Editing Betti number (section)

== Examples ==

=== Betti numbers of a graph ===
Consider a [[Topological graph theory|topological graph]] ''G'' in which the set of vertices is ''V'', the set of edges is ''E'', and the set of connected components is ''C''. As explained in the page on [[graph homology]], its homology groups are given by:
: <math>H_k(G) = \begin{cases} 
  \mathbb Z^{|C|}         & k=0 \\
  \mathbb Z^{|E|+|C|-|V|} & k=1 \\
  \{0\}                   & \text{otherwise}
\end{cases}</math>

This may be proved straightforwardly by [[mathematical induction]] on the number of edges. A new edge either increments the number of 1-cycles or decrements the number of connected components.

Therefore, the "zero-th" Betti number ''b''<sub>0</sub>(''G'') equals |''C''|, which is simply the number of connected components.<ref name="Hage1996">{{cite book|author=Per Hage|url=https://books.google.com/books?id=ZBdLknuP0BYC&pg=PA49|title=Island Networks: Communication, Kinship, and Classification Structures in Oceania|publisher=Cambridge University Press|year=1996|isbn=978-0-521-55232-5|page=49}}</ref>

The first Betti number ''b''<sub>1</sub>(''G'') equals |''E''| + |''C''| - |''V''|. It is also called the [[cyclomatic number]]—a term introduced by [[Gustav Kirchhoff]] before Betti's paper.<ref name="Kotiuga2010">{{cite book|author=Peter Robert Kotiuga|url=https://books.google.com/books?id=mqLXi0FRIZwC&pg=PA20|title=A Celebration of the Mathematical Legacy of Raoul Bott|publisher=American Mathematical Soc.|year=2010|isbn=978-0-8218-8381-5|page=20}}</ref> See [[cyclomatic complexity]] for an application to [[software engineering]].

All other Betti numbers are 0.

=== Betti numbers of a simplicial complex ===
[[File:Simplicialexample.png|160x320px|alt=Example|right]]
Consider a [[simplicial complex]] with 0-simplices: a, b, c, and d, 1-simplices: E, F, G, H and I, and the only 2-simplex is J, which is the shaded region in the figure. There is one connected component in this figure (''b''<sub>0</sub>); one hole, which is the unshaded region (''b''<sub>1</sub>); and no "voids" or "cavities" (''b''<sub>2</sub>).

This means that the rank of <math>H_0</math> is 1, the rank of <math>H_{1}</math> is 1 and the rank of <math>H_2</math> is 0.

The Betti number sequence for this figure is 1, 1, 0, 0, ...; the Poincaré polynomial is <math>1 + x\,</math>.

=== Betti numbers of the projective plane ===
The homology groups of the [[projective plane]] ''P'' are:
: <math>H_k(P) = \begin{cases} \mathbb Z & k=0 \\ \mathbb Z _ 2 & k=1 \\ \{0\} & \text{otherwise} \end{cases}</math>

Here, '''Z'''<sub>2</sub> is the [[cyclic group]] of order 2. The 0-th Betti number is again 1. However, the 1-st Betti number is 0. This is because ''H''<sub>1</sub>(''P'') is a finite group - it does not have any infinite component. The finite component of the group is called the '''torsion coefficient''' of ''P''. The (rational) Betti numbers ''b''<sub>''k''</sub>(''X'') do not take into account any [[torsion subgroup|torsion]] in the homology groups, but they are very useful basic topological invariants. In the most intuitive terms, they allow one to count the number of ''holes'' of different dimensions.