Mapping class groups of nonorientable surfaces
Let \(S\) be a closed connected nonorientable surface of genus \(g\geq 1\) with \(n\) distinguished points and boundary \(\partial S\). The mapping class group \({\mathcal M}_{g,n}\) consists of isotopy classes of diffeomorphisms \(S\to S\) which take \(\partial S\) to itself. The pure mapping class group \({\mathcal P}{\mathcal M}_{g,n}\) consists of isotopy classes of diffeomorphisms which fix \(\partial S\). If \(z_n\) is one of the punctures and \(R\) the surface with \(n-1\) punctures obtained from \(S\) by forgetting that \(z_n\) is a pucture, there is an exact sequence originally discovered by Birman in the orientable case: \[ \cdots\to \pi(R,z_n)\to {\mathcal P}{\mathcal M}_{g,n}\to {\mathcal P}{\mathcal M}_{g,n-1}\to 1. \] This is used to define a finite set of generators for \({\mathcal P}{\mathcal M}_{g,n}\) with \(n\geq 1\), consisting of Dehn twists, crosscap slides and boundary slides and, adding elementary braids, for \({\mathcal M}_{g,n}\) with \(n\geq 2\). The cases of the punctured projective plane, punctured Klein bottle and surfaces with genus at least 3 are handled separately. After computing the first homology group of the mapping class group and certain subgroups, the paper ends with an application of the results. It is shown that the image of any homomorphism of \({\mathcal M}_{g,n}\), \(g\geq 9\), to the group of orientation preserving real-analytic diffeomorphisms of the circle is either trivial or \(\mathbb{Z}_2\).
scientific article; zbMATH DE number 1747812