# Stable complex structures on vector bundles over manifolds

## Seminar/Forum

107
Peter Hall
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 $$\mathbb{Z}/2$$ coefficients.. 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.