Dehn-Twist
In topology, a branch of mathematics, are Dehn twists certain self-maps of surfaces. Dehn twists were introduced by Max Dehn, who originally called the " Screw holes ".
Definition
Be an orientable surface and a simple closed curve. Be a tubular neighborhood of, ie we have a homeomorphism on the maps. We use this homeomorphism to parametrisiereren by coordinates.
We then define a mapping by
Because corresponds to the identity, we can continue to it continuously by means of the identity mapping and thus obtain a homeomorphism, is referred to as stretch- twist to the curve c.
Note: The above- defined mapping is dependent on the selected environment and the selected parameterization. For other environments and other parameterizations but you get another homotopic pictures with this construction. The homotopy class (Figure Class ) of is so well defined.
Examples
We identify with the torus. Each matrix from then corresponds to a self-map of the torus. ( The matrix acts linearly and forms after the other. One can show that any orientation- preserving homeomorphism of the torus homotopic to be such a figure.)
The matrices correspond to the stretching, and then twists and meridian of longitude (ie the images of the x-and y -axis. )
Mapping class group
Be the closed orientable surface of genus and its mapping class group. For ( torus ) and it can be proved by means of Euclid's algorithm that is generated from the matrices and, hence, stretching of the twists to longitude and meridian. Max Dehn proved to all that the mapping class group of Dehn twists is generated. Lickorish showed that shown in the image on the right Dehn twists generate the mapping class group. Humphries proved that is generated for the mapping class group of Dehn twists and that this is the smallest possible number of producers.
Generalized Dehn twists
Let be a symplectic manifold and a Lagrangian sphere. By a theorem of Weinstein there is a neighborhood of which is symplektomorph to a neighborhood of the cotangent bundle ( with the canonical symplectic structure). It is therefore sufficient generalized Dehn twists for environments to be defined by in.
The function is smooth outside the zero - section, its Hamiltonian flow is the normalized geodesic flow. The figure can be continued to the zero - section, because all geodesics of length have the same endpoint. The so- defined mapping is a Symplektomorphismus and you can modify so that it is the identity outside a compact environment. For is homotopic to the identity, while for ( ie for Dehn twists on surfaces ), the Dehn twists infinite order in the mapping class group.