Betti number

In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some points onward (Betti numbers vanish above the dimension of a space), and they are all finite.

The n^th Betti number represents the rank of the n^thhomology group, denoted H_n, which tells us the maximum amount of cuts that must be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. These numbers are used today in fields such as simplicial homology, computer science, digital images, etc.

The term "Betti numbers" was coined by Henri Poincaré after Enrico Betti.

Definitions

Informally, the kth Betti number refers to the number of k-dimensional holes on a topological surface. The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional simplicial complexes: