Characteristic classes of complexified bundles

We examine the topological characteristic cohomology classes of complexified vector bundles. In particular, all the classes coming from real vector bundles are computed. We use characteristic classes with the axioms of Milnor and Stasheff [6]. Introduction Definition (Real generator bundles). We consider a real vector bundle F → B and a complex vector bundle E → B over the same base space B. If the fibre-wise constructed complexification F ⊗R C =: FC is isomorphic to E, we’ll call F a real generator bundle of E. We want to attribute topological characteristic classes c(F ) of the real generator bundles to the complexified bundles FC. Not every complex vector bundle admits a real generator bundle, as we shall see in a moment. So, supplementary cohomological information might be gathered when restricting attention to the subcategory of complex vector bundles that admit one. Obstruction to real generator bundles. Consider a real vector bundle F → B. By reflection on the real axes given by F , FC is isomorphic to its complex conjugate bundle FC. So, any complex bundle E → B that admits F as a real generator bundle must be isomorphic to its own conjugate bundle: E ∼= FC ∼= FC ∼= E. The odd Chern classes c2k+1 have the property c2k+1(E) = −c2k+1(Ē) ([6, lemma 14.9]), so c2k+1(E) = −c2k+1(Ē) = −c2k+1(E) 2 H(B,Z) ⇒ 2c2k+1(E) = 0. Consequently, no complex vector bundle with some nonzero and non-torsion odd Chern class can admit a real generator bundle. We are interested in all attributions of topological characteristic classes c(F ) of the real generator bundles to the complexified bundles FC. For such an attribution to be well-defined, we need that real generator bundles F , G of the same complex bundle provide the same class c(F ) = c(G). For short, we get the Basic requirement FC ∼= GC ⇒ c(F ) = c(G). 12000 Mathematics Subject Classification. 55R50.