| Abstrakt: |
Let n ≥ 1, and let ι n : F n (M) → ∏ 1 n M be the natural inclusion of the nth configuration space of M in the n-fold Cartesian product of M with itself. In this paper, we study the map ιn, the homotopy fibre In of ιn and its homotopy groups, and the induced homomorphisms (ιn)#k on the kth homotopy groups of Fn(M) and ∏ 1 n M for all k ≥ 1, where M is the 2-sphere S 2 or the real projective plane ℝP2.It is well known that the group πk(In) 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 (ιn)#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 In has the homotopy type of K (R n − 1 , 1) × Ω (∏ 1 n − 1 S 2 ) , where Rn−1 is the (n − 1)th Artin pure braid group if M = S 2 , and is the fundamental group Gn−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 = ℝP2, 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 = ℝP2 and n ≥ 2), we show that Ker((ιn)#1) is isomorphic to the quotient of Rn−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 Pn (M) that is reminiscent of the combing operation for the Artin pure braid groups, as well as decompositions obtained in [GG5]. [ABSTRACT FROM AUTHOR] |
Full text from SpringerLink