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