Open book decompositions for contact structures on Brieskorn manifolds

In this paper, we give an open book decomposition for the contact structures on some Brieskorn manifolds, in particular for the contact structures of Ustilovsky. The decomposition uses right-handed Dehn twists as conjectured by Giroux.

     

Invariants of contact structures from open books

In this note we define three invariants of contact structures in terms of open books supporting the contact structures. These invariants are the support genus (which is the minimal genus of a page of a supporting open book for the contact structure), the binding number (which is the minimal number of binding components of a supporting open book for the contact structure with minimal genus pages) and the norm (which is minus the maximal Euler characteristic of a page of a supporting open book).

     

Surgery diagrams and open book decompositions of contact 3-manifolds

Summary We show an algorithmic way for finding a compatible open book decomposition on a contact 3-manifold given by contact (±1)-surgery.     

Five-Dimensional Contact SU(2)- and SO(3)-Manifolds and Dehn Twists

In the first part of this paper the five-dimensional contact SO(3)-manifolds are classified up to equivariant coorientation preserving contactomorphisms. The construction of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an application that all simply connected 5-manifolds with singular orbits are realized by a Brieskorn manifold with exponents (k,2,2,2). The standard contact structure on such a manifold gives right-handed Dehn twists, and a second contact structure defined in the article gives left-handed twists. In an appendix we also describe the classification of five-dimensional contact SU(2)-manifolds.     

On the classification of certain planar contact structures

We focus on contact structures supported by planar open book decompositions. We study right-veering diffeomorphisms to keep track of overtwistedness property of contact structures under some monodromy changes. As an application we give infinitely many examples of overtwisted contact structures supported by open books whose pages are the four-punctured sphere, and also we prove that a certain family is Stein fillable using lantern relation.     

On the Heegaard genus of contact 3-manifolds

It is well-known that the Heegaard genus is additive under connected sum of 3-manifolds. We show that the Heegaard genus of contact 3-manifolds is not necessarily additive under contact connected sum. We also prove some basic properties of the contact genus (a.k.a. open book genus [Rubinstein J.H., Comparing open book and Heegaard decompositions of 3-manifolds, Turkish J. Math., 2003, 27(1), 189–196]) of 3-manifolds, and compute this invariant for some 3-manifolds.     

Closed minimal surfaces in cusped hyperbolic three-manifolds

We introduce essential open book foliations by refining open book foliations, and develop technical estimates of the fractional Dehn twist coefficient (FDTC) of monodromies and the FDTC for closed braids, which we introduce as well. As applications, we quantitatively study the ‘gap’ between overtwisted contact structures and non-right-veering monodromies. We give sufficient conditions for a 3-manifold to be irreducible and atoroidal. We also show that the geometries of a 3-manifold and the complement of a closed braid are determined by the Nielsen–Thurston types of the monodromies of their open book decompositions.     

Khovanov homology, open books, and tight contact structures

We define the reduced Khovanov homology of an open book (S,?), and identify a distinguished “contact element” in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S,?). Our construction generalizes the relationship between the reduced Khovanov homology of a link and the Heegaard Floer homology of its branched double cover. As an application, we give combinatorial proofs of tightness for several contact structures which are not Stein-fillable. Lastly, we investigate a comultiplication structure on the reduced Khovanov homology of an open book which parallels the comultiplication on Heegaard Floer homology defined in Baldwin (2008) [4].     

Contact open books with exotic pages

We consider a fixed contact 3-manifold that admits infinitely many compact Stein fillings which are all homeomorphic but pairwise non-diffeomorphic. Each of these fillings gives rise to a closed contact 5-manifold described as a contact open book whose page is the filling at hand and whose monodromy is the identity symplectomorphism. We show that the resulting infinitely many contact 5-manifolds are all diffeomorphic but pairwise non-contactomorphic. Moreover, we explicitly determine these contact 5-manifolds.     

Limits of families of Brieskorn lattices and compactified classifying spaces

We investigate variations of Brieskorn lattices over non-compact parameter spaces, and discuss the corresponding limit objects on the boundary divisor. We study the associated variation of twistors and the corresponding limit mixed twistor structures. We construct a compact classifying space for regular singular Brieskorn lattices and prove that its pure polarized part carries a natural hermitian structure and that the induced distance makes it into a complete metric space.     

Computing the Thurston–Bennequin invariant in open books

We give explicit formulas and algorithms for the computation of the Thurston–Bennequin invariant of a nullhomologous Legendrian knot on a page of a contact open book and on Heegaard surfaces in convex position. Furthermore, we extend the results to rationally nullhomologous knots in arbitrary 3-manifolds.     

The intersection matrix of Brieskorn singularities

To each isolated singularity of a hypersurface of dimensionn, one associates the local fundamental groupG of the moduli space minus the discriminant locus, and a representation σ:G→Aut(H), whereH is then-homology group, with integer coefficients, of the non singular fibre. Although, in general it is very difficult to determine even a presentation ofG, we show that the image of σ can be computed rather easily, by exploiting some relations in a first aproximate presentation ofG, in the case of Brieskorn polynomials namely, polynomials of the type \(x_0^{a_0 } + \cdot \cdot \cdot + x_0^{a_n } \) . In this way we solve an open problem stated by Brieskorn [1] and Pham [8].     

Right-veering diffeomorphisms of compact surfaces with boundary

We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary. The monoid strictly contains the monoid of products of positive Dehn twists. We explain the relationship to tight contact structures and open book decompositions. Mathematics Subject Classification (1991) Primary 57M50, secondary 53C15     

Full Lutz twist along the binding of an open book

Let T denote a binding component of an open book (S, f){(\Sigma, \phi)} compatible with a closed contact 3-manifold (M, ξ). We describe an explicit open book (S￠, f￠){(\Sigma', \phi')} compatible with (M, ζ), where ζ is the contact structure obtained from ξ by performing a full Lutz twist along T. Here, (S￠, f￠){(\Sigma', \phi')} is obtained from (S, f){(\Sigma, \phi)} by a local modification near the binding.     

On the contact boundaries of normal surface singularities

The abstract boundary M of a normal complex-analytic surface singularity is canonically equipped with a contact structure. We show that if M is a rational homology sphere, then this contact structure is uniquely determined by the topological type of M. An essential tool is the notion of open book carrying a contact structure, defined by E. Giroux. To cite this article: C. Caubel, P. Popescu-Pampu, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Brieskorn modules and Gauss Manin systems for non-isolated hypersurface singularities

We study the Brieskorn modules associated to a germ of a holomorphicfunction with non-isolated singularities and show that the Brieskornmodule has naturally the structure of a module over the ringof microdifferential operators of non-positive degree, and thatthe kernel of the morphism to the Gauss–Manin system coincideswith the torsion part for the action of t and also with thatfor the action of the inverse of the Gauss–Manin connection.This torsion part is not finitely generated in general, anda sufficient condition for the finiteness is given here. A Thom–Sebastiani-typetheorem for the sheaf of Brieskorn modules is also proved whenone of two functions has an isolated singularity.     

Milnor open books and Milnor fillable contact 3-manifolds

We say that an oriented contact manifold (M,ξ) is Milnor fillable if it is contactomorphic to the contact boundary of an isolated complex-analytic singularity (X,x). In this article we prove that any three-dimensional oriented manifold admits at most one Milnor fillable contact structure up to contactomorphism. The proof is based on Milnor open books: we associate an open book decomposition of M with any holomorphic function f:(X,x)→(C,0), with isolated singularity at x and we verify that all these open books carry the contact structure ξ of (M,ξ)—generalizing results of Milnor and Giroux.     

The links of 3‐dimensional singularities defined by Brieskorn polynomials

In this paper, we completely determine the diffeomorphism types of the 5‐dimensional links of 3‐dimensional log‐canonical singularities defined by Brieskorn polynomials. Moreover, we show that if k is an integer with 1 ≤ k < 611, then there is no link K defined by a Brieskorn polynomial in ?⁴ such that the order of H₂(K) is 6k. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Über die kritischen Werte gewisser holomorpher Abbildungen

In this note, it is proved that a proper open holomorphic mapping of a three-dimensional complex manifold onto a two-dimensional complex manifold cannot have isolated critical values, if the generic fibre is not the Riemann sphere. The main tools used are the universal property of the Teichmüller family of compact Riemann surfaces, and a theorem of Moiezon asserting that the indeterminacies of certain meromorphic mappings can be resolved by a succession of monoidal transformations with non-singular centres.

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Der Referent hat mich darauf aufmerksam gemacht, daß diese Aussage auch bei Brieskorn (Singularitäten von Hyperflächen, Habilitationsschrift, Bonn 1967) bewiesen worden ist.