examples of mapping class group

An 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?

 ${\cal{M}}(K)={\mathbb{Z}}_{2}$
 ${\cal{M}}^{*}(K)={\mathbb{Z}}_{2}\oplus{\mathbb{Z}}_{2}$
Title examples of mapping class group ExamplesOfMappingClassGroup 2013-03-22 15:41:19 2013-03-22 15:41:19 juanman (12619) juanman (12619) 8 juanman (12619) Example msc 57R50 first homeotopy group isotopy group Group Isotopy