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The term Euler characteristic is defined for several objects.
If is a finite simplicial complex of dimension , let be the number of simplexes of dimension . The Euler characteristic of is defined to be
Next, if is a finite CW complex, let be the number of i-cells in . The Euler characteristic of is defined to be
If is a finite polyhedron, with triangulation , a simplicial complex, then the Euler characteristic of is . It can be shown that all triangulations of have the same value for so that this is well-defined.
Finally, if is a finitely generated graded group, then the Euler characteristic of is defined to be
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