Equivariant cohomology g 7→ ag, in such a way that the action map and the Cartan model (1) a: G × M → M is continuous. An important special case of G-spaces Eckhard Meinrenken are principal G-bundles E → B, i.e. G-spaces locally University of Toronto isomorphic to products U × G. Definition . classifying bundle 1. Introduction 2.1 A for G is a princi- pal G-bundle EG → BG, with the following universal If a compact Lie group G acts on a manifold M, the property: For any principal G-bundle E → B, there is space M/G of orbits of the action is usually a singu- a map f : B → BG, unique up to homotopy, such that lar space. Nonetheless, it is often possible to develop E is isomorphic to the pull-back bundle f ∗EG. The a ’differential geometry’ of the orbit space in terms map f is known as a classifying map of the principal of appropriately defined equivariant objects on M. In bundle. this article, we will be mostly concerned with ’differen- To be precise, the base spaces of the principal bun- tial forms on M/G’. A first idea would be to work with dles considered here must satisfy some technical con- the complex of ’basic’ forms on M, but for many pur- dition. For a careful discussion, see Husemoller [18]. poses this complex turns out to be too small. A much Classifying bundles exist for all G (by a construction more useful complex of equivariant differential forms due to Milnor [22]), and are unique up to G-homotopy on M was introduced by H. Cartan in 1950, in [2, Sec- equivalence. tion 6]. In retrospect, Cartan’s approach presented a It is a basic fact that principal G-bundles with con- differential form model for the equivariant cohomol- tractible total space are classifying bundles. ogy of M, as defined by A. Borel [8] some ten years R R Z 1 later. Borel’s construction replaces the quotient M/G Examples 2.2. (a) The bundle → / = S is a Z by a better behaved (but usually infinite-dimensional) classifying bundle for G = . (b) Let H be a separable complex Hilbert space, dim H = homotopy quotient MG, and Cartan’s complex should ∞. It is known that unit sphere S(H) is contractible. be viewed as a model for forms on MG. One of the features of equivariant cohomology are It is thus a classifying U(1)-bundle, with base the pro- the localization formulas for the integrals of equivari- jective space P (H). More generally, the Stiefel mani- ant cocycles. The first instance of such an integration fold St(k, H) of unitary k-frames is a classifying U(k)- formula was the ’exact stationary phase formula’, dis- bundle, with base the Grassmann manifold Gr(k, H) covered by Duistermaat-Heckman [12] in 1980. This of k-planes. formula was quickly recognized, by Berline-Vergne [5] (c) Any compact Lie group G arises as a closed sub- and Atiyah-Bott [3], as a localization principle in equi- group of U(k), for k sufficiently large. Hence, the variant cohomology. Today, equivariant localization is Stiefel manifold St(k, H) also serves as a model for a basic tool in mathematical physics, with numerous EG. applications. (d) The based loop group G = L0K of a connected Lie In this article, we will begin with Borel’s topological group K acts by gauge transformations on the space 1 1 1 definition of equivariant cohomology. We then proceed of connections A(S ) = Ω (S , k). This is a classify- to describe H. Cartan’s more algebraic approach, and ing bundle for L0K, with base K. The quotient map conclude with a discussion of localization principles. takes a connection to its holonomy. As additional references for the material covered For any commutative ring R (e.g. Z, R, Z2) let here, we particularly recommend the books by Berline- H(·; R) denote the (singular) cohomology with coef- Getzler-Vergne [4] and Guillemin-Sternberg [17]. ficients in R. Recall that H(·; R) is a graded commu- tative ring under cup product. Definition 2.3. The equivariant cohomology HG(M) = 2. Borel’s model of HG(M) HG(M; R) of a G-space M is the cohomology ring of its homotopy quotient M = EG × M, Let G be a topological group. A G-space is a topo- G G logical space M on which G acts by transformations (2) HG(M; R) = H(MG; R). 1 2 Equivariant cohomology is a contravariant functor points is an isomorphism for k > 0. Since the pull- from the category of G-spaces to the category of R- back map is a ring homomorphism, we conclude that 2 modules. The G-map M → pt induces an algebra HU(1)(S ; R) is the commutative ring generated by homomorphism from HG(pt) = H(BG) to HG(M). two elements x± of degree 2, subject to a single re- In this way, HG(M) is a module over the ring H(BG). lation x+x− = 0. Example 2.4. (Principal G-bundles) Suppose E → B 3. g-differential algebras is a principal G-bundle. The homotopy quotient EG Let G be a Lie group, with Lie algebra g. A G- may be viewed as a bundle E×G EG over B. Since the fiber is contractible, there is a homotopy equivalence manifold is a manifold M together with a G-action such that the action map (1) is smooth. We would (3) EG ≃ B, like to introduce a concept of equivariant differential and therefore HG(E) = H(B). forms on M. This complex should play the role of dif- ferential forms on the infinite-dimensional space MG. Example 2.5. (Homogeneous spaces) If K is a closed In Cartan’s approach, the starting point is an alge- subgroup of a Lie group G, the space EG may be braic model for the differential forms on the classifying viewed as a model for EK, with BK = EG/K = bundle EG. EG ×K (G/K). Hence, The algebraic machinery will only depend on the (4) HG(G/K) = H(BK). infinitesimal action of G. It is therefore convenient to introduce the following concept. Let us briefly describe two of the main techniques Definition . g for computing HG(M). 3.1 Let be a finite-dimensional Lie al- gebra. A g-manifold is a manifold M, together with (1) Leray spectral sequences. If R is a field, the equi- a Lie algebra homomorphism a: g → X(M), ξ 7→ aξ variant cohomology may be computed as the E∞ term into the Lie algebra of vector fields on M, such that g of the spectral sequence for the fibration MG → BG. the map × M → T M, (ξ,m) 7→ aξ(m) is smooth. If BG is simply connected (as is the case for all com- Any G-manifold M becomes a g-manifold, by tak- pact connected Lie groups), the E -term of the spec- 2 ing aξ to be the generating vector field tral sequence reads d p,q p q (6) aξ(m) := aexp(−tξ)(m). (5) E2 = H (BG) ⊗ H (M). dt t=0 Conversely, if G is simply connected, and M is a g- (2) Mayer-Vietoris sequences. If M = U1 ∪ U2 is a manifold for which all of the vector fields aξ are com- union of two G-invariant open subsets, there is a long plete, the g-action integrates uniquely to an action of exact sequence the group G. k k k The de Rham algebra (Ω(M), d) of differential forms · · · → HG(M) → HG(U1) ⊕ HG(U2) → on a g-manifold M carries graded derivations Lξ = k k+1 → HG(U1 ∩ U2) → HG (M) →··· L(aξ) (Lie derivatives, degree 0) and ιξ = ι(aξ) (con- More generally, associated to any G-invariant open tractions, degree −1). One has the following graded commutation relations, cover there is a spectral sequence converging to HG(M). (7) [d, d]=0, [L , d]=0, [ι , d] = L , Example 2.6. Consider the standard U(1)-action on ξ ξ ξ 2 2 S by rotations. Cover S by two open sets U±, given (8) [ιξ,ιη] = 0, [Lξ, Lη] = L[ξ,η]g , [Lξ,ιη] = ι[ξ,η]g . as the complement of the south- and north pole, re- More generally, we define: spectively. Since U+ ∩ U− retracts onto the equa- torial circle, on which U(1) acts freely, its equivari- Definition 3.2. A g-differential algebra (g − da) is a ∞ n ant cohomology vanishes except in degree 0. On the commutative graded algebra A = n=0 A , equipped other hand, U± retract onto the poles p±. Hence, with graded derivations d, Lξ,ιξ Lof degrees 1, 0, −1 k 2 ∼ by the Mayer-Vietoris sequence the map HU(1)(S ) = (where Lξ,ιξ depend linearly on ξ ∈ g), satisfying the k k graded commutation relations (7), (8). HU(1)(p+) ⊕ HU(1)(p−) given by pull-back to the fixed 3 Definition 3.3. For any g − da A one defines the hori- Theorem 4.2. W g is acyclic. That is, the inclusion R zontal subalgebra Ahor = ξ ker(ιξ), the invariant sub- → W g is a homotopy equivalence. algebra Ag = ker(L T), and the basic subalgebra ξ ξ Acyclicity of W g corresponds to the contractibility T g Abasic = Ahor ∩A . of the total space of EG. Note that the basic subalgebra is a differential sub- The basic subalgebra of W g is equal to (Sg∗)g, and complex of A.