betti

Betti may refer to persons with the surname Betti or to mathematical concepts named after the Italian mathematician Enrico Betti. In mathematics, Betti numbers are a sequence of integers that describe the topological features of a space. For a topological space X with coefficients in a field, the k-th Betti number b_k is the rank of the k-th homology group H_k(X). Intuitively, b0 equals the number of connected components, b1 counts independent one-dimensional holes or tunnels, and b2 counts independent two-dimensional voids, and so on. The Betti numbers are invariants under homeomorphism, and form the Betti sequence that characterizes basic shape features across dimensions. They are central in algebraic topology and appear in various branches of geometry and data analysis, including persistent homology where they are tracked across scales.

In algebraic geometry and commutative algebra, a related concept appears as graded Betti numbers in minimal free resolutions, organizing the generators and relations of a graded module. The term "Betti" thus spans both topological invariants and algebraic invariants named after Enrico Betti.

The surname Betti is of Italian origin and is borne by individuals in various fields. The name's association with the mathematical concept is primarily historical, named to honor Betti's early contributions to topology.