Orbit configuration spaces and the homotopy groups of the pair \((\prod_1^n {M,{F_n}} (M))\) for \(M\) either \({\mathbb{S}^2}\) or \(\mathbb{R} P^2\) (Q6588709)

Let \(\prod_1^n M = M \times M \times \cdots \times M\) be the \(n\)-fold Cartesian product of a connected surface \(M\) with itself. The \textit{\(n^{\mathrm{th}}\) configuration space} of \(M\) is defined by \N\[ \NF_n (M) = \{ (x_1, x_2, \ldots, x_n) \in \prod_1^n M \mid x_i \neq x_j \text{ for all } 1 \leq i,j \leq n, ~ i \neq j \}. \N\] \NIn [Math. Scand. 10, 111--118 (1962; Zbl 0136.44104)], \textit{E. Fadell} and \textit{L. Neuwirth} proved that the fundamental group \(\pi_1 (F_n (M))\) of \(F_n (M)\) is isomorphic to the pure braid group \(P_n (M)\) of \(M\) on \(n\) strings, and in [Commun. Pure Appl. Math. 22, 41--72 (1969; Zbl 0157.30904)], \textit{J. S. Birman} studied the homomorphism \({\iota_n}_{\sharp} : P_n (M) \rightarrow \pi_1 (\prod_1^n M)\) between fundamental groups induced by the inclusion map \(\iota_n : F_n (M) \rightarrow \prod_1^n M\) for \(n \geq 1\). Let \(C = \mathbb{S}^2 \backslash \{ \tilde z_0, -\tilde z_0 \}\) be the open cylinder for \(\tilde z_0 \in \mathbb{S}^2\), and let \(\tau : \mathbb{S}^2 \rightarrow \mathbb{S}^2\) be the antipodal map as well as its restriction to \(C\). Let \(G_n = \pi_1 (F_n^{\langle \tau \rangle} (C))\) be the fundamental group of the \(n^{\mathrm{th}}\) orbit configuration space \(F_n^{\langle \tau \rangle} (C)\) of \(C\) with respect to the group \(\langle \tau \rangle\). Let \((\tilde \lambda_{\tilde x_0}, \tilde \lambda_{\tilde x_1}, \ldots, \tilde \lambda_{\tilde x_{n-3}}, \tilde \lambda_{\tilde z_0}, \tilde \lambda_{-\tilde z_0})\) and \((\lambda_{x_0}, \lambda_{x_1}, \ldots, \lambda_{x_{n-2}}, \lambda_{z_0})\) be bases of the free abelian groups \(\pi_2 (\prod_1^n \mathbb{S}^2)\) and \(\pi_2 (\prod_1^n \mathbb{R}P^2)\), respectively.\N\NIn the paper under review, the authors develop the inclusion map \(\iota_n\), the homotopy fibre \(I_n\) of \(\iota_n\), its homotopy groups, and the induced homomorphisms \({(\iota_n)}_{\sharp k}\) on the \(k^{\mathrm{th}}\) homotopy groups of \(F_n (M)\) and \(\prod_1^n M\) for all \(k \geq 1\), where \(M\) is the \(2\)-sphere \(\mathbb{S}^2\) or the real projective plane \(\mathbb{R}P^2\). More precisely, the authors show that the homotopy fiber \(I_n\) of the inclusion map \(\iota_n : F_n (M) \rightarrow \prod_1^n M\) has the same homotopy type of \(F_{n-1}(\mathbb{D}^2) \times \Omega (\prod_1^{n-1} \mathbb{S}^2)\) and \(F_{n-1}^{\langle \tau \rangle} (C) \times \Omega (\prod_1^{n-1} \mathbb{S}^2)\) for \(n \geq 2\) if \(M = \mathbb{S}^2\) and \(M = \mathbb{R}P^2\), respectively. The authors also prove that the image of the connecting homomorphism \(\partial_n : \pi_2 (\prod_1^n M) \rightarrow \pi_1 (I_n)\) is equal to \(\langle \partial_n (\tilde \lambda_{\tilde x_0}), \ldots, \partial_n (\tilde \lambda_{\tilde x_{n-3}}), \partial_n (\tilde \lambda_{\tilde z_0}), \hat \tau_n^2 \rangle\) and \(\langle \partial_n (\lambda_{x_0}), \ldots, \partial_n ( \lambda_{x_{n-2}}), \hat \tau_n^2 \rangle\) if \(M = \mathbb{S}^2\) and \(M = \mathbb{R}P^2\), respectively, where \(\hat \tau_n\) is a homotopy class of \(\pi_1 (I_n)\).