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# 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

$\chi(C)=\sum_{{q\geq 0}}(-1)^{q}rank(C_{q}).$ |

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Reference

## Mathematics Subject Classification

55N99*no label found*

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