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].

设n ≥ 1，且令\({\iota _n}:{F_n}(M) \to \prod\nolimits_1^n M \)为M在n重笛卡尔中的第 n^个配置空间的自然包含M与其自身的乘积。在本文中，我们研究了映射ι _n、ι _n的同伦纤维I _n及其同伦群，^以及_F n ₍ M )_和\ ( \ prod\nolimits_1^n M \)对于所有k ≥ 1，其中M是 2-球面\({\mathbb{S}^2}\)或实射影平面 ℝ P ²。众所周知，该群π _k ( I _n ) 是所有k ≥ 0的同伦群\({\pi _{k + 1}}(\prod\nolimits_1^n {M,{F_n}} (M))\)。如果k ≥ 2，我们证明同态 ( ι _n ⁾ _#k是单射且对角的，但n = k = 2 和\(M = {\mathbb{S}^2}\)的情况除外，其中反对角线。然后我们证明I _n的同伦类型为\(K({R_{n - 1}},1) \times \Omega (\prod\nolimits_1^{n - 1} {{\mathbb{S}^2 }} )\)，其中R _{n −1}是第 ( n − 1)^个Artin 纯编织群 if \(M = {\mathbb{S}^2}\)，并且是基本群G _{n −1}开圆柱体的第 ( n −1)^{个轨道配置空间}\({\mathbb{S}^2}\backslash \{ {\widetilde z_0}, - {\widetilde z_0}\} \)相对于作用如果M = ℝ P ^{2 ，则}\({\mathbb{S}^2}\)的对映图，其中\({\widetilde z_0} \in {\mathbb{S}^2}\)。这使我们能够描述同伦纤维的长精确序列\({I_n} \to {F_n}(M)\buildrel {{\iota _n}} \over\longrightarrow \prod\nolimits_1^n M \)用几何术语来说，特别是边界同态的图像\({\pi _{k + 1}}(\prod\nolimits_1^n M ) \to {\pi _k}({I_n})\)。由此看来，如果\(M = {\mathbb{S}^2}\)且n ≥ 3 （分别为M = ℝ P ²且n ≥ 2），我们证明 Ker(( ι _n ) _#1 ) 与商同构R _{n −1}乘以其中心的平方，以及自由群与由P _n ( M ) 中心生成的 2 阶子群的迭代半直积，这让人想起了Artin纯编织组，以及[GG5]中获得的分解。