Open book
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==Open Books== | ==Open Books== | ||
<wikitex>; | <wikitex>; | ||
− | The simplest general definition of an open book, as the concept first appeared explicitly in the literature \ | + | 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. | 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 | + | {{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 $1$ 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}} | {{endrem}} | ||
Line 12: | Line 12: | ||
of each page. | of each page. | ||
− | Sometimes for certain homological and/or analytic applications (for example, \cite{Barreto&Medrano&Verjovsky2013|p. 3}, \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 \{ | + | Sometimes for certain homological and/or analytic applications (for example, \cite{Barreto&Medrano&Verjovsky2013|p. 3}, \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{Ranicki1998|p. 616}, and \cite{Winkelnkemper1973}. |
The first general open book existence theorem was obtained in 1972: | The first general open book existence theorem was obtained in 1972: | ||
− | {{beginthm|Theorem|\ | + | {{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 $0$. | If $n > 6$ and $M^n$ is simply connected, then $M^n$ is an open book if and only if its signature is $0$. | ||
{{endthm}} | {{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. | + | 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, González-Acuña 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 $2$-disk with holes. |
</wikitex> | </wikitex> | ||
==Applications== | ==Applications== | ||
<wikitex>; | <wikitex>; | ||
− | Open book decompositions have had numerous applications (see the Appendix in \ | + | 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 $3$, open books are crucial for proving Giroux's theorem that, roughly put, on any closed, orientable $3$-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 $3$-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 González-Acuña: in dimension $3$ 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 $3$-manifold. Thus, the whole theory of closed, orientable $3$-manifolds is encoded in the purely discrete, group-theoretic theory of Artin presentations \ | + | Another notable feature was discovered by González-Acuña: in dimension $3$ 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 $3$-manifold. Thus, the whole theory of closed, orientable $3$-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}. |
</wikitex> | </wikitex> | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Definitions]] | [[Category:Definitions]] |
Latest revision as of 15:09, 7 October 2013
The users responsible for this page are: Jack Calcut, Horst Winkelnkemper. No other users may edit this page at present. |
This page has not been refereed. The information given here might be incomplete or provisional. |
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 be a smooth, compact -manifold with . Let be a diffeomorphism that restricts to the identity on . The mapping torus of , denoted , has as boundary. By identifying with for each and , we obtain a smooth, closed -manifoldTex syntax error. Let denote the image of under the identification map.
Tex syntax error. The fibers of , where , are codimension submanifolds of
Tex syntax errorwhose images are the pages of the open book. The image of the closed, codimension 2 submanifold of
Tex syntax erroris called the binding of the open book.
Tex syntax error, every point lies on one and only one page, and 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 and is simply connected, then is an open book if and only if its signature is .
In general, if is orientable and , then the existence of open book structures was settled by T. Lawson when is odd: there is no obstruction, and for any by F. Quinn: when is even, the obstruction lies in the asymmetric Witt group of , generalizing the Wall surgery obstruction [Ranicki1998, p. 360]. In 1975, González-Acuña used fundamental work of Alexander and Lickorish to prove: if , then every closed, orientable is an open book; furthermore, the page can be chosen to have a connected boundary (that is, every such contains a fibered knot) or be as simple as possible: a compact -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 , 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) [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 [Thurston&Winkelnkemper1975].
Another notable feature was discovered by González-Acuña: 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 [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 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, TQFT's and the braid groups, J. Knot Theory Ramifications 11 (2002), no.2, 223–275. MR1895372 (2003b:57045) Zbl 0994.57026
Tex syntax error. Let denote the image of under the identification map.
Tex syntax error. The fibers of , where , are codimension submanifolds of
Tex syntax errorwhose images are the pages of the open book. The image of the closed, codimension 2 submanifold of
Tex syntax erroris called the binding of the open book.
Tex syntax error, every point lies on one and only one page, and 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 and is simply connected, then is an open book if and only if its signature is .
In general, if is orientable and , then the existence of open book structures was settled by T. Lawson when is odd: there is no obstruction, and for any by F. Quinn: when is even, the obstruction lies in the asymmetric Witt group of , generalizing the Wall surgery obstruction [Ranicki1998, p. 360]. In 1975, González-Acuña used fundamental work of Alexander and Lickorish to prove: if , then every closed, orientable is an open book; furthermore, the page can be chosen to have a connected boundary (that is, every such contains a fibered knot) or be as simple as possible: a compact -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 , 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) [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 [Thurston&Winkelnkemper1975].
Another notable feature was discovered by González-Acuña: 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 [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 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, TQFT's and the braid groups, J. Knot Theory Ramifications 11 (2002), no.2, 223–275. MR1895372 (2003b:57045) Zbl 0994.57026
Tex syntax error. Let denote the image of under the identification map.
Tex syntax error. The fibers of , where , are codimension submanifolds of
Tex syntax errorwhose images are the pages of the open book. The image of the closed, codimension 2 submanifold of
Tex syntax erroris called the binding of the open book.
Tex syntax error, every point lies on one and only one page, and 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 and is simply connected, then is an open book if and only if its signature is .
In general, if is orientable and , then the existence of open book structures was settled by T. Lawson when is odd: there is no obstruction, and for any by F. Quinn: when is even, the obstruction lies in the asymmetric Witt group of , generalizing the Wall surgery obstruction [Ranicki1998, p. 360]. In 1975, González-Acuña used fundamental work of Alexander and Lickorish to prove: if , then every closed, orientable is an open book; furthermore, the page can be chosen to have a connected boundary (that is, every such contains a fibered knot) or be as simple as possible: a compact -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 , 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) [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 [Thurston&Winkelnkemper1975].
Another notable feature was discovered by González-Acuña: 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 [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 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, TQFT's and the braid groups, J. Knot Theory Ramifications 11 (2002), no.2, 223–275. MR1895372 (2003b:57045) Zbl 0994.57026