# 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

 $\chi(K)=\sum_{i\geq 0}(-1)^{i}\alpha_{i}.$

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

 $\chi(C)=\sum_{q\geq 0}(-1)^{q}rank(C_{q}).$
Title Euler characteristic EulerCharacteristic 2013-03-22 16:12:51 2013-03-22 16:12:51 Mathprof (13753) Mathprof (13753) 13 Mathprof (13753) Definition msc 55N99