Betti Number

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 $n$th Betti number is the rank of the $n$th Homology Group. Let $p_r$ be the Rank of the Homology Group $H_r$ of a Topological Space $K$. For a closed, orientable surface of Genus $g$, the Betti numbers are $p_0=1$, $p_1=2g$, and $p_2=1$. For a nonorientable surface with $k$ Cross-Caps, the Betti numbers are $p_0=1$, $p_1=k-1$, and $p_2=0$.

See also Euler Characteristic, Poincaré Duality