In
geometric topology, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Dehn twist is a certain type of
self-homeomorphism of a
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 ...
(two-dimensional
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
).
Definition
Suppose that ''c'' is a
simple closed curve
In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior ...
in a closed,
orientable
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 ...
surface ''S''. Let ''A'' be a
tubular neighborhood
In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.
The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the pla ...
of ''c''. Then ''A'' is an
annulus
Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to:
Human anatomy
* ''Anulus fibrosus disci intervertebralis'', spinal structure
* Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...
,
homeomorphic
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 isomorphi ...
to the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
of a circle and a
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. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...
''I'':
:
Give ''A'' coordinates (''s'', ''t'') where ''s'' is a complex number of the form
with
and .
Let ''f'' be the map from ''S'' to itself which is the identity outside of ''A'' and inside ''A'' we have
:
Then ''f'' is a Dehn twist about the curve ''c''.
Dehn twists can also be defined on a non-orientable surface ''S'', provided one starts with a
2-sided simple closed curve ''c'' on ''S''.
Example
Consider the
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
represented by a
fundamental polygon In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal eq ...
with edges ''a'' and ''b''
:
Let a closed curve be the line along the edge ''a'' called
.
Given the choice of gluing homeomorphism in the figure, a tubular neighborhood of the curve
will look like a band linked around a doughnut. This neighborhood is homeomorphic to an
annulus
Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to:
Human anatomy
* ''Anulus fibrosus disci intervertebralis'', spinal structure
* Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...
, say
:
in the complex plane.
By extending to the torus the twisting map
of the annulus, through the homeomorphisms of the annulus to an open cylinder to the neighborhood of
, yields a Dehn twist of the torus by ''a''.
:
This self homeomorphism acts on the closed curve along ''b''. In the tubular neighborhood it takes the curve of ''b'' once along the curve of ''a''.
A homeomorphism between topological spaces induces a natural isomorphism between their
fundamental groups
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
. Therefore one has an automorphism
: