- Source: 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 A be a tubular neighborhood of c. Then A is an annulus, homeomorphic to the Cartesian product of a circle and a unit interval I:
c
⊂
A
≅
S
1
×
I
.
{\displaystyle c\subset A\cong S^{1}\times I.}
Give A coordinates (s, t) where s is a complex number of the form
e
i
θ
{\displaystyle e^{i\theta }}
with
θ
∈
[
0
,
2
π
]
,
{\displaystyle \theta \in [0,2\pi ],}
and t ∈ [0, 1].
Let f be the map from S to itself which is the identity outside of A and inside A we have
f
(
s
,
t
)
=
(
s
e
i
2
π
t
,
t
)
.
{\displaystyle f(s,t)=\left(se^{i2\pi t},t\right).}
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 represented by a fundamental polygon with edges a and b
T
2
≅
R
2
/
Z
2
.
{\displaystyle \mathbb {T} ^{2}\cong \mathbb {R} ^{2}/\mathbb {Z} ^{2}.}
Let a closed curve be the line along the edge a called
γ
a
{\displaystyle \gamma _{a}}
.
Given the choice of gluing homeomorphism in the figure, a tubular neighborhood of the curve
γ
a
{\displaystyle \gamma _{a}}
will look like a band linked around a doughnut. This neighborhood is homeomorphic to an annulus, say
a
(
0
;
0
,
1
)
=
{
z
∈
C
:
0
<
|
z
|
<
1
}
{\displaystyle a(0;0,1)=\{z\in \mathbb {C} :0<|z|<1\}}
in the complex plane.
By extending to the torus the twisting map
(
e
i
θ
,
t
)
↦
(
e
i
(
θ
+
2
π
t
)
,
t
)
{\displaystyle \left(e^{i\theta },t\right)\mapsto \left(e^{i\left(\theta +2\pi t\right)},t\right)}
of the annulus, through the homeomorphisms of the annulus to an open cylinder to the neighborhood of
γ
a
{\displaystyle \gamma _{a}}
, yields a Dehn twist of the torus by a.
T
a
:
T
2
→
T
2
{\displaystyle T_{a}:\mathbb {T} ^{2}\to \mathbb {T} ^{2}}
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. Therefore one has an automorphism
T
a
∗
:
π
1
(
T
2
)
→
π
1
(
T
2
)
:
[
x
]
↦
[
T
a
(
x
)
]
{\displaystyle {T_{a}}_{\ast }:\pi _{1}\left(\mathbb {T} ^{2}\right)\to \pi _{1}\left(\mathbb {T} ^{2}\right):[x]\mapsto \left[T_{a}(x)\right]}
where [x] are the homotopy classes of the closed curve x in the torus. Notice
T
a
∗
(
[
a
]
)
=
[
a
]
{\displaystyle {T_{a}}_{\ast }([a])=[a]}
and
T
a
∗
(
[
b
]
)
=
[
b
∗
a
]
{\displaystyle {T_{a}}_{\ast }([b])=[b*a]}
, where
b
∗
a
{\displaystyle b*a}
is the path travelled around b then a.
Mapping class group
It is a theorem of Max Dehn that maps of this form generate the mapping class group of isotopy classes of orientation-preserving homeomorphisms of any closed, oriented genus-
g
{\displaystyle g}
surface. W. B. R. Lickorish later rediscovered this result with a simpler proof and in addition showed that Dehn twists along
3
g
−
1
{\displaystyle 3g-1}
explicit curves generate the mapping class group (this is called by the punning name "Lickorish twist theorem"); this number was later improved by Stephen P. Humphries to
2
g
+
1
{\displaystyle 2g+1}
, for
g
>
1
{\displaystyle g>1}
, which he showed was the minimal number.
Lickorish also obtained an analogous result for non-orientable surfaces, which require not only Dehn twists, but also "Y-homeomorphisms."
See also
Fenchel–Nielsen coordinates
Lantern relation
References
Andrew J. Casson, Steven A Bleiler, Automorphisms of Surfaces After Nielsen and Thurston, Cambridge University Press, 1988. ISBN 0-521-34985-0.
Stephen P. Humphries, "Generators for the mapping class group," in: Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), pp. 44–47, Lecture Notes in Math., 722, Springer, Berlin, 1979. MR0547453
W. B. R. Lickorish, "A representation of orientable combinatorial 3-manifolds." Ann. of Math. (2) 76 1962 531—540. MR0151948
W. B. R. Lickorish, "A finite set of generators for the homotopy group of a 2-manifold", Proc. Cambridge Philos. Soc. 60 (1964), 769–778. MR0171269
Kata Kunci Pencarian:
- Homeomorfisme
- Presentasi grup
- Dehn twist
- Max Dehn
- Mapping class group of a surface
- Mapping class group
- Torus bundle
- Lickorish–Wallace theorem
- Homeomorphism
- List of geometric topology topics
- Computational topology
- Geometrization conjecture
