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 \(\Gamma\).


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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