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. Informally, the Betti number is the maximum number of cuts that can be made without dividing a surface into two separate pieces (Gardner 1984, pp. 9-10). Formally, the th Betti number is the rank of the th homology group of a topological space.

The first Betti number of a graph is commonly known as its circuit rank (or cyclomatic number).

The following table gives the Betti number of some common surfaces.

Let be the group 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 .

The Betti number of a finitely generated Abelian group is the (uniquely determined) number such that

where , ..., are finite cyclic groups (see Kronecker decomposition theorem).

The Betti numbers of a finitely generated module over a commutative Noetherian local unit ring are the minimal numbers for which there exists a long exact sequence

which is called a minimal free resolution of . The Betti numbers are uniquely determined by requiring that be the minimal number of generators of for all . These Betti numbers are defined in the same way for finitely generated positively graded -modules if is a polynomial ring over a field.