A presentation for the mapping class group of a closed orientable surface

Abstract

THE CENTRAL objects of study in this paper are collections {C,, . . . , C,} of g disjoint circles on a closed orientable surface M of genus g, whose complement M-(C, u . . . U C,) is a 2g-punctured sphere. We call an isotopy class of such collections a cut system. Of course, any two cut systems are related by a diffeomorphism of M, by the classification of surfaces. We show that any two cut systems are also joined by a finite sequence of simple moues, in which just one Cj changes at a time, to a circle intersecting it transversely in one point and disjoint from the other Ci’s. Furthermore, we find a short list of relations between sequences of simple moves, sufficient to pass between any two sequences of simple moves joining the same pair of cut systems. From these properties of cut systems it is a routine matter to read off a finite presentation for the mapping class group of M, the group of isotopy classes of orientation preserving self-diffeomorphisms of M. Unfortunately, the presentation so obtained is rather complicated, and stands in need of considerable simplification before much light will be shed on the structure of the mapping class group. Qualitatively, one can at least deduce from the presentation that all relations follow from relations supported in certain subsurfaces of M, finite in number, of genus at most two. This may be compared with the result of Dehn [3] and Lickorish [4] that the mapping class group is generated by diffeomorphisms supported in finitely many annuli. A finite presentation in genus two was obtained by Birman-Hilden[2], completing a program begun by Bergau-Mennicke [l]. For higher genus the existence of finite presentations was shown by McCool[ lo], using more algebraic techniques. For another approach to finite presentations, see [12], and for general background on mapping class groups, see [ 111. Our methods apply also to maximal systems of disjoint, non-contractible, nonisotopic circles on M. This is discussed briefly in an appendix.