Related papers: On the stable equivalence of open books in three-m…
We provide a complete set of two moves that suffice to relate any two open book decompositions of a given 3-manifold. One of the moves is the usual plumbing with a positive or negative Hopf band, while the other one is a special local…
We prove that every stable Hamiltonian structure on a closed oriented three-manifold is stably homotopic to one which is supported (with suitable signs) by an open book.
We study open books on three manifolds which are compatible with an overtwisted contact structure. We show that the existence of certain arcs, called sobering arcs, is a sufficient condition for an open book to be overtwisted, and is…
We build handle decompositions of n-manifolds that encode given open book decompositions and describe handle slides that reveal new open book decompositions on the same underlying manifold, for $n \geq 3$. This recovers known stabilization…
We show that every open book decomposition of a contact 3-manifold can be represented (up to isotopy) by a smooth R-invariant family of pseudoholomorphic curves on its symplectization with respect to a suitable stable Hamiltonian structure.…
We prove that the homology of the mapping class group of any 3-manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3-manifold when both manifolds are compact and orientable. The stabilization also holds for…
We study open books (or open book decompositions) of a closed oriented 3-manifold which support overtwisted contact structures. We focus on a simple closed curve along which one can perform Stallings twist, called ``twisting loop''. We show…
For any link in the $3$-sphere, we give a visual construction of a stable map $f$ from the $3$-sphere into the real plane enjoying the following properties; $f$ has no cusp point, the set of definite fold points of $f$ is isotopic to the…
A stable map of a closed orientable $3$-manifold into the real plane is called a stable map of a link in the manifold if the link is contained in the set of definite fold points. We give a complete characterization of the hyperbolic links…
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…
We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact…
We use contact fiber sums of open book decompositions to define an infinite hierarchy of filling obstructions for contact 3-manifolds, called planar k-torsion for nonnegative integers k, all of which cause the contact invariant in Embedded…
A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. In this article we study open book decompositions on smooth real 3-manifolds that are compatible with the real…
This note explains how to relate some contact geometric operations, such as surgery, to operations on an underlying contact open book. In particular, we shall give a simple proof of the fact that stabilizations of contact open books yield…
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…
This paper presents a new proof of the Giroux Correspondence for tight contact $3$-manifolds using techniques from Heegaard splittings and convex surface theory. We introduce tight Heegaard splittings, which generalise the Heegaard…
The open book decompositions of the 3-sphere whose pages are pairs of pants have been fully understood for some time, through the lens of contact geometry. The purpose of this note is to exhibit a purely topological derivation of the…
We show rational homological stability for the homotopy automorphisms and block diffeomorphims of iterated connected sums of products of spheres. The spheres can have different dimension, but need to satisfy a certain connectivity…
We extend the theory of relative trisections of smooth, compact, oriented $4$-manifolds with connected boundary given by Gay and Kirby to include $4$-manifolds with an arbitrary number of boundary components. Additionally, we provide…
Emmanuel Giroux showed that every contact structure on a closed three dimensional manifold is supported by an open book decomposition. We will extend this result by showing that the open book decomposition can be chosen in such a way that…