  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.
