examples of mapping class groupExampleOfMappingClassGroup2006-02-17 23:14:542006-06-03 02:58:01Examplemapping class groupDehn's twistsurface% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands hereAn example of this concept is to take the 2-sphere $S^2$, then one can calculate that
$${\cal{M}}(S^2)=1,$$
but
$${\cal{M}}^*(S^2)={\mathbb{Z}}_2.$$
For the genus one orientable surface, i.e. the torus $T=S^1\times S^1$, it is known that its (extended) mapping class group
$${\cal{M}}^*(T)=GL_2({\mathbb{Z}}),$$
but usually by the (non-extended) mapping class group, that is, the group of isotopy classes of homeomorphisms that preserve orientations (the Dehn's twists) is just
$${\cal{M}}(T)=SL_2({\mathbb{Z}}).$$
In these two examples we see that $\cal{M}^*$ is an extension of $\cal{M}$ by ${\mathbb{Z}}_2$, trivial for the 2-sphere and non trivial for the torus.
For the projective plane ${\mathbb{R}}P^2$ we have
$${\cal{M}}({\mathbb{R}}P^2)={\cal{M}}^*({\mathbb{R}}P^2)=1$$
And what about the Klein bottle?
$${\cal{M}}(K)={\mathbb{Z}}_2$$
$${\cal{M}}^*(K)={\mathbb{Z}}_2\oplus{\mathbb{Z}}_2$$