Orbit configuration spaces and the homotopy groups of the pair $$(\prod\nolimits_1^n {M,{F_n}} (M))$$ for M either $${\mathbb{S}^2}$$ or ℝP2

Abstract

Let n ≥ 1, and let \({\iota _n}:{F_n}(M) \to \prod\nolimits_1^n M \) be the natural inclusion of the n^th configuration space of M in the n-fold Cartesian product of M with itself. In this paper, we study the map ι_n, the homotopy fibre I_n of ι_n and its homotopy groups, and the induced homomorphisms (ι_n)_#k on the k^th homotopy groups of F_n(M) and \(\prod\nolimits_1^n M \) for all k ≥ 1, where M is the 2-sphere \({\mathbb{S}^2}\) or the real projective plane ℝP².It is well known that the group π_k(I_n) is the homotopy group \({\pi _{k + 1}}(\prod\nolimits_1^n {M,{F_n}} (M))\) for all k ≥ 0. If k ≥ 2, we show that the homomorphism (ι_n⁾_#k is injective and diagonal, with the exception of the case n = k = 2 and \(M = {\mathbb{S}^2}\), where it is anti-diagonal. We then show that I_n has the homotopy type of \(K({R_{n - 1}},1) \times \Omega (\prod\nolimits_1^{n - 1} {{\mathbb{S}^2}} )\), where R_n−1 is the (n − 1)^th Artin pure braid group if \(M = {\mathbb{S}^2}\), and is the fundamental group G_n−1 of the (n−1)^th orbit configuration space of the open cylinder \({\mathbb{S}^2}\backslash \{ {\widetilde z_0}, - {\widetilde z_0}\} \) with respect to the action of the antipodal map of \({\mathbb{S}^2}\) if M = ℝP², where \({\widetilde z_0} \in {\mathbb{S}^2}\). This enables us to describe the long exact sequence in homotopy of the homotopy fibration \({I_n} \to {F_n}(M)\buildrel {{\iota _n}} \over\longrightarrow \prod\nolimits_1^n M \) in geometric terms, and notably the image of the boundary homomorphism \({\pi _{k + 1}}(\prod\nolimits_1^n M ) \to {\pi _k}({I_n})\). From this, if \(M = {\mathbb{S}^2}\) and n ≥ 3 (resp. M = ℝP² and n ≥ 2), we show that Ker((ι_n)_#1 ) is isomorphic to the quotient of R_n−1 by the square of its centre, as well as to an iterated semi-direct product of free groups with the subgroup of order 2 generated by the centre of P_n (M) that is reminiscent of the combing operation for the Artin pure braid groups, as well as decompositions obtained in [GG5].