Editing Betti number (section)

{{Short description|Roughly, the number of k-dimensional holes on a topological surface}}
In [[algebraic topology]], the '''Betti numbers''' are used to distinguish [[topological space]]s based on the connectivity of ''n''-dimensional [[simplicial complex]]es. For the most reasonable finite-dimensional [[topological space|space]]s (such as [[compact manifold]]s, finite [[simplicial complexes]] or [[CW complexes]]), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite.

The ''n''th Betti number represents the [[Rank of a group|rank]] of the ''n''th [[homology group]], denoted ''H''<sub>''n''</sub>, which tells us the maximum number of cuts that can be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc.<ref>{{cite web|last=Barile, and Weisstein|first=Margherita and Eric|title=Betti number|url=http://mathworld.wolfram.com/BettiNumber.html|publisher=From MathWorld--A Wolfram Web Resource.}}</ref> For example, if <math>H_n(X) \cong 0</math> then <math>b_n(X) = 0</math>, if <math>H_n(X) \cong \mathbb{Z}</math> then <math>b_n(X) = 1</math>, if <math>H_n(X) \cong \mathbb{Z} \oplus \mathbb{Z}</math> then <math>b_n(X) = 2</math>, if <math>H_n(X) \cong \mathbb{Z} \oplus \mathbb{Z}\oplus \mathbb{Z}</math> then <math>b_n(X) = 3</math>, etc. Note that only the ranks of infinite groups are considered, so for example if  <math>H_n(X) \cong \mathbb{Z}^k \oplus \mathbb{Z}/(2)</math>, where <math>\mathbb{Z}/(2)</math> is the [[finite cyclic group]] of order 2, then <math>b_n(X) = k</math>. These finite components of the homology groups are their [[torsion subgroup]]s, and they are denoted by '''torsion coefficients'''.

The term "Betti number" was coined by [[Henri Poincaré]] after [[Enrico Betti]]. The modern formulation is due to [[Emmy Noether#Second epoch (1920–1926): Contributions to topology|Emmy Noether]]. Betti numbers are used today in fields such as [[simplicial homology]], [[computer science]] and [[digital images]].