First homology group of mapping class groups of nonorientable surfaces

Abstract

Recall that the first homology group H1(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 H1([Mscr ]F) is again trivial. This result is due to Powell [P]. The group H1([Mscr ]F) is Z10 if the genus of F is two, proved by Mumford [Mu], and Z12 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.