Mapping class group of a non-orientable surface and moduli space of Klein surfaces
The main result of the paper under review states that for \(g\geq 3\) the mapping class group \(\text{Mod}_g\) of a compact nonorientable surface of genus \(g\) with empty boundary is generated by involutions. This extends to the nonorientable case the classical result of \textit{J. McCarthy} and \textit{A. Papadopoulus} [Enseign. Math., II. Sér. 33, 275-290 (1987; Zbl 0655.57005)] for orientable surfaces. Notice that the moduli space \(M_g\) of compact nonorientable Klein surfaces of genus \(g\) with empty boundary is the quotient \(M_g=T_g/ \text{Mod}_g\) of the Teichmüller space \(T_g\) under the action of \(\text{Mod}_g\), and this last acts properly and discontinuously on \(T_g\). It was proved by \textit{A. M. Macbeath} and \textit{D. Singerman} [Proc. Lond. Math. Soc., III. Ser. 31, 211-256 (1975; Zbl 0314.32012)] that the stabilizer of each point of \(T_g\) is finite, and that each element of finite order in \(\text{Mod}_g\), in particular involutions, has a fixed point in \(T_g\). Hence, it follows from a general result by \textit{M. A. Armstrong} [Proc. Camb. Philos. Soc. 64, 299-301 (1968; Zbl 0159.33002)], that \(M_g\) is a simply-connected space. This way, the author extends to the nonorientable case the analogous result for orientable surfaces due to \textit{C. Maclachlan} [Proc. Am. Math. Soc. 29, 85-86 (1971; Zbl 0222.32012)]. Each compact nonorientable Klein surface \(X\) of genus \(g\) with empty boundary is the quotient \(X=Y/\sigma\) of a unique (up to isomorphism) compact Riemann surface \(Y\) of genus \(g-1\) under its antianalytic involution \(\sigma\). Thus, there exists a mapping \(\pi:M_g\to M_g^+ :[X]\to[Y]\) between the moduli spaces of compact nonorientable Klein surfaces of genus \(g\) and compact Riemann surfaces of genus \(g-1\), and one could try to prove the simply-connectedness of \(M_g\) from the one of \(M_g^+\). However, this map \(\pi\) is not well understood yet, and so the author has developped a completely different approach, which is based on some results by \textit{J. S. Birman} and \textit{D. R. J. Chillingworth} [Proc. Camb. Philos. Soc. 71, 437-448 (1971; Zbl 0232.57001)]. This short note is transparently written, with very clear and clever proofs.
scientific article; zbMATH DE number 1921114