Editing Mapping class group (section)

==== Non-orientable surfaces ====
Some [[orientability|non-orientable]] surfaces have mapping class groups with simple presentations.  For example, every homeomorphism of the [[real projective plane]] <math>\mathbf{P}^2(\R)</math> is isotopic to the identity:

:<math> \operatorname{MCG}(\mathbf{P}^2(\R)) = 1. </math>

The mapping class group of the [[Klein bottle]] ''K'' is:

:<math> \operatorname{MCG}(K)= \Z_2 \oplus  \Z_2.</math>

The four elements are the identity, a [[Dehn twist]] on a two-sided curve which does not bound a [[Möbius strip]], the [[y-homeomorphism]] of [[Lickorish]], and the product of the twist and the y-homeomorphism.  It is a nice exercise to show that the square of the Dehn twist is isotopic to the identity.

We also remark that the closed [[genus (mathematics)|genus]] three non-orientable surface ''N''<sub>3</sub> (the connected sum of three projective planes) has:

:<math> \operatorname{MCG}(N_3) = \operatorname{GL}(2,\Z). </math>

This is because the surface ''N'' has a unique class of one-sided curves such that, when ''N'' is cut open along such a curve ''C'', the resulting surface <math>N\setminus C</math> is ''a torus with a disk removed''. As an unoriented surface, its mapping class group is <math>\operatorname{GL}(2,\Z)</math>. (Lemma 2.1<ref>{{cite journal |first=Martin |last=Scharlemann |title=The complex of curves on nonorientable surfaces |journal=Journal of the London Mathematical Society |volume=s2-25 |issue=1 |pages=171–184 |date=February 1982 |doi=10.1112/jlms/s2-25.1.171 |citeseerx=10.1.1.591.2588}}</ref>).