  We establish the equality of two definitions of an Euler class in algebraic
geometry: the first definition is as a "characteristic class" with values in
Chow-Witt theory, while the second definition is as an "obstruction class."
Along the way, we refine Morel's relative Hurewicz theorem in A^1-homotopy
theory, and show how to define (twisted) Chow-Witt groups for geometric
classifying spaces.
