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

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 $\Gamma$.