Euler class as pullback of the Thom class

Let $\pi:E\to B$ be an oriented topological real vector bundle of rank $r$ over a paracompact Hausdorff base, let $u_E\in H^r(E,E^\times;\mathbb Z)$ be its Thom class, and let $0_E:B\to E$ be its continuous zero section $b\mapsto0_b$ , and put $E^\times=E\setminus0_E(B)$ . The orientation is the integral orientation used to choose $u_E$ . The Euler class of $E$ is

e(E):=0_E^*u_E\in H^r(B;\mathbb Z),

where $0_E$ is regarded as a map of pairs $(B,\varnothing)\to(E,E^\times)$ . This definition applies in every rank and uses only the orientation and topology of $E$ ; it is natural under orientation-preserving pullback.

Zeros of sections

If $B$ is a closed oriented smooth manifold and $E$ is smooth, a smooth section $s:B\to E$ that is transverse to the zero section has a zero set is a codimension- $r$ submanifold whose Poincaré dual is $e(E)$ , with the induced orientation. This geometric interpretation follows from the Thom construction.

In particular, a nowhere-zero section has empty zero locus and forces $e(E)=0$ . The converse is false in general: vanishing of the primary Euler obstruction need not produce a nowhere-zero section without additional dimension or obstruction-theoretic hypotheses.

Relation to other definitions

For an oriented even-rank smooth bundle with a metric connection, the image of this integral class in real cohomology agrees with the Chern–Weil Euler class represented by the normalized Pfaffian of the curvature. Thus the zero-section construction supplies the integral topological class, while Chern–Weil theory supplies a differential-form representative of its real image.

For a closed oriented manifold $B$ , the tangent-bundle Euler class satisfies

\langle e(TB),[B]\rangle=\chi(B),

the cohomological form of the Poincaré–Hopf theorem.

Examples and conventions

For the trivial oriented bundle $B\times\mathbb R^r$ with $r>0$ , a constant nonzero section shows that $e(E)=0$ . For the tangent bundle of the oriented sphere $S^{2m}$ , evaluation on the fundamental class gives $2$ , so the Euler class is nonzero. Reversing the orientation of $E$ changes the sign of $e(E)$ .

References

John W. Milnor and James D. Stasheff, Characteristic Classes, Annals of Mathematics Studies 76, Princeton University Press, 1974. DOI record. Relevant: Chapter 9, Euler classes, Thom classes, and zeros of sections.
Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. DOI record. Relevant: Chapter 6, Thom isomorphism and Euler class.