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abelian group B(PL B(PLn barycentric subdivision base space c.s.s. complex c.s.s. group complex class of maps classifying space completes the proof composition countable number defined Definition diagram differentiable structure easily verified Euclidean space exist neighborhoods fibre dimension q finite complex finite simplicial complex follows form an abelian functor germ given homomorphism homotopy class homotopy commutative homotopy type identity map inclusion map induced bundle inverse isomorphic k-simplex kpL(B kQ(B Let f locally finite simplicial M X M m-universal manifold map f monotone simplicial map Munkres natural transformation normal microbundle obstruction theory open subset orthogonal group 0(n piecewise linear PL-homeomorphism PL-manifold of dimension PL-map f Pontrjagin classes product M X R Proof of Theorem proves Lemma restricted s-isomorphic to f t(M simplex simplicial approximation subcomplex tangent bundle tangent microbundle tangent vector bundle total space triangulation trivial bundle trivial normal bundle unique universal bundle theorem vector bundle t(M Whitney sum