Equivalence of two kinds of orbifold Euler characteristic

摘要

Orbifold is a generalization of manifold. There are three definitions of the orbifold Euler characteristic: the alternating sum of inverses of the orders of isotropy subgroups of simplices, the alternating sum of the indices of some vector field at its isolated singularities, and the alternating sum of multiplicities of some transversal multisection at its regular zero points. Satake has shown that the first two numbers are equal. In this note, we prove that the latter two are also equal, and this result is verified in the case of n-teardrop.