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Let $\xi$ have real rank $r$, choose a bundle metric, and write $D(\xi ),S(\xi )$ for its disk and sphere bundles. In a multiplicative generalized cohomology theory, a Thom class in a generalized cohomology theory is a classwhose restriction to each fiber pair $(D^r,S^{r-1})$ is the suspension of the coefficient unit $1\in h^0(\mathrm{pt})$ under the chosen oriented identification. Equivalently, it is a fiberwise generator over the coefficient ring compatible with the $h$-orientation. Cup product with this class gives the Thom isomorphism theoremwhere $X$ is the base.

Let $z:X\to D(\xi )$ be the zero section and let $ȷ:h^r(D(\xi ),S(\xi ))\to h^r(D(\xi ))$ forget relative supports. The Euler class in a generalized cohomology theory isEquivalently, identify relative cohomology with reduced cohomology of the Thom space and pull its Thom class back along the zero section followed by the quotient map.

Suppose $s$ is a nowhere-zero section. Normalize it to $\hat{s}(x)=s(x)/\Vert s(x)\Vert \in S(\xi {)}_x$. The mapsgive a homotopy within $D(\xi )$ from $z$ to the sphere-valued section $\hat{s}$. The image $ȷ(u_h(\xi ))$ restricts to zero on $S(\xi )$, by the exact sequence of the pair. Homotopy invariance therefore yieldsThus a nowhere-zero section annihilates generalized Euler classes:The argument uses the actual nowhere-zero section and is valid for any chosen Thom orientation in the multiplicative theory.