On the second nilpotent quotient of higher homotopy groups, for hypersolvable arrangements
| Autor: | Macinic, Daniela Anca, Matei, Daniel, Papadima, Stefan |
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| Rok vydání: | 2013 |
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| Zdroj: | Int. Math. Res. Notices vol. 2015, no.24 (2015), 13194--13207 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1093/imrn/rnv080 |
| Popis: | We examine the first non-vanishing higher homotopy group, $\pi_p$, of the complement of a hypersolvable, non--supersolvable, complex hyperplane arrangement, as a module over the group ring of the fundamental group, $\Z\pi_1$. We give a presentation for the $I$--adic completion of $\pi_p$. We deduce that the second nilpotent $I$--adic quotient of $\pi_p$ is determined by the combinatorics of the arrangement, and we give a combinatorial formula for the second associated graded piece, $\gr^1_I \pi_p$. We relate the torsion of this graded piece to the dimensions of the minimal generating systems of the Orlik--Solomon ideal of the arrangement $\A$ in degree $p+2$, for various field coefficients. When $\A$ is associated to a finite simple graph, we show that $\gr^1_I \pi_p$ is torsion--free, with rank explicitly computable from the graph. Comment: 11 pages, updated references |
| Databáze: | arXiv |
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