Mapping class group

The mapping class group of a room is the group of “symmetries” (classes of mapping) of this room. Images that can be continuously deformed into one another are each viewed as a class of images.

In a more formal way, one considers all homeomorphisms (continuous self-mapping that have a continuous inverse mapping) of a space . It is said that two homeomorphisms belong to the same isotope class or are isotopic if there is a continuous mapping with for all and . These isotope classes of homeomorphisms form a group with the ( well-defined ) linkage of homeomorphisms and this is referred to as a mapping group or . ${\displaystyle f\colon S\to S}$ ${\displaystyle S}$ ${\displaystyle f_{0},f_{1}}$ ${\displaystyle F\colon S\times \left[0,1\right]\to S}$ ${\displaystyle F(s,i)=f_{i}(s)}$ ${\displaystyle s\in S}$ ${\displaystyle i\in \left\{0,1\right\}}$ ${\displaystyle Mod(S)}$ ${\displaystyle MCG(S)}$

In the context of orientable manifolds , only the isotope classes of orientation-preserving homeomorphisms are considered. (The group of all isotope classes is then referred to as the extended mapping class group.) In the case of manifolds with a boundary, only those homeomorphisms are considered that leave the boundary point-wise fixed and only such isotopes are allowed. You can then define it formally

${\displaystyle Mod(S):=\pi _{0}(Homeo^{+}(S,\partial S))}$,

where the compact-open topology bears and denotes the -th homotopy set (i.e. the set of path-related components ). ${\displaystyle Homeo^{+}(S,\partial S))}$ ${\displaystyle \pi _{0}}$ ${\displaystyle 0}$

Mostly, especially in the context of group theory , mapping class groups of orientable surfaces are meant when “mapping class groups” are mentioned.

In the following, this article only deals with mapping class groups of orientable surfaces.

Examples

Circular disc

From the Alexander trick it follows : every self-image of the circular disc that fixes the edge point by point is homotopic to represent identity . ${\displaystyle Mod(D^{2})=0}$

sphere

It also follows from the Alexander trick : every self-representation of the sphere is homotopic to represent identity. ${\displaystyle Mod(S^{2})=0}$

trousers

The trousers' mapping class group is also trivial. If you look at images that do not necessarily fix the edge, you get the symmetrical group as the mapping class group . ${\displaystyle \Sigma _{3}}$

Circular ring

The mapping class group of the circular ring is cyclically of infinite order, a generator is the stretch twist on the core curve.

Orientable closed area by gender 3

Torus

The mapping class group of the torus is the modular group : . The same applies to the torus with a hole . ${\displaystyle Mod(T^{2})=SL(2,\mathbb {Z} )}$

Surfaces of the higher sex

For are the connected components of contractible , i.e. homotopy equivalent to . (Hamstrom) ${\displaystyle S\not =S^{2},\mathbb {R} ^{2},D^{2},T^{2}}$ ${\displaystyle Homeo^{+}(S,\partial S)}$ ${\displaystyle Homeo^{+}(S,\partial S)}$ ${\displaystyle Mod(S)}$

Because the mapping class groups of sphere, circular disk, circular ring and torus are easy to describe, in this article we will only deal with the mapping class groups of surfaces with negative Euler characteristics . ${\displaystyle \chi (S)<0}$

Presentation class group presentations

Stretching twists on these 3g-1 curves create the mapping class group.

Producer

The mapping class group of a surface of Dehn twists generated . As Lickorish has proven, the stretching twists on the 3g-1 curves shown in the picture on the right are sufficient to generate the mapping class group. Another generating system with only 2g-1 stretching twists was given by Humphries.

Relations

The 7 curves occurring in the lantern relation.

There are a number of relationships between stretch twists. In the following we list a few examples, with the stretching twist on the curve and ${\displaystyle \tau _{\alpha }}$ ${\displaystyle \alpha }$

${\displaystyle i(\alpha ,\beta ):=min\left\{\mid \alpha ^{\prime }\cap \beta ^{\prime }\mid \colon \alpha ^{\prime }\ {\mbox{isotop zu}}\ \alpha ,\beta ^{\prime }\ {\mbox{isotop zu}}\ \beta ,\alpha ^{\prime }\pitchfork \beta ^{\prime }\right\}}$

denotes the geometric intersection of the curves . ${\displaystyle \alpha ,\beta }$

Disjoint relation : If the number of cuts is, then . ${\displaystyle i(\alpha ,\beta )=0}$ ${\displaystyle \left[\tau _{\alpha },\tau _{\beta }\right]=1}$

Braid relation : If the number of cuts is, then . ${\displaystyle i(\alpha ,\beta )=1}$ ${\displaystyle \tau _{\alpha }\tau _{\beta }\tau _{\alpha }=\tau _{\beta }\tau _{\alpha }\tau _{\beta }}$

Lantern relation : If 7 curves are arranged as in the picture on the right, then the following applies , where the curves in the picture denote blue and those in the picture red curves. ${\displaystyle \tau _{A}\tau _{B}\tau _{C}=\tau _{R}\tau _{S}\tau _{T}\tau _{U}}$ ${\displaystyle A,B,C}$ ${\displaystyle R,S,T,U}$

On the other hand, the stretching twists create a free group . ${\displaystyle i(\alpha ,\beta )\not \in \left\{0,1\right\}}$ ${\displaystyle \tau _{\alpha },\tau _{\beta }}$

Presentations

There are several ways to give explicit presentations of the mapping class groups, such as the Wajnryb presentation or the Gervais presentation .

For example, has the presentation and has the Birman Hilden presentation . ${\displaystyle Mod(S_{1})=Mod(T^{2})}$ ${\displaystyle Mod(T^{2})=\langle \tau _{l},\tau _{m}\mid \tau _{\alpha }\tau _{\beta }\tau _{\alpha }=\tau _{\beta }\tau _{\alpha }\tau _{\beta },(\tau _{\alpha }\tau _{\beta })^{6}=1\rangle }$ ${\displaystyle Mod(S_{2})}$

Algebraic properties of the mapping class groups

The center of is trivial for . ${\displaystyle Mod(S_{g})}$ ${\displaystyle g\geq 3}$

The first group homology is trivial for . For one has and for the torus . ${\displaystyle H_{1}(Mod(S_{g});\mathbb {Z} )}$ ${\displaystyle g\geq 3}$ ${\displaystyle g=2}$ ${\displaystyle H_{1}(Mod(S_{2}),\mathbb {Z} )=\mathbb {Z} /10\mathbb {Z} }$ ${\displaystyle H_{1}(Mod(S_{1};\mathbb {Z} )=\mathbb {Z} /12\mathbb {Z} }$

The second group homology was calculated by Harer, for has one . ${\displaystyle g\geq 4}$ ${\displaystyle H_{2}(Mod(S_{g});\mathbb {Z} )=\mathbb {Z} }$

The mapping class groups are residual finite .

Curves on surfaces

Let be simple closed curves on a connected surface . We use to denote the areas created by cutting along or from . We say that a separating or non-separating curve is when is disconnected or connected. ${\displaystyle \alpha ,\beta }$ ${\displaystyle S}$ ${\displaystyle S_{\alpha },S_{\beta }}$ ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle S}$ ${\displaystyle \alpha }$ ${\displaystyle S_{\alpha }}$

The following property is called the change of coordinates principle for the effect of the mapping class group:

There is a homeomorphism with then and only if ${\displaystyle \phi \colon S\to S}$ ${\displaystyle \phi (\alpha )=\beta }$

• either and both are non-separating curves, ${\displaystyle \alpha }$ ${\displaystyle \beta }$
• or both are separating curves and the (disconnected) surfaces and are homeomorphic to each other. ${\displaystyle S_{\alpha }}$ ${\displaystyle S_{\beta }}$

The complex of curves is an important aid when studying the mapping class group .

Dehn-Nielsen-Baer theorem

The mapping class group is a subgroup of index 2 in the “extended mapping class group” of the isotope classes of all (non-orientation- preserving ) homeomorphisms. ${\displaystyle Mod(S)}$ ${\displaystyle Mod^{\pm }(S)}$

The theorem of Dehn-Nielsen-Baer (named after Max Dehn , Jakob Nielsen and Reinhold Baer ) states that there is an isomorphism

${\displaystyle Mod^{\pm }(S)=Out(\pi _{1}S)}$

where the fundamental group of the surface and its outer automorphism group denotes. ${\displaystyle \pi _{1}S}$ ${\displaystyle S}$ ${\displaystyle Out(\pi _{1}S):=Aut(\pi _{1}S)/Inn(\pi _{1}S)}$

For every aspherical space the group of self- homotopy equivalences modulo homotopy is isomorphic to . From the Dehn-Nielsen-Baer theorem it follows that the mapping class group could equivalently also be defined as a group of self-homotopy equivalences (instead of just homeomorphisms) modulo homotopy. ${\displaystyle M}$ ${\displaystyle Out(\pi _{1}M)}$

Torelli group

The effect of on is given by the sectional shape , which is a symplectic shape . This gives a surjective homomorphism ${\displaystyle Mod(S_{g})}$ ${\displaystyle H_{1}(S_{g};\mathbb {R} )}$

${\displaystyle Mod(S_{g})\to Sp(2g;\mathbb {Z} )}$,

whose core is called the Torelli group .

Johnson theorem: For , the Torelli group is generated by for pairs that border a double dotted torus . ${\displaystyle g\geq 3}$ ${\displaystyle \tau _{\alpha }\tau _{\beta }^{-1}}$ ${\displaystyle (\alpha ,\beta )}$

All 3-dimensional spheres of homology can be obtained by Heegaard decompositions , the glued image of which represents an element of the Torelli group.

Areas with marked points

For a closed, orientable area of gender with points one defines ${\displaystyle S}$ ${\displaystyle g}$ ${\displaystyle r}$ ${\displaystyle x_{1},\ldots ,x_{r}\in S}$

${\displaystyle \Gamma _{g,r}=Mod(S_{g,r})}$

as the group of homotopy classes of homeomorphisms with , whereby the homotopias should also fix the points . ${\displaystyle \phi \colon S\to S}$ ${\displaystyle \phi (x_{1})=x_{1},\ldots ,\phi (x_{r})=x_{r}}$ ${\displaystyle x_{1},\ldots ,x_{r}}$

The mapping class groups for different ones are related to one another via Birman sequences . ${\displaystyle \Gamma _{g,r}}$ ${\displaystyle r}$

In different contexts it is easier instead to examine, such as the calculation of the stable homology of mapping class groups ( set of Madsen-Weiss ). ${\displaystyle \Gamma _{g,1}}$ ${\displaystyle \Gamma _{g,0}}$

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 . ( online ; pdf)
• Nikolai Ivanov: Mapping class groups. In: RJ Daverman (Ed.): Handbook of geometric topology. North-Holland, Amsterdam 2002, ISBN 0-444-82432-4 , pp. 523-633.

Web links

• Yair Minsky: A brief introduction to mapping class groups.
• Gwenaël Massuyeau: A short introduction to mapping class groups.

Individual evidence

1. ↑ For surfaces can be the mapping class group equivalent by Isotopieklassen of diffeomorphisms define: . Furthermore, one of Baer's theorem states that homeomorphisms of surfaces are isotopic if and only if they are homotopic , which is why the mapping class group of surfaces could also be defined by homotopy classes of homeomorphisms. This also applies accordingly to areas with a border, where two homeomorphisms that fix the border point by point are isotopic if and only if they are homotopic (with regard to isotopias or homotopies that fix the border point by point). ${\displaystyle Mod(S):=\pi _{0}(Diff^{+}(S,\partial S))}$
2. ↑ All examples in this section can be found in Chapter 2 of Farb-Margalit (op.cit.).
3. ↑ H. Zieschang, E. Vogt, H.-D. Coldewey: surfaces and planar discontinuous groups. (= Lecture Notes in Mathematics. Vol. 122). Springer-Verlag, Berlin / New York 1970, ISBN 3-540-04911-8 .
4. ↑ Dennis L. Johnson: homeomorphisms of a surface Which act trivially on homology. In: Proc. Amer. Math. Soc. 75, no. 1, 1979, pp. 119-125.