Extensions of the Euler-Satake characteristic determine point singularities of orientable 3-orbifolds (Q1951674)

This paper discusses the extensions of the Euler-Satake characteristics of orbifolds. For a finitely generated discrete group \(\Gamma\), the \(\Gamma\)-extensions of the Euler-Satake characteristic, or the \(\Gamma\)-Euler-Satake characteristic, of a closed, effective, orientable 3-orbifold \(Q\) is defined to be the Euler-Satake characteristic of the associated orbifold \(\tilde Q_\Gamma\) of \(\Gamma\)-sectors of \(Q\), see \textit{C. Farsi} and \textit{C. Seaton} [Algebr. Geom. Topol. 11, No. 1, 523--551 (2011 Zbl 1213.22005)] for details about the definition of \(\Gamma\)-Euler-Satake characteristic. In this paper, for nonnegative integer \(l\), the authors compute \(\mathbf Z^l\)- and \(F_l\)-Euler-Satake characteristics of a closed, effective, orientable 3-orbifold, and show that the first type characteristics are determined by the stringy orbifold Euler characteristics, and the last type determine totally the numbers and types of point singularities of \(Q\).