Equivariant Characteristic Classes in the Cartan Model

This note shows the compatibility of the differential geometric and the topological formulations of equivariant characteristic classes for a compact connected Lie group action. Suppose G and S are two compact Lie groups, and P and M are manifolds. A principal G-bundle π : P → M is said to be S-equivariant if S acts on the left on both P and M in such a way that a) the projection map π is S-equivariant: π(s.p) = s.π(p) for all s ∈ S and p ∈ P ; b) the left action of S commutes with the right action of G: s.(p.g) = (s.p).g for all s ∈ S, p ∈ P, and g ∈ G. An S-equivariant principal G-bundle π : P → M induces a principal G-bundle πS : PS → MS over the homotopy quotient MS . In the topological category, the equivariant characteristic classes of P → M are defined to be the corresponding ordinary characteristic classes of PS → MS. Thus, the equivariant characteristic classes are elements of the equivariant cohomology ring H S(M). There is also a differential geometric definition of equivariant characteristic classes in terms of the curvature of a connection on P ([3])([4]). However, there does not seem to be an explanation or justification in the literature bridging the two approaches. The modest purpose of this note is to show the compatibility of the usual differential geometric formulation of equivariant characteristic classes with the topological one. We have also tried to be as self-contained as possible, which partly explains the length of this article. Let us first recall the situation for ordinary characteristic classes. Here the famous Chern-Weil construction represents the ordinary characteristic classes of a principal G-bundle π : P → M by differential forms as follows. Fix a connection Θ on P . Then its curvature K is a 2-form on P with values in the Lie algebra g of G. For each AdG-invariant polynomial f on g, one shows that f(K) is a basic form on P and so is the pullback π∗Λf of a form Λf on M . Moreover, Λf is Date: January 31, 2001.