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 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 [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)
- [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)
- [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 ()
- [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)
- [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