| Full text | |
| Author(s): |
Goncalves, Daciberg Lima
;
Guaschi, John
Total Authors: 2
|
| Document type: | Journal article |
| Source: | PACIFIC JOURNAL OF MATHEMATICS; v. 287, n. 1, p. 71-99, MAR 2017. |
| Web of Science Citations: | 2 |
| Abstract | |
Let S be a surface, perhaps with boundary, and either compact or with a finite number of points removed from the interior of the surface. We consider the inclusion iota : F-n(S) -> Pi(n)(1) S of the n-th configuration space F-n(S) of S into the n-fold Cartesian product of S, as well as the induced homomorphism iota(\#) : P-n(S) -> Pi(n)(1) pi(1)(S) where P-n(S) is the n-string pure braid group of S. Both iota and iota(\#) were studied initially by J. Birman, who conjectured that Ker (iota(\#)) is equal to the normal closure of the Artin pure braid group P-n in P-n(S) The conjecture was later proved by C. Goldberg for compact surfaces without boundary different from the 2-sphere S-2 and the projective plane RP2. In this paper, we prove the conjecture for S-2 and RP2. In the case of RP2, we prove that Ker (iota(\#)) is equal to the commutator subgroup of P-n(RP2), we show that it may be decomposed in a manner similar to that of P-n(S-2) as a direct sum of a torsion-free subgroup L-n and the finite cyclic group generated by the full twist braid, and we prove that L-n may be written as an iterated semidirect product of free groups. Finally, we show that the groups B-n(S-2) and P-n(S-2) (resp. B-n(RP2) and P-n(RP2)) have finite virtual cohomological dimension equal to n - 3 (resp. n - 2), where B-n(S) denotes the full n-string braid group of S-2 This allows us to determine the virtual cohomological dimension of the mapping class groups of S-2 and RP2 with marked points, which in the case of S-2 reproves a result due to J. Harer. (AU) | |
| FAPESP's process: | 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 |
| Grantee: | Daciberg Lima Gonçalves |
| Support Opportunities: | Regular Research Grants |