scholarly journals CORRELATORS AND DESCENDANTS OF SUBCRITICAL STEIN MANIFOLDS

2013 ◽  
Vol 24 (02) ◽  
pp. 1350004
Author(s):  
JIAN HE
Keyword(s):  
Exact Sequence ◽  
Contact Manifold ◽  
Full Potential ◽  
Contact Homology ◽  
Symplectic Homology ◽  
Stein Fillings ◽  
Symplectic Field Theory ◽  
Homology Algebra ◽  
First Chern Class ◽  
Invent Math

We determine the contact homology algebra of a subcritical Stein-fillable contact manifold whose first Chern class vanishes. We also compute the genus-0 one point correlators and gravitational descendants of compactly supported closed forms on their subcritical Stein fillings. This is a step towards determining the full potential function of the filling as defined in [Y. Eliashberg, A. Givental and H. Hofer. Introduction to symplectic field theory, Geom. Funct. Anal.Special Volume (2000) 560–673]. These invariants also give a canonical presentation of the cylindrical contact homology. With respect to this presentation, we determine the degree-2 differential in the Bourgeois–Oancea exact sequence of [F. Bourgeois and A. Oancea. An exact sequence for contact and symplectic homology, Invent. Math.175(3) (2009) 611–680]. As a further application, we proved that if a Kähler manifold M2n admits a subcritical polarization and c1 vanishes in the subcritical complement, then M is uniruled.

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 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 CONTACT HOMOLOGY OF LEFT-HANDED STABILIZATIONS AND PLUMBING OF OPEN BOOKS

2010 ◽  
Vol 12 (02) ◽  
pp. 223-263 ◽  
Author(s):  
FRÉDÉRIC BOURGEOIS ◽  
OTTO VAN KOERT
Keyword(s):  
Contact Manifold ◽  
Finite Energy ◽  
Unit Element ◽  
Contact Homology ◽  
Connected Sums ◽  
Open Books ◽  
Left Handed ◽  
Closed Contact ◽  
Homology Algebra ◽  
Energy Plane

We show that on any closed contact manifold of dimension greater than 1 a contact structure with vanishing contact homology can be constructed. The basic idea for the construction comes from Giroux. We use a special open book decomposition for spheres. The page is the cotangent bundle of a sphere and the monodromy is given by a left-handed Dehn twist. In the resulting contact manifold, we exhibit a closed Reeb orbit that bounds a single finite energy plane like in the computation for the overtwisted case. As a result, the unit element of the contact homology algebra is exact and so the contact homology vanishes. This result can be extended to other contact manifolds by using connected sums. The latter is related to the plumbing or 2-Murasugi sum of contact open books. We shall give a possible description of this construction and some conjectures about the plumbing operation.

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.

scholarly journals Local contact homology and applications

2015 ◽  
Vol 07 (02) ◽  
pp. 167-238 ◽  
Author(s):  
Umberto L. Hryniewicz ◽  
Leonardo Macarini
Keyword(s):  
Field Theory ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Local Version ◽  
Contact Homology ◽  
Local Contact ◽  
Symplectic Field Theory ◽  
Uniformly Bounded

We introduce a local version of contact homology for an isolated periodic orbit of the Reeb flow and prove that its rank is uniformly bounded for isolated iterations. Several applications are obtained, including a generalization of Gromoll–Meyer's theorem on the existence of infinitely many simple periodic orbits, resonance relations and conditions for the existence of non-hyperbolic periodic orbits. Most of the results of this paper remain conjectural until the foundational issues of Symplectic Field Theory are resolved.

scholarly journals Erratum to: An exact sequence for contact- and symplectic homology

2015 ◽  
Vol 200 (3) ◽  
pp. 1065-1076 ◽  
Author(s):  
Frédéric Bourgeois ◽  
Alexandru Oancea
Keyword(s):  
Exact Sequence ◽  
Symplectic Homology

scholarly journals Inverse System in The Category of Intuitionistic Fuzzy Soft Modules

2018 ◽  
Vol 14 (1) ◽  
pp. 7486-7502
Author(s):  
S. E. Abdullayev ◽  
Sadi Bayramov
Keyword(s):  
Exact Sequence ◽  
Inverse Limit ◽  
Inverse System ◽  
Derived Functor ◽  
Intuitionistic Fuzzy ◽  
Homology Algebra ◽  
Basic Concepts ◽  
Exact Sequences

This paper begins with the basic concepts of soft module. Later, we introduce inverse system in the category of intutionistic fuzzy soft modules and prove that its limit exists in this category. Generally, limit of inverse system of exact sequences of intutionistic fuzzy soft modules is not exact. Then we define the notion  which is first derived functor of the inverse limit functor. Finally, using methods of homology algebra, we prove that the inverse system limit of exact sequence of intutionistic fuzzy soft modules is exact.

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 Sub-leading asymptotics of ECH capacities

2020 ◽  
Vol 26 (5) ◽  
Author(s):  
Dan Cristofaro-Gardiner ◽  
Nikhil Savale
Keyword(s):  
Contact Homology ◽  
Embedded Contact Homology ◽  
Invent Math

AbstractIn previous work (Cristofaro-Gardiner et al. in Invent Math 199:187–214, 2015), the first author and collaborators showed that the leading asymptotics of the embedded contact homology spectrum recovers the contact volume. Our main theorem here is a new bound on the sub-leading asymptotics.

Sign in / Sign up

Export Citation Format

Share Document