In mathematics, an open book decomposition (or simply an open book) is a

decomposition Decomposition or rot is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. The process is a part of the nutrient cycle and is ...

of a closed

oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds ...

''M'' into a union of

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...

s (necessarily with boundary) and solid tori. Open books have relevance to

contact geometry In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution m ...

, with a famous theorem of Emmanuel Giroux (given below) that shows that contact geometry can be studied from an entirely topological viewpoint.

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'' → ''S''¹ is a

fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all map ...

of the

complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...

of ''B'' such that for each θ ∈ ''S''¹, 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 In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

which is the identity near the boundary, we can construct an open book by first forming the

mapping torus In mathematics, the mapping torus in topology of a homeomorphism ''f'' of some topological space ''X'' to itself is a particular geometric construction with ''f''. Take the cartesian product of ''X'' with a closed interval ''I'', and glue the bound ...

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; one torus for each boundary component. To complete the construction, solid tori are glued to fill in the boundary tori so that each circle ''S''¹ × ⊂ ''S''¹×∂''D''² is identified with the boundary of a page. In this case, the binding is the collection of ''n'' cores ''S''¹×{q} of the ''n'' solid tori glued into the mapping torus, for arbitrarily chosen ''q'' ∈ ''D''². 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 A Rolodex is a rotating card file device used to store business contact information. Its name, a portmanteau of the words ''rolling'' and ''index'', has become somewhat genericized (usually as ''rolodex'') for any personal organizer performing th ...

-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 In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field. Definition Fix a field ''k'' of characteristic not equal to two. All vector spaces ...

obstruction is 0, by a 1979 theorem of Frank Quinn.

Giroux correspondence

In 2002, Emmanuel Giroux published the following result: Theorem. Let ''M'' be a compact oriented 3-manifold. Then there is a bijection between the set of oriented

contact structure In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution m ...

s on ''M''

up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...

isotopy and the set of open book decompositions of ''M'' up to positive stabilization. ''Positive stabilization'' consists of modifying the page by adding a 2-dimensional 1-handle and modifying the monodromy by adding a positive

Dehn twist In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold). Definition Suppose that ''c'' is a simple closed curve in a closed, orientable surface ''S''. Let ...

along a curve that runs over that handle exactly once. Implicit in this theorem is that the new open book defines the same contact 3-manifold. Giroux's result has led to some breakthroughs in what is becoming more commonly called contact topology, such as the classification of contact structures on certain classes of 3-manifolds. Roughly speaking, a contact structure corresponds to an open book if, away from the binding, the contact distribution is isotopic to the tangent spaces of the pages through confoliations. One imagines smoothing the contact planes (preserving the contact condition almost everywhere) to lie tangent to the pages.

References

*Etnyre, John B. ''Lectures on open book decompositions and contact structures''
ArXiv
*Ranicki, Andrew, ''High-dimensional knot theory'', Springer (1998) *Ranicki, Andrew, ''Mapping torus of an automorphism of a manifold''
Springer Online Encyclopedia of Mathematics
Topology 3-manifolds Structures on manifolds Contact geometry