Obstructions to almost complex structures following Massey

Abstract

We provide proofs of two theorems stated by Massey in 1961, concerning the obstructions to finding complex structures on real vector bundles. In addition, we determine the second obstruction to a complex structure on a rank six orientable real vector bundle. The obstructions are fractional parts of integral Stiefel–Whitney classes, and a fourth of an appropriate combination of Pontryagin, Chern, and Euler classes.