Introduction to Equivariant Cohomology Theory 11

INTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY

YOUNG-HOON KIEM

1. Definitions and Basic Properties 1.1. Lie group. Let G be a Lie group (i.e. a manifold equipped with diﬀerentiable group operations mult : G × G → G, inv : G → G, id ∈ G satisfying the usual group axioms). We shall be concerned only with linear groups (i.e. a subgroup of GL(n) = GL(n, C) for some n) such as the unitary group U(n), the special unitary group SU(n). A connected compact Lie group G is called a torus if it is abelian. Explicitly, they are products U(1)n of the circle group U(1) = {eiθ | θ ∈ R} = S1. Since a complex linear (reductive) group is homotopy equivalent1 to its maximal compact subgroup, it suﬃces to consider only compact groups. For instance, the equivariant cohomology for SL(n) (GL(n), C∗, resp.) is the same as the equivariant cohomology for SU(n) (U(n), S1, resp.).

1.2. Classifying space. Suppose a compact Lie group G acts on a topological space X continuously. We say the group action is free if the stabilizer group Gx = {g ∈ G | gx = x} of every point x ∈ X is the trivial subgroup. A topological space X is called contractible if there is a homotopy equivalence with a point (i.e. ∃h : X × [0, 1] → X such that h(x, 0) = x0, h(x, 1) = x for x ∈ X).

Theorem 1. For each compact Lie group G, there exists a contractible topological space EG on which G acts freely. Proof: Since G ⊂ GL(n) for some n, it suﬃces to construct a contractible free space for G = GL(n). Let Am be the space of n × m matrices with complex entries of rank n for m > n. Of course, G acts freely on Am but Am is not contractible. We claim the limit limm→∞ Am is contractible and this is the desired space. The set n×m n×m M of all n × m matrices is contractible and the complement M − Am is a union of locally closed submanifolds whose codimensions grow to inﬁnity. Use Gysin sequence2 and Whitehead’s theorem3 to conclude that the limit is contracitble. ¤

Date: 2008.2.27-28; Ajou University. 1Two spaces X and Y are homotopy equivalent if ∃f : X → Y, g : Y → X such that g ◦ f : X → X and f ◦ g : Y → Y are homotopic (=continuously deformable) to the identity maps. Ordinary cohomology groups are invariant under homotopy equivalence. 2Gysin sequence: Let F ⊂ M be a submanifold of codimension c whose normal bundle is oriented. Then we have an exact sequence ··· → Hi−c(F) → Hi(M) → Hi(M − F) → Hi+1−c(F) → Hi+1(M) → Hi+1(M − F) → ··· .

3Whitehead’s theorem: If f : X → Y is continuous map of reasonably nice spaces such that f∗ : Hi(Y) → Hi(X) is an isomorphism for all i (with integer coeﬃcients), then f is a homotopy equivalence. 1 2 YOUNG-HOON KIEM

Deﬁnition. The classifying space of a group G is the quotient space BG = EG/G of EG. It is unique up to homotopy equivalence.4 Since the action of G on EG is free, each ﬁber of the quotient map π : EG → BG is an orbit Gx, homeomorphic to G.

Examples. 1 ∞ 1 ∞ 1 2n+1 1 ∞ (1) ES = S , BS = S /S = limn→∞ S /S = CP . (Exercise: Prove ∞ ∞ ∞ that S is contractible. Prove that B(Z2) = S /Z2 = RP .) It is well known that H∗(CPn; R) = R[t]/(tn+1). Hence, H∗(BS1) = R[t] by taking inverse limit.5 (2) For G = G1 × G2, EG = EG1 × EG2 and BG = BG1 × BG2. In particular, B(S1 × · · · × S1) = CP∞ × · · · × CP∞ and H∗(B(S1 × · · · × S1)) = H∗(BS1 × 1 ∼ ∗ 1 ∗ 1 ∼ · · · × BS ) = H (BS ) ⊗ · · · ⊗ H (BS ) = R[t1, ··· , tn]. (3) Suppose H is a subgroup of G. Then BH =∼ EG/H and we have a map BH → BG whose ﬁbers are G/H.

1.3. Equivariant cohomology. Suppose a topological group G acts on a topological space X continuously. When the group action is not free, the quotient X/G may be terribly ugly. The idea of Borel is to replace X by a free G-space X† which is homotopically equivalent to X. The obvious choice is X† = X × EG on which G acts diagonally g · (x, e) = (gx, ge).

Deﬁnition. The homotopy quotient of a G-space X is XG = X×GEG = (X×EG)/G. ∗ ∗ The equivariant cohomology of X is HG(X) = H (XG). We shall use real coeﬃcients R for convenience.

Borel’s diagram: X EG EG o X × EG / X

G G Gx o / BG X ×G EG X/G X BGx

1.4. Basic properties. Almost all basic properties follow directly from the deﬁ- nition.

(1) Suppose G acts freely on X. Then the ﬁbers of XG = (X×EG)/G → X/G are contractible EG, and hence the map XG → X/G is a homotopy equivalence. ∗ ∗ ∼ ∗ In particular, HG(X) = H (XG) = H (X/G). (2) Suppose a normal subgroup K of G acts freely on X. Then S = G/K acts ∼ ∼ ∗ ∼ ∗ on Y = X/K (= X ×K EK = X ×K EG). We have HG(X) = HS(Y). (Proof: EG×ES is a contractible space acted on freely by G via the quotient homomorphism G → S. Then the obvious map XG = X ×G (EG × ES) → X ×G ES = Y ×S ES = YS is a homotopy equivalence with contractible ﬁber EG.)

4We will prove this later. 5For convenience, we will use real coeﬃcients for cohomology groups in this lecture course but there is no privilege for R in equivariant cohomology theory. Most authors prefer Q coeﬃcients. INTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY 3

(3) Suppose G acts trivially on X. Then XG = X ×G EG = X × BG and hence ∗ ∼ ∗ ∗ ∗ ∗ HG(X) = H (X) ⊗ H (BG). In particular, HG(X) is a free H (BG)-module. (The module structure comes from the map XG → BG.) (4) Suppose K is a subgroup of G. Let Y be a K-space. Then X = G ×K Y = ∗ ∗ (G×Y)/K admits an obvious action of G. We have HG(X) = HK(Y) because XG = (G ×K Y) ×G EG = Y ×K EG = Y ×K EK = YK. 4 YOUNG-HOON KIEM

2. Cartan model and spectral sequence Let G be a compact Lie group acting on a smooth compact manifold M diﬀer- entiably. Let G be the Lie algebra of G, i.e. the tangent space of G at the identity. −1 There is a natural action of G on G by conjugation g · v = limt→0 gγ(t)g where d ∗ j ∗ dt |t=0γ(t) = v. Let S(G ) = ⊕j≥0S (G ) be the polynomial algebra generated by the dual space G∗ of G, i.e. the set of polynomial functions on the vector space G. Of course, there is an induced G-action on S(G∗). The diﬀerential forms on M form a complex

d d d 0 / Ω0(M) / Ω1(M) / Ω2(M) / ··· . A theorem of de Rham says that the cohomology H∗(M) with real coeﬃcients is isomorphic to the cohomology of (Ω(M), d). When there is an action of G on M, we get an induced action on Ω(M). The main theorem of this section is the following due to H. Cartan.

i j ∗ i−2j G G Theorem 2. Let ΩG(M) = ⊕j[S (G ) ⊗ Ω (M)] where [−] denotes the 6 i i+1 G-invariant subspace. Let dG : ΩG(M) → ΩG (M) be deﬁned as follows: For a P 7 basis {ξa} of G and its dual basis {fa}, dG(F ⊗ σ) = F ⊗ dσ − a faF ⊗ ıξa σ. Then ∗ the equivariant cohomology HG(M) of M is the cohomology of the complex

/ 0 dG / 1 dG / 2 dG / 0 ΩG(M) ΩG(M) ΩG(M) ··· .

Proof (sketch): The de Rham complex of the quotient MG = (M × EG)/G is embedded into the de Rham complex of M × EG as G-invariant subspaces. So it suffices to find the de Rham complex for M×EG and take the invariant (horizontal) part. For M, we have the ordinary de Rham complex Ω(M) but what is the de Rham complex for the infinite dimensional contractible space EG? Recall that EG = limm→∞ Am and hence Ω(EG) = lim← Ω(AM). Then one finds that Ω(EG) can be chopped down to a more economical complex, namely the Koszul complex W := S(G∗) ⊗ ∧(G∗) of S(G∗)-modules. So we obtain ∗ ∗ ∼ ∗ HG(M) = H (M ×G EG) = H ([Ω(M) ⊗ W]hor). Then by pure algebra, we obtain an isomorphism ∗ ∼ ∗ H ([Ω(M) ⊗ W]hor) = H (ΩG(M)). ¤

∗ ∼ ∗ ∼ ∗ K Corollary (homogeneous spaces). HG(G/K) = H (BK) = S(K ) is the space of K-invariant polynomial functions on K. A nonzero element of K∗ corresponds to a degree 2 class, i.e. H2j(BK) =∼ Sj(K∗)K and Hodd(BK) = 0.

Examples. (1) If G is the n-dimensional torus (S1)n, then G =∼ Rn and G acts ∗ ∗ n trivially on G. Hence HG(pt) = H (BG) = S(R ) = R[t1, ··· , tn] as expected. (2) For G = U(n), G is the space of skew-hermitian n × n complex matrices. For A ∈ G, we consider the coeﬃcients ci(A) of the characteristic polynomial

6 We think of an element in ΩG(M) as a differential form valued polynomial function defined on G∗. For f ∈ G∗, deg f = 2. 7 ıξa denotes the interior product by the vector field on M generated by ξa, i.e. (ξa)x = d ξat dt |t=0e x. INTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY 5

P i n−i det(λ−A) = (−1) ci(A)λ . In particular, c1(A) = tr(A) and cn(A) = det(A). ∗ G It is well known that S(G ) = R[c1, c2, ··· , cn] and hence ∗ ∗ HG(pt) = H (BG) = R[c1, c2, ··· , cn].

The ci is called the i-th Chern class. (3) For G = O(n), G is the space of skew-symmetric n × n real matrices. Since det(λ − A) = det(λ − At) = det(λ + A), we have the vanishing of odd degree coefficients ci(A) = 0 for i odd. The even coefficients pi(A) = c2i(A) generate the ∗ G invariant subring S(G ) . The pi is called the i-th Pontryagin class. (4) For G = SO(2n), we have an additional class, called the Pfaffian. A skew- symmetric matrix gives us a skew-symmetric bilinear form which can be put into a standard form. The determinant turns out to be the square of an invariant polynomial Pfaff.

An indispensable tool for equivariant cohomology theory is spectral sequence. ∗ G 8 Note that ΩG(M) = [S(G ) ⊗ Ω(M)] is bigraded and the differentialP dG is the sum of its vertical part d := 1 ⊗ d and horizontal part δ := − fa ⊗ ıξa . p,q A double complex is a bigraded vector space C = ⊕p,q≥0C with operators d : Cp,q → Cp,q+1, δ : Cp,q → Cp+1,q satisfying d2 = 0, δ2 = 0 and dδ + δd = 0. n p,q The total complex of C is defined by C = ⊕p+q=nC with differential d + δ : Cn → Cn+1. The goal is to calculate the cohomology of the total complex by approximation. p,q p,q Let E0 = C . Inductively, we can construct bigraded complexes Er for r ≥ 0 p,q p+r,q−r+1 p,q together with differential δr : Er → Er such that Er+1 is the cohomology p,q of (Er, δr) at (p, q)-th place. Furthermore, in the limit E∞, the sum ⊕p+q=nE∞ is isomorphic to the n-th cohomology of the total complex of C. We say the spectral ∼ sequence collapses at Er if Er = E∞. This happens when δr vanishes and so do all higher differentials. 9 For practical use, it suffices to know only the first two terms E1 and E2. For E1, you ignore δ and take the cohomology at each place with only the vertial differential d. Then the E2 term is given by taking cohomology of E1 with the horizontal differential δ. For the Cartan model, the E1 term is p,q p ∗ q−p G E1 = [S (G ) ⊗ H (M)] . If G is connected, then G acts trivially on H∗(M) because a continuous deformation of cycles doesn’t change the (co)homology classes. Hence, if G is connected, we have p,q p ∗ G q−p 2p q−p E1 = S (G ) ⊗ H (M) = H (BG) ⊗ H (M) ∗,∗ ∼ ∗ ∗ or E1 = H (BG) ⊗ H (M). For many friendly spaces, the spectral sequence ∗ G degenerates at E1 and so we obtain an isomorphism of S(G ) -modules ∗ ∼ ∗ ∗ HG(M) = H (BG) ⊗ H (M) for connected G. We make this into a definition.

Deﬁnition. A manifold M on which G acts is called equivariantly formal if the ∗ ∼ ∗ ∗ G spectral sequence collapses at E1, i.e. HG(M) = [S(G ) ⊗ H (M)] .

8A differetial means nothing but a sequence of homomorphisms such that the composition of any two consecutive maps is zero. 9 If you can’t finish the calculation by E2, it is often too difficult to continue. 6 YOUNG-HOON KIEM

Here are some criteria for equivariant formality. Theorem 3. (1) If G is connected and Hodd(M) = 0, then M is equivariantly formal.10 (2) Every compact symplectic manifold on which G acts in a Hamiltonian fash- ion11 is equivariantly formal. In partucular, any linear action on a smooth projective variety is equivariantly formal.

The ﬁrst item is completely obvious but the proof of the second item requires the equivariant Morse theory (a.k.a. Atiyah-Bott-Kirwan theory). The point is that if you live with only compact symplectic or algebraic manifolds, equivariant formality is almost always there.

Example. Suppose a connected group G acts on the complex projective space Pn. Since all the odd cohomology groups for Pn vanish, it is equivariantly formal and hence we have an isomorphism of S(G∗)G-modules ∗ n ∼ ∗ ∗ n HG(P ) = H (BG) ⊗ H (P ). More generally, we have the same isomorphism for arbitrary smooth projective varieties (with linear actions), such as Grassmannians and toric varieties.12

Warning. The above isomorphisms are not isomorphisms of rings!!

When G is not connected, the E1 term of the spectral sequence is the invariant subspace ∗ ∗ π0(G) [H (BG0) ⊗ H (M)] where G0 is the identity component of G and π0(G) = G/G0.

10 Obvioiusly, δr for r ≥ 1 vanish. 11Whatever it is, this is a very weak requirement. 12Another way to obtain such an isomorphism is to use the Leray spectral sequence: When there is a fiber bundle π : P → B with fiber F (i.e. B is covered by open sets U such that π−1(U) =∼ U×F and π|π−1(U) is the projection to U) and π1(B) = 1, there is a spectral sequences whose E2-term p,q p q is E2 = H (B)⊗H (F). If F is a compact Kählermanifold, the Leray spectral sequence collapses at E2 by Deligne’s criterion. INTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY 7

3. Classifying maps and characteristic classes 3.1. Classifying maps. A principal G-bundle on a topological space Y is a continuous map π : X → Y such that there is an open cover {Ui} of Y such that −1 ∼ −1 ϕi : π (Ui) = Ui × G and over Ui ∩ Uj, (ϕj ◦ ϕi )(y, g) = (y, aij(y)g) for continuous functions aij : Ui ∩ Uj → G. Such ϕi are called local trivialization and aij are called transition functions. The obvious (right) actions of G on Ui × G glue to give us a free G-action on X and we have X/G = Y. Similarly, we define a complex vector bundle on Y of rank n as a continuous map −1 ∼ π : X → Y such that there is an open cover {Ui} of Y such that ϕi : π (Ui) = n −1 Ui × C and over Ui ∩ Uj, (ϕj ◦ ϕi )(y, v) = (y, aij(y)v) for continuous functions aij : Ui ∩Uj → GL(n). Note that a complex vector bundle of rank n and a principal GL(n)-bundle are given by the same data {aij}. Examples. (1) When a compact Lie group G acts on a manifold X freely, the quotient space X/G is also a manifold and the map X → X/G is a principal G- bundle (fibers are all G). (2) Let π : X → Y be a principal G-bundle and let ρ : G → GL(n) be a rep- n resentation. Then X ×G C → Y is a vector bundle of rank n on Y. Conversely, given a vector bundle, the frame bundle (i.e. the set of all bases of fibers) gives us a principal bundle. In other words, {aij} determine the principal GL(n)-bundle. Topological classification of principal G-bundles is attained by means of the classifying space BG. Theorem 4. Suppose X is a free G-space. Then any section s : X/G → XG = (X × EG)/G (i.e. its composition with XG → X/G is identity on X/G) uniquely determines a G-equivariant map13 h : X → EG which makes the commutative diagram h X / EG

π ρ X/G / BG. f Any two sections are homotopic and the homotopy class of (f, h) is independent of the section s.

Proof (sketch): An element of XG is an orbit {(gx, ge) | g ∈ G} ⊂ Gx × EG ⊂ X × EG which is the graph of an equivariant map Gx → EG. Hence

tGx∈X/Gs(Gx) ⊂ X × EG determines an equivariant map h : X → EG given by s(G · x) = G · (x, h(x)) for all x ∈ X.

Since the ﬁbers of XG → X/G is contractible EG, a section s always exists and any two sections are homotopic.14 ¤

Corollary. Let E1 and E2 be two contractible G-spaces. Then there exist G- equivariant maps φ : E1 → E2 and ψ : E2 → E1 such that φ ◦ ψ and ψ ◦ φ are

13A map f : X → Y of G-spaces is equivariant if f(g · x) = g · f(x). 14An excellent reference for basics on principal bundles is Steenrod’s classic book on ﬁber bundles. 8 YOUNG-HOON KIEM

∗ ∼ ∗ homotopic to identity maps. In particular, H (X ×G E1) = H (X ×G E2) for any G-space X and BG is unique up to homotopy equivalence.

3.2. Characteristic classes. By Whitehead’s theorem, the homotopy class of a continuous map f : Y → BG is determined by the induced map f∗ : H∗(BG) → H∗(Y). (1) G = U(n) (or GL(n)): A principal G-bundle is the same thing as a complex vector bundle of rank n by taking orthonomal frames. From pre- ∗ vious lecture, H (BG) is freely generated by invariant polynomials c1 = tr, c2, c3, ··· , cn = det. Hence, principal G-bundles (or vector bundles) on ∗ Y are determined by the images of c1, ··· , cn in H (Y). The image of ci is called the i-th Chern class of the principal bundle. (2) G = O(n): This case corresponds to real vector bundles of rank n. Principal ∗ G-bundles are determined by the images of p1, p2, ··· in H (Y). We call them the Pontryagin classes of the principal bundle. (3) G = SO(2n): This case corresponds to real oriented vector bundles of rank 2n. We have in addition the pullback of Pfaﬃan. This is called the Euler ∗ class e. By simple linear algebra, we have cn = e via H (BSO(2n)) → H∗(BU(n)). The top Chern class is the Euler class! 3.3. Whitney sum formula. The total Chern class of a complex vector bundle E → Y of rank n is deﬁned as

c(E) = 1 + c1(E) + ··· + cn(E).

WhitneyP sum formula says if E = E1 ⊕ E2 then c(E) = c(E1) · c(E2), i.e. ck(E) = c (E )c (E ). In particular, if E = L ⊕· · ·⊕L for L line bundles (=vector i+j=k i 1 j 2 Q 1 n i bundles of rank 1), then c(E) = (1 + c1(Li)). Example. Let us calculate the Chern classes of the tangent bundle of complex projective space Pn. By the famous Euler sequence ⊕n+1 0 → C → L → TPn → 0 where C = Pn × C is the trivial line bundle and L → Pn is the line bundle whose ﬁber over a point ξ in Pn is the dual space of the line in Cn+1 represented by ξ. (Recall that Pn is the space of lines through 0 in Cn+1.) By Whitney’s formula, we obtain X µ ¶ n+1 n + 1 k c(T n ) = (1 + c (L)) = α P 1 k k 2 n ∗ n where α = c1(L) ∈ H (P ) is the generator of H (P ). In particular, c1(TPn ) = n n+1 (n + 1)α and e(TPn ) = cn(TPn ) = (n + 1)α . Note α = 0. 3.4. Equivariant characteristic classes. Let G and K be compact Lie groups and K × G act on a manifold P. Suppose K = K × {1G} acts freely on P. Then we get an induced action of G on the quotient X = P/K and the quotient map P → X is a G-equivariant principal K-bundle. So we obtain a principal K-bundle

P ×G EG = PG → XG = X ×G EG ∗ ∗ and its characteristic classes in HG(X) = H (XG) are called the equivariant characteristic classes of the principal K-bundle P → X. We will be only interested in the K = U(n) case, i.e. complex vector bundles. INTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY 9

Let π : E → X be a complex vector bundle of rank n which is G-equivariant. Then EG → XG is a vector bundle of rank n and thus we obtain the equivariant Chern classes G ∗ ∗ ci (E) := ci(EG) ∈ H (XG) = HG(X). G G The equivariant Euler class is the top equivariant Chern class e (E) = cn (E). Example (point case). The equivariant Chern classes are not trivial even if the bundle E is trivial (with nontrivial group action). For example, consider a vector bundle E → pt over a point on which a group G acts. This is nothing but a vector space Cn together with a homomorphism G → U(n) = K. This homomorphism induces ∼ ∗ K ∗ G ∼ ∗ R[c1, ··· , cn] = S(K ) → S(G ) = HG(pt). ∗ The equivariant Chern classes of E are the images of cj in HG(pt). If the G-action is not trivial, then the equivariant Chern classes are not trivial. Furthermore, if G is a torus, then any C-representation E of G is the direct sum of ∗ 1-dimensionalQ representations with weights αj ∈ G . The equivariant Chern classes ∼ n are given by j(1 + αj). More concretely, if G = U(1) and G acts on E = C as iθ im θ imnθ e · (v1, ··· , vn) = (e 1 v1, ··· , e vn), then the total equivariant Chern class Qn is j=1(1 + mjt). In particular, the equivariant Euler class is nonzero iff mj 6= 0 for all j. 3.5. Equivariant integration. For compact manifold M acted on by a compact group G, we have a homomorphism Z ∗ ∗ ∗ : HG(M) → HG(pt) = H (BG) M by integrating the differential form part for any dG-closed element in ΩG(M) = [S(G∗) ⊗ Ω(M)]G. We call this homomorphism equivariant integration. 3.6. Reduction to torus. Let G be a compact connected Lie group and T be a maximal torus in G. For example, when G = U(n) (resp. SU(n)), T =∼ U(1)n (resp. U(1)n−1). In many respects, tori are easier than arbitrary compact Lie groups. For a G-manifold M, we have a convenient isomorphism ∗ ∼ ∗ W ∗ HG(M) = HT (M) ⊂ HT (M) where W = N(T)/T is the Weyl group. For instance, W = Sn (resp. Sn−1 for G = U(n) (resp. SU(n)). The above isomorphism follows from the Leray spectral sequence associated to the map MT = M ×T EG → M ×G EG = MG whose fiber is G/T. The spectral ∗ ∼ ∗ ∗ sequence collapses at E2 and thus HT (M) = HG(M) ⊗ H (G/T). Then take W- invariant subspace. When M is a point, we obtain H∗(BG) =∼ S(G∗)G =∼ S(T∗)W =∼ H∗(BT)W ⊂ H∗(BT). 10 YOUNG-HOON KIEM

4. Localization theorem In this section, G is a torus U(1)r.

4.1. Cohomology with compact support. Let M be a manifold and X be a closed submanifold. The relative cohomology H∗(M, X) also has a de Rham theo- retic interpretation: Let Ω(M − X)c ⊂ Ω(M − X) be the subcomplex of differential forms with compact support. We call the cohomology of this subcomplex coho- ∗ mology with compact support and denote it by H (M − X)c. Then we have an isomorphism ∗ ∼ ∗ H (M, X) = H (M − X)c. and a long exact sequence k k k k+1 ··· → H (M − X)c → H (M) → H (X) → H (M − X)c → ··· When there is an action of G, we have an obvious analogue for the equivariant cohomology (by approximation by finite dimensional manifolds) k k k k+1 ··· → HG(M − X)c → HG(M) → HG(X) → HG (M − X)c → ··· . 4.2. Support of S(G∗)-module. Recall that S(G∗) is isomorphic to the poly- ∗ nomial ring R = R[t1, ··· , tr] = H (BG) in r variables and every G-equivariant cohomology is an S(G∗)-module. Suppose A is a finitely generated R-module and let IA be the annihilator ideal IA = {f ∈ R | fA = 0}. Then the support of A is defined as n suppA = {x ∈ C | f(x) = 0 for all f ∈ IA}

Suppose A → B → C is an exact sequence of R-modules. Then IB ⊃ IA ∩ IC and hence suppB ⊂ suppA ∪ suppC. (If not, ∃x ∈ suppB such that f(x) 6= 0, g(x) 6= 0 for some f ∈ IA, g ∈ IC. But then fg ∈ IA ∩ IC ⊂ IB and fg does not vanish at x. Contradiction.)

4.3. Localization theorem. Let M be a compact G-manifold. Our goal is to prove Theorem 5. Let MG be the G-ﬁxed point set and ı : MG → M be the inclusion. ∗ ∼ ∗ ∗ ∗ ∗ G If M is equivariantly formal HG(M) = H (M) ⊗R R, then ı : HG(M) → HG(M ) is injective.

Lemma. Let K be a closed subgroup of G and φ : M → G/K be a G-equivariant ∗ ∗ C n map. Then suppHG(M) and suppHG(M)c are contained in K := K ⊗R C ⊂ C = GC. φ Proof: From M −→ G/K → pt, we get homomorphisms of rings ∗ ∗ ∗ ∗ R = HG(pt) → HG(G/K) = S(K ) → HG(M). ∗ ∗ Hence the R-module structure on HG(M) comes from the S(K )-module structure and hence I ∗ contains all polynomials vanishing on K. ¤ HG(M)

For x ∈ M and any G-invariant tubular neighborhood U,15 we have the obvious projection U → Gx = G/K, where K = Gx is the stabilizer. By the lemma above, ∗ C n we see that the support of HG(U) is contained in K ⊂ C .

15Invariant tubular nbd always exists for compact G by slice theorem. INTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY 11

Suppose M is compact. Then there are only ﬁnitely many closed subgroups which may be the stabilizer group of a point (Mostow’s theorem). Let {Ki | i = 1, 2, ··· , l} be the list of stabilizers.

∗ ∗ ∗ G C Lemma. The kernel of ı : HG(M) → HG(M ) has support in the set ∪iKi . Proof: By the long exact sequence k G k k G k+1 G HG(M − M )c → HG(M) → HG(M ) → HG (M − M )c, ∗ G C it suﬃces to prove that the support of HG(M − M )c is contained in ∪iKi . Delete a small tubular neighborhood U of MG so that M − U is compact. Then ¯ we can cover M − U by invariant tubular neighborhoods Uj of ﬁnitely many orbits

Gxj with stabilizer group Gxj = Ki for some i. By the Mayer-Vietrois sequence k−1 k k k HG (V1 ∩ V2) → HG(V1 ∪ V2) → HG(V1) ⊕ HG(V2) ∗ ¯ ∼ ∗ G C and induction, we see that HG(M − U) = HG(M − M ) has support in ∪iKi . ¤ ∗ ∼ ∗ ∗ When M is equivariantly formal, HG(M) = R ⊗ H (M) and so HG(M) is a free G ∗ G R-module. On the other hand, since G acts trivially on M , HG(M ) is also a ∗ C free R-module. Now the support of the kernel of ı is contained in ∪iKi and thus ∗ ∗ the kernel is a torsion submodule of HG(M). Since HG(M) is free, the torsion submodule has to be zero, i.e. ı∗ is injective. ¤

n Example. Suppose G = U(1) acts on M = P with distinct weights m0, ··· mn, iθ im θ imnθ G i.e. e · (x0 : x1 : ··· : xn) = (e 0 x0 : ··· : e xn). The fixed point set M consists of (n + 1) points, p0, ··· , pn where pi = (0 : ··· : 0 : 1 : 0 ··· : 0) with 1 ∗ ∗ n at i-th place. As a R = R[t]-module, HG(M) = R ⊗ H (P ) is a free R-module of ∗ n rank n + 1. Let ξ = c1(L) be the generator of H (P ). Its restriction to the point pi is −mit because the restriction of L to pi has weight −mi. By the localization Q ∗ n theorem,Q i(ξ + mit) is a relation in HG(P ) and this is the only relation because R[ξ]/( (ξ+mit)) is already a free module of rank n+1. In summary, we obtained an isomorphism of rings Yn ∗ n HU(1)(P ) = R[t, ξ]/( (ξ + mit)). i=0 4.4. Integration formula. As an application of the localization theorem, we obtain a useful integration formula, which basically says all integrals are concentrated at the fixed point set. Let F be the set of fixed point components of the G-action on M. Each F ∈ F is a submanifold and suppose the normal bundle NF to F is oriented of even rank. 16 k−c k In this case, we have the Gysin map ı∗ : HG (F) → HG(M) where c is the real ∗ k k codimension of F. When composed with the restriction ı : HG(M) → HG(F), it gives us multiplication by the equivariant Euler class of the normal bundle ∗ k−c k ı ◦ ı∗ = e(NF): HG (F) → HG(F). R ∗ ∗ Recall that we have the equivariant integration M : HG(M) → HG(pt) = H∗(BG) for compact M.

16 This isR obtained byR multiplying a diﬀerential form α supported in a small neighborhood of F such that M η ∧ α = F η for all closed diﬀerential form η on F. 12 YOUNG-HOON KIEM

Theorem 6 is the most important tool for Gromov-Witten theory and geometry of symplectic quotients.

Wisdom of the area. For many problems with equivariant cohomology, two things are quite useful: ∗ ∗ W (1) reduction to torus (HG(M) = HT (M) , Martin’s trick,...), (2) reduction to ﬁxed point set (localization theorem). Good luck!!

References 1. M. Atiyah and R. Bott. Yang-Mills equations on Riemann surfaces. Proc. Royal Acad. Sci. 1983. 2. V. Gullemin. Supersymmetry and equivariant de Rham theory.

Department of Mathematics and Research Institute of Mathematics, Seoul National University, Seoul 151-747, Korea E-mail address: [email protected]