Characteristic classes and transversality

Abstract

Let ξ be a smooth vector bundle over a differentiable manifold M. Let h : ε n − i + 1 → ξ be a generic bundle morphism from the trivial bundle of rank n − i + 1 to ξ. We give a geometric construction of the Stiefel–Whitney classes when ξ is a real vector bundle, and of the Chern classes when ξ is a complex vector bundle. Using h we define a differentiable closed manifold Z ˜ ( h ) and a map ϕ : Z ˜ ( h ) → M whose image is the singular set of h. The ith characteristic class of ξ is the Poincaré dual of the image, under the homomorphism induced in homology by ϕ, of the fundamental class of the manifold Z ˜ ( h ) . We extend this definition for vector bundles over a paracompact space, using that the universal bundle is filtered by smooth vector bundles.