Betti numbers are topological objects which were proved to be invariants by Poincaré, and used by him to extend the Polyhedral Formula to higher dimensional spaces. The th Betti number is the rank of the th Homology Group. Let be the Rank of the Homology Group of a Topological Space . For a closed, orientable surface of Genus , the Betti numbers are , , and . For a nonorientable surface with Cross-Caps, the Betti numbers are , , and .
See also Euler Characteristic, Poincaré Duality