ON THE COHOMOLOGY OF THE REAL GRASSMANN COMPLEXES AND THE CHARACTERISTIC CLASSES OF w-PLANE BUNDLES^)

where fc is the complexification of the bundle f (see [4, XII] for details). Thus, pi(Z)EH"(B; Z), where B is the base of j\ For the results which follow we will need the following relationship between the Pontrjagin and Chern classes. Suppose thatco" is a complex re-plane bundle. Each fibre of wM is then an re-dimensional complex vector space. By restricting attention to the real numbers as scalars, we see that each fibre is also a 2re-dimensional real vector space. Thus, co" gives rise to a real 2«plane bundle, co% We then have the following result relating the classes of these two bundles (see Wu [9] and [4, §XII]):