scholarly journals On Open Book Embedding of Contact Manifolds in the Standard Contact Sphere

2019 ◽  
Vol 63 (4) ◽  
pp. 755-770 ◽  
Author(s):  
Kuldeep Saha
Keyword(s):  
Large Class ◽  
Alternative Proof ◽  
Open Book ◽  
Constructive Approach ◽  
Contact Manifolds ◽  
Book Embedding ◽  
Book Embeddings ◽  
Standard Contact

AbstractWe prove some open book embedding results in the contact category with a constructive approach. As a consequence, we give an alternative proof of a theorem of Etnyre and Lekili that produces a large class of contact 3-manifolds admitting contact open book embeddings in the standard contact 5-sphere. We also show that all the Ustilovsky $(4m+1)$-spheres contact open book embed in the standard contact $(4m+3)$-sphere.

On the non-regularity of tangent sphere bundles

1978 ◽  
Vol 82 (1-2) ◽  
pp. 13-17 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Riemannian Manifold ◽  
Large Class ◽  
Constant Curvature ◽  
Positive Constant ◽  
Contact Structure ◽  
Compact Riemannian Manifold ◽  
Contact Structures ◽  
Sphere Bundle ◽  
Contact Manifolds ◽  
Tangent Sphere

SynopsisClassically the tangent sphere bundles have formed a large class of contact manifolds; their contact structures are not in general regular, however. Specifically we prove that the natural contact structure on the tangent sphere bundle of a compact Riemannian manifold of non-positive constant curvature is not regular.

A NOTE ON OPEN BOOK EMBEDDINGS OF -MANIFOLDS IN

2021 ◽  
pp. 1-8
Author(s):  
SUHAS PANDIT ◽  
SELVAKUMAR A
Keyword(s):  
Open Book ◽  
Open Book Decomposition ◽  
Book Embeddings

Abstract In this note, we show that given a closed connected oriented $3$ -manifold M, there exists a knot K in M such that the manifold $M'$ obtained from M by performing an integer surgery admits an open book decomposition which embeds into the trivial open book of the $5$ -sphere $S^5.$

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.

On contact embeddings of contact manifolds in the odd dimensional Euclidean spaces

2015 ◽  
Vol 26 (07) ◽  
pp. 1550045 ◽  
Author(s):  
Naohiko Kasuya
Keyword(s):  
Contact Structure ◽  
Euclidean Spaces ◽  
Contact Manifolds ◽  
Standard Contact ◽  
Simply Connected

We prove that a closed co-oriented contact (2m + 1)-manifold (M2m + 1, ξ) can be a contact submanifold of the standard contact structure on ℝ4m + 1, if it satisfies one of the following conditions: (1) m is odd (m ≥ 3) and H1(M2m + 1; ℤ) = 0, (2) m is even (m ≥ 4) and M2m + 1 is 2-connected, (3) m = 2 and M5 is simply-connected.

scholarly journals On Subcritically Stein Fillable 5-manifolds

2018 ◽  
Vol 61 (1) ◽  
pp. 85-96 ◽  
Author(s):  
Fan Ding ◽  
Hansjörg Geiges ◽  
Guangjian Zhang
Keyword(s):  
Fundamental Group ◽  
Contact Structure ◽  
Contact Structures ◽  
Homotopy Classification ◽  
Contact Manifolds ◽  
Almost Contact Structure ◽  
Diffeomorphism Type ◽  
Standard Contact ◽  
Simply Connected

AbstractWe make some elementary observations concerning subcritically Stein fillable contact structures on 5-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case where the fundamental group is finite cyclic, and we show that on the 5-sphere, the standard contact structure is the unique subcritically ?llable one. More generally, it is shown that subcritically fillable contact structures on simply connected 5-manifolds are determined by their underlying almost contact structure. Along the way, we discuss the homotopy classification of almost contact structures.

scholarly journals On manifolds with infinitely many fillable contact structures

2020 ◽  
Vol 31 (13) ◽  
pp. 2050108
Author(s):  
Alexander Fauck
Keyword(s):  
Large Class ◽  
Chern Class ◽  
Contact Structures ◽  
Finitely Generated ◽  
Contact Manifolds ◽  
Symplectic Homology ◽  
Famous Result ◽  
Cotangent Bundles ◽  
First Chern Class ◽  
Invariance Theorem

We introduce the notion of asymptotically finitely generated contact structures, which states essentially that the Symplectic Homology in a certain degree of any filling of such contact manifolds is uniformly generated by only finitely many Reeb orbits. This property is used to generalize a famous result by Ustilovsky: We show that in a large class of manifolds (including all unit cotangent bundles and all Weinstein fillable contact manifolds with torsion first Chern class) each carries infinitely many exactly fillable contact structures. These are all different from the ones constructed recently by Lazarev. Along the way, the construction of Symplectic Homology is made more general. Moreover, we give a detailed exposition of Cieliebak’s Invariance Theorem for subcritical handle attaching, where we provide explicit Hamiltonians for the squeezing on the handle.

THE SUPPORT GENERA OF CERTAIN LEGENDRIAN KNOTS

2012 ◽  
Vol 21 (11) ◽  
pp. 1250105 ◽  
Author(s):  
YOULIN LI ◽  
JIAJUN WANG
Keyword(s):  
Contact Structure ◽  
Open Book ◽  
Legendrian Knots ◽  
Three Sphere ◽  
Open Book Decompositions ◽  
Standard Contact ◽  
Trefoil Knots ◽  
Positive Support

In this paper, the support genera of all Legendrian right-handed trefoil knots and some other Legendrian knots are computed. We give examples of Legendrian knots in the three-sphere with the standard contact structure which have positive support genera with arbitrarily negative Thurston–Bennequin invariant. This answers a question in [S. Onaran, Invariants of Legendrian knots from open book decompositions, Int. Math. Res. Not.10 (2010) 1831–1859].

Open book embeddings of closed non-orientable $3$-manifolds

2019 ◽  
Vol 49 (4) ◽  
pp. 1143-1168
Author(s):  
Abhijeet Ghanwat ◽  
Suhas Pandit ◽  
Selvakumar A
Keyword(s):  
Open Book ◽  
Book Embeddings
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