Open book
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− | {{Authors|Jack Calcut | + | {{Authors|Jack Calcut|Horst Winkelnkemper}}{{Stub}} |
==Open Books== | ==Open Books== | ||
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− | 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 '''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. | {{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}} | ||
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For an open book $M$, every point $x\in M-N$ lies on one and only one page, and $N$ is the boundary | 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. | of each page. | ||
− | Sometimes for certain homological and/or analytic applications (for example, \cite | + | 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}. |
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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 . 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 $2$-disk with holes. | + | 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 $2$-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 | + | 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 == | ||
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[[Category:Definitions]] | [[Category:Definitions]] |
Revision as of 15:03, 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 \{cite|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 -manifold . Let denote the image of under the identification map.
Definition 1.1. A closed manifold is an open book provided it is diffeomorphic to . The fibers of , where , are codimension submanifolds of whose images are the pages of the open book. The image of the closed, codimension 2 submanifold of is called the binding of the open book.
For an open book , 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 \{cite|Giroux2005}, [Ranicki1998, p. 616], and \{cite|Winkelnkemper1973}.
The first general open book existence theorem was obtained in 1972:
Theorem 1.2 \{cite.} 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, Gonz{\'a}lez-Acu{\ n}a 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 \{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) [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 \{cite|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
- [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
- [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