A Poincar\'e-Hopf theorem for n-valued vector fields

TL;DR

This paper extends the classical Poincaré-Hopf theorem to n-valued vector fields, providing a generalized topological invariant for sections of vector bundles over closed manifolds.

Contribution

It introduces a Poincaré-Hopf theorem for n-valued vector fields, broadening the scope of the classical theorem to multi-valued sections.

Findings

Generalization of Poincaré-Hopf theorem to n-valued sections

Interpretation of line field results as a special case

Framework applicable to closed manifolds of the same dimension

Abstract

The Poincar\'e-Hopf theorem for line fields, as described in a paper of Crowley and Grant, is interpreted as a special case of a Poincar\'e-Hopf theorem for $n$-valued sections of a vector bundle over a closed manifold of the same dimension.