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首页> 外文期刊>Topology and its applications >The braid group B_(n,m)(RP~2) and the splitting problem of the generalised Fadell-Neuwirth short exact sequence
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The braid group B_(n,m)(RP~2) and the splitting problem of the generalised Fadell-Neuwirth short exact sequence

机译：The braid group B_(n,m)(RP~2) and the splitting problem of the generalised Fadell-Neuwirth short exact sequence

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Let n,m ∈ N, and let B_(n,m)(RP~2) be the set of (n + m)-braids of the projective plane whose associated permutation lies in the subgroup S_n × S_m of the symmetric group S_(n+m). We study the splitting problem of the following generalisation of the Fadell-Neuwirth short exact sequence: 1 → B_m(RP~2 {x_1,...,x_n}) → B_(n,m)(RP~2) q → B_n(RP~2) →1, where the map q can be considered geometrically as the epimorphism that forgets the last m strands, as well as the existence of a section of the corresponding fibration q : F_(n+m)(RP~2)/S_n × S_m → F_n(RP~2)/S_n, where we denote by F_n(RP~2) the nth ordered configuration space of the projective plane RP~2. Our main results are the following: if n = 1 the homomorphism q and the corresponding fibration q admits no section, while if n = 2, then q and q admit a section. For n ≥ 3, we show that if q and q admit a section then m = 0, (n - 1)~2 mod n(n-1). Moreover, using geometric constructions, we show that the homomorphism q and the fibration q admit a section for m = kn(2n - 1)(2n - 2), where k ≥ 1, and for m = 2n(n - 1). In addition, we show that for m ≥ 3, B_m(RP~2{x_1,...,x_n}) is not residually nilpotent and for m ≥ 5, it is not residually solvable.

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