Betti number

English Etymology A calque of en fr nombre de Betti, coined in 1892 by Henri Poincaré; named after Italian mathematician Enrico Betti in recognition of an 1871 paper. Noun en topology algebraic topology Any 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; en formally the rank of the nth homology group, Hn, of K. en Poincaré proved that Betti numbers are invariants and used them to extend Euler's polyhedral formula to higher dimensional spaces. 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 . For different choices of degrees and generic weights, the moduli spaces of parabolic Higgs bundles have the same Betti numbers. en 2012 Guillaume Damiand; Alexandre Dupas 12: Combinatorial Maps for Image Segmentation Valentin E. Brimkov; Reneta P. Barneva Digital Geometry Algorithms Springer https://books.google.com.au/books?id=lFOQEr-L9pMC&pg=PA380&dq=%22Betti+number%22%7C%22Betti+numbers%22&hl=en&sa=X&ved=0ahUKEwiSgNvFltveAhWEGDQIHXGlAbgQ6AEIbTAN#v=onepage&q=%22Betti%20number%22%7C%22Betti%20numbers%22&f=false 380 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 in a 2D image partition is equal to the first Betti number . Usage notes The dimensionality of a hole (as used in the definition) is that of its enclosing boundary: a torus, for example, has a central 1-dimensional hole and a 2-dimensional hole (a "void" or "cavity") enclosed by its ring. Informally, the Betti number represents the maximum number of cuts needed to separate K into two pieces (-cycles). can be interpreted as the number of components in . Translations number that characterises a topological space with respect to a given dimension Danish: da Bettital n French: fr nombre de Betti m German: de Bettizahl f Italian: it numero di Betti m Polish: pl liczba Bettiego f See also en Poincaré polynomial Further reading Homology (mathematics) Euler characteristic Topological data analysis Betti number on Encyclopedia of Mathematics Betti number on NLab nLab Betti Number on MathWorld Wolfram MathWorld en eponyms