Surface bundle over the circle
Encyclopedia
In mathematics
, a surface bundle over the circle is a fiber bundle
with base space a circle
, and with fiber space a surface
. Therefore the total space has dimension 2 + 1 = 3. In general, fiber bundle
s over the circle are a special case of mapping tori
.
Here is the construction: take the Cartesian product
of a surface with the unit interval
. Glue the two copies of the surface, on the boundary, by some homeomorphism. This homeomorphism is called the monodromy
of the surface bundle. It is possible to show that the homeomorphism type of the bundle obtained depends only on the conjugacy class
, in the mapping class group
, of the gluing homeomorphism chosen.
This construction is an important source of examples both in the field of low-dimensional topology
as well as in geometric group theory
. In the former we find that the geometry
of the three-manifold is determined by the dynamics of the homeomorphism. This is the fibered part of Thurston's geometrization theorem for Haken manifolds, whose proof requires the Nielsen–Thurston classification for surface homeomorphisms as well as deep results in the theory of Kleinian group
s. In geometric group theory the fundamental group
s of such bundles give an important class of HNN-extensions: that is, extension
s of the fundamental group of the fiber (a surface) by the integer
s.
A simple special case of this construction (considered in Poincaré
's foundational paper) is that of a torus bundle
.
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...
, a surface bundle over the circle is a fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
with base space a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
, and with fiber space a 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...
. Therefore the total space has dimension 2 + 1 = 3. In general, fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
s over the circle are a special case of mapping tori
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...
.
Here is the construction: take the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of a surface with the unit interval
Unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
. Glue the two copies of the surface, on the boundary, by some homeomorphism. This homeomorphism is called the monodromy
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
of the surface bundle. It is possible to show that the homeomorphism type of the bundle obtained depends only on the conjugacy class
Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
, in the mapping class group
Mapping class group
In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...
, of the gluing homeomorphism chosen.
This construction is an important source of examples both in the field of low-dimensional topology
Low-dimensional topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology.A number of...
as well as in geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
. In the former we find that the geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
of the three-manifold is determined by the dynamics of the homeomorphism. This is the fibered part of Thurston's geometrization theorem for Haken manifolds, whose proof requires the Nielsen–Thurston classification for surface homeomorphisms as well as deep results in the theory of Kleinian group
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
s. In geometric group theory the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
s of such bundles give an important class of HNN-extensions: that is, extension
Group extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence...
s of the fundamental group of the fiber (a surface) by the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s.
A simple special case of this construction (considered in Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
's foundational paper) is that of a torus bundle
Torus bundle
In mathematics, in the sub-field of geometric topology, a torus bundle is a kind of surface bundle over the circle, which in turn are a class of three-manifolds.-Construction:To obtain a torus bundle: let f be an...
.