Nonvanishing vector fields on orbifolds

TL;DR

This paper introduces a new obstruction, based on Euler-Satake characteristics of $ ext{Gamma}$-sectors, to determine when a closed orbifold admits a nonvanishing vector field, extending to orbifolds with boundary and open suborbifolds.

Contribution

It constructs the space of $ ext{Gamma}$-sectors for orbifolds and defines a cohomological obstruction, the $ ext{Gamma}$-Euler-Satake class, providing a complete criterion for nonvanishing vector fields.

Findings

Obstruction expressed via Euler-Satake characteristics of $ ext{Gamma}$-sectors.

Complete obstruction for orbifolds with boundary.

Extension of obstruction to open suborbifolds.

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.