W-triviality of low dimensional manifolds

Abstract

A space X is W-trivial if for every real vector bundle \(\alpha \) over X the total Stiefel-Whitney class \(w(\alpha )\) is 1. It follows from a result of Milnor that if X is an orientable closed smooth manifold of dimension 1, 2, 4 or 8, then X is not W-trivial. In this note we completely characterize W-trivial orientable connected closed smooth manifolds in dimensions 3, 5 and 6. In dimension 7, we describe necessary conditions for an orientable connected closed smooth 7-manifold to be W-trivial.