A note on stable complex structures on real vector bundles over manifolds

Let M be an n-dimensional closed oriented smooth manifold with n equivalent to 0 mod 8, xi be a real vector bundle over M. Suppose that xi admits a stable complex structure over the (n 1)-skeleton of M. Then the necessary and sufficient conditions for xi to admit a stable complex structure over M are given in terms of the characteristic classes of xi and M. As an application, we obtain the criteria to determine which real vector bundle over 8-dimensional manifold admits a stable complex structure. (C) 2015 Elsevier B.V. All rights reserved.