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 \(M\).