Betti numbers
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 point onward (Betti numbers vanish above the dimension of a space), and they are all finite.
The nth Betti number represents the rank of the nthhomology group, denoted Hn, 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.
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:
Thus, for example, a torus has one connected surface component so b0 = 1, a circular (two-dimensional) hole at each end of the central space so b1 = 2 and a single cavity enclosed within the surface so b2 = 1.
Closely related to the Betti numbers of a topological surface is the Poincaré polynomial of that surface. The Poincaré polynomial of a surface is defined to be the generating function of its Betti numbers. For example, the Betti numbers of the torus are 1, 2, and 1; thus its Poincaré polynomial is . The same definition applies to any topological space which has a finitely generated homology.
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