Singular open book structures from real mappings

We define open book structures with singular bindings. Starting with an extension of Milnor’s results on local fibrations for germs with nonisolated singularity, we find classes of genuine real analytic mappings which yield such open book structures.     

Equisingularity of families of map germs between curves

Given a finite map germ f : (X, 0) → (Y, 0) between complex analytic reduced space curves, we look at invariants which control the topological triviality and the Whitney equisingularity in families of this type of map germs. In the case that (Y, 0) is smooth, the main invariant is the Milnor number of a function on a curve. We deduce some applications to the equisingularity of families of finitely determined map germs ${f : (\mathbb{C}^2, 0) \to (\mathbb{C}^2, 0)}$ and ${f : (\mathbb{C}^2, 0) \to (\mathbb{C}^3, 0)}$ .     

Open book decompositions and stable Hamiltonian structures

We show that every open book decomposition of a contact 3-manifold can be represented (up to isotopy) by a smooth R$\mathbb{R}$-invariant family of pseudoholomorphic curves on its symplectization with respect to a suitable stable Hamiltonian structure. In the planar case, this family survives small perturbations, and thus gives a concrete construction of a stable finite energy foliation that has been used in various applications to planar contact manifolds, including the Weinstein conjecture (Abbas et al., 2005) [2] and the equivalence of strong and Stein fillability (Wendl, to appear) [20].     

Open book structures on semi-algebraic manifolds

Given a C ² semi-algebraic mapping \({F} : {\mathbb{R}^N \rightarrow \mathbb{R}^p}\), we consider its restriction to \({W \hookrightarrow \mathbb{R^{N}}}\) an embedded closed semi-algebraic manifold of dimension \({n-1 \geq p \geq 2}\) and introduce sufficient conditions for the existence of a fibration structure (generalized open book structure) induced by the projection \({\frac{F}{\Vert F \Vert}:W{\setminus} F^{-1}(0) \to S^{p-1}}\). Moreover, we show that the well known local and global Milnor fibrations, in the real and complex settings, follow as a byproduct by considering W as spheres of small and big radii, respectively. Furthermore, we consider the composition mapping of F with the canonical projection \({\pi: \mathbb{R}^{p} \to \mathbb{R}^{p-1}}\) and prove that the fibers of \({\frac{F}{\Vert F \Vert}}\) and \({\frac{\pi \circ F}{\Vert \pi \circ F \Vert}}\) are homotopy equivalent. We also show several formulae relating the Euler characteristics of the fiber of the projection \({\frac{F}{\Vert F \Vert}}\) and \({W \cap F^{-1}(0)}\). Similar formulae are proved for mappings obtained after composition of F with canonical projections.     

Topological {\mathcal{K}} -classification of finitely determined map germs

We consider smooth finitely C ⁰- ${\mathcal{K}}$ -determined map germs ${f : (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)}$ and we look at the classification under C ⁰- ${\mathcal{K}}$ -equivalence. The main tool is the homotopy type of the link, which is obtained by intersecting the image of f with a small enough sphere centered at the origin. When f ^?1(0) = {0}, the link is a smooth map between spheres and f is C ⁰- ${\mathcal{K}}$ -equivalent to the cone of its link. When f ^?1(0) ≠ {0}, we consider a link diagram, which contains some extra information, but again f is C ⁰- ${\mathcal{K}}$ -equivalent to the generalized cone. As a consequence, we deduce some known results due to Nishimura (for n = p) or the first named author (for n < p). We also prove some new results of the same nature.     

Stabilizing the monodromy of an open book decomposition

We prove that any mapping class on a compact oriented surface with non-empty boundary can be made pseudo-Anosov and right-veering after a sequence of positive stabilizations.      

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.     

Versal unfolding of smooth map germs under a subgroup of the left-right equivalence group

定义了左右等价群的一个子群,给出了在这个子群下映射芽的等价及其开折的同构等概念.利用乘积积分理论,讨论在该子群下映射芽的通用开折,证明了相应开折的平凡性引理,几何引理和代数引理,给出了映射茅的开折是通用开折的充要条件.     

Formal and analytic linearizability of families of conformal map germs (ℂ, 0) → (ℂ, 0)

We consider a nondegenerate one-parameter family of germs of conformal maps of (?, 0) into (?, 0). We prove that such a family is analytically linearizable whenever it is formally linearizable. In this case, the linearizing coordinate change analytically depends on the parameter.     

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.     

Links in an open book decomposition and in the standard contact structure

We study a relationship between arc presentations of links in and Legendrian links in with the standard tight contact structure. We determine the arc indices of torus knots.

     

The space of strong types and an open map theorem

We define a new topology on the space of strong types on a subsetA of a model of a given theory and prove that either . We also deduce an open map theorem.     

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.