#### Vol. 6, No. 3, 2013

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Extensions of the Euler–Satake characteristic for nonorientable $3$-orbifolds and indistinguishable examples

### Ryan Carroll and Christopher Seaton

Vol. 6 (2013), No. 3, 345–368
##### Abstract

We compute the ${\mathbb{F}}_{\ell }$-Euler–Satake characteristics of an arbitrary closed, effective $3$-dimensional orbifold $Q$ where ${\mathbb{F}}_{\ell }$ is a free group with $\ell$ generators. We focus on the case of nonorientable orbifolds, extending previous results for the case of orientable orbifolds. Using these computations, we determine examples of distinct $3$-orbifolds $Q$ and ${Q}^{\prime }$ such that ${\chi }_{\Gamma }^{ES}\left(Q\right)={\chi }_{\Gamma }^{ES}\left({Q}^{\prime }\right)$ for every finitely generated discrete group $\Gamma$.

##### Keywords
orbifold, $3$-orbifold, Euler–Satake characteristic, orbifold Euler characteristic
##### Mathematical Subject Classification 2010
Primary: 57R18, 57R20
Secondary: 22A22, 57S17