Cyclic polytopes, oriented matroids and intersections of quadrics

Let M be a manifold that is given by the intersection of k quadrics in Rn with the unit sphere, such that it is symmetric with respect to the n coordinate hyperplanes. Let P be the quotient of this manifold by the action of Z2n (as group of reflections). P is a simple polytope and M is determined by P, so the homology groups of M are determined by the combinatorial structure of P. And P is associated to an oriented matroid. In this work, we explore the relation between the topes of this oriented matroid and the topology of M. Then we consider the case in which P is the dual polytope of a cyclic polytope Q. When k= 3 , we prove that M is a connected sum of sphere products, including the four-dimensional case. Finally, we calculate the Betti numbers of M by means of a cell decomposition of the plane associated to the oriented matroid, looking for new bridges between different areas of the mathematics. © 2016, Sociedad Matemática Mexicana.