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 ofsimplexes 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-cellsin $K$. 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 shownthat all triangulations of $X$ have the same value for $\chi (K)$ so thatthis is well-defined.

Finally, if $C=\{C_q\}$ is a finitely generated graded group, thenthe Euler characteristic of $C$ is defined to be

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