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A nowhere-zero vector field forces zero Euler characteristic

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $M$ be a closed smooth $n$ -manifold, $n \geq 1$ , and suppose $M$ admits a smooth vector field $X$ with no zeros at all. Then $χ (M) = 0$ (Euler characteristic of a compact manifold).

Facts & Assumptions

Given: A closed smooth $n$ -manifold $M$ with a nowhere-zero smooth vector field $X$ .

[F1]

A vector field with no zeros has only isolated zeros, so Poincare-Hopf applies: $\sum_{p : X (p) = 0} {ind}_{p} X = χ (M)$ (Poincare-Hopf for closed manifolds, Isolated zero and local index of a vector field).

Proof

1.1F1algebra

The zero set of $X$ is empty by hypothesis, so it consists of isolated zeros vacuously and the hypothesis of [F1] is satisfied; the index sum $\sum_{p : X (p) = 0} {ind}_{p} X$ is the empty sum $0$ .

2.1F1step 1.1algebra∎

Poincare-Hopf [F1] now gives $χ (M) = \sum_{p : X (p) = 0} {ind}_{p} X = 0$ .

Depends on

Used by

The hairy-ball theorem for even spheres Example
Converse Poincare-Hopf for nowhere-zero fields Theorem

Dependency tree · two levels

36 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John W. Milnor, Topology from the Differentiable Viewpoint (complete 76-page PDF, including the appendix Classifying 1-manifolds) (standard reference, not scraped)
Victor Guillemin and Alan Pollack, Differential Topology (Prentice-Hall 1974; complete PDF) (standard reference, not scraped)