Betti number

plural Betti numbers

1. algebraic topologyAny of a sequence of numbers, denoted bₙ, which characterise a given topological space K by giving, for each dimension, the number of holes in K of said dimension; (formally) the rank of the nth homology group, Hₙ, of K.

   Poincaré proved that Betti numbers are invariants and used them to extend Euler's polyhedral formula to higher dimensional spaces.

   3 more examples

   1979 [W. H. Freeman & Company], Michael Henle, A Combinatorial Introduction to Topology, 1994, Dover, page 163, Prove that, for compact surfaces, the zeroth Betti number is the number of components of the surface, where a component is a connected subset of the surface, such that any larger containing subset is not connected.

   2007, Oscar García-Prada, Peter Beier Gothen, Vicente Muñoz, Betti Numbers of the Moduli Space of Rank 3 Parabolic Higgs Bundles, American Mathematical Society, page 7, PROPOSITION 2.1. Fix the rank r. For different choices of degrees and generic weights, the moduli spaces of parabolic Higgs bundles have the same Betti numbers.

   The goal is to compute Betti numbers in 2D and 3D image partitions using the practical definition of Betti numbers. Thus, depending on the dimension of the topological map, we count the number of connected components, the number of tunnels, and the number of cavities to obtain the Betti numbers.[…]The number of connected components of region r in a 2D image partition is equal to the first Betti number b#95;0(r).

   2012, Guillaume Damiand, Alexandre Dupas, “12: Combinatorial Maps for Image Segmentation”, in Valentin E. Brimkov, Reneta P. Barneva, editors, Digital Geometry Algorithms, Springer, page 380