Fundamental group of Desargues configuration spaces (Q2859345)

The authors compute the fundamental groups of several spaces of labelled point configurations in \(\mathbb C \mathbb P ^n\) having restricted underlying incidence structure. Let \({\mathcal D}^{3,n}\) denote the space of labelled configurations of six distinct points on three concurrent non-coplanar lines in \(\mathbb C \mathbb P ^n\). Similarly, \({\mathcal D}_I^{3,n}\) is the space of configurations of six distinct points on three non-coplanar lines containing a fixed point \(I \in \mathbb C \mathbb P ^n\). Analogous spaces \({\mathcal D}^{2,n}\) and \({\mathcal D}_I^{2,n}\), in which the specified lines are coplanar, are also considered. These spaces are related to realization spaces of matroids and hyperplane arrangements. For instance, \({\mathcal D}^{3,n}\) can be identified with the space of realizations in \(\mathbb C \mathbb P ^n\) of seven-point, rank-four matroids having three three-point lines with a point in common; by duality and de-homogenization, \({\mathcal D}_I^{2,n}\) can be identified with the space of affine hyperplane arrangements in \({\mathbb C}^2\) consisting of three pairs of parallel hyperplanes.NEWLINENEWLINEThe fundamental groups of these spaces are computed using some natural fibrations along with elementary facts about the first and second homotopy groups of Grassmannians and of configuration spaces of two or three points in the plane or sphere. The arguments use only elementary techniques and are very explicit: loops, lifts, homotopies, and local trivializations are given in coordinates, based on natural geometric constructions. The nontrivial fundamental groups are: \(\pi_1({\mathcal D}_I^{2,2})\cong {\mathbb Z}^3\), \(\pi_1({\mathcal D}^{2,2})\cong \pi_1({\mathcal D}_I^{2,n})\cong {\mathbb Z}^2\), \(\pi_1({\mathcal D}_I^{3,3})\cong \pi_1({\mathcal D}^{2,n })\cong{\mathbb Z}\) for \(n\geq 3\), and \(\pi_1({\mathcal D}^{3,3})\cong {\mathbb Z}_4\). Generators are given by natural motions, such as full twists of pairs of points on the lines they span, or full twists of pairs of lines, and are also described explicitly by equations and illustrations.