On the structure of surface pure braid groups. (Q5916336)

Let \(M\) be a connected surface. For \(n\in\mathbb{N}\), let \(F_n(M)\) denote the \(n\)-th configuration space of \(M\). Then the \(n\)-string pure braid group of \(M\) is \(P_n(M)=\pi_1(F_n(M),(x_1,\dots,x_n))\). Fadell and Neuwirth (1962) showed that if \(M\) is closed and is neither \(S^2\) nor \(\mathbb{R} P^2\), then the following ``pure braid group sequence'' (PBS) is exact: \[ 1\to\pi_1(M\setminus\{x_1,\dots,x_{n-1}\},x_n)\to P_n(M)\to P_{n-1}(M)\to 1. \] In 1969, Birman showed that if \(M=T^2\), then for all \(n\geq 2\), PBS splits, and asked if this is true for any closed orientable surface of genus at least 2. The authors prove that if \(M\) is a closed orientable surface of genus at least two, then PBS splits if and only if \(n=2\). In fact, they prove the following generalization: Theorem. Let \(M\) be a closed, connected orientable surface of genus at least two, and let \(n>m\) be positive integers. Then the following short exact sequence splits if and only if \(m=1\): \[ 1\to\pi_1(F_{n-m}(M\setminus\{x_1,\dots,x_{m}\},\{x_{m+1}, \dots,x_n))\to P_n(M)\to P_m(M)\to 1. \] They also have analogous results for non-orientable and noncompact surfaces: Theorem. Let \(M\) be a closed, connected surface different from \(S^2\) and \(\mathbb{R} P^2\). Then for all \(n\in\mathbb{N}\), the following short exact sequence splits: \[ 1\to\pi_1(F_{n-m}(M\setminus\{x_n\},\{x_1,\dots,x_{n-1}))\to P_n(M)\to P_1(M)\to 1. \] Theorem. Let \(M\) be a closed, connected surface different from \(S^2\) and \(\mathbb{R} P^2\), but with a finite, non-zero number of points deleted. Then for all \(n\geq 2\), PBS splits. Finally, the authors prove that if \(M\) is a closed, connected surface different from \(S^2\) and \(\mathbb{R} P^2\), then \(P_n(M)\) is a repeated semi-direct product of free groups and a single one-relator group.