Knowlpedia

Theorem

Euler class obstruction to a nowhere-zero section

A nowhere-zero section forces the Euler class to vanish, and in base dimension equal to the bundle rank this is the complete obstruction.

Statement

Let E→XE\to X be an oriented real rank-nn over a CW complex. If EE admits a nowhere-zero , then

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

More precisely, e(E)e(E) is the primary obstruction to a section of the unit sphere bundle S(E)→XS(E)\to X. Consequently, when dim⁡X≤n\dim X\leq n, the Euler class is the complete obstruction: such a section exists if and only if e(E)=0e(E)=0. For bases of dimension greater than nn, higher obstruction classes may remain after e(E)e(E) vanishes.

Obstruction-theoretic mechanism

A turns a into a section of the S(E)S(E), whose fiber is Sn−1S^{n-1}. Since Sn−1S^{n-1} is (n−2)(n-2)-connected, a section extends through the (n−1)(n-1)-skeleton. The first possible obstruction lies in degree nn, with coefficients πn−1(Sn−1)≅Z\pi_{n-1}(S^{n-1})\cong\mathbb Z. The orientation trivializes this local coefficient system, and the resulting class is e(E)e(E). There are no cells on which a higher obstruction could live when dim⁡X≤n\dim X\le n Milnor–Stasheff, discussion of the Euler obstruction.

Geometric interpretation

For a smooth section transverse to the , its zero set represents the Poincaré dual of e(E)e(E). When the base and bundle both have dimension nn, the signed count of isolated zeros equals

⟨e(E),[X]⟩.\langle e(E),[X]\rangle.

Applied to E=TME=TM on a closed oriented manifold, this is the Poincaré–Hopf theorem: the total index of a is χ(M)\chi(M). Thus the even sphere has no nowhere-zero tangent vector field.

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 .

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