Generators for the Mapping Class Group of the Nonorientable Surface

We show that Szepietowski’s system of generators [8] for the mapping class group of the non-orientable surface is the minimal generating set by Dehn twists and Y -homemorphisms. Let Ng be a non-orientable surface which is a connected sum of g projective planes. We assume that g ≥ 4. Let M(Ng) be the group of isotopy class of homeomorphisms over Ng, i.e, the mapping class group of Ng. We introduce some elements of M(Ng). A simple closed curve γ1 (resp. γ2) in Ng is two-sided (resp. one-sided) if the regular neighborhood of γ1 (resp. γ2) is an annulus (resp. Möbius band). For a one-sided simple closed curve m and a two-sided simple closed curve a which intersect transversely in one point, let K ⊂ Ng be a regular neighborhood of m ∪ a, which is a union of the tubular neighborhood of m and that of a and then is homeomorphic to the Klein bottle with a hole. Let M be a regular neighborhood of m. We denote by Ym,a a homeomorphism over Ng which is described as the result of pushing M once along a keeping the boundary of K fixed (see Figure 1). We call Ym,a a Y -homeomorphism. For a two-sided simple closed curve γ on Ng, we denote by Tγ a Dehn twist about γ. Lickorish showed that M(Ng) is generated by Dehn twists and Y -homeomorphisms [6], and that M(Ng) is not generated by Dehn twists [7]. Furthermore, Chillingworth [2] found a finite system of generators for M(Ng). Birman and Chillingworth [1] obtained the finite system of generators by using the argument on the orientable two fold covering of Ng. Szepietowski [8] reduced the system of Chillingworth’s generators for M(Ng) and showed: This research was supported by Grant-in-Aid for Scientific Research (C) (No. 24540096), Japan Society for the Promotion of Science.