Obstructions to almost complex structures following Massey

TL;DR

This paper revisits classical theorems by Massey, providing proofs and new insights into the obstructions to complex structures on real vector bundles, including a detailed analysis for rank six bundles.

Contribution

It offers rigorous proofs of Massey's theorems and determines the second obstruction for rank six bundles, clarifying the role of characteristic classes.

Findings

Obstructions are fractional parts of Stiefel-Whitney classes.

Second obstruction for rank six bundles identified.

Obstructions involve Pontryagin, Chern, and Euler classes.

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.