Vanishing of Stiefel-Whitney classes

Abstract

Atiyah and Hirzebruch proved in [1] that for any vector bundle $ over the 9-fold iterated suspension of a finite CW-complex, the mod 2 StiefelWhitney class wj($) is zero for all i>O. A complement to this for 2-, 3-, and 5-fold suspensions is included in the following: REMARK. Let $ be a sphericalfibring over the (2k + 1)-fold suspension of a CW-complex (k any nonnegative integer). Then wi($)=O for i#O mod 2k+1. The Hopf bundles over spheres show that, when k 3; but an example in ?4 of [2] shows that it is best possible for sphericalfibrings when k=3. In fact that example generalises to any k for which the Whitehead product [t, t] can be halved, where t generates V-T(Sn) and n=2k+1_ 1. The remark follows from the Wu formulae (see [3]). For given i# O mod 2k+1, we may write i=2r+l.m+2T, where m, r are nonnegative integers and r?k. Now put s=2r.m, t=2r(m+1), and consider the Wu formula for Sqswt. Using the vanishing of cup-products on a suspension, this yields Sqswt=wi. But Sqswt=O since we are on a (t-s+1)-fold