Euler classes of vector bundles over iterated suspensions of real projective spaces

Abstract

Abstract We show that when k ≠ 2, 4, 8 the Euler class of any vector bundle over Σ k ℝℙ m is zero if the rank of the bundle is not m + k, provided that m ≠ 3 when k = 6. If k = 2, 4, 8 we show that the Euler class of any vector bundle over Σ k ℝℙ m is zero whenever the rank of the bundle is not kr + k, provided that m ≠ 6, 7 when k = 2, where r is the largest integer such that kr ≤ m.