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 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.

Definition 1.1. 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.

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, [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 and M^n is simply connected, then M^n is an open book if and only if its signature is 0.

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 [Ranicki1998, p. 360]. 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.

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, 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) [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 [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 [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

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