A HOMOTOPY THEOREM ON ORIENTED MATROIDS

Consider a finite family of hyperplanes H = {H-1,...,H(n)} in the finite-dimensional vector space R(d). We call chambers (determined by H) the connected components of R(d)\union i=1(n) H(i). Galleries are finite families of chambers (C0, C1,...,C(m)), where exactly one hyperplane separates C(i+1) from C(i), for O less-than-or-equal-to m, and exactly m hyperplanes separate C0 from C(m). Using oriented matroid theory, we prove that any two galleries with the same extremities can be derived from each other by a finite number of deformations of the same kind (elementary deformations). When the chambers are simplicial cones, this is a result of Deligne (1972). Our theorem generalizes also a result of Salvetti (1987).