Euler class obstruction to a nowhere-zero section

Statement

Let $E\to X$ be an oriented real rank- $n$ topological vector bundle with $n\ge2$ over a CW complex. If $E$ admits a continuous section $s:X\to E$ satisfying $\pi s=\operatorname{id}_X$ and $s(x)\ne0_x$ for all $x$ , then

e(E)=0\in H^n(X;\mathbb Z).

Here $e(E)$ is the integral Euler class. More precisely, $e(E)$ is the primary obstruction to a section of the unit sphere bundle $S(E)=\{v:\|v\|=1\}\to X$ for any continuous positive-definite fiber metric (the topological unit sphere is defined by the displayed norm equation). Consequently, when $\dim X\leq n$ , the Euler class is the complete obstruction: such a section exists if and only if $e(E)=0$ . For bases of dimension greater than $n$ , higher obstruction classes may remain after $e(E)$ vanishes.

Obstruction-theoretic mechanism

A continuous positive-definite fiber metric turns a nowhere-zero section into a section of the unit sphere bundle $S(E)$ , whose fiber is $S^{n-1}$ . Since $S^{n-1}$ is $(n-2)$ -connected, a section extends through the $(n-1)$ -skeleton. The first possible obstruction lies in degree $n$ , with coefficients $\pi_{n-1}(S^{n-1})\cong\mathbb Z$ . The orientation trivializes this local coefficient system, and the resulting class is $e(E)$ . There are no cells on which a higher obstruction could live when $\dim X\le n$ .

Geometric interpretation

For a smooth bundle over a closed oriented smooth base, a smooth section transverse to the zero section has a zero set representing the Poincaré dual of $e(E)$ . When the base and bundle both have dimension $n$ , the signed count of isolated zeros equals

\langle e(E),[X]\rangle.

Applied to $E=TM$ on a closed oriented manifold, this is the Poincaré–Hopf theorem: the total index of a vector field is $\chi(M)$ . Thus $S^{2m}$ , for $m\ge1$ , has no nowhere-zero tangent vector field.

Rank zero and rank one

An oriented real line bundle over a CW complex is trivial, so it has a nowhere-zero section and Euler class zero in every base dimension. The displayed obstruction argument using $\pi_{n-1}(S^{n-1})\cong\mathbb Z$ is for $n\ge2$ . A rank-zero bundle over a nonempty base has no nowhere-zero section and has Euler class $1\in H^0(X;\mathbb Z)$ .

Scope and near-misses

For a nonorientable bundle, the primary obstruction lives in cohomology with the orientation local system rather than ordinary integral cohomology. A section that vanishes somewhere is not a counterexample: the required condition is nowhere-zero, not merely nonzero as an element of the section module.

References

John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974. DOI record. Relevant: Euler classes, zero sections, and the obstruction interpretation.
Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: sphere bundles, cross-sections, and obstruction theory for vector bundles.