Hello, you have come here looking for the meaning of the word
Betti number. In DICTIOUS you will not only get to know all the dictionary meanings for the word
Betti number, but we will also tell you about its etymology, its characteristics and you will know how to say
Betti number in singular and plural. Everything you need to know about the word
Betti number you have here. The definition of the word
Betti number will help you to be more precise and correct when speaking or writing your texts. Knowing the definition of
Betti number, as well as those of other words, enriches your vocabulary and provides you with more and better linguistic resources.
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.
Poincaré proved that Betti numbers are invariants and used them to extend Euler's polyhedral formula to higher dimensional spaces.
- 1979 , 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.
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 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
See also
Further reading