Betti number

== English == === Etymology === A calque of French nombre de Betti, coined in 1892 by Henri Poincaré; named after Italian mathematician Enrico Betti in recognition of an 1871 paper. === Noun === Betti number (plural Betti numbers) (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; (formally) the rank of the nth homology group, Hn, of K. 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 {\displaystyle r} . For different choices of degrees and generic weights, the moduli spaces of parabolic Higgs bundles have the same Betti numbers. ==== 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 b n {\displaystyle b_{n}} represents the maximum number of cuts needed to separate K into two pieces ( n {\displaystyle n} -cycles). b 0 {\displaystyle b_{0}} can be interpreted as the number of components in K {\displaystyle K} . ==== Translations ==== === See also === Poincaré polynomial === Further reading === Homology (mathematics) on Wikipedia.Wikipedia Euler characteristic on Wikipedia.Wikipedia Topological data analysis on Wikipedia.Wikipedia Betti number on Encyclopedia of Mathematics Betti number on nLab Betti Number on Wolfram MathWorld