# Euler characteristic

The term *Euler characteristic ^{}* is defined for several objects.

If $K$ is a finite simplicial complex^{} of dimension^{} $m$, let ${\alpha}_{i}$ be the number of
simplexes of dimension $i$. The *Euler characteristic* of $K$
is defined to be

$$\chi (K)=\sum _{i=0}^{m}{(-1)}^{i}{\alpha}_{i}.$$ |

Next, if $K$ is a finite CW complex, let ${\alpha}_{i}$ be the number of i-cells
in $K$. The *Euler characteristic* of $K$
is defined to be

$$\chi (K)=\sum _{i\ge 0}{(-1)}^{i}{\alpha}_{i}.$$ |

If $X$ is a finite polyhedron, with triangulation $K$, a simplicial complex,
then the *Euler characteristic* of $X$ is $\chi (K)$. It can be shown
that all triangulations of $X$ have the same value for $\chi (K)$ so that
this is well-defined.

Finally, if $C=\{{C}_{q}\}$ is a finitely generated^{} graded group, then
the *Euler characteristic* of $C$ is defined to be

$$\chi (C)=\sum _{q\ge 0}{(-1)}^{q}rank({C}_{q}).$$ |

Title | Euler characteristic |
---|---|

Canonical name | EulerCharacteristic |

Date of creation | 2013-03-22 16:12:51 |

Last modified on | 2013-03-22 16:12:51 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 13 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 55N99 |