Secondary characteristic classes and the Euler class.
Résumé
We discuss secondary (and higher) characteristic classes for algebraic vector bundles with trivial top Chern class. We then show that if $X$ is a smooth affine scheme of dimension $d$ over a field $k$ of finite $2$-cohomological dimension (with $\mathrm{char}(k)\neq 2$) and $E$ is a rank $d$ vector bundle over $X$, vanishing of the Chow-Witt theoretic Euler class of $E$ is equivalent to vanishing of its top Chern class and these higher classes. We then derive some consequences of our main theorem when $k$ is of small $2$-cohomological dimension.