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