Minimal generating and normally generating sets for the braid and mapping class groups of the disc, the sphere and the projective plane

TL;DR

This paper determines the minimal (normal) generating sets for braid groups on the sphere and projective plane, extending previous results and analyzing their implications for related mapping class groups and homological actions.

Contribution

It establishes the minimal cardinality of generating sets for braid groups on S^2 and RP^2, including finite order elements, and applies these findings to mapping class groups and homological actions.

Findings

Minimal generating sets for B_{n}(S^2) and P_{n}(S^2) determined.

Minimal generating sets for B_{n}(RP^2) and P_{n}(RP^2) determined.

Action of braid groups on H_3 of the universal cover is trivial.

Abstract

We consider the (pure) braid groups B_{n}(M) and P_{n}(M), where M is the 2-sphere S^2 or the real projective plane RP^2. We determine the minimal cardinality of (normal) generating sets X of these groups, first when there is no restriction on X, and secondly when X consists of elements of finite order. This improves on results of Berrick and Matthey in the case of S^2, and extends them in the case of RP^2. We begin by recalling the situation for the Artin braid groups. As applications of our results, we answer the corresponding questions for the associated mapping class groups, and we show that for M=S^2 or RP^2, the induced action of B_n(M) on H_3 of the universal covering of the n th configuration space of M is trivial.