Abelian quotients of subgroups of the mapping class group of surfaces
The author continues his detailed study of the algebraic structure of the mapping class groups of closed orientable surfaces, that is the groups of isotopy classes of orientation preserving diffeomorphisms (fixing an embedded disk). The present paper can be considered as a sequel to the author's paper [Invent. Math. 111, No. 1, 197-224 (1993; Zbl 0787.57008)]. The Torelli group is the subgroup generated by diffeomorphisms operating trivially on homology. Extending earlier work of Sullivan, D. Johnson obtained fundamental results concerning the structure of the Torelli group. In the above mentioned paper, the author generalized the work of Sullivan and Johnson on the Torelli group in the context of the whole mapping class group. Using the lower central series of the fundamental group of the surface (a free group), he obtained certain ``approximations'' of the mapping class group and defined what he called Johnson homomorphisms on the pieces of this approximation. The present work is devoted to a more detailed study of these Johnson homomorphisms. As the results are somewhat technical to state and depend strongly on the previous paper, we have to refer the interested reader to the introduction of the paper for an exact statement of the constructions and results.
mapping class groups of closed orientable surfaces
lower central series of the fundamental group of the surface
scientific article; zbMATH DE number 558981
scientific article