Stretch twist

In topology , a branch of mathematics , stretch twists are certain self-mapping of surfaces . Stretching twists were introduced by Max Dehn , who originally referred to them as "screwing".

definition

Let be an orientable surface and a simple closed curve . Be a tubular neighborhood of , that is, we have a homeomorphism , which on maps. We use this homeomorphism to parameterize with coordinates . ${\displaystyle S}$ ${\displaystyle c\subset S}$ ${\displaystyle U}$ ${\displaystyle c}$ ${\displaystyle U\cong S^{1}\times \left[-1,1\right]}$ ${\displaystyle c}$ ${\displaystyle S^{1}\times \left\{0\right\}}$ ${\displaystyle U}$ ${\displaystyle (e^{i\theta },t)}$ ${\displaystyle \theta \in \left[0,2\pi \right),t\in \left[-1,1\right]}$

We then define a mapping through ${\displaystyle t_{c}:U\rightarrow U}$

${\displaystyle t_{c}(e^{i\theta },t)=(e^{i(\theta +2\pi t)},t)}$.

Because up corresponds to the identity, we can continue it steadily through the identity mapping and thus obtain a homeomorphism , which is called the stretch twist on curve c . ${\displaystyle t_{c}}$ ${\displaystyle \partial U\cong S^{1}\times \left\{-1,1\right\}}$ ${\displaystyle S\setminus U}$ ${\displaystyle t_{c}:S\rightarrow S}$

Note: The figure defined above depends on the selected environment and the selected parameterization. For other environments and other parameterizations one gets homotopic mappings to one another with this construction . The homotopy class ( mapping class ) of is thus well defined. ${\displaystyle t_{c}:S\rightarrow S}$ ${\displaystyle t_{c}}$

Examples

Longitude and meridian of the torus.

We identify the torus with . Each matrix from then corresponds to a self-mapping of the torus. (The matrix has a linear effect and maps to . One can show that every orientation- preserving homeomorphism of the torus is homotopic to such a map.) ${\displaystyle \mathbb {R} ^{2}/\mathbb {Z} ^{2}}$ ${\displaystyle SL(2,\mathbb {Z} )}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {Z} ^{2}}$ ${\displaystyle \mathbb {Z} ^{2}}$

The matrices and then correspond to the stretching twists on the longitude and meridian (i.e. on the images of the x and y axes.) ${\displaystyle \left({\begin{array}{cc}1&1\\0&1\end{array}}\right)}$ ${\displaystyle \left({\begin{array}{cc}1&0\\1&1\end{array}}\right)}$

Mapping class group

The stretching twists on these 3g-1 curves (here for g = 3) create the mapping class group.

Let the closed, orientable surface be from the gender and its mapping class group . For (the torus) it is and one can prove with the help of the Euclidean algorithm that is generated from the matrices and , i.e. from the stretching twists at the longitude and meridian. Max Dehn also proved for everyone that the mapping class group is created by Dehn Twists. Lickorish showed that the stretch twists shown in the picture on the right create the mapping class group. Humphries proved that stretch twists are generated for the mapping class group and that this is the smallest possible number of generators. ${\displaystyle S_{g}}$ ${\displaystyle g}$ ${\displaystyle MCG(S_{g})}$ ${\displaystyle g=1}$ ${\displaystyle MCG(S_{1})\cong SL(2,\mathbb {Z} )}$ ${\displaystyle SL(2,\mathbb {Z} )}$ ${\displaystyle \left({\begin{array}{cc}1&1\\0&1\end{array}}\right)}$ ${\displaystyle \left({\begin{array}{cc}1&0\\1&1\end{array}}\right)}$ ${\displaystyle g\geq 2}$ ${\displaystyle MCG(S_{g})}$ ${\displaystyle 3g-1}$ ${\displaystyle g\geq 2}$ ${\displaystyle 2g-1}$

Generalized stretching twists

Let be a symplectic manifold and a Lagrangian sphere . According to Weinstein's theorem, there is a neighborhood of that is symplectomorphic to a neighborhood of in the cotangent bundle (with the canonical symplectic structure ). It is therefore sufficient to define generalized stretching twists for surroundings of in . ${\displaystyle M}$ ${\displaystyle V\subset M}$ ${\displaystyle U\subset M}$ ${\displaystyle V}$ ${\displaystyle S^{n}}$ ${\displaystyle T^{*}S^{n}=\left\{(u,v)\in \mathbb {R} ^{n+1}\times \mathbb {R} ^{n+1}\colon |u|=1,\langle u,v\rangle =0\right\}}$ ${\displaystyle \sum _{j=1}^{n+1}du_{j}dv_{j}}$ ${\displaystyle S^{n}}$ ${\displaystyle T^{*}S^{n}}$

The function is smoothly outside the zero cut, its Hamiltonian flow is the normalized geodetic flow . The mapping can be continued to the zero intersection because all geodesics of the length have the same end point. The mapping defined in this way is a symplectomorphism and can be modified so that it is the identity outside of a compact environment. For is homotopic to identity, while for (i.e. for stretch twists on surfaces) the stretch twists have infinite order in the mapping class group. ${\displaystyle \mu (u,v)=|v|}$ ${\displaystyle \phi _{t}^{\mu }}$ ${\displaystyle \phi _{\pi }^{\mu }}$ ${\displaystyle \pi }$ ${\displaystyle \tau _{V}:M\rightarrow M}$ ${\displaystyle n\geq 2}$ ${\displaystyle \tau _{V}^{2}}$ ${\displaystyle n=1}$

supporting documents

1. ↑ M. Dehn: The group of the illustration classes. The arithmetic field on surfaces. In: Acta Math. 69, no. 1, 1938, pp. 135-206.
2. ^ P. Seidel: Floer homology and the symplectic isotopy problem. Oxford 1997. (www-math.mit.edu; pdf)

Web links

Video showing stretching twists on the torus

literature

• Benson Farb, Dan Margalit: A primer on mapping class groups. (= Princeton Mathematical Series. 49). Princeton University Press, Princeton, NJ, 2012, ISBN 978-0-691-14794-9 .