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}