A generalized Desargues configuration and the pure braid group
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.
scientific article; zbMATH DE number 955148
scientific article