# Open book

## 1 Open Books

The simplest general definition of an open book, as the concept first appeared explicitly in the literature [Winkelnkemper1973], is the following.

Let $V$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}V$ be a smooth, compact $(n-1)$$(n-1)$-manifold with $\partial V \neq \emptyset$$\partial V \neq \emptyset$. Let $h:V\to V$$h:V\to V$ be a diffeomorphism that restricts to the identity on $\partial V$$\partial V$. The mapping torus of $h$$h$, denoted $V_{h}$$V_{h}$, has $\partial V \times S^1$$\partial V \times S^1$ as boundary. By identifying $(x,t)$$(x,t)$ with $(x,t')$$(x,t')$ for each $x\in\partial V$$x\in\partial V$ and $t,t'\in S^1$$t,t'\in S^1$, we obtain a smooth, closed $n$$n$-manifold $M$$M$. Let $N$$N$ denote the image of $\partial V \times S^1$$\partial V \times S^1$ under the identification map.

Definition 1.1. A closed manifold is an open book provided it is diffeomorphic to $M$$M$. The fibers $V \times \left\{ t \right\}$$V \times \left\{ t \right\}$ of $V_{h}$$V_{h}$, where $t \in S^1$$t \in S^1$, are codimension $1$$1$ submanifolds of $M$$M$ whose images are the pages of the open book. The image of the closed, codimension 2 submanifold $N$$N$ of $M$$M$ is called the binding of the open book.

For an open book $M$$M$, every point $x\in M-N$$x\in M-N$ lies on one and only one page, and $N$$N$ is the boundary of each page.

Sometimes for certain homological and/or analytic applications (for example, [Barreto&Medrano&Verjovsky2013, p. 3], [Quinn1979, p. 55], and [Ranicki1998, p. 252]) one uses more elaborate, but equivalent, definitions. Furthermore, one can generalize the open book definition to compact manifolds with nonempty boundaries [Ranicki1998, p. 353], [Quinn1979, p. 55]. For nice figures, see [Giroux2005], [Ranicki1998, p. 616], and [Winkelnkemper1973].

The first general open book existence theorem was obtained in 1972:

Theorem 1.2 [Winkelnkemper1973] . If $n > 6$$n > 6$ and $M^n$$M^n$ is simply connected, then $M^n$$M^n$ is an open book if and only if its signature is $0$$0$.

In general, if $M^n$$M^n$ is orientable and $n > 5$$n > 5$, then the existence of open book structures was settled by T. Lawson when $n$$n$ is odd: there is no obstruction, and for any $n > 3$$n > 3$ by F. Quinn: when $n$$n$ is even, the obstruction lies in the asymmetric Witt group of $\mathbb{Z}[\pi_{1}(M^n)]$$\mathbb{Z}[\pi_{1}(M^n)]$, generalizing the Wall surgery obstruction [Ranicki1998, p. 360]. In 1975, González-Acuña used fundamental work of Alexander and Lickorish to prove: if $n=3$$n=3$, then every closed, orientable $M^3$$M^3$ is an open book; furthermore, the page can be chosen to have a connected boundary (that is, every such $M^3$$M^3$ contains a fibered knot) or be as simple as possible: a compact $2$$2$-disk with holes.

## 2 Applications

Open book decompositions have had numerous applications (see the Appendix in [Ranicki1998]). More recent applications are to: diffusion processes and the averaging principle [Freidlin&Wentzell2004], contact forms on moment angle manifolds [Barreto&Medrano&Verjovsky2013], and Haefliger's singular foliations [Laudenbach&Meigniez2012]. Most notably, in dimension $3$$3$, open books are crucial for proving Giroux's theorem that, roughly put, on any closed, orientable $3$$3$-manifold there is a one-to-one correspondence between contact forms (Geometry) and open books (Topology) [Etnyre2006]. In other words, every contact form on a closed, orientable $3$$3$-manifold arises via an open book as in the construction of contact forms by Thurston and Winkelnkemper [Thurston&Winkelnkemper1975].

Another notable feature was discovered by González-Acuña: in dimension $3$$3$ and when the page is planar, the monodromy homeomorphism of the page can be substituted by a purely discrete presentation, an Artin presentation of the fundamental group of the $3$$3$-manifold. Thus, the whole theory of closed, orientable $3$$3$-manifolds is encoded in the purely discrete, group-theoretic theory of Artin presentations [Winkelnkemper2002], [Ranicki1998, p. 617]. This theory is now an autonomous one with purely group-theoretic analogues of the Casson invariant [Calcut2005] and Donaldson's Theorem [Ranicki1998, p. 621], [Winkelnkemper2002, p. 240].