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Introduction to Equivariant Cohomology Theory 11

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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 di®erentiable group operations mult : G £ G ! G, inv : G ! G, id 2 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) = feiθ j θ 2 Rg = S1. Since a complex linear (reductive) group is homotopy equivalent1 to its maximal compact subgroup, it su±ces 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 = fg 2 G j gx = xg of every point x 2 X is the trivial subgroup. A topological space X is called contractible if there is a homotopy equivalence with a point (i.e. 9h : X £ [0; 1] ! X such that h(x; 0) = x0, h(x; 1) = x for x 2 X). Theorem 1. For each compact Lie group G, there exists a contractible topological space EG on which G acts freely. Proof: Since G ½ GL(n) for some n, it su±ces 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!1 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 in¯nity. 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 9f : 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 ½ 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 coe±cients), then f is a homotopy equivalence. 1 2 YOUNG-HOON KIEM De¯nition. 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 ¯ber of the quotient map ¼ : EG ! BG is an orbit Gx, homeomorphic to G. Examples. 1 1 1 1 1 2n+1 1 1 (1) ES = S , BS = S =S = limn!1 S =S = CP . (Exercise: Prove 1 1 1 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) = CP1 £ ¢ ¢ ¢ £ CP1 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 ¯bers are G=H. 1.3. Equivariant cohomology. Suppose a topological group G acts on a topo- logical 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 Xy which is homotopically equivalent to X. The obvious choice is Xy = X £ EG on which G acts diagonally g ¢ (x; e) = (gx; ge). De¯nition. 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 coe±cients R for convenience. Borel's diagram: X EG EG o X £ EG / X G G Gx ² ² ² o / BG X £G EG X=G X BGx 1.4. Basic properties. Almost all basic properties follow directly from the de¯- nition. (1) Suppose G acts freely on X. Then the ¯bers 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 ¯ber EG.) 4We will prove this later. 5For convenience, we will use real coe±cients for cohomology groups in this lecture course but there is no privilege for R in equivariant cohomology theory. Most authors prefer Q coe±cients. 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 di®er- 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 jt=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 di®erential forms on M form a complex d d d 0 / ­0(M) / ­1(M) / ­2(M) / ¢ ¢ ¢ : A theorem of de Rham says that the cohomology H¤(M) with real coe±cients is isomorphic to the cohomology of (­(M); d). When there is an action of G on M, we get an induced action on ­(M). The main theorem of this section is the following due to H. Cartan. i j ¤ i-2j G G Theorem 2. Let ­G(M) = ©j[S (G ) ­ ­ (M)] where [-] denotes the 6 i i+1 G-invariant subspace. Let dG : ­G(M) ! ­G (M) be de¯ned as follows: For a P 7 basis f»ag of G and its dual basis ffag, dG(F ­ σ) = F ­ dσ - a faF ­ {»a σ. Then ¤ the equivariant cohomology HG(M) of M is the cohomology of the complex / 0 dG / 1 dG / 2 dG / 0 ­G(M) ­G(M) ­G(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 su±ces to ¯nd the de Rham complex for M£EG and take the invariant (horizontal) part. For M, we have the ordinary de Rham complex ­(M) but what is the de Rham complex for the in¯nite dimensional contractible space EG? Recall that EG = limm!1 Am and hence ­(EG) = limà ­(AM). Then one ¯nds that ­(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 ([­(M) ­ W]hor): Then by pure algebra, we obtain an isomorphism ¤ » ¤ H ([­(M) ­ W]hor) = H (­G(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 2 G, we consider the coe±cients ci(A) of the characteristic polynomial 6 We think of an element in ­G(M) as a di®erential form valued polynomial function de¯ned on G¤.
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