First homology group of mapping class groups of nonorientable surfaces

Abstract

Recall that the first homology group H 1 ( G ) of a group G is the derived quotient G /[ G , G ]. The first homology groups of the mapping class groups of closed orientable surfaces are well known. Let F be a closed orientable surface of genus g . Recall that the extended mapping class group [Mscr ]* F of the surface F is the group of the isotopy classes of self-homeomorphisms of F . The mapping class group [Mscr ] F of F is the subgroup of [Mscr ]* F consisting of the isotopy classes of orientation-preserving self-homeomorphisms of F . It is well known that [Mscr ] F is trivial if F is a sphere. Hence the first homology group of the mapping class group of a sphere is trivial. If the genus of F is at least three, then H 1 ([Mscr ] F ) is again trivial. This result is due to Powell [ P ]. The group H 1 ([Mscr ] F ) is Z 10 if the genus of F is two, proved by Mumford [ Mu ], and Z 12 if F is a torus. When a problem about orientable surfaces is solved, it is natural to ask the corresponding problem for nonorientable surfaces. This is our motivation for the present paper.