Open book decomposition

In mathematics, an open book decomposition (or simply an open book) is a decomposition of a closed oriented 3-manifold M into a union of surfaces (necessarily with boundary) and solid tori. Open books have relevance to contact geometry, with a famous theorem of Emmanuel Giroux (given below) that shows that contact geometry can be studied from an entirely topological viewpoint. In mathematics, an open book decomposition (or simply an open book) is a decomposition of a closed oriented 3-manifold M into a union of surfaces (necessarily with boundary) and solid tori. Open books have relevance to contact geometry, with a famous theorem of Emmanuel Giroux (given below) that shows that contact geometry can be studied from an entirely topological viewpoint. Definition. An open book decomposition of a 3-dimensional manifold M is a pair (B, π) where This is the special case m = 3 of an open book decomposition of an m-dimensional manifold, for any m. When Σ is an oriented compact surface with n boundary components and φ: Σ → Σ is a homeomorphism which is the identity near the boundary, we can construct an open book by first forming the mapping torus Σφ. Since φ is the identity on ∂Σ, ∂Σφ is the trivial circle bundle over a union of circles, that is, a union of tori; one torus for each boundary component. To complete the construction, solid tori are glued to fill in the boundary tori so that each circle S1 × {p} ⊂ S1×∂D2 is identified with the boundary of a page. In this case, the binding is the collection of n cores S1×{q} of the n solid tori glued into the mapping torus, for arbitrarily chosen q ∈ D2. It is known that any open book can be constructed this way. As the only information used in the construction is the surface and the homeomorphism, an alternate definition of open book is simply the pair (Σ, φ) with the construction understood. In short, an open book is a mapping torus with solid tori glued in so that the core circle of each torus runs parallel to the boundary of the fiber. Each torus in ∂Σφ is fibered by circles parallel to the binding, each circle a boundary component of a page. One envisions a rolodex-looking structure for a neighborhood of the binding (that is, the solid torus glued to ∂Σφ)—the pages of the rolodex connect to pages of the open book and the center of the rolodex is the binding. Thus the term open book. It is a 1972 theorem of Elmar Winkelnkemper that for m > 6, a simply-connected m-dimensional manifold has an open book decomposition if and only if it has signature 0. In 1977 Terry Lawson proved that for odd m > 6, every m-dimensional manifold has an open book decomposition. For even m > 6, an m-dimensional manifold has an open book decomposition if and only if an asymmetric Witt group obstruction is 0, by a 1979 theorem of Frank Quinn.