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.
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