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)

1. (topology, algebraic topology) Any of a sequence of numbers, denoted b_n, 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, H_n, of K.

   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 $r$. For different choices of degrees and generic weights, the moduli spaces of parabolic Higgs bundles have the same Betti numbers.

   • 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:

     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 $b_{0}(r)$.

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 $b_{n}$ represents the maximum number of cuts needed to separate K into two pieces ($n$-cycles).
• $b_{0}$ can be interpreted as the number of components in $K$.

Translations

number that characterises a topological space with respect to a given dimension

• Danish: Bettital n
• French: nombre de Betti m
• German: Bettizahl f
• Italian: numero di Betti m

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