## Extensions of the Euler-Satake characteristic for nonorientable 3-orbifolds and indistinguishable examples.(English)Zbl 1286.57025

Taking the point of view of an orbifold $$Q$$ as presented (as the orbit space) by a proper étale Lie groupoid $$\mathcal{G}$$, and given any finitely generated discrete group $$\Gamma$$, the orbifold of $$\Gamma$$-sectors $$\widetilde{Q}_{\Gamma}$$ of $$Q$$ is presented by the proper étale Lie groupoid $$\mathcal{G} \ltimes \text{HOM}(\Gamma, \mathcal{G})$$. The orbifold of $$\Gamma$$-sectors of $$Q$$ is a collection of orbifolds that contains $$Q$$ along with other orbifolds of differing dimensions each of which is some finite singular cover of a part of the singular locus of $$Q$$. The $$\Gamma$$-Euler-Satake characteristic $$\chi_{\Gamma}^{\text{ES}}(Q)$$ of $$Q$$, an orbifold invariant, is defined as the usual Euler-Satake characteristic $$\chi^{\text{ES}}(\widetilde{Q}_{\Gamma})$$ of the orbifold of $$\Gamma$$-sectors of $$Q$$.
In this paper, the authors build on previous work [R. Carroll and C. Seaton, Kodai Math. J. 36, No. 1, 179–188 (2013; Zbl 1272.57019)] to show that, for an arbitrary closed, effective 3-dimensional orbifold $$Q$$, $$\chi_{\Gamma}^{\text{ES}}(Q)$$ depends only on the number and type of the point singularities of $$Q$$ and Euler characteristic of the topological boundary of its underlying space, obviating the need to determine anything about the (generally difficult to calculate) 2-dimensional sectors of $$Q$$. They also compute $$\chi_{\mathbb{F}_{l}}^{\text{ES}}(Q)$$ for an arbitrary closed, effective 3-dimensional orbifold $$Q$$ with $$\Gamma = \mathbb{F}_{l}$$ the free group on $$l$$ generators. This computation is used to show that no collection of the $$\chi_{\mathbb{F}_{l}}^{\text{ES}}(Q)$$ can determine the point singularities of a closed, effective non-orientable 3-orbifold $$Q$$. This result is in contrast to the orientable case, in which any infinite collection of the $$\chi_{\mathbb{F}_{l}}^{\text{ES}}$$ determines the number and type of the point singularities [ibid.], and the authors conclude the paper with a construction of two distinct closed, effective 3-orbifolds whose $$\Gamma$$-Euler-Satake characteristics coincide for all $$\Gamma$$.

### MSC:

 57R18 Topology and geometry of orbifolds 57R20 Characteristic classes and numbers in differential topology 22A22 Topological groupoids (including differentiable and Lie groupoids) 57S17 Finite transformation groups

Zbl 1272.57019
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