The Euler class as a cohomology generator (Q1850235)

A geometric proof that the Euler class of any vector bundle over \(S^n\) for \(n\neq 2\), \(4\) or \(8\) must be an even multiple of a cohomology generator of \(H^n(S^n)\) is given. Consequences: 1) the Stiefel-Whitney class \(w(\xi)\) of any vector bundle \(\xi\) over \(S^n\) is trivial; 2) any rank \(4n\) bundle over \(S^{4n}\) with trivial Pontryagin class is equivalent to a pullback \(f^*\tau\) of the tangent bundle \(\tau\), for some map \(f: S^{4n}\to S^{4n}\); 3) in case of the base manifold not being the sphere, it is proved that under some conditions a cohomology class exists that cannot be realized as the Euler class of any bundle over a simply and rationally connected manifold of \(M\). See also \textit{L. Guijasso} and \textit{G. Walschap} [J. Differ. Geom. 52, 189-202 (1999; Zbl 1035.53041)].