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Nonvanishing vector fields on orbifolds

We introduce a complete obstruction to the existence of nonvanishing vector fields on a closed orbifold Q. Motivated by the inertia orbifold, the space of multi-sectors, and the generalized orbifold Euler characteristics, we construct for each finitely generated group Γ an orbifold called the space of Γ-sectors of Q. The obstruction occurs as the Euler-Satake characteristics of the Γ-sectors for an appropriate choice of Γ; in the case that Q is oriented, this obstruction is expressed as a cohomology class, the Γ-Euler-Satake class. We also acquire a complete obstruction in the case that Q is compact with boundary and in the case that Q is an open suborbifold of a closed orbifold.