Betti Number

A nonnegative integer equal to the rank of a homology group in a given dimension; it counts independent k-dimensional cycles modulo boundaries (the rank over Z or over a chosen coefficient field).

A numerical invariant equal to the rank (dimension over a field) of a homology or cohomology group in a given degree, or equivalently the rank of a free module appearing in a minimal resolution; it counts independent cycles or relations in that degree.