Orbit configuration spaces and the homotopy groups of the pair (∏1nM,Fn(M)) for M either S2 or ℝP2

Let n ≥ 1, and let ι n : F n ( M ) → ∏ 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 ∏ 1 n M for all k ≥ 1, where M is the 2-sphere S 2 or the real projective plane ℝ P 2 .It is well known that the group π k ( I n ) is the homotopy group π k + 1 ( ∏ 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 = S 2 , where it is anti-diagonal. We then show that I n has the homotopy type of K ( R n − 1 , 1 ) × Ω ( ∏ 1 n − 1 S 2 ) , where R n −1 is the ( n − 1) th Artin pure braid group if M = S 2 , and is the fundamental group G n −1 of the ( n −1) th orbit configuration space of the open cylinder S 2 \ { z ˜ 0 , − z ˜ 0 } with respect to the action of the antipodal map of S 2 if M = ℝ P 2 , where z ˜ 0 ∈ S 2 . This enables us to describe the long exact sequence in homotopy of the homotopy fibration I n → F n ( M ) → ι n ∏ 1 n M in geometric terms, and notably the image of the boundary homomorphism π k + 1 ( ∏ 1 n M ) → π k ( I n ) . From this, if M = S 2 and n ≥ 3 (resp. M = ℝ P 2 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 ].