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 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 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 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 Open books and exact symplectic cobordisms

2018 ◽  
Vol 29 (04) ◽  
pp. 1850026 ◽  
Author(s):  
Mirko Klukas
Keyword(s):  
Disjoint Union ◽  
Contact Manifold ◽  
Higher Dimensions ◽  
Fiber Bundles ◽  
Open Book ◽  
Contact Manifolds ◽  
Open Books ◽  
Symplectic Cobordism ◽  
The Given ◽  
Symplectic Cobordisms

Given two open books with equal pages, we show the existence of an exact symplectic cobordism whose negative end equals the disjoint union of the contact manifolds associated to the given open books, and whose positive end induces the contact manifold associated to the open book with the same page and concatenated monodromy. Using similar methods, we show the existence of strong fillings for contact manifolds associated with doubled open books, a certain class of fiber bundles over the circle obtained by performing the binding sum of two open books with equal pages and inverse monodromies. From this we conclude, following an outline by Wendl, that the complement of the binding of an open book cannot contain any local filling obstruction. Given a contact [Formula: see text]-manifold, according to Eliashberg there is a symplectic cobordism to a fibration over the circle with symplectic fibers. We extend this result to higher dimensions recovering a recent result by Dörner–Geiges–Zehmisch. Our cobordisms can also be thought of as the result of the attachment of a generalized symplectic [Formula: see text]-handle.

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 Non-fillable invariant contact structures on principal circle bundles and left-handed twists

2016 ◽  
Vol 27 (03) ◽  
pp. 1650024 ◽  
Author(s):  
River Chiang ◽  
Fan Ding ◽  
Otto van Koert
Keyword(s):  
Circle Action ◽  
Contact Structures ◽  
Contact Manifolds ◽  
Open Books ◽  
Dehn Twists ◽  
Left Handed ◽  
Circle Bundles

We define symplectic fractional twists, which subsume Dehn twists and fibered twists and use these in open books to investigate contact structures. The resulting contact structures are invariant under a circle action, and share several similarities with the invariant contact structures that were studied by Lutz and Giroux. We show that left-handed fractional twists often give rise to “algebraically overtwisted” contact manifolds, a certain class of non-fillable contact manifolds.

scholarly journals Connected sums and finite energy foliations I: Contact connected sums

2018 ◽  
Vol 16 (6) ◽  
pp. 1639-1748
Author(s):  
Joel W. Fish ◽  
Richard Siefring
Keyword(s):  
Finite Energy ◽  
Connected Sums

scholarly journals A note on overtwisted contact structures

2011 ◽  
Vol 48 (1) ◽  
pp. 130-134
Author(s):  
El.if Yilmaz
Keyword(s):  
Recent Work ◽  
Contact Structures ◽  
Open Book ◽  
Open Book Decomposition ◽  
Dehn Twists ◽  
Left Handed ◽  
Closed Contact

In this note, we use the recent work of Honda-Kazez-Matić [3] to prove that a closed contact 3-manifold admitting a compatible open book decomposition with a nontrivial monodromy which can be presented as a product of left handed Dehn twists is overtwisted.

scholarly journals Rigid constellations of closed Reeb orbits

2017 ◽  
Vol 153 (11) ◽  
pp. 2394-2444 ◽  
Author(s):  
Ely Kerman
Keyword(s):  
Contact Manifold ◽  
Existence Results ◽  
Rigidity Theorem ◽  
Contact Form ◽  
Multiplicity Results ◽  
Tight Contact ◽  
Reeb Flows ◽  
Closed Orbits ◽  
Closed Contact ◽  
Standard Contact

We use Hamiltonian Floer theory to recover and generalize a classic rigidity theorem of Ekeland and Lasry. That theorem can be rephrased as an assertion about the existence of multiple closed Reeb orbits for certain tight contact forms on the sphere that are close, in a suitable sense, to the standard contact form. We first generalize this result to Reeb flows of contact forms on prequantization spaces that are suitably close to Boothby–Wang forms. We then establish, under an additional nondegeneracy assumption, the same rigidity phenomenon for Reeb flows on any closed contact manifold. A natural obstruction to obtaining sharp multiplicity results for closed Reeb orbits is the possible existence of fast closed orbits. To complement the existence results established here, we also show that the existence of such fast orbits cannot be precluded by any condition which is invariant under contactomorphisms, even for nearby contact forms.

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.

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