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: $$1\to B_m(\mathbb{R}{P}^2\setminus \{x_1,\dots ,x_n\})\to B_{n,m}(\mathbb{R}{P}^2)\stackrel{\bar{q}}{\to }{B}_n(\mathbb{R}{P}^2)\to 1,$$ 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 $n\ge 3$, we show that if $\bar{q}$ and $q$ admit a section then $m\equiv 0,(n-1{)}^2\text{mod}n(n-1)$. 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 $k\ge 1$, and for $m=2n(n-1)$. In addition, we show that for $m\ge 3$, $B_m(\mathbb{R}{P}^2\setminus \{x_1,\dots ,x_n\})$ is not residually nilpotent and for $m\ge 5$, it is not residually solvable.

arXiv comment34 pages, 15 figures

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