On Chern classes of stably fibre homotopic trivial bundles

Abstract

Let ξ be a stably fibre homotopic trivial vector bundle. A classical result of Thorn states that the Stiefel-Whitney classes of ξ vanish, and one way to prove this is as follows. Letube the Thorn class of ξ in mod 2 cohomology. Thenuis stably spherical by [2] and therefore all stable cohomology operations vanish onu, showing thatwi(ξ)u= Sqiu= 0. In this note we shall apply this same method using complex cobordism and Landweber-Novikov operations to study relations among Chern classes of a stably fibre homotopic trivial complex vector bundle. We will thus obtain in a unified way certain strong modpconditions for every primep.