Brunnian subgroups of mapping class groups and braid groups

Abstract

This is the second of a pair of papers on the Delta-group structure on the braid and mapping class groups of a surface. We obtain a description of the homotopy groups of these Delta-groups, and generalize to an arbitrary surface the Berrick-Cohen-Wong-Wu exact sequence relating the Brunnian braid groups of the 2-sphere to its homotopy groups. We prove a similar result for Brunnian mapping class groups. 1 Introduction and statement of results Recent work on braid groups has shown that varying the number of strands can give rise to a combinatorial structure, that of a -group, that contains remarkable information. An example is given by the group of Brunnian braids Brun(Pk+1(M)) of a Riemann surface M comprising those pure braids on k+1 strands that become trivial when any strand is deleted. The paper [2] obtains striking results about these Brunnian braid groups when the surfaceM is either the disc D or sphere S. In general the Brunnian groups are free groups of in nite rank, and would therefore seem a rather crude instrument for calculating the fundamental building blocks of homotopy theory, the nite abelian groups that are the homotopy groups of spheres. The following theorem reveals how this counterintuitive phenomenon occurs. Theorem ([2]). Whenever k 4, there is an exact sequence 1! Brun(Pk+2(S))! Brun(Pk+1(D)) c ! Brun(Pk+1(S))! k(S)! 1: The above result has been described as “one of those mysterious alluring connections which makes mathematics worth doing” [27]. Part of its allure is the possibility it raises of relating seemingly purely geometric data to topological maps of spheres. For example, [2, Subsection 7.2] demonstrates how the Hopf bration that provides the generator of the stable 1-stem in homotopy groups of spheres corresponds to the Brunnian 3-strand braid with which young women have for centuries plaited their hair, and whose closure forms the classical Borromean rings. In general, a -group G comprises an in nite sequence of groups G0; G1; : : : related by face maps. These face maps di : Gn ! Gn 1 (i = 0; : : : ; n) are 2010 Mathematics Subject Classi cation. Primary 20F36; Secondary 55Q40, 55R80, 55U10, 57M07, 57S05. Key words and phrases: braid groups, mapping class groups, con guration spaces, simplicial structures, homotopy groups of spheres, Brunnian groups.