310 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS
3.1. The Cartan model
There is a De Rham description of equivariant cohomology, which is due to Cartan
([11]: see also [6]). Let G be a compact connected Lie group. We introduce the
complex
D(;(M) = (D*(M) © S(g*))ewhere D(M) is the ordinary De Rham complex of Mand S(g) is the symmetric
algebra on g* (its elements can be thought of as polynomial functions on g). The
superscript G denotes the invariant part under the G-action defined by the given
action of G on M and the adjoint action on g. We have in particular that
D(; = D(;(pt) = S(g*f.
An element f E D(;(M) may be thought of as a G-equivariant map f: g __, D*(M),
where the dependence of f(X) E D*(M) on X E g is polynomial. The grading
on D(;(M) is defined by deg(!) = n + 2p if X f-+ f(X) is p-linear in X and
f(X) E nn(M). We may define a differential
D: D(;(M) __, D(;(M)by(Df)(X) = d(f(X)) - ix#f(X)
where X# is the vector field on M generated by the action of X E g and i denotes
the interior product. Then Do D = 0 and D increases the degree in D(;(M) by 1.Theorem 3.2. (Cartan [11]) H(;(M) is naturally isomorphic to the cohomology
H*(D(;(M), D) of this complex.In particular, since D = 0 on D(;, we see that H(; = H(;(pt) = S(g*)e.3.2. Equivariant characteristic classes
Define the pushforward map 7f: D(;(M) __, D(; by 1f(17)(X) =JM 17(X).
Stokes' Theorem for equivariant cohomology in the Cartan model tells us thatif Mis a G-manifold with boundary and G: 8M __, 8M (where the action of G on
8M is locally free) and 17 E DG(M), then
r (D17)(X) = r 17(X).
JM laMIt follows that the pushforward map 7f* induces a map H(;(M) __, H(;.
Definition 3.3. Suppose E is a (complex) vector bundle on a manifold M
equipped with a Hamiltonian action of a group G which lifts t he action of G on M.
The equivariant Chern classes c;? ( E) are given by
c?(E) = cr(E Xe EG __, M Xe EG).Likewise the equivariant Euler class of E is given byee(E) = e(E Xe EG __, M Xe EG).