GENERALIZING THE LOCALIZATION FORULA IN EQUIVARIANT COHOOLOGY Abstract. We give a generalization of the Atiyah-Bott-Berline-Vergne localization theorem for the equivariant cohomology of a torus action. We replace the manifold having a torus action by an equivariant map of a compact connected Lie group. This provides a systematic method for calculating the Gysin homomorphism in ordinary cohomology of an equivariant map. As an example, we recover a formula of Akyildiz-Carrell for the Gysin homomorphism of flag manifolds. Suppose is a compact oriented manifold on which a torus T acts. The Atiyah- Bott-Berline-Vergne localization formula calculates the integral of an equivariant cohomology class on in terms of an integral over the fixed point set T. This formula has found many applications, for example, in analysis, topology, symplectic geometry, and algebraic geometry (see [2], [8], [10], [14]. Similar, but not entirely analogous, formulas exist in K-theory ([3], cobordism theory ([12], and algebraic geometry ([9]. Taking cues from the work of Atiyah and Segal in K-theory [3], we state and prove a localization formula for a compact connected Lie group in terms of the fixed point set of a conjugacy class in the group. As an application, the formula can be used to calculate the Gysin homomorphism in ordinary cohomology of an equivariant map. For a compact connected Lie group G with maximal torus T and a closed subgroup H containing T, we work out as an example the Gysin homomorphism of the canonical projection f : G/T G/H, a formula first obtained by Akyildiz and Carrell [1]. The application to the Gysin map in this article complements that of [14]. The previous article [14] shows how to use the ABBV localization formula to calculate the Gysin map of a fiber bundle. This article shows how to use the relative localization formula to calculate the Gysin map of an equivariant map. We thank Alberto Arabia, Fulton Gonzalez, Fabien orel, and ichèle Vergne for many helpful discussions. 1. Borel-type localization formula for a conjugacy class Suppose a compact connected Lie group G acts on a manifold. For g G, define g to be the fixed point set of g: g = {x g x = x}. The set g is not G-invariant. The G-invariant subset it generates is h G h ( g = h G hgh 1 = k C(g k Date: November 30, 2004. 2000 athematics Subject Classification. Primary: 55N25, 57S15; Secondary: 1415. Key words and phrases. Atiyah-Bott-Berline-Vergne localization formula, push-forward, Gysin map, equivariant cohomology, group action. The first author was supported in part by the Pedroza Foundation. The second author acknowledges the hospitality and support of the Institut Henri Poincaré the Institut de athématiques de Jussieu, Paris. 1
2 where C(g is the conjugacy class of g. This suggests that for every conjugacy class C in G, we consider the set C of elements of that are fixed by at least one element of the conjugacy class C: C = g C g. Then C is a closed G-subset of ([3], footnote 1, p. 532; however it is not always smooth. From now on we make the assumption that C is smooth. Remark 1.1. If T is a maximal torus in the compact connected Lie group G and dim T = l, then H (BG = H (BT WG = Q[u 1,..., u l ] WG. Thus, H (BG is an integral domain. Let Q be its field of fractions. For any H (BG-module W, we define the localization of W with respect to the zero ideal in H (BG to be Ŵ := W H (BG Q. It is easily verified that W is H (BG-torsion if and only if Ŵ = 0. For a G- manifold, we call ĤG ( the localized equivariant cohomology of. Proposition 1.2. Let be a T -manifold with a finite number of orbit types and no fixed points. Then HT ( is torsion. Proof. For p in let t p be the Lie algebra of the stabilizer of p. Since the T action on has finitely many orbit types and has no fixed points, it follows that only a finite number of proper subspaces t 1, t 2,... t r of t can occur as t p for p in. Let v be a vector in t not in the union of the t i. Then the induced vector field v in is nowhere zero. Consider the invariant 1-form η on defined by η(x = v, X. Thus η(v 1 and d C (η = dη u 1. Therefore and Note that 1 = 1 dη u 1 = d C ( 1 dη 1 = dη + u 1 u 1 ( ηu1 } ( {d C ( ηu1 1 dη 1. u 1 ( 2 ( k dη dη + +. Since d C (dη = u j ι j dη = u j dι j η = u j d(δ j 1 = 0 it follows that 1 = d C { η ( 1 dη } 1. u 1 u 1 u 1 u 1 Hence HT ( is torsion. Lemma 1.3. If H T ( is H (BT -torsion, then H G ( is H (BG-torsion. Proof. Recall that the map ψ : HG ( H T ( is injective. Since H T ( is H (BT -torsion, there is a H (BT such that a 1 H T ( = 0. Consider the average of a over the Weyl group W of G with respect to T, ã = 1 W (a + ω 1a + + ω r a H (BG.
LOCALIZATION FORULAS IN EQUIVARIANT COHOOLOGY 3 Under ψ, the element ã 1 H G ( goes to 1 W (ω 1a + + ω r a1 H T (. But (ω j a1 H T ( = ω j (a1 H T ( = 0 for any j. Thus ã 1 H G ( = 0 in H G (. Proposition 1.4. Let G be a compact connected Lie group acting on a compact manifold. Let U be a G-invariant set such that U C is empty, then the equivariant cohomology H G (U is H (BG-torsion. Proof. The set U C can be described U C = {p U : hsh 1 G p for some h G} where S is the closure of the group generated by some g C. Let T a maximal torus containing S. For every p, T is not contained in hg p h 1 for all h G. Therefore the fixed point set of with respect to T is empty. Since is compact, U has only a finite number of orbit types. Thus from Prop. 1.2 and Lemma 1.3 it follows that HG (U is torsion. In the rest of this section, torsion will mean H (BG-torsion. Theorem 1.5 (Borel-type localization formula for a conjugacy class. Let G be a compact connected Lie group acting on a compact space, and C a conjugacy class in G. Then the inclusion i : C induces an isomorphism in localized equivariant cohomology i : Ĥ G( Ĥ G( C. Proof. Let U be a G-invariant tubular neighborhoods of C. Then {U, C } is a G-invariant open cover of. oreover, H G (U H G ( C because U has the G-homotopy type of C. By Prop. 1.4, H G ( C and H G (U ( C are torsion. Then in the localized ayer-vietoris sequence... Ĥ 1 G (U ( C Ĥ G ( Ĥ G ( C Ĥ G (U Ĥ G (U ( C..., all the terms except Ĥ G ( and Ĥ G (U are zero. It follows that is an isomorphism of H (BG-modules. Ĥ G( Ĥ G(U Ĥ G( C When the group is a torus T, a conjugacy class C consist of a single element t T. If t is generator, it follows that the fixed point set of t is the same as the fixed point set of the whole group T, C = t = T. Observe that in this case C is smooth. Thus Borel s localization theorem follows from Theorem 1.5 by taking the conjugacy class C = {t} in T. 2. The equivariant Euler class Suppose a compact connected Lie group G acts on a smooth manifold. Let C be a conjugacy class in G, and C as before. From now on we assume that C is smooth with oriented normal bundle. Denote by i : C the inclusion map and by e H G ( C the equivariant Euler class of the normal bundle of C in.
4 Proposition 2.1. Let be a compact connected oriented G-manifold. Then the equivariant Euler class e of the normal bundle of C in is invertible in ĤG ( C. Proof. Fix a G-invariant riemannian metric on. Then the normal bundle ν C is a G-equivariant vector bundle. Let ν 0 be the normal bundle minus the zero section. Since ν 0 is equivariantly diffeomorphic to an open set in C, by Prop. 1.4 Ĥ G (ν 0 vanishes. From the Gysin long exact sequence in localized equivariant cohomology Ĥ G (ν 0 Ĥ G ( C e Ĥ G ( C Ĥ G (ν 0 it follows that multiplication by the equivariant Euler class gives an automorphism of Ĥ G ( C. Thus e has an inverse in Ĥ G ( C. Recall that the inclusion map i : C satisfies the identity i i (x = xe. in equivariant cohomology. From this identity and Prop. 2.1, the map ϕ : ĤG ( C Ĥ G ( given by ( x ϕ(x = i is the inverse of the restriction map i : Ĥ G ( Ĥ G ( C of Theorem 1.5. e 3. Relative localization formula Let N be a G-manifold, the equivariant Euler class of the normal bundle of N C, and f : N a G-equivariant map. There is a commutative diagram of maps i C (1 Let f C N C N. i N (f G : Ĥ G( Ĥ G(N, f C : Ĥ G( C Ĥ G(N C be the push-forward maps in localized equivariant cohomology. Theorem 3.1 (Relative localization formula. Let and N be compact oriented manifolds on which a compact connected Lie group G acts, and f : N a G-equivariant map. For a HG (, (f G a = (i N 1 f C C i a where the push-forward and restriction maps are in localized equivariant cohomology. Proof. The commutative diagram (1, induces a commutative diagram in localized equivariant cohomology Ĥ G ( C (f C (f G (2 ĤG (N C ĤG (N. f e i i N Ĥ G (
LOCALIZATION FORULAS IN EQUIVARIANT COHOOLOGY 5 By Prop. 2.1 and the commutativity of the diagram (2, 1 (f G a = (f G i i e (a ( 1 = i N (f C i e a. Hence, ( 1 i N (f G a = i N i N f C i e a ( 1 = f C i e a = (f C C i e a (projection formula. By Theorem 1.5, i N is an isomorphism in localized equivariant cohomology, (f G a = (i N 1 (f C C i e a. Suppose a torus T acts on compact oriented manifolds and N, and f : N is a T -equivariant map. The map f induces a map f T : T N T of the fixed point sets. Recall that for a singular element t in T, t = T. In this case, T is always a manifold. Let i : T be the inclusion and e the equivariant Euler class of the normal bundle to T in, and similarly for i N and. Corollary 3.2 (Relative localization formula for a torus action. Let and N be manifolds on which a torus T acts, and f : N a T -equivariant map with compact oriented fibers. For a Ĥ T (, (f T a = (i N 1 (f T T i e a, where the push-forward and restriction maps are in localized equivariant cohomology. This formula appears in the work of Lian, Liu and Yau in [11]. 4. Applications to the Gysin homomorphism Let G be a compact connected Lie group. For a G-manifold, let h : G be the inclusion of as a fiber of the bundle G BG and i : G the inclusion of the fixed point set G in. The map h induces a homomorphism in cohomology h : H G( H (. The inclusion i induces a homomorphism in equivariant cohomology i : H G( H G( G. A cohomology class a H ( is said to have an equivariant extension ã H G ( under the G action if under the restriction map h : H G ( H (, the equivariant class ã restricts to a. Suppose f : N is a G-equivariant map of compact oriented G-manifolds. In this section we show that if a class in H ( has an equivariant extension, then its image under the Gysin map f : H ( H (N in ordinary cohomology can be computed from the relative localization formulas (Cor. 3.2 or Th. 3.1.
6 We consider first the case of an action by a torus T. Let f T : T N T be the induced map of homotopy quotients and f T : T N T the induced map of fixed point sets. As before, e denotes the equivariant Euler class of the normal bundle of the fixed point set T in. Proposition 4.1. Let f : N be a T -equivariant map of compact oriented T -manifolds. If a cohomology class a H ( has an equivariant extension ã H T (, then its image under the Gysin map f : H ( H (N is, 1 in terms of equivariant integration over : f a = h Nf T ã, 2 in terms of equivariant integration over the fixed point set T : f a = h N (i N 1 (f T T i e ã. Proof. The inclusions h : T and h N : N N T fit into a commutative diagram h f N N T. h N This diagram is Cartesian in the sense that is the inverse image of N under f T. Hence, the push-pull formula f h = h N f T holds. Then T f T f a = f h ã = h N f T ã. 2 follows from 1 and the relative localization formula for a torus action (Cor. 3.2. Using the relative localization formula for a conjugacy class, one obtains analogously a push-forward formula in terms of the fixed point sets of a conjugacy class. Now h and i are the inclusion maps h : G, i : C, e is the equivariant Euler class of the normal bundle of C in, and f C : C N C is the induced map on the fixed point sets of the conjugacy class C. Proposition 4.2. Let f : N be a G-equivariant map of compact oriented G- manifolds. Assume that the fixed point sets C and N C are smooth with oriented normal bundle. For a class a H ( that has an equivariant extension ã HG (, f a = h N (i N 1 (f C C i e ã. 5. Example: the Gysin homomorphism of flag manifolds Let G be a compact connected Lie group with maximal torus T, and H a closed subgroup of G containing T. In [1] Akyildiz and Carrell compute the Gysin homomorphism for the canonical projection f : G/T G/H. In this section we deduce the formula of Akyildiz and Carrell from the relative localization formula in equivariant cohomology. Let N G (T be the normalizer of the torus T in the group G. The Weyl group W G of T in G is W G = N G (T /T. We use the same letter w to denote an element
LOCALIZATION FORULAS IN EQUIVARIANT COHOOLOGY 7 of the Weyl group W G and a lift of the element to the normalizer N G (T. The Weyl group acts on G/T by (gt w = gwt for gt G/T and w W G. This induces an action of W G on the cohomology ring H (G/T. We may also consider the Weyl group W H of T in H. By restriction the Weyl group W H acts on G/T and on H (G/T. To each character γ of T with representation space C γ, one associates a complex line bundle L γ := G T C γ over G/T. Fix a set + (H of positive roots for T in H, and extend + (H to a set + of positive roots for T in G. Theorem 5.1 ([1]. The Gysin homomorphism f : H (G/T H (G/H is given by, for a H (G/T, w W f a = H ( 1 w w a α + (H c 1(L α. Remark 5.2. There are two other ways to obtain this formula. First, using representation theory, Brion [7] proves a push-forward formula for flag bundles that includes Th. 5.1 as a special case. Secondly, since G/T G/H is a fiber bundle with equivariantly formal fibers, the method of [14] using the ABBV localization theorem also applies. To deduce Th. 5.1 from Prop. 4.1 we need to recall a few facts about the cohomology and equivariant cohomology of G/T and G/H (see [13], [14]. Cohomology ring of BT. Let ET BT be the universal principal T -bundle. To each character γ of T, one associates a complex line bundle S γ over BT : S γ := ET T C γ. For definiteness, fix a basis χ 1,..., χ l for the character group ˆT, where we write the characters additively, and set u i = c 1 (S χi H 2 (BT, z i = c 1 (L χi H 2 (G/T. Let R = Sym( ˆT be the symmetric algebra over Q generated by ˆT. γ c 1 (S γ induces an isomorphism R = Sym( ˆT H (BT = Q[u 1,..., u l ]. The map γ c 1 (L γ induces an isomorphism R = Sym( ˆT Q[z 1,..., z l ]. The map The Weyl groups W G and W H act on the characters of T and hence on R: for w W G and γ ˆT, (w γ(t = γ(w 1 tw. Cohomology rings of flag manifolds. The cohomology rings of G/T and G/H are described in [5]: H (G/T H (G/H R + Q[z 1,..., z l ] (R+ WG, (R WG RWH (R WG + Q[z 1,..., z l ]WH (R+ WG, where (R WG + denotes the ideal generated by the W G-invariant homogeneous polynomials of positive degree.
8 The torus T acts on G/T and G/H by left multiplication. Their equivariant cohomology rings are (see [6], [13] HT (G/T = Q[u 1,..., u l, y 1,..., y l ], J HT (G/H = Q[u 1,..., u l ] (Q[y 1,..., y l ] WH, J where J denotes the ideal generated by q(y q(u for q R WG +. Fixed point sets. The fixed point sets of the T -action on G/T and on G/H are the Weyl group W G and the coset space W G /W H respectively. Since these are finite sets of points, H T (W G = w WG H T ({w} w WG R, H T (W G/W H = wwh W G/W H R. Thus, we may view an element of HT (W G as a function from W G to R, and an element of HT (W G/W H as a function from W G /W H to R. Let h : T be the inclusion of as a fiber in the fiber bundle T BT and i : T the inclusion of the fixed point set T in. Note that i is T -equivariant and induces a homomorphism in T -equivariant cohomology, i : H T ( H T ( T. In order to apply Prop. 4.1, we need to know how to calculate the restriction maps h : H T ( H ( and i : H T ( H T ( T as well as the equivariant Euler class e of the normal bundle to the fixed point set T, for = G/T and G/H. This is done in [13]. Restriction and equivariant Euler class formulas for G/T. Since h : HT ( H ( is the restriction to a fiber of the bundle T BT, and the bundle K χi = (L χi T on T pulls back to L χi on, (3 h (u i = 0, h (y i = h (c 1 (K χi = c 1 (L χi = z i. Let i w : {w} G/T be the inclusion of the fixed point w W G and i w : H T (G/T H T ({w} = R the induced map in equivariant cohomology. By ([13], Prop. 2, for p(y HT (G/T, (4 i wu i = u i, i wp(y = w p(u, i wc 1 (K γ = w c 1 (S γ. Thus, the restriction of p(y to the fixed point set W G is the function i p(y : W G R whose value at w W G is (5 (i p(y(w = w p(u. The equivariant Euler class of the normal bundle to the fixed point set W G assigns to each w W G the equivariant Euler class of the normal bundle ν w at w; thus, it is also a function e : W G R. By ([13], Prop. 6, (6 e (w = e T (ν w = w c 1 (S α = ( 1 w c 1 (S α. α + α +
LOCALIZATION FORULAS IN EQUIVARIANT COHOOLOGY 9 Restriction and equivariant Euler class formulas for G/H. For the manifold = G/H, the formulas for the restriction maps h N and i N are the same as in (3 and (4, except that now the polynomial p(y must be W H -invariant. In particular, (7 h N (u i = 0, h N p(y = p(z, h N (c 1(K γ = c 1 (L γ, and (8 (i N p(y(ww H = w p(u. If γ 1,..., γ m are characters of T such that p(c 1 (K γ1,..., c 1 (K γm is invariant under the Weyl group W H, then (9 (i Np(c 1 (K γ1,..., c 1 (K γm (ww H = w p(c 1 (S γ1,..., c 1 (S γm. The equivariant Euler class of the normal bundle of the fixed point set W G /W H is the function : W G /W H R given by (10 (ww H = w c 1 (S α. α + + (H Proof of Th. 5.1. With = G/T and N = G/H in Prop. 4.1, let p(z H (G/T = Q[z 1,..., z l ]/(R WG +. It is the image of p(y H T (G/T under the restriction map h : H T (G/T H (G/T. By Prop. 4.1, (11 f p(z = f h p(y = h N f T p(y and and f T p(y = (i N 1 (f T T By Eq. (5, (6, and (10, for w W G, e (i p(y(w = i wp(y = w p(u, i p(y. T ( (w = (f T ( (ww H α = w + + (H c 1(S α e e (w α c 1(S + α 1 = (. w α + (H c 1(S α To simplify the notation, define temporarily the function k : W G R by ( p(u k(w = w α + (H c. 1(S α Then (12 f T p(y = (i N 1 (f T (k. Now (f T (k HT (W G/W H is the function: W G /W H R whose value at the point ww H is obtained by summing over the fiber of f T : W G W G /W H above ww H. Hence, ((f T k(ww H = ( p(u wv wv ww H α + (H c 1(S α ( p(u = w v v W H α + (H c. 1(S α
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