We compute the extensions of the Euler-Satake characteristic of a closed, effective, orientable 3-orbifold corresponding to free and free abelian groups in terms of the number and type of point singularities of the orbifold. Using these computations, we show that the free Euler-Satake characteristics determine the number and type of point singularities, and that it takes an infinite collection of free Euler-Satake characteristics to do so. Additionally, we show that the stringy orbifold Euler characteristic determines all of the free abelian Euler-Satake characteristics for an orbifold in this class.
"Extensions of the Euler-Satake characteristic determine point singularities of orientable 3-orbifolds." Kodai Math. J. 36 (1) 179 - 188, March 2013. https://doi.org/10.2996/kmj/1364562729