scholarly journals Contact blow up and cylindrical contact homology of toric contact manifolds of Reeb type

2017 ◽  
Vol 102 (116) ◽  
pp. 61-71
Author(s):  
Aleksandra Marinkovic
Keyword(s):  
Symplectic Manifold ◽  
Blow Up ◽  
Contact Manifold ◽  
Contact Homology ◽  
Contact Manifolds ◽  
Special Cases ◽  
Toric Contact Manifolds

Let (V,?) be a toric contact manifold of Reeb type that is a prequantization of a toric symplectic manifold (M,?). A contact blow up of (V,?) is the prequantization of a symplectic blow up of (M,?). Thus, a contact blow up of (V,?) is a new toric contact manifold of Reeb type. In some special cases we are able to compute the cylindrical contact homology for the contact blowup using only the cylindrical contact homology of the contact manifold we started with.

scholarly journals Contact homology of good toric contact manifolds

2011 ◽  
Vol 148 (1) ◽  
pp. 304-334 ◽  
Author(s):  
Miguel Abreu ◽  
Leonardo Macarini
Keyword(s):  
Homotopy Class ◽  
Chern Class ◽  
Einstein Metrics ◽  
Contact Manifold ◽  
Infinite Family ◽  
Contact Structures ◽  
Contact Homology ◽  
Contact Manifolds ◽  
First Chern Class ◽  
Toric Contact Manifolds

AbstractIn this paper we show that any good toric contact manifold has a well-defined cylindrical contact homology, and describe how it can be combinatorially computed from the associated moment cone. As an application, we compute the cylindrical contact homology of a particularly nice family of examples that appear in the work of Gauntlett et al. on Sasaki–Einstein metrics. We show in particular that these give rise to a new infinite family of non-equivalent contact structures on S2×S3 in the unique homotopy class of almost contact structures with vanishing first Chern class.

scholarly journals Two closed orbits for non-degenerate Reeb flows

2020 ◽  
pp. 1-36
Author(s):  
MIGUEL ABREU ◽  
JEAN GUTT ◽  
JUNGSOO KANG ◽  
LEONARDO MACARINI
Keyword(s):  
Contact Manifold ◽  
Contact Manifolds ◽  
Symplectic Homology ◽  
Connected Sums ◽  
Reeb Flows ◽  
Closed Orbits ◽  
Finite Quotient ◽  
Closed Contact ◽  
First Chern Class ◽  
Toric Contact Manifolds

Abstract We prove that every non-degenerate Reeb flow on a closed contact manifold M admitting a strong symplectic filling W with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of W satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any contact finite quotient of M. Several examples of such contact manifolds are provided, like displaceable ones, unit cosphere bundles, prequantisation circle bundles, Brieskorn spheres and toric contact manifolds. We also show that this condition on the equivariant symplectic homology is preserved by boundary connected sums of Liouville domains. As a byproduct of one of our applications, we prove a sort of Lusternik–Fet theorem for Reeb flows on the unit cosphere bundle of not rationally aspherical manifolds satisfying suitable additional assumptions.

scholarly journals Toric Contact Geometry in Arbitrary Codimension

2018 ◽  
Vol 2020 (8) ◽  
pp. 2436-2467 ◽  
Author(s):  
Vestislav Apostolov ◽  
David M J Calderbank ◽  
Paul Gauduchon ◽  
Eveline Legendre
Keyword(s):  
Symplectic Manifold ◽  
Contact Geometry ◽  
Contact Manifolds ◽  
Arbitrary Codimension ◽  
Toric Contact Manifolds

Abstract We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold.

scholarly journals On the Mean Euler Characteristic of Gorenstein Toric Contact Manifolds

2018 ◽  
Vol 2020 (14) ◽  
pp. 4465-4495 ◽  
Author(s):  
Miguel Abreu ◽  
Leonardo Macarini
Keyword(s):  
Euler Characteristic ◽  
Chern Class ◽  
Contact Manifold ◽  
Contact Manifolds ◽  
Toric Diagram ◽  
The Mean ◽  
Toric Contact Manifolds ◽  
Symplectic Filling

Abstract We prove that the mean Euler characteristic of a Gorenstein toric contact manifold, that is, a good toric contact manifold with zero 1st Chern class, is equal to half the normalized volume of the corresponding toric diagram and give some applications. A particularly interesting one, obtained using a result of Batyrev and Dais, is the following: twice the mean Euler characteristic of a Gorenstein toric contact manifold is equal to the Euler characteristic of any crepant toric symplectic filling, that is, any toric symplectic filling with zero 1st Chern class.

scholarly journals Sub-Riemannian Limit of the Differential Form Heat Kernels of Contact Manifolds

2020 ◽  
Author(s):  
Pierre Albin ◽  
Hadrian Quan
Keyword(s):  
Blow Up ◽  
Riemannian Metrics ◽  
Contact Manifold ◽  
Heat Kernels ◽  
Analytic Torsion ◽  
Contact Manifolds ◽  
Spectral Invariants ◽  
Eta Invariant ◽  
Hodge Laplacian ◽  
Contact Distribution

Abstract We study the behavior of the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the $\eta $-invariant and the determinant of the Laplacian. In particular, we prove that contact versions of the relative $\eta $-invariant and the relative analytic torsion are equal to their Riemannian analogues and hence topological.

QUANTUM TYPE COHOMOLOGIES ON CONTACT MANIFOLDS

2013 ◽  
Vol 10 (05) ◽  
pp. 1350012 ◽  
Author(s):  
YONG SEUNG CHO
Keyword(s):  
Symplectic Manifold ◽  
Moduli Space ◽  
Homology Class ◽  
Contact Manifold ◽  
Integral Homology ◽  
Two Dimensional ◽  
Contact Manifolds ◽  
The One ◽  
Dimensional Integral

We extend the notion of a pseudoholomorphic map in a symplectic manifold to the one of an almost coholomorphic map on a contact manifold M of an odd dimension. We study the moduli space of stable almost coholomorphic maps that represent a two-dimensional integral homology class of M, Gromov–Witten type invariants, quantum type products and quantum type cohomologies.

scholarly journals Legendrian contact homology and topological entropy

2019 ◽  
Vol 11 (01) ◽  
pp. 53-108 ◽  
Author(s):  
Marcelo R. R. Alves
Keyword(s):  
Growth Rate ◽  
Topological Entropy ◽  
Infinite Family ◽  
Contact Structures ◽  
Legendrian Knots ◽  
Contact Homology ◽  
Contact Manifolds ◽  
3 Dimensional ◽  
Reeb Flows ◽  
Legendrian Contact Homology

In this paper we study the growth rate of a version of Legendrian contact homology, which we call strip Legendrian contact homology, in 3-dimensional contact manifolds and its relation to the topological entropy of Reeb flows. We show that: if for a pair of Legendrian knots in a contact 3-manifold [Formula: see text] the strip Legendrian contact homology is defined and has exponential homotopical growth with respect to the action, then every Reeb flow on [Formula: see text] has positive topological entropy. This has the following dynamical consequence: for all Reeb flows (even degenerate ones) on [Formula: see text] the number of hyperbolic periodic orbits grows exponentially with respect to the period. We show that for an infinite family of 3-manifolds, infinitely many different contact structures exist that possess a pair of Legendrian knots for which the strip Legendrian contact homology has exponential growth rate.

scholarly journals Computation of Quantum Cohomology From Fukaya Categories

2020 ◽  
Author(s):  
Fumihiko Sanda
Keyword(s):  
Symplectic Manifold ◽  
Symplectic Structure ◽  
Blow Up ◽  
Quantum Cohomology ◽  
Fukaya Category ◽  
Fukaya Categories ◽  
Compact Symplectic Manifold

Abstract Assume the existence of a Fukaya category $\textrm{Fuk}(X)$ of a compact symplectic manifold $X$ with some expected properties. In this paper, we show $\mathscr{A} \subset \textrm{Fuk}(X)$ split generates a summand $\textrm{Fuk}(X)_e \subset \textrm{Fuk}(X)$ corresponding to an idempotent $e \in QH^{\bullet }(X)$ if the Mukai pairing of $\mathscr{A}$ is perfect. Moreover, we show $HH^{\bullet }(\mathscr{A}) \cong QH^{\bullet }(X) e$. As an application, we compute the quantum cohomology and the Fukaya category of a blow-up of $\mathbb{C} P^2$ at four points with a monotone symplectic structure.

scholarly journals Reeb periodic orbits after a bypass attachment

2013 ◽  
Vol 35 (2) ◽  
pp. 615-672
Author(s):  
ANNE VAUGON
Keyword(s):  
Periodic Orbits ◽  
Contact Structure ◽  
Convex Surface ◽  
Three Dimensional ◽  
Contact Manifold ◽  
Contact Structures ◽  
Contact Homology ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Manifold With Boundary

AbstractOn a three-dimensional contact manifold with boundary, a bypass attachment is an elementary change of the contact structure consisting in the attachment of a thickened half-disc with a prescribed contact structure along an arc on the boundary. We give a model bypass attachment in which we describe the periodic orbits of the Reeb vector field created by the bypass attachment in terms of Reeb chords of the attachment arc. As an application, we compute the contact homology of a product neighbourhood of a convex surface after a bypass attachment, and the contact homology of some contact structures on solid tori.

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