| Texto completo | |
| Autor(es): |
Goncalves, Daciberg Lima
;
Guaschi, John
Número total de Autores: 2
|
| Tipo de documento: | Artigo Científico |
| Fonte: | Israel Journal of Mathematics; v. 261, n. 1, p. 53-pg., 2023-11-29. |
| Resumo | |
Let n >= 1, and let tau(n) : F-n(M) -> Pi(n)(1) 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 tau, the homotopy fibre I-n of tau(n) and its homotopy groups, and the induced homomorphisms (tau(n))(#k) on the kth homotopy groups of Fn(M) and Pi(n)(1) 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 pi(k)(I-n) is the homotopy group pi(k+1)(Pi(n)(1) M, F-n(M)) for all k >= 0. If k >= 2, we show that the homomorphism (tau(n)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 In has the homotopy type of K(Rn-1, 1) x Omega(Pi(n-1)(1) S-2), where Rn-1 is the (n - 1)th Artin pure braid group if M = S2, and is the fundamental group G(n-1) of the (n-1)th orbit configuration space of the open cylinder S-2 \{(z) over tilde (0),-(z) over tilde (0)} with respect to the action of the antipodal map of S-2 if M = RP2, where (z) over tilde (0) is an element of S-2. This enables us to describe the long exact sequence in homotopy of the homotopy fibration I-n -> F-n(M) in(sic) Pi(n)(1) in geometric terms, and notably the image of the boundary homomorphism pi(k+1)(Pi(n)(1) M) -> pi(k)(I-n). From this, if M = S-2 and n >= 3 (resp. M = RP2 and n >= 2), we show that Ker((tau(n))(#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 P-n(M) that is reminiscent of the combing operation for the Artin pure braid groups, as well as decompositions obtained in [GG5]. (AU) | |
| Processo FAPESP: | 14/50131-7 - Algebraic and topological properties of the braid groups of the real projective plane, sphere, disk, orbit configuration spaces, and relations with crystallographic groups |
| Beneficiário: | Daciberg Lima Gonçalves |
| Modalidade de apoio: | Auxílio à Pesquisa - Regular |
| Processo FAPESP: | 16/50354-1 - Generalisations of configuration spaces, relations between braid and almost-crystallographic groups, and applications to the study of the Borsuk-Ulam property and multi-valued maps |
| Beneficiário: | Daciberg Lima Gonçalves |
| Modalidade de apoio: | Auxílio à Pesquisa - Regular |
| Processo FAPESP: | 16/24707-4 - Topologia algébrica, geométrica e diferencial |
| Beneficiário: | Daciberg Lima Gonçalves |
| Modalidade de apoio: | Auxílio à Pesquisa - Temático |