examples of mapping class group
An example of this concept is to take the 2sphere ${S}^{2}$, then one can calculate that

$$\mathcal{M}({S}^{2})=1,$$ 

but

$${\mathcal{M}}^{*}({S}^{2})={\mathbb{Z}}_{2}.$$ 

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^{}

$${\mathcal{M}}^{*}(T)=G{L}_{2}(\mathbb{Z}),$$ 

but usually by the (nonextended) mapping class group, that is, the group of isotopy^{} classes of homeomorphisms^{} that preserve orientations^{} (the Dehn’s twists) is just

$$\mathcal{M}(T)=S{L}_{2}(\mathbb{Z}).$$ 

In these two examples we see that ${\mathcal{M}}^{*}$ is an extension of $\mathcal{M}$ by ${\mathbb{Z}}_{2}$, trivial for the 2sphere and non trivial for the torus.
For the projective plane^{} $\mathbb{R}{P}^{2}$ we have

$$\mathcal{M}(\mathbb{R}{P}^{2})={\mathcal{M}}^{*}(\mathbb{R}{P}^{2})=1$$ 

And what about the Klein bottle?

$$\mathcal{M}(K)={\mathbb{Z}}_{2}$$ 


$${\mathcal{M}}^{*}(K)={\mathbb{Z}}_{2}\oplus {\mathbb{Z}}_{2}$$ 
