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

TL;DR

This paper investigates the splitting problem of a generalized Fadell-Neuwirth short exact sequence for braid groups on the real projective plane, revealing conditions for the existence of sections and properties of associated groups.

Contribution

It provides new results on when the sequence admits a section, depending on parameters n and m, and explores algebraic properties of related braid groups.

Findings

No section exists for n=1

Sections exist for n=2 and specific m values

Certain braid groups are not residually nilpotent or solvable for m≥3 and m≥5

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.