Orientability of vector bundles and formulae for Stiefel-Whitney classes

Abstract

It is proved in §12 of J. W. Milnor and J. D. Stasheff's book: Characteristic classes (1974; Zbl 0298.57008) that if ξ is an orientable real vector bundle over a paracompact base space B then w1(ξ)=0. That the converse is also true for CW-complexes is left as an exercise (Problem 12A). Exercise H on page 281 of E. H. Spanier's book: Algebraic topology (1982; Zbl 0477.55001) deals with Stiefel- Whitney classes of sphere bundles, and 3(d) of this exercise states that a sphere bundle n is orientable if and only if w₁(n). In the case ξ possesses a Euclidean metric, one can apply this result to the associated sphere bundle of ξ. But when B is not paracompact, it is not true in general that ξ possesses a Euclidean metric. In this paper, we adopt a slightly weaker definition of orientability of a vector bundle and prove that a real vector bundle ξ over an arbitrary base space is orientable if and only if w₁(ξ).