Is there a way to figure out if a 4-polytope is a tiling of the 3-sphere based on only the number of vertices, edges, faces, and cells? Here is a specific example. Say that there are two "polytopes" - one is a tesseract, and one is a disjoint union of two tesseracts. Both will have an Euler characteristic of 0, but single tesseract is homeomorphic to a 3-sphere and the pair of tesseracts is not. Is there a way to figure out this fact based only on the number of vertices, edges, faces, and cells in each polytope? In three dimensions the analogous problem could be easily solved because the single polyhedron would have an Euler characteristic of 2 and the disjoint union would have a characteristic of 4. I'm not quite sure my terminology is correct but I gave it a try.