Generalized combinatorial cells and facet splitting

The similarity between triangulations of the sphere and simplicial polytopes makes cells with triangulated boundaries natural generalizations of simplicial polytopes. In this paper we extend this generalization to cells whose boundaries are broken up into more general structures than just simplices. These structures are called gee's. In doing so we get a generalization of the d-polytope. We shall investigate a method of constructing these structures, called facet splitting. We show that almost all d-gec's with up to 3 + d facets can be constructed by facet splitting, and we construct a simple 4-gcc with 10 facets that cannot be constructed in this way.