Vector bundles over (8k + 1)-dimensional manifolds

Abstract

We obtain a necessary and sufficient condition for an orientable n-plane bundle η over a manifold Mn of dimension n= 8k + 1 with k > 1 satisfying certain conditions to have span(η) ≥ 5 or 6. Using a method of least indeterminacy due to Browder when η is the tangent bundle and M is a spin manifold satisfying w4(M)=0 and v4k(M)=0 when k is even, we show that the top-dimensional stable obstruction to the existence to five or six linearly independent vector fields is trivial. We also obtain a variant of the Browder-Dupont invariant which might be a candidate for a new invariant for a spin manifold M. In particular, when dim M=n is congruent to 9 mod 16 and n > 9, if M is 3-connected mod 2 with w4(M)=0, then span M ≥ 4 implies span M ≥ 6. © 1994.