Open book decomposition
Encyclopedia
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 surface
s (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.
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. 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.
Theorem. Let M be a compact oriented 3-manifold. Then there is a bijection
between the set of oriented contact structures on M up to
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
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an open book decomposition (or simply an open book) is a decomposition
Manifold decomposition
In topology, a branch of mathematics, a manifold M may be decomposed or split by writing M as a combination of smaller pieces. When doing so, one must specify both what those pieces are and how they are put together to form M....
of a closed
Closed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
oriented 3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
M into a union of surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
s (necessarily with boundary) and solid tori
Solid torus
In mathematics, a solid torus is a topological space homeomorphic to S^1 \times D^2, i.e. the cartesian product of the circle with a two dimensional disc endowed with the product topology. The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary...
. Open books have relevance to contact geometry
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 and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'...
, 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 linkLink (knot theory)In mathematics, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory...
in M, called the binding of the open book; - π: M \ B → S1 is a fibrationFibrationIn topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
of the complementComplement (set theory)In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
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
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
which is the identity near the boundary, we can construct an open book by first forming the mapping torus
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...
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. To complete the construction, solid tori
Solid torus
In mathematics, a solid torus is a topological space homeomorphic to S^1 \times D^2, i.e. the cartesian product of the circle with a two dimensional disc endowed with the product topology. The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary...
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
Rolodex
A Rolodex is a rotating file device used to store business contact information currently manufactured by Newell Rubbermaid. The Rolodex holds specially shaped index cards; the user writes the contact information for one person or company on each card...
-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
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. All vector spaces will be assumed to be finite-dimensional...
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
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
between the set of oriented contact structures on M up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
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
Dehn twist
In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface .-Definition:...
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.