Secondary characteristic classes and the Euler class (Q273848)

Summary: 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.