Topology of the intersections of quadrics II

Abstract

We study the topology of the affine real variety given by the intersection with the unit sphere of the zero set in $${\mathbb R}^n$$ of a pair of quadratic forms. We give a complete topological description of this variety in the generic case: when non-empty, it is always diffeomorphic to either the unit tangent bundle of a sphere, the product of two or three spheres, or the connected sum of an odd number of manifolds, each of them a product of two spheres. With this we complete the partial description given in (Lopez de Medrano, 1989). Starting with the cases described in that article and other elementary ones, the proofs are based on the study of three geometric operations that give new of these varieties from simpler ones. For each operation, the topology of the new variety can be described, under appropriate conditions, from that of the old one. The global structure of the proof consists in an elaborate partial ordering of all the cases, in such a way that at each step those conditions are satisfied.