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A Poincaré-Hopf theorem for n-valued vector fields

M. C. Crabb

TL;DR

This work generalizes the Poincaré-Hopf theorem from single-valued vector fields to $n$-valued sections of real or complex vector bundles over closed manifolds. It introduces a degree theory for $n$-valued maps built on configuration-space methods and reduces $n$-valued sections of projective bundles to sphere- or lens-space data, enabling a unified index formula. The main theorem shows that the sum of local indices of an $n$-valued nowhere-zero section equals $n$ times the classical invariant $e(\xi)[M]$ (or $n$ times $w_m(\xi)[M]$ in the mod $2$ setting), with explicit complex- and real-geometry formulas. The approach covers real and complex projective bundles and lens-space variants, and connects to the fixed-point theory of $n$-valued maps, offering a broad framework for multivalued index theory in differential topology.

Abstract

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

A Poincaré-Hopf theorem for n-valued vector fields

TL;DR

This work generalizes the Poincaré-Hopf theorem from single-valued vector fields to -valued sections of real or complex vector bundles over closed manifolds. It introduces a degree theory for -valued maps built on configuration-space methods and reduces -valued sections of projective bundles to sphere- or lens-space data, enabling a unified index formula. The main theorem shows that the sum of local indices of an -valued nowhere-zero section equals times the classical invariant (or times in the mod setting), with explicit complex- and real-geometry formulas. The approach covers real and complex projective bundles and lens-space variants, and connects to the fixed-point theory of -valued maps, offering a broad framework for multivalued index theory in differential topology.

Abstract

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

Paper Structure

This paper contains 5 sections, 6 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Sections of real projective bundles
  3. Sections of complex projective bundles
  4. The degree of an $n$-valued map
  5. A Poincaré-Hopf theorem for $n$-valued sections of a vector bundle

Key Result

Theorem 1.1

(Poincaré-Hopf). Let $\xi$ be an $m$-dimensional real vector bundle over $M$. Suppose that $s$ is a section of $\xi$ which is non-zero outside a finite set $\Sigma$. (a). One can assign to each point of $\Sigma$ a local index of $s$ in ${\mathbb Z} /2{\mathbb Z}$, and the sum of the local indices is

Theorems & Definitions (21)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Definition 4.1
  • ...and 11 more