The braid group \(B_{n, m}(\mathbb{R} P^2)\) and the splitting problem of the generalised Fadell-Neuwirth short exact sequence (Q2165564)

Surface braid groups are both a generalisation, to any connected surface, of the fundamental group of the surface and of the braid groups of the plane, known as Artin braid groups and defined by Artin in 1925. They were initially introduced by Zariski in 1936. During the 1960's, Fox gave an equivalent definition of surface braid groups from a topological point of view in terms of configuration spaces Let \(n, m \in \mathbb{N}\), and let \(B_{n,m}(\mathbb{R}P^2)\) be the set of \((n + m)\)-braids in the projective plane whose associated permutation lies in the subgroup \(S_n \times S_m\) of the symmetric group \(S_{n+m}\). In the present paper the author studies the splitting problem of the following generalisation of the Fadell-Neuwirth short exact sequence: \[ 1 \to B_m(\mathbb{R}P^2 \setminus \{x_1, \ldots, x_n\}) \to B_{n,m}(\mathbb{R}P^2) \xrightarrow{\bar{q}} B_n(\mathbb{R}P^2) \to 1, \] 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 \[ q \colon F_{n+m}(\mathbb{R}P^2)/S_n \times S_m \to F_n(\mathbb{R}P^2)/S_n, \] where \(F_n(\mathbb{R}P^2)\) is the \(n^{th}\) ordered configuration space of the projective plane \(\mathbb{R}P^2\). The main results are the following: if \(n = 1\) the homomorphism \(\bar{q}\) and the corresponding fibration \(q\) admit no section, while if \(n = 2\), then \(\bar{q}\) and \(q\) admit a section. For \(n \geq 3\), it is shown that if \(\bar{q}\) and \(q\) admit a section then \(m \equiv 0, (n-1)^2\) mod \(n(n-1)\). Moreover, using geometric constructions, it is shown that the homomorphism \(\bar{q}\) and the fibration \(q\) admit a section for \(m = kn(2n - 1)(2n - 2)\), where \(k \geq 1\), and for \(m = 2n(n - 1)\). In addition, it is shown that for \(m \geq 3\), \(B_m(\mathbb{R}P^2 \setminus \{x_1, \ldots, x_n\}) \) is not residually nilpotent and for \(m \geq 5\), it is not residually solvable.