The braid groups $B_{n,m}(\mathbb{R}P^2)$ and the splitting problem of the generalised Fadell-Neuwirth short exact sequence

Abstract

Let $n,m\in \mathbb{N}$, and let $B_{n,m}(\mathbb{R}P^2)$ be the set of $(n + m)$-braids of the projective plane whose associated permutation lies in the subgroup $S_n\times S_m$ of the symmetric group $S_{n+m}$. We study the splitting problem of the following generalisation of the Fadell-Neuwirth short exact sequence: $$1\rightarrow B_m(\mathbb{R}P^2 \setminus \{x_1,\dots,x_n\})\rightarrow B_{n,m}(\mathbb{R}P^2)\xrightarrow{\bar{q}} B_n(\mathbb{R}P^2)\rightarrow 1,$$ where the map $\bar{q}$ can be considered geometrically as the epimorphism that forgets the last $m$ strands, as well as the existence of a section of the corresponding fibration $q:F_{n+m}(\mathbb{R}P^2)/S_n\times S_m\to F_{n}(\mathbb{R}P^2)/S_n$, where we denote by $F_n(\mathbb{R}P^2)$ the $n^{th}$ ordered configuration space of the projective plane $\mathbb{R}P^2$. Our main results are the following: if $n=1$ the homomorphism $\bar{q}$ and the corresponding fibration $q$ admits no section, while if $n=2$, then $\bar{q}$ and $q$ admit a section. For $n\geq 3$, we show that if $\bar{q}$ and $q$ admit a section then $m\equiv 0, (n-1)^2\ \textrm{mod}\ n(n-1)$. Moreover, using geometric constructions, we show that the homomorphism $\bar{q}$ and the fibration $q$ admit a section for $m=kn(2n-1)(2n-2)$, where $ k\geq1$, and for $m=2n(n-1)$. In addition, we show that for $m\geq3$, $B_m(\mathbb{R}P^2\setminus\{x_1,\dots,x_n\})$ is not residually nilpotent and for $m\geq 5$, it is not residually solvable.