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.
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. The definition for general m is similar, except that the surface with boundary (Σ, B) is replaced by an (m − 1)-manifold with boundary (P, ∂P).
Equivalently, the open book decomposition can be thought of as a homeomorphism of M to the quotient space
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.
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.
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.
Quinn also showed that for even m > 6, an m-dimensional manifold has an open book decomposition if and only if an asymmetric Witt group obstruction is 0.