Open book decomposition 3 manifold

We prove that every closed, orientable 3 manifold has an open book decomposition with connected binding. The pair s,h can be interpreted as an open book decomposition of some 3manifold. Here, s is a surface, h is the monodromy of the abstract open book, and k is the fibered link that makes up the binding of the ob. An open book decomposition of a 3dimensional manifold m is a pair b. On the alexander polynomials of open book decompositions of 3. Equivalence of heegaard floer homology and embedded. A closed manifold has an open book decomposition if it. We prove that every closed, orientable 3manifold has an open book decomposition with connected binding.

Open book decompositions of 3manifolds take f x 0, 1. Definitions for open book decomposition open book decomposition. Open book decompositions of 3manifolds pdf book manual. Due to alexander, every closed oriented 3manifold has an open book decomposition. We say that an open book decomposition for mis a pair b where the binding bis a smooth link in m. In the present paper we will show that for every link l in a stein fillable 3manifold there exists an additional knot l. Two open books for a manifold m are said to be equivalent if there is an ambient. The little that exists of the 3 manifolds book see below for a table of contents is rather crude and unpolished, and doesnt cover a lot of material, but it does contain a few things that may not be readily available elsewhere, like the elementary form of the jacoshalenjohannson torus decomposition theorem. An open book decomposition of a closed, connected, oriented 3manifold m consists of a smooth family f f.

Positive open book decompositions of stein fillable 3. Is there a direct topological proof that shows that every 3manifold admits an open book decomposition. Open book decompositions versus prime factorizations of. Piergallini that a closed, oriented, smooth 3 manifold is stein fillable if and only if it has a positive open book decomposition. In particular, in order to show that two contact structures on a given 3 manifold are isomorphic, it is enough to show that they are carried by isomorphic open books. An open book decomposition of a closed, connected, oriented 3manifold mconsists of a smooth family f ff g 2s 1 of compact, connected surfaces f. Also, special properties of the heegaard splitting associated to an open book decomposition are given. Open books have relevance to contact geometry, with a famous theorem of emmanuel giroux given below that shows that contact geometry can be studied from an entirely topological viewpoint. Namely, we can determine the binding, the page and the monodromy of the constructed planar open book decomposition for m. It is known by loi and piergallini that a closed, oriented, smooth 3manifold is stein fillable if and only if it has a positive open book decompositi. The existence of an open book decomposition for any closed, orientable 3 manifold was proved by j. An open book decomposition of m is said to be positive if its monodromy h consists of a. An open book decomposition of a closed, oriented, smooth 3 manifold is the following.

On legendrian embbeddings into open book decompositions. Trisections of 4manifolds with boundary nickolas a. Ideally, such a proof might start with a heegaard decomposition or a surgery description of a given 3 manifold, and then proceed to give an algorithm for converting this to an open book decomposition. Let s be a compact, oriented, smooth twodimensional manifold with boundary s and h an automorphism of s, which is the identity on s. Then m has an open book decomposition with binding f xa. Recall that a rational open book decomposition of a closed 3manifold m is a pair k. Special moves for open book decompositions of 3manifolds. Information and translations of open book decomposition in the most comprehensive dictionary definitions resource on the web. Given a handle decomposition of a 4manifold with boundary, and an open book decomposition of the boundary, we show how to produce a trisection diagram of a trisection of the 4manifold inducing the given open book. In 17, markov gave an equivalent condition for two braids in r3 to be isotopic. Let n 6 and m be a closed, simplyconnected nmanifold. It is a wellknown result in 3dimensional topology that every closed 3dimensional orientable manifold admits an open book decomposition. S 1 such that, for each t 2s, we get a compact oriented surface. Prepare an open book decomposition of m and assume by plumbing positive hopf bands that the binding is connected.

We assume that the given orientation on bcoincides with the boundary orientation induced by the pages. M \ b s 1 is a fibration of the complement of b such that for each. Surgery diagrams and open book decompositions of contact 3. Braids and open book decompositions elena pavelescu. It shows the existence of planar open book decompositions for 3manifolds. Let b be an open book decomposition of a contact 3manifold y, then cb 6 0. Comparing open book and heegaard decompositions of 3manifolds. Abstractit is known by loi and piergallini that a closed, oriented, smooth 3manifold is stein fillable if and only if it has a positive open book decomposition. Ishikawa topology 45 2006 325342 by identifying hx,0 and x,1 for x. Open book decompositions let k be a link in a closed oriented 3manifold m.

We prove that for any contact 3 manifold supported by a spinal open book decomposition with planar pages, there is a universal bound on the euler characteristic and signature of its minimal. An open book decomposition of a 3manifold m is a pair b. More precisely, the heegard decomposition is made of the three manifold we get by. The cassonwalker invariant of 3manifolds with genus one.

Equivalence of heegaard floer homology and embedded contact. Given a handle decomposition of a 4manifold with boundary and an open book decomposition of the boundary, we show how to produce a trisection diagram of a trisection of the 4manifold inducing the given open book. F where kis an oriented link called the binding of the open book and mnk. This is the first of a series of papers devoted to proving the equivalence of heegaard floer homology and embedded contact homology abbreviated ech. In mathematics, an open book decomposition or simply an open book is a decomposition of a closed oriented 3 manifold m into a union of surfaces necessarily with boundary and solid tori. Since every contact 3manifold admits a supporting open book decomposition with pseudoanosov monodromy, connected binding, and k. The theme of this book is the role of the fundamental group in determining the topology of a given 3 manifold. The topology and geometry of contact struc tures in. A decomposition of a 3 manifold into a link and a fibration over the circle by surfaces bounded by that link. Openbookdecomposition dictionary definition openbook. Open book decompositions of 3manifolds robert myers1 abstract. Embedded contact homology and open book decompositions. We provide a complete set of two moves that suffice to relate any two open book decompositions of a given 3manifold.

An open book decomposition of a closed, oriented, connected 3 manifold y is an oriented link. Moreover, two open book decompositions s,h,k and s. Open book decompositions of 3 manifolds robert myers1 abstract. M, called the pages of the open book, having the same boundary. Castro developed this case further, in particular showing the importance of this case by. Open book decompositions versus prime factorizations of closed, oriented 3manifolds paolo ghiggini paolo lisca let m be a closed, oriented, connected 3manifold and. The latter has proven a very useful way to study manifolds. The following proof is from rolfsens book, theorem 10k1. Open book decomposition in finnish englishfinnish dictionary. Suppose that for a link l in a 3manifold y the com plement y l fibers as 1r. We do this by making the original proof of the existence of relative trisections more explicit in terms of handles. We show an algorithmic way for finding a compatible open book decomposition on a contact 3manifold given by contact 1surgery. Moreover, for open book decompositions of a given 3 manifold m, the moves can be realized as embedded in m and the resulting equivalence of open books can be thought up to ambient isotopy in m, not just up to di.

Also, special properties of the heegaard splitting associated to. Citeseerx document details isaac councill, lee giles, pradeep teregowda. In this article, we describe an explicit abstract open book decomposition adapted to the contact3manifoldst. Ideally, such a proof might start with a heegaard decomposition or a surgery description of a given 3manifold, and then proceed to give an algorithm for converting this to an open book decomposition. Recently, honda, kazez and matic 9 described an adapted partial open book decomposition of a compact contact 3manifold with convex boundary by generalizing the work of giroux in the closed case. Every closed oriented 3manifold has an open book decomposition. Even though the field has grown tremendously during that time, the book remains one of the best and most popular introductions to the subject. In mathematics, an open book decomposition or simply an open book is a decomposition of a closed oriented 3manifold m into a union. We only assume very basic topology and mostly appeal to the intuition of the readers. It is a wellknown result in three dimensional topology that every closed three dimensional orientable manifold admits an open book decomposition. Such open book decompositions of mx are called milnor open book decompositions. When developing the foundations of the study of trisections of 4manifolds in ref. Stein fillable 3manifolds admit positive open book.

We use the following presentation of a closed orientable 3 manifold m as a branched covering of s3 to construct an open book decomposition of m with connected binding. In this article, we describe an explicit abstract open book decomposition adapted to the contact 3manifoldst. In the present paper we will show that for every link l in a stein fillable 3 manifold there exists an additional knot l to l such that the union of the links l and l is the binding of a positive open book decomposition of the stein. Moreover, for open book decompositions of a given 3manifold m, the moves can be realized as embedded in m and the resulting equivalence of open books can be thought up to ambient isotopy in m, not just up to di. Open book decompositions of 3manifold and associated heegard. We then present an algorithmic way for determining the monodromy of this planar open book decomposition. Open book decomposition encyclopedia of mathematics. A manifold is prime if it cannot be presented as a connected sum of more than one manifold, none of which is the sphere of the same dimension. He suggested that the binding may be assumed connected, but the first published proof of this fact was given by r. Definition of open book decomposition in the definitions. We use the following presentation of a closed orientable 3manifold m as a branched covering of s3 to construct an open book decomposition of m with connected binding.