Generators for the mapping class group of a nonorientable surface (Q725643)

Let \(N_g\) denote the connected sum of \(g\) projective planes, and let \(\mathcal M(N_g)\) denote the mapping class group of \(N_g\). Let \(t_c\) denote the right-handed Dehn twist about a two-sided simple closed curve \(c\) in \(N_g\). \textit{J. S. Birman} and \textit{D. R. J. Chillingworth} [Proc. Camb. Philos. Soc. 71, 437--448 (1972; Zbl 0232.57001)] obtained an explicit finite generating set for \(\mathcal M(N_g)\) comprising \(g\) Dehn twists and a single crosscap slide. Moreover, it is a well known result of \textit{S. P. Humphries} [Lect. Notes Math. 722, 44--47 (1979; Zbl 0732.57004)] that the mapping class group of a closed orientable surface of genus \(g\) is generated by no fewer than \(2g+1\) Dehn twists. The main theorem in this paper is a non-orientable analog of this result: Theorem 2. If for \(g \geq 4\), \(\mathcal M(N_g)\) is generated by the set \(\{t_{c_1},\ldots,t_{c_n}, Y_1, \ldots, Y_k\}\), where the \(c_i\) are two-sided simple closed curves, and the \(Y_i\) are crosscap slides, then \(n \geq g\) and \(k \geq 1\). In particular, the theorem above implies that the generating set derived in [Birman and Chillingworth, loc. cit.] is minimal. The author uses the notion of \(\mathbb{Z}_4\)-quadratic forms over \(H_1(N_g;\mathbb{Z}_2)\), and a result of \textit{W. B. R. Lickorish} [Proc. Camb. Philos. Soc. 60, 769--778 (1964; Zbl 0131.20801)] that \(\mathcal M(N_g)\) is not generated by Dehn twists, to obtain a proof of the main theorem.