Twisted cohomology and enumeration of vector bundles

In the present paper we give a technique for completely enumerating real 4-plane bundles over a 4-dimensional space, real 5-plane bundles over a 5-dimensional space, and real 6plane bundles over a 6-dimensional space. We give a complete table of real and complex vector bundles over real projective space Pk, for k ^ 5. Some interesting results are: