A 5-polytope is a closed five-dimensional figure with vertices, edges, faces, and cells, and 4-faces.
Furthermore, the following requirements must be met: The topology of any given 5-polytope is defined by its Betti numbers and torsion coefficients.
This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.
[1] Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.
Regular 5-polytopes can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} polychoral facets around each face.