Minimal generating and normally generating sets for the braid and mapping class groups of $\mathbb{D }^{2}$, $\mathbb S ^{2}$ and $\mathbb{R P^2}$

Abstract

We consider the (pure) braid groups $B_{n}(M)$ and $P_{n}(M)$, where $M$ is the $2$-sphere $\mathbb S ^{2}$ or the real projective plane $\mathbb R P^2$. 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 $\mathbb S ^{2}$, and extends them in the case of $\mathbb R P^2$. We begin by recalling the situation for the Artin braid groups ($M=\mathbb{D }^{2}$). As applications of our results, we answer the corresponding questions for the associated mapping class groups, and we show that for $M=\mathbb S ^{2}$ or $\mathbb R P^2$, the induced action of $B_n(M)$ on $H_3(\widetilde{F_n(M)};\mathbb{Z })$ is trivial, $F_{n}(M)$ being the $n^\mathrm{th}$ configuration space of $M$.