Nonvanishing vector fields on orbifolds

Abstract

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 $\Gamma$ an orbifold called the space of $\Gamma$-sectors of $Q$. The obstruction occurs as the Euler-Satake characteristics of the $\Gamma$-sectors for an appropriate choice of $\Gamma$; in the case that $Q$ is oriented, this obstruction is expressed as a cohomology class, the $\Gamma$-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.

28 pages; edited for clearer exposition, fixed Example 2.12, added Example 4.6

References in corpus (6)

• Orbifolds as Groupoids: an Introduction
• Stringy Geometry and Topology of Orbifolds
• Characteristic Classes of Bad Orbifold Vector Bundles
• Orbifolds and stable homotopy groups
• A Complete Obstruction to the Existence of Nonvanishing Vector Fields on Almost-Complex, Closed, Cyclic Orbifolds
• The index of a vector field on an orbifold with boundary

Cited by in corpus (5)

• Generalized twisted sectors of orbifolds
• Generalized orbifold Euler characteristics for general orbifolds and wreath products
• Classifying closed 2-orbifolds with Euler characteristics
• Orbifold Euler characteristics of non-orbifold groupoids
• Functional equations for orbifold wreath products