V - Characteristic Classes

Summary

In this chapter we mimic as closely as possible [Milnor and Stasheff 1974], itself an expos´e of the Chern–Weil construction of characteristic classes in terms of curvature forms. See also [Dupont 1978; Husemoller 1975; Lawson and Michelsohn 1989; de Rham 1955].

The Chern–Weil procedure begins with a vector bundle with a certain structural group G. In our situation we consider complex (tangentially smooth) bundles with structural group GL(n,ℂ) real vector bundles with structural group GL(n, ℝ), and oriented real even-dimensional vector bundles with structural group SO.(2n). Choose a tangential connection ∇, see below, that respects the structure. The associated curvature form K determines a closed tangential 2- form whose tangential cohomology class is independent of choice of the connection. Then any polynomial or formal power series P which is G-invariant determines a characteristic form. In the case X = M is a manifold with FX = TM then this yields the usual characteristic classes in de Rham cohomology H*M.

We shall assume throughout that all bundles over foliated spaces are tangentially smooth and that leaf-preserving maps between foliated spaces are also tangentially smooth; this is not a real restriction, in view of our smoothing results (Proposition 2.16). We use Milnor and Stasheff’s sign conventions for characteristic classes.