A generalized Desargues configuration and the pure braid group
type: Item
instance of: scholarly article
title: A generalized Desargues configuration and the pure braid group
zbMATH Open document ID: 0871.52010
DOI: 10.1016/0012-365X(95)00152-M
author: Raul Cordovil
author: António Guedes de Oliveira
author: Michel Las Vergnas
published in: Discrete Mathematics
publication date: 1997-01-14T00:00:00Z
review text: For a hyperplane arrangement in \(\mathbb{R}^d\), it is of interest to examine the topology of the open manifold \({\mathbf M}\) obtained by complexifying the complement of the arrangement. Extending a result of Salvetti, the first two authors showed in 1992 how to compute the fundamental group of \({\mathbf M}\) using the oriented matroid associated with the hyperplane arrangement. This oriented matroid, in fact, contains enough information to reconstruct the entire topology of \({\mathbf M}\); but the fundamental group may be calculated using a rank 3 truncation of the oriented matroid. For the arrangement \({\mathcal A}_d\) consisting of the hyperplanes \(H_{ij}= \{(x_1, \dots, x_d): x_1=x_j\}\), the fundamental group of \({\mathbf M} ({\mathcal A}_d)\) is the pure braid group on \(d\) strings, \(P_d\). The associated oriented matroid is the cycle matroid of the complete graph \(K_d\) on \(d\) points (with edges oriented as a total ordering on the points). In this paper, the authors construct a family \(\{{\mathcal D}_d: d=4,5, \dots\}\) of configurations of points and lines in the plane, which generalizes the well-known Desargues configuration \({\mathcal D}_5\). These are rank 3 truncations of \(K_d\), and thus allow a presentation of \(P_d\) to be computed in an elegant way, as the fundamental group of \({\mathbf M} ({\mathcal A}_d)\). The presentation obtained is similar, though not identical, to the usual one.
reviewed by: Robert Dawson
Mathematics Subject Classification ID: 52B40
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 05B30
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 20F36
zbMATH DE Number: 955148
zbMATH Keywords: hyperplane arrangement
zbMATH Keywords: pure braid group
zbMATH Keywords: oriented matroid
zbMATH Keywords: Desargues configuration
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 955148