Abelian quotients of subgroups of the mapping class group of surfaces
type: Item
instance of: scholarly article
title: Abelian quotients of subgroups of the mapping class group of surfaces
zbMATH Open document ID: 0801.57011
DOI: 10.1215/S0012-7094-93-07017-2
author: Shigeyuki Morita
published in: Duke Mathematical Journal
publication date: 1994-04-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 558981
zbMATH Keywords: mapping class groups of closed orientable surfaces
zbMATH Keywords: Torelli group
zbMATH Keywords: lower central series of the fundamental group of the surface
zbMATH Keywords: Johnson homomorphisms
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
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full work available at URL: s0012-7094-93-07017-2
OpenAlex ID: W1482125263
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description: scientific article; zbMATH DE number 558981