  There is a one-to-one correspondence between geometric lattices and the
intersection lattices of arrangements of homotopy spheres. When the
arrangements are essential and fully partitioned, Zaslavsky's enumeration of
the cells of the arrangement still holds. An application of the theory shows
that all minimal cellular resolutions of matroid Steiner ideals are bounded
subcomplexes of homotopy sphere arrangements of the given matroid. As a result
the Betti numbers of the ideal are computed and seen to be equivalent to
Stanley's formula in the special case of face ideals of independence complexes
of matroids.

Combinatorics;Commutative Algebra;Geometric Topology