Konstantin Rybnikov - Oriented matroids from liftings and stresses

An oriented matroid is a combinatorial abstraction which naturally appears in the study of combinatorial and algebraic properties of hyperplane arrangements, convex polytopes, polyhedral scenes, directed graphs, Delaunay decompositions, lattice points, zonotopes, and other objects of discrete geometry.

A realization of vertices of an abstract incidence structure (V,F,I) in R^d gives rise to an oriented matroid of liftings. A polyhedral complex in R^d gives rise to oriented matroids of k-stresses, . For example, a 2-stress is simply a Maxwell stress on the framework formed by the 1-skeleton of the complex. Matroids from polyhedral scenes and stresses were studied by Crapo, Edmonds, Lovasz, White, Whiteley et al. When (V,F,I) is the incidence structure of a homology manifold realized in R^d, there is a natural relationship between dependent sets of the matroid of liftings and dependent sets of the matroid of d-stresses. We prove that the oriented matroid of liftings is isomorphic to the oriented matroid of d-stresses when , and investigate some properties of this matroid.