On trivialities of Stiefel–Whitney classes of vector bundles over highly connected complexes

It is a classical theorem of Milnor that for every vector bundle over Sn, all the Stiefel–Whitney classes vanish if and only if n≠1,2,4,8. We describe a space B as W-trivial (except for one dimension) if for every vector bundle over B, all the Stiefel–Whitney classes vanish (except for a single fixed dimension). We establish theorems which state that certain high-connectivities of B imply these trivialities as well as a theorem which states that there are infinitely many “W-trivial except for one dimension” spaces.