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1-1 correspondence admits base space basepoint bundle associated bundle map bundle with fibre c(ir cell complex Chern classes classes of maps classifying space compact complex line bundle complex n-plane bundle continuous map coordinate neighborhood covering homotopy CP(E define f double cover element equivalence classes Example ffi2 fibre bundle fibre F follows G-equivalent G-structure group G group of homeomorphisms Hausdorff Hence homeomorphism homotopy classes homotopy f homotopy groups homotopy theorem identity map inner product isomorphism k-frame k-plane bundle Lemma let f Let tt map f matrix Note nullhomotopic numerable G-bundle numerable principal G-bundle obtain open cover orthogonal p'PU p(ir paracompact space partition of unity Pontrjagin class principal G-space proof of Theorem provisional definition smooth manifold space BG structure subbundle supp tangent bundle tangent plane Theorem 6.6 topology total space transition functions trivial iff trivial map Whitney sum write