Betti number

English

Etymology

A calque of Template:Der, coined in 1892 by Template:W; named after Italian mathematician Template:W in recognition of an 1871 paper.

Noun

1. Template:Lb 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; Template:Lb the rank of the nth homology group, H_n, of K.

   Template:Ux

   • 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, Template:W, 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.

   • Template:Quote-book

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

• Danish: Template:T
• French: Template:T
• German: Template:T
• Italian: Template:T

See also

• Template:L

Further reading

• Template:Pedia
• Template:Pedia
• Template:Pedia
• Betti number on Template:W
• Betti number on Template:W
• Betti Number on Template:W