Open Book Decomposition - Definition and Construction

Definition and Construction

Definition. An open book decomposition of a 3-dimensional manifold M is a pair (B, π) where

• B is an oriented link in M, called the binding of the open book;
• π: M \ B → S1 is a fibration of the complement of B such that for each θ ∈ S1, π−1(θ) is the interior of a compact surface Σ ⊂ M whose boundary is B. The surface Σ is called the page of the open book.

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 Σ_φ. Since φ is the identity on ∂Σ, ∂Σ_φ is the trivial circle bundle over a union of circles, that is, a union of tori. 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 ∂Σ_φ 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 ∂Σ_φ)—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.