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Secondary characteristic classes and the Euler class. (English) Zbl 1349.14078
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.

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 14C15 (Equivariant) Chow groups and rings; motives 13C10 Projective and free modules and ideals in commutative rings 55S20 Secondary and higher cohomology operations in algebraic topology
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