# Index of a singular point of a vector field or of a 1-form on an orbifold

## TL;DR

This paper introduces a universal index for isolated singular points of vector fields or 1-forms on orbifolds, extending classical index theory and establishing an orbifold analogue of the Poincaré-Hopf theorem.

## Contribution

It defines the universal index of singular points on orbifolds as an element of a ring, generalizing classical indices and proving an orbifold version of the Poincaré-Hopf theorem.

## Key findings

- Universal index takes values in a ring generated by finite group classes
- An orbifold Poincaré-Hopf theorem analogue is established
- The index generalizes classical singular point indices to orbifolds

## Abstract

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincar\'e--Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related with corresponding analogues of indices of singular points. Earlier there was defined a notion of the universal Euler characteristic of an orbifold. It takes values in a ring R, as an abelian group freely generated by the generators, corresponding to the isomorphism classes of finite groups. Here we define the universal index of an isolated singular point of a vector field or of a 1-form on an orbifold as an element of the ring R. For this index, an analogue of the Poincar\'e-Hopf theorem holds.