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
Canonical name	ExamplesOfMappingClassGroup
Date of creation	2013-03-22 15:41:19
Last modified on	2013-03-22 15:41:19
Owner	juanman (12619)
Last modified by	juanman (12619)
Numerical id	8
Author	juanman (12619)
Entry type	Example
Classification	msc 57R50
Synonym	first homeotopy group
Related topic	isotopy
Related topic	group
Related topic	Group
Related topic	Isotopy