Two-plane sub-bundles of nonorientable real vector-bundles

TLDR: In this paper, the tangent bundle of a closed, connected, non-orientable smooth manifold is embedded as a sub-bundle of a 2-plane bundle over a CW complex of dimension m or less.

Abstract: Let ζ be a nonorientable m-plane bundle over a CW complex X of dimension m or less Given a 2-plane bundle η over X, we wish to know whether η can be embedded as a sub-bundle of ζ The bundle η need not be orientable When ζ is even-dimensional there is the added complication of twisted coefficients In that case, we use Postnikov decomposition of certain nonsimple fibrations in order to describe the obstructions for the embedding problem Emery Thomas [11] and [12] treated this problem for ζ and η both orientable The results found here are applied to the tangent bundle of a closed, connected, nonorientable smooth manifold, as a special case