-Yau manifolds shall be formulated as an isomorphism of cohomology F-manifolds. O.ANote that here we view the small quantum product as a product on the tangent space to Q at q = 0,Algebraically the big quantum (cup) product can equivalently be viewed as product on the space of vector fields T (1,0) Q on Q,In other words, it is an element in the space T (1,2) Q of (1, 2)-tensor fields on Q. It is the key ingredient of the Frobenius manifold structure on Q ∼ = QH * (M ) defined by Dubrovin, see [22], [11].Generalizing from closed to open symplectic manifolds M , it is known that quantum cohomology needs to be replaced by the symplectic cohomology SH * (M ) of M , defined using Floer theory. While it is well-known that the small quantum product on QH * (M ) generalizes to the pair-of-pants product, it is the main goal of this paper to define the generalization of the big quantum product and the resulting Frobenius manifold structure. After defining a big pair-of-pants product on symplectic cohomology, we will prove Theorem 0.1. Generalizing Dubrovin's definition of Frobenius manifolds for closed symplectic manifolds, the symplectic cohomology of a Liouville manifold, Calabi-Yau or not, can be equipped with the structure of a cohomology F-manifold in such a way that, on the tangent space at zero, we recover the ring structure given by the pair-of-pants product.Cohomology F-manifolds are generalizations of Frobenius manifolds defined by Merkulov in [20], [21]. Among other things, the vector field product ⋆ now just lives on a differential graded manifold instead of a graded linear space and one drops the requirement for an underlying potential. In other words, a cohomology F-manifold is a differential graded manifold (Q, X) equipped with a graded commutative and associative product for vector fields ⋆ : T (1,0) Q X ⊗ T (1,0) Q X → T (1,0) Q X .Following [8] and [19] 1 , a differential graded manifold is given by a pair of a graded linear space Q (more generally, a formal pointed graded manifold

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Contact homology of Hamiltonian mapping tori

Cited by 10 publications

References 10 publications

Obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory

Obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory

Symplectic Floer homology of area-preserving surface diffeomorphisms

Higher algebraic structures in Hamiltonian Floer theory

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