The index of a vector field on an orbifold with boundary

TL;DR

This paper establishes a Poincaré-Hopf type theorem for compact orbifolds with boundary, linking vector field indices to orbifold Euler characteristics and boundary contributions.

Contribution

It extends the Poincaré-Hopf theorem to orbifolds with boundary, incorporating boundary orbifold terms and inertia orbifold index relations.

Findings

Proves a Poincaré-Hopf theorem for orbifolds with boundary.

Expresses boundary contributions via Euler characteristics of orbifold tangency and exit regions.

Relates the index sum on inertia orbifolds to underlying topological space characteristics.

Abstract

A Poincar\'{e}-Hopf theorem in the spirit of Pugh is proven for compact orbifolds with boundary. The theorem relates the index sum of a smooth vector field in generic contact with the boundary orbifold to the Euler-Satake characteristic of the orbifold and a boundary term. The boundary term is expressed as a sum of Euler characteristics of tangency and exit-region orbifolds. As a corollary, we express the index sum of the vector field induced on the inertia orbifold to the Euler characteristics of the associated underlying topological spaces.