Arrangements of Spheres and Projective Spaces

We develop the theory of arrangements of spheres. Consider a finite collection of codimension-$1$ subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of intersections. We also associate a topological space: the complement of the union of tangent bundles of these subspheres in the tangent bundle of the ambient sphere. We call this space the tangent bundle complement. As in the case of hyperplane arrangements the aim of this new notion is to understand the interaction between the combinatorics of the intersections and the topology of the tangent bundle complement. In the present paper we find a closed form formula for the homotopy type of the complement and express some of its topological invariants in terms of the associated combinatorial information.