Stable complex structures on vector bundles over manifolds


Let \(M \) be an \(n\)-dimensional closed oriented smooth manifold, \(\xi\)  be a real vector bundle over \(M \). The obstructions for \(\xi\)  to admit a stable complex structure lie in the cohomology groups of \(M \) with \( \mathbb{Z} \), or \(\mathbb{Z}/2 \) coefficients.. The obstructions with \( \mathbb{Z} \)-coefficient are investigated by Massey in 1961. The first obstruction with \( \mathbb{Z}/2 \)-coefficient was determined by Thomas in 1967 in terms of a second cohomology operation.

In this talk, under the assumption that \(n= 0 mod 8\), the final obstruction with \( \mathbb{Z}/2 \)-coefficient will be given in terms of the characteristic classes of \(M \) and \(\xi \). As an application, the necessary and sufficient conditions for \(\xi \)  to admit a stable complex structure over \( M\) with \(dimM = 8\) will be obtained.


  • Dr Huijun Yang
    Dr Huijun Yang, University of Melbourne