Let \(\xi\) be an n-dimensional real vector bundle over a finite complex X. A necessary condition for the existence of an r-field on \(\xi\), that is, r linearly independent cross-sections, is the vanishing of the stable cohomotopy Euler class \(\gamma\) (H\(\otimes \xi)\) of the tensor product of \(\xi\) and the Hopf line bundle H over real projective space \(P({\mathbb{R}}^ r)\). The Hurewicz image in integral cohomology of the stable cohomotopy Euler class \(\gamma\) (H\(\otimes \xi)\) is the classical obstruction to the existence of an r-field.
This paper contains a study of \({\mathbb{Z}}/2\)-equivariant KO-theoretic methods in obstruction theory. In general, we are concerned with the work of M. F. Atiyah and J. L. Dupont [Acta Math. 128, 1-40 (1972; Zbl 0233.57010)]. Using the Adams operations and, specifically, the \({\mathbb{Z}}/2\)-e-invariant, the author obtains a KO-theory criterion for an n-dimensional real vector bundle to admit an r-field.