Betti number

In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, 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 the n^th homology group, denoted H_n, 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.^[1] For example, if ${\displaystyle H_{n}(X)\cong 0}$ then ${\displaystyle b_{n}(X)=0}$, if ${\displaystyle H_{n}(X)\cong \mathbb {Z} }$ then ${\displaystyle b_{n}(X)=1}$, if ${\displaystyle H_{n}(X)\cong \mathbb {Z} \oplus \mathbb {Z} }$ then ${\displaystyle b_{n}(X)=2}$, if ${\displaystyle H_{n}(X)\cong \mathbb {Z} \oplus \mathbb {Z} \oplus \mathbb {Z} }$ then ${\displaystyle b_{n}(X)=3}$, etc. Note that only the ranks of infinite groups are considered, so for example if ${\displaystyle H_{n}(X)\cong \mathbb {Z} ^{k}\oplus \mathbb {Z} /(2)}$, where ${\displaystyle \mathbb {Z} /(2)}$ is the finite cyclic group of order 2, then ${\displaystyle b_{n}(X)=k}$. These finite components of the homology groups are their torsion subgroups, and they are denoted by torsion coefficients.

The term "Betti numbers" was coined by Henri Poincaré after Enrico Betti. The modern formulation is due to Emmy Noether. Betti numbers are used today in fields such as simplicial homology, computer science and digital images.