Open book

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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$${{Authors|Jack Calcut, Horst Winkelnkemper}}{{Stub}} ==Open Books== <wikitex>; The simplest general definition of an open book, as the concept first appeared explicitly in the literature {{cite|Winkelnkemper1973}}, is the following. Let $V$ be a smooth, compact $(n-1)$-manifold with $\partial V \neq \emptyset$. Let $h:V\to V$ be a diffeomorphism that restricts to the identity on $\partial V$. The mapping torus of $h$, denoted $V_{h}$, has $\partial V \times S^1$ as boundary. By identifying $(x,t)$ with $(x,t')$ for each $x\in\partial V$ and $t,t'\in S^1$, we obtain a smooth, closed $n$-manifold $M$. Let $N$ denote the image of $\partial V \times S^1$ under the identification map . {{beginrem|Definition|}} A closed manifold is an '''open book''' provided it is diffeomorphic to $M$. The fibers $V \times \left\{ t \right\}$ of $V_{h}$, where $t \in S^1$, are codimension V$ be a smooth, compact $(n-1)$$(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, Gonz{\'a}lez-Acu{\ n}a 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 Gonz{\'a}lez-Acu{\ n}a: 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].

3 References

• [Barreto&Medrano&Verjovsky2013] Y. Barreto, S. López de Medrano and A. Verjovsky, Open Book Structures on Moment-Angle Manifolds $Z^{\mathbb C}(\Lambda)$$Z^{\mathbb C}(\Lambda)$ and Higher Dimensional Contact Manifolds, (2013). Available at the arXiv:1303.2671.
• [Calcut2005] J. S. Calcut, Knot theory and the Casson invariant in the Artin presentation theory, Fundamental and Applied Mathematics 11 No. 4, L. V. Keldysh Memorial Proceedings, Moscow, (2005), 119--126. MR2192960 (2006m:57013)
• [Etnyre2006] J. B. Etnyre, Lectures on open book decompositions and contact structures, in Floer homology, gauge theory, and low-dimensional topology, 103–141, Clay Math. Proc., 5, Amer. Math. Soc., (2006). MR2249250 (2007g:57042) Zbl 1108.53050
• [Freidlin&Wentzell2004] M. I. Freidlin and A. D. Wentzell, Diffusion processes on an open book and the averaging principle, Stochastic Process. Appl. 113 (2004), no.1, 101–126. MR2078539 (2005e:60178) Zbl 1075.60100
• [Giroux2005] E. Giroux, What is...an Open Book?, Notices of the Amer. Math. Soc. 52 (2005), 42–43.
• [Laudenbach&Meigniez2012] F. Laudenbach and G. Meigniez, Haefliger's codimension-one singular foliations, open books and twisted open books in dimension 3, Bull. Braz. Math. Soc. (N.S.) 43 (2012), 347–373. MR3024060 Zbl 1263.57022
• [Quinn1979] F. Quinn, Open book decompositions, and the bordism of automorphisms, Topology 18 (1979), no.1, 55–73. MR528236 (81e:57027) Zbl 0425.57010
• [Ranicki1998] A. Ranicki, High-dimensional knot theory, Springer-Verlag, 1998. MR1713074 (2000i:57044) Zbl 1059.19003
• [Thurston&Winkelnkemper1975] W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975), 345–347. MR0375366 (51 #11561) Zbl 0312.53028
• [Winkelnkemper1973] H. E. Winkelnkemper, Manifolds as open books, Bull. Amer. Math. Soc. 79 (1973), 45–51. MR0310912 (46 #10010) Zbl 0269.57011
• [Winkelnkemper2002] H. E. Winkelnkemper, Artin presentations. I. Gauge theory, $3 1$$3 1$ TQFT's and the braid groups, J. Knot Theory Ramifications 11 (2002), no.2, 223–275. MR1895372 (2003b:57045) Zbl 0994.57026
• [[Template:{Barreto&Medrano&Verjovsky2013|[{Barreto&Medrano&Verjovsky2013]]] {{{Barreto&Medrano&Verjovsky2013}}

4 References

• [Barreto&Medrano&Verjovsky2013] Y. Barreto, S. López de Medrano and A. Verjovsky, Open Book Structures on Moment-Angle Manifolds $Z^{\mathbb C}(\Lambda)$$Z^{\mathbb C}(\Lambda)$ and Higher Dimensional Contact Manifolds, (2013). Available at the arXiv:1303.2671.
• [Calcut2005] J. S. Calcut, Knot theory and the Casson invariant in the Artin presentation theory, Fundamental and Applied Mathematics 11 No. 4, L. V. Keldysh Memorial Proceedings, Moscow, (2005), 119--126. MR2192960 (2006m:57013)
• [Etnyre2006] J. B. Etnyre, Lectures on open book decompositions and contact structures, in Floer homology, gauge theory, and low-dimensional topology, 103–141, Clay Math. Proc., 5, Amer. Math. Soc., (2006). MR2249250 (2007g:57042) Zbl 1108.53050
• [Freidlin&Wentzell2004] M. I. Freidlin and A. D. Wentzell, Diffusion processes on an open book and the averaging principle, Stochastic Process. Appl. 113 (2004), no.1, 101–126. MR2078539 (2005e:60178) Zbl 1075.60100
• [Giroux2005] E. Giroux, What is...an Open Book?, Notices of the Amer. Math. Soc. 52 (2005), 42–43.
• [Laudenbach&Meigniez2012] F. Laudenbach and G. Meigniez, Haefliger's codimension-one singular foliations, open books and twisted open books in dimension 3, Bull. Braz. Math. Soc. (N.S.) 43 (2012), 347–373. MR3024060 Zbl 1263.57022
• [Quinn1979] F. Quinn, Open book decompositions, and the bordism of automorphisms, Topology 18 (1979), no.1, 55–73. MR528236 (81e:57027) Zbl 0425.57010
• [Ranicki1998] A. Ranicki, High-dimensional knot theory, Springer-Verlag, 1998. MR1713074 (2000i:57044) Zbl 1059.19003
• [Thurston&Winkelnkemper1975] W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975), 345–347. MR0375366 (51 #11561) Zbl 0312.53028
• [Winkelnkemper1973] H. E. Winkelnkemper, Manifolds as open books, Bull. Amer. Math. Soc. 79 (1973), 45–51. MR0310912 (46 #10010) Zbl 0269.57011
• [Winkelnkemper2002] H. E. Winkelnkemper, Artin presentations. I. Gauge theory, $3 1$$3 1$ TQFT's and the braid groups, J. Knot Theory Ramifications 11 (2002), no.2, 223–275. MR1895372 (2003b:57045) Zbl 0994.57026
• [[Template:{Barreto&Medrano&Verjovsky2013|[{Barreto&Medrano&Verjovsky2013]]] {{{Barreto&Medrano&Verjovsky2013}}

$ submanifolds of $M$ whose images are the '''pages''' of the open book. The image of the closed, codimension 2 submanifold $N$ of $M$ is called the '''binding''' of the open book. {{endrem}} For an open book $M$, every point $x\in M-N$ lies on one and only one page, and $N$ is the boundary of each page. Sometimes for certain homological and/or analytic applications (for example, \cite[p . 3]{Barreto&Medrano&Verjovsky2013}, \cite{Quinn1979|p .55}, and \cite{Ranicki1998|p . 252}) one uses more elaborate, but equivalent, definitions. Furthermore, one can generalize the open book definition to compact manifolds with nonempty boundaries \cite[p . 353]{Ranicki1998}, \cite{Quinn1979|p. 55}. For nice figures, see {{cite|Giroux2005}}, \cite[p . 616]{Ranicki1998}, and {{cite|Winkelnkemper1973}}. The first general open book existence theorem was obtained in 1972: {{beginthm|Theorem|[{{cite|Winkelnkemper1973}}]}} If $n > 6$ and $M^n$ is simply connected, then $M^n$ is an open book if and only if its signature is $V$ 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, Gonz{\'a}lez-Acu{\ n}a 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 Gonz{\'a}lez-Acu{\ n}a: 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].

3 References

• [Barreto&Medrano&Verjovsky2013] Y. Barreto, S. López de Medrano and A. Verjovsky, Open Book Structures on Moment-Angle Manifolds $Z^{\mathbb C}(\Lambda)$$Z^{\mathbb C}(\Lambda)$ and Higher Dimensional Contact Manifolds, (2013). Available at the arXiv:1303.2671.
• [Calcut2005] J. S. Calcut, Knot theory and the Casson invariant in the Artin presentation theory, Fundamental and Applied Mathematics 11 No. 4, L. V. Keldysh Memorial Proceedings, Moscow, (2005), 119--126. MR2192960 (2006m:57013)
• [Etnyre2006] J. B. Etnyre, Lectures on open book decompositions and contact structures, in Floer homology, gauge theory, and low-dimensional topology, 103–141, Clay Math. Proc., 5, Amer. Math. Soc., (2006). MR2249250 (2007g:57042) Zbl 1108.53050
• [Freidlin&Wentzell2004] M. I. Freidlin and A. D. Wentzell, Diffusion processes on an open book and the averaging principle, Stochastic Process. Appl. 113 (2004), no.1, 101–126. MR2078539 (2005e:60178) Zbl 1075.60100
• [Giroux2005] E. Giroux, What is...an Open Book?, Notices of the Amer. Math. Soc. 52 (2005), 42–43.
• [Laudenbach&Meigniez2012] F. Laudenbach and G. Meigniez, Haefliger's codimension-one singular foliations, open books and twisted open books in dimension 3, Bull. Braz. Math. Soc. (N.S.) 43 (2012), 347–373. MR3024060 Zbl 1263.57022
• [Quinn1979] F. Quinn, Open book decompositions, and the bordism of automorphisms, Topology 18 (1979), no.1, 55–73. MR528236 (81e:57027) Zbl 0425.57010
• [Ranicki1998] A. Ranicki, High-dimensional knot theory, Springer-Verlag, 1998. MR1713074 (2000i:57044) Zbl 1059.19003
• [Thurston&Winkelnkemper1975] W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975), 345–347. MR0375366 (51 #11561) Zbl 0312.53028
• [Winkelnkemper1973] H. E. Winkelnkemper, Manifolds as open books, Bull. Amer. Math. Soc. 79 (1973), 45–51. MR0310912 (46 #10010) Zbl 0269.57011
• [Winkelnkemper2002] H. E. Winkelnkemper, Artin presentations. I. Gauge theory, $3 1$$3 1$ TQFT's and the braid groups, J. Knot Theory Ramifications 11 (2002), no.2, 223–275. MR1895372 (2003b:57045) Zbl 0994.57026
• [[Template:{Barreto&Medrano&Verjovsky2013|[{Barreto&Medrano&Verjovsky2013]]] {{{Barreto&Medrano&Verjovsky2013}}

4 References

• [Barreto&Medrano&Verjovsky2013] Y. Barreto, S. López de Medrano and A. Verjovsky, Open Book Structures on Moment-Angle Manifolds $Z^{\mathbb C}(\Lambda)$$Z^{\mathbb C}(\Lambda)$ and Higher Dimensional Contact Manifolds, (2013). Available at the arXiv:1303.2671.
• [Calcut2005] J. S. Calcut, Knot theory and the Casson invariant in the Artin presentation theory, Fundamental and Applied Mathematics 11 No. 4, L. V. Keldysh Memorial Proceedings, Moscow, (2005), 119--126. MR2192960 (2006m:57013)
• [Etnyre2006] J. B. Etnyre, Lectures on open book decompositions and contact structures, in Floer homology, gauge theory, and low-dimensional topology, 103–141, Clay Math. Proc., 5, Amer. Math. Soc., (2006). MR2249250 (2007g:57042) Zbl 1108.53050
• [Freidlin&Wentzell2004] M. I. Freidlin and A. D. Wentzell, Diffusion processes on an open book and the averaging principle, Stochastic Process. Appl. 113 (2004), no.1, 101–126. MR2078539 (2005e:60178) Zbl 1075.60100
• [Giroux2005] E. Giroux, What is...an Open Book?, Notices of the Amer. Math. Soc. 52 (2005), 42–43.
• [Laudenbach&Meigniez2012] F. Laudenbach and G. Meigniez, Haefliger's codimension-one singular foliations, open books and twisted open books in dimension 3, Bull. Braz. Math. Soc. (N.S.) 43 (2012), 347–373. MR3024060 Zbl 1263.57022
• [Quinn1979] F. Quinn, Open book decompositions, and the bordism of automorphisms, Topology 18 (1979), no.1, 55–73. MR528236 (81e:57027) Zbl 0425.57010
• [Ranicki1998] A. Ranicki, High-dimensional knot theory, Springer-Verlag, 1998. MR1713074 (2000i:57044) Zbl 1059.19003
• [Thurston&Winkelnkemper1975] W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975), 345–347. MR0375366 (51 #11561) Zbl 0312.53028
• [Winkelnkemper1973] H. E. Winkelnkemper, Manifolds as open books, Bull. Amer. Math. Soc. 79 (1973), 45–51. MR0310912 (46 #10010) Zbl 0269.57011
• [Winkelnkemper2002] H. E. Winkelnkemper, Artin presentations. I. Gauge theory, $3 1$$3 1$ TQFT's and the braid groups, J. Knot Theory Ramifications 11 (2002), no.2, 223–275. MR1895372 (2003b:57045) Zbl 0994.57026
• [[Template:{Barreto&Medrano&Verjovsky2013|[{Barreto&Medrano&Verjovsky2013]]] {{{Barreto&Medrano&Verjovsky2013}}

$. {{endthm}} In general, if $M^n$ is orientable and $n > 5$, then the existence of open book structures was settled by T. Lawson when $n$ is odd: there is no obstruction, and for any $n > 3$ by F. Quinn: when $n$ is even, the obstruction lies in the asymmetric Witt group of $\mathbb{Z} \pi_{1}(M^n)$, generalizing the Wall surgery obstruction \cite[p . 360]{Ranicki1998}. In 1975, Gonz{\'a}lez-Acu{\ n}a used fundamental work of Alexander and Lickorish to prove: if $n=3$, then every closed, orientable $M^3$ is an open book; furthermore, the page can be chosen to have a connected boundary (that is, every such $M^3$ contains a fibered knot) or be as simple as possible: a compact $-disk with holes. ==Applications== ; Open book decompositions have had numerous applications (see the Appendix in {{cite|Ranicki1998}}). More recent applications are to: diffusion processes and the averaging principle {{cite|Freidlin&Wentzell2004}}, contact forms on moment angle manifolds {{cite|{Barreto&Medrano&Verjovsky2013}}, and Haefliger's singular foliations {{cite|Laudenbach&Meigniez2012}}. Most notably, in dimension $, open books are crucial for proving Giroux's theorem that, roughly put, on any closed, orientable $-manifold there is a one-to-one correspondence between contact forms (Geometry) and open books (Topology) {{cite|Etnyre2006}}. In other words, every contact form on a closed, orientable $-manifold arises via an open book as in the construction of contact forms by Thurston and Winkelnkemper \cite{Thurston&Winkelnkemper1975}. Another notable feature was discovered by Gonz{\'a}lez-Acu{\ n}a: in dimension $ 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 $-manifold. Thus, the whole theory of closed, orientable $-manifolds is encoded in the purely discrete, group-theoretic theory of Artin presentations {{cite|Winkelnkemper2002}}, \cite{Ranicki1998|p .617}. This theory is now an autonomous one with purely group-theoretic analogues of the Casson invariant {{cite|Calcut2005}} and Donaldson's Theorem \cite[p . 621]{Ranicki1998}, \cite{Winkelnkemper2002|p .240}. == References == {{#RefList:}} [[Category:Manifolds]] == References == {{#RefList:}} [[Category:Definitions]]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, Gonz{\'a}lez-Acu{\ n}a 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 Gonz{\'a}lez-Acu{\ n}a: 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].

3 References

• [Barreto&Medrano&Verjovsky2013] Y. Barreto, S. López de Medrano and A. Verjovsky, Open Book Structures on Moment-Angle Manifolds $Z^{\mathbb C}(\Lambda)$$Z^{\mathbb C}(\Lambda)$ and Higher Dimensional Contact Manifolds, (2013). Available at the arXiv:1303.2671.
• [Calcut2005] J. S. Calcut, Knot theory and the Casson invariant in the Artin presentation theory, Fundamental and Applied Mathematics 11 No. 4, L. V. Keldysh Memorial Proceedings, Moscow, (2005), 119--126. MR2192960 (2006m:57013)
• [Etnyre2006] J. B. Etnyre, Lectures on open book decompositions and contact structures, in Floer homology, gauge theory, and low-dimensional topology, 103–141, Clay Math. Proc., 5, Amer. Math. Soc., (2006). MR2249250 (2007g:57042) Zbl 1108.53050
• [Freidlin&Wentzell2004] M. I. Freidlin and A. D. Wentzell, Diffusion processes on an open book and the averaging principle, Stochastic Process. Appl. 113 (2004), no.1, 101–126. MR2078539 (2005e:60178) Zbl 1075.60100
• [Giroux2005] E. Giroux, What is...an Open Book?, Notices of the Amer. Math. Soc. 52 (2005), 42–43.
• [Laudenbach&Meigniez2012] F. Laudenbach and G. Meigniez, Haefliger's codimension-one singular foliations, open books and twisted open books in dimension 3, Bull. Braz. Math. Soc. (N.S.) 43 (2012), 347–373. MR3024060 Zbl 1263.57022
• [Quinn1979] F. Quinn, Open book decompositions, and the bordism of automorphisms, Topology 18 (1979), no.1, 55–73. MR528236 (81e:57027) Zbl 0425.57010
• [Ranicki1998] A. Ranicki, High-dimensional knot theory, Springer-Verlag, 1998. MR1713074 (2000i:57044) Zbl 1059.19003
• [Thurston&Winkelnkemper1975] W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975), 345–347. MR0375366 (51 #11561) Zbl 0312.53028
• [Winkelnkemper1973] H. E. Winkelnkemper, Manifolds as open books, Bull. Amer. Math. Soc. 79 (1973), 45–51. MR0310912 (46 #10010) Zbl 0269.57011
• [Winkelnkemper2002] H. E. Winkelnkemper, Artin presentations. I. Gauge theory, $3 1$$3 1$ TQFT's and the braid groups, J. Knot Theory Ramifications 11 (2002), no.2, 223–275. MR1895372 (2003b:57045) Zbl 0994.57026
• [[Template:{Barreto&Medrano&Verjovsky2013|[{Barreto&Medrano&Verjovsky2013]]] {{{Barreto&Medrano&Verjovsky2013}}

4 References

• [Barreto&Medrano&Verjovsky2013] Y. Barreto, S. López de Medrano and A. Verjovsky, Open Book Structures on Moment-Angle Manifolds $Z^{\mathbb C}(\Lambda)$$Z^{\mathbb C}(\Lambda)$ and Higher Dimensional Contact Manifolds, (2013). Available at the arXiv:1303.2671.
• [Calcut2005] J. S. Calcut, Knot theory and the Casson invariant in the Artin presentation theory, Fundamental and Applied Mathematics 11 No. 4, L. V. Keldysh Memorial Proceedings, Moscow, (2005), 119--126. MR2192960 (2006m:57013)
• [Etnyre2006] J. B. Etnyre, Lectures on open book decompositions and contact structures, in Floer homology, gauge theory, and low-dimensional topology, 103–141, Clay Math. Proc., 5, Amer. Math. Soc., (2006). MR2249250 (2007g:57042) Zbl 1108.53050
• [Freidlin&Wentzell2004] M. I. Freidlin and A. D. Wentzell, Diffusion processes on an open book and the averaging principle, Stochastic Process. Appl. 113 (2004), no.1, 101–126. MR2078539 (2005e:60178) Zbl 1075.60100
• [Giroux2005] E. Giroux, What is...an Open Book?, Notices of the Amer. Math. Soc. 52 (2005), 42–43.
• [Laudenbach&Meigniez2012] F. Laudenbach and G. Meigniez, Haefliger's codimension-one singular foliations, open books and twisted open books in dimension 3, Bull. Braz. Math. Soc. (N.S.) 43 (2012), 347–373. MR3024060 Zbl 1263.57022
• [Quinn1979] F. Quinn, Open book decompositions, and the bordism of automorphisms, Topology 18 (1979), no.1, 55–73. MR528236 (81e:57027) Zbl 0425.57010
• [Ranicki1998] A. Ranicki, High-dimensional knot theory, Springer-Verlag, 1998. MR1713074 (2000i:57044) Zbl 1059.19003
• [Thurston&Winkelnkemper1975] W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975), 345–347. MR0375366 (51 #11561) Zbl 0312.53028
• [Winkelnkemper1973] H. E. Winkelnkemper, Manifolds as open books, Bull. Amer. Math. Soc. 79 (1973), 45–51. MR0310912 (46 #10010) Zbl 0269.57011
• [Winkelnkemper2002] H. E. Winkelnkemper, Artin presentations. I. Gauge theory, $3 1$$3 1$ TQFT's and the braid groups, J. Knot Theory Ramifications 11 (2002), no.2, 223–275. MR1895372 (2003b:57045) Zbl 0994.57026
• [[Template:{Barreto&Medrano&Verjovsky2013|[{Barreto&Medrano&Verjovsky2013]]] {{{Barreto&Medrano&Verjovsky2013}}