The Mod 3 Homology of the Space of Loops on the Exceptional Lie Groups and the Adjoint Action

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1 The Mod Homology of the Space of Loops on the Exceptional Lie Groups and the Adjoint Action Hiroaki HAMANAKA Department of Mathematics, Kyoto University and Shin-ichiro HARA Nagaoka University of Technology. July 10, Introduction Let p be a prime number and G be a compact, connected, simply connected and simple Lie group. Let ΩG be the loop space of G. Bott showed H (ΩG; Z/p) is a finitely generated bicommutative Hopf algebra concentrated in even degrees, and determined it for classical groups G ([1]). Here, let G be an exceptional Lie group, that is, G = G, F 4, E 6, E 7, E 8. In [], K.Kozima and A.Kono determined H (ΩG; Z/) as a Hopf algebra over A, where A p is the mod p Steenrod Algebra and acts on it dually. Let Ad : G G G and ad : G ΩG ΩG be the adjoint actions of G on G and ΩG respectively. In [], the cohomology maps of these adjoint actions are studied and it is shown that H (ad; Z/p) = H (p ; Z/p) where p is the second projection if and only if H (G; Z) is p-torsion free. For p =, and 5, some exceptional Lie groups have p-torsions on its homology. Partially supported by JSPS Research Fellowships for Young Scientists. 1

2 Moreover in [8, 9] mod p homology map of ad is determined for (G, p) = (G, ), (F 4, ), (E 6, ), (E 7, ) and (E 8, 5). This result is applied to compute the A 5 module structure of H (ΩE 8 ; Z/5) and H (E 8 ; Z/5) in [9]. For a compact and connected Lie group G, the free loop group of G is denoted by LG(G), i.e. the space of free loops on G equipped with multiplication as φ ψ(t) = φ(t) ψ(t), and has ΩG as its normal subgroup. Then LG(G)/ΩG = G, and identifying elements of G with constant maps from S 1 to G, LG(G) is equal to the semi-direct product of G and ΩG. This means that the homology of LG(G) is determined by the homology of G and ΩG as module and the algebra structure of H (LG(G); Z/p) depends on H (ad; Z/p) where ad : G ΩG ΩG is the adjoint map. Since the next diagram commutes where λ,λ and µ are the multiplication maps of ΩG, LG(G) and G respectively and ω is the composition (1 ΩG T 1 G ) (1 ΩG G ad 1 G ) (1 ΩG G 1 ΩG G ), ΩG G ΩG G ω ΩG ΩG G G λ µ ΩG G = = = LG(G) LG(G) λ LG(G) we can determine directly the algebra structure of H (LG(G); Z/p) by the knowledge of the Hopf algebra structures of H (G; Z/p), H (ΩG; Z/p) and induced homology map H (ad; Z/p). See Theorem 6.1 of [8] for detail. In this paper we determine the Hopf algebra structure over A of the homology group H (ΩG; Z/) for G = F 4, E 6, E 7 and E 8 by using adjoint action and determine the mod homology map of ad for them. The result is shown in. This paper is organized as follows. We refer to the results of [4, 5, 6] for the structure of H (G) and compute H (ΩG) for the lower dimensions and their cohomology operations are partially determined. This is done in.

3 In 4 we turn to their homology rings. We determine the algebra structure of H (ΩG ; Z/) and we partly determine the Hopf algebra structure and cohomology operations on H (ΩG; Z/). Finally in 5 the homology map of the adjoint action and the rest of the Hopf algebra structure and cohomology operations are determined. The computations are completely algebraic. Results Let G(l) be the compact, connected, simply connected and simple exceptional Lie group of rank l where l = 4, 6, 7 or 8. The exponents of G(l) are the integers n(1) < n() < < n(l) which are given by the following table : l n(1), n(),..., n(l) Put E(l) = {n(1),, n(l)} and φ(t) = (t) (t t) where is the diagonal map. P k is the dual of the Steenrod operation P k. Then the results are following : Therorem 1. As a Hopf Algebra over A, H (ΩG(l); Z/) = { Z/[tj j E(l) {}]/(t ), if l = 4, 6, 7 Z/[t j j E(8) {, 9}]/(t, t 6 ), if l = 8 where t j = j. φ(t j ) = P r 0, if j, 9, t t t t, if j =, t t 6 t + t t 6 t t 6 t 6 t t 6 t t 6 t t 6 t t 6 t 6 t 6 + t t t 6 + t t t 6, if j = 9, t j = 0, if r, P 9 t j = { t, if j = 9, 0, otherwise. P 1 t j and P t j are given by the following table:

4 t j t t 6 t 8 t 10 t 14 t 16 t 18 t t 6 t 4 t 8 t 46 t 58 P t 1 j 0 t 0 0 t 10 0 ɛt 14 t t 6 κt 6 ɛt ɛt 10 ɛt 4 ɛt 14 t 18 P t j t 6 0 t 14 t t 6 t 4 0 where ɛ and κ are 1 or 1. (Remark) In Theorem 1, if t j does not exist in H (ΩG(l) ; Z/), we regard t j as 0 for such j. Let Ad : G G G and ad : G ΩG ΩG be the adjoint actions of a Lie group G defined by Ad(g, h) = ghg 1 and ad(g, l)(t) = gl(t)g 1 where g, h G, l ΩG and t [0, 1]. These induce the homology maps Ad : H (G; Z/) H (G; Z/) H (G; Z/) ad : H (G; Z/) H (ΩG ; Z/) H (ΩG ; Z/). Therorem. There are generators y 8 in H (G(l); Z/) for l = 4, 6, 7 and y 8 and y 0 in H (E 8 ; Z/). We can choose these generators so that ad (y i t j ) (i = 8, 0) is given by the following table. t j ad (y 8 t j ) ad (y 0 t j ) t j ad (y 8 t j ) ad (y 0 t j ) t t 10 ɛt t t 10 t 14 t 6 t 14 t 10 t t 6 ɛt t t 6 t 4 t 46 t 8 t 16 t 4 t 14 ɛt 18 t 10 κt 6 t 8 t 46 t 58 t 14 t t 4 t 46 ɛt 18 ɛt t 16 δt 8 t 58 ɛt t 6 t 18 t 6 + t 10 t 6 t t 14 t 6 t 8 + ɛt t 6 t t 6 t 6 where δ, ɛ Z/Z and ɛ 0. For other generators y i H (G(l); Z/), ad (y i t j ) = 0 for all j. The mod cohomology groups We recall the results of [4, 5, 6] for the structure of H (G(l); Z/) as the Hopf algebra over A. 4

5 Therorem. There is an isomorphism : H (G(l); Z/) Λ(x j+1 j E(l) {} {11}) Z/[x 8 ]/(x 8 ), if l = 4, 6, 7, = Λ(x j+1 j E(8) {, 9} {11, 9}) Z/[x 8, x 0 ]/(x 8, x 0 ), if l = 8, the coproduct is given by : x i ϕx i x 11 x 8 x x 15 x 8 x 7 x 17 x 8 x 9 x 7 x 8 x 19 + x 0 x 7 x 5 x 8 x 7 x 8 x 19 + x 0 x 15 + x 8 x 0 x 7 x 9 x 0 x 19 x 47 x 8 x 9 x 0 x 7 x 0 x 8 x 19 + x 0 x 7 others 0 and the cohomology operations are determined by the following table: x i x x 7 x 8 x 9 x 11 x 15 x 17 x 19 x 0 x 7 x 5 x 9 x 47 βx i 0 x x 8 0 x 0 0 x 8 x 0 x 8 x 0 x 0 x 8 x 0 P 1 x i x x 15 ɛx ɛx P x i 0 x 19 x x x 9 x where ɛ is 1 or -1. If r > 1 then P r x i = 0. (Remark) We consider x i in these tables as 0 when x i / H. Recall a Serre fibration: ΩG(l) G(l). (A) First, we compute H (ΩG(l); Z/) by the Serre spectral sequence associated with the fibration (A). This spectral sequence has a Hopf algebra structure. We can proceed to compute it using degree-reason and Kudo s transgression theorem ([7]) from the previous theorem. For j E(l) {9, 11, 9}, there are universally transgressive elements a j H (ΩG(l); Z/), such that τa j = x j+1. Thus we can show that for j = 9, 11, 15, 1, 7 and 9, there are a j such that satisfy 5

6 d 7 (1 a 18 ) = x 7 a 6, for l = 4, 6, 7, d 11 (1 a 0 ) = x 11 a 10, for l = 4, 6, 7, d 15 (1 a 4 ) = x 15 a 14, for l = 8, d 19 (1 a ) = x x 8 a, for l = 4, 6, 7, 8, d 19 (1 a 54 ) = x 19 a 18, for l = 8, d 47 (1 a 58 ) = x 7 x 0 a 6, for l = 8. a j s are generators of the cohomology group in the low dimensions. The results are the following: Proposition 4. For the dimensions less than n(l) +, the next isomorphism holds : H (ΩG(l); Z/) = Z/[a j j E(l) {9}]/(a 9 ), if l = 4, 6, Z/[a j j E(7) {15}]/(a 10 ), if l = 7, Z/[a j j E(8) {1, 7}]/(a 7, a 14 ), if l = 8. Now we start to determine the cohomology operations and the coproducts on a j. Therorem 5. For j E(l) {9, 11, 9} a j H (ΩG(l); Z/) is primitive and cohomology operations are determined by a j a a 8 a 10 a 14 a 16 a 6 a 4 a 8 a 46 P 1 a j a 0 a 14 9 ɛa 0 0 ɛa P a j a 6 0 a 8 a If r > 1 then P r a j = 0. Proof. For j E(l) {9, 11, 9}, a j is transgressive, therefore P i a j = P i σx j+1 = σp i x j+1. Thus this can be determined by Theorem. For the investigation of a j which is not transgressive we start from the following theorem. In the next theorem, ψ means the coproduct of H (ΩG; Z/) and we set ψ(a) = ψ(a) (a a). 6

7 Therorem 6. For j = 9, 15, 1, 7, ψa j is given by the following formula: ψa j = a a 6 + a 6 a, if j = 9, a 10 a 10 + a 10 a 10, if j = 15, a 14 a 14 + a 14 a 14, if j = 1, a 9 a 18 + a 18 a 9, if j = 7. Proof. To begin with, we investigate the element a 18. Let a be the generator of H (ΩF 4 ; Z). H (ΩF 4 ; Z) has no torsion and is a commutative Hopf algebra over Z. Since a 9 = 0, there is a 18 such that a 9 = a 18 and ρa 18 0, where ρ is modulo reduction. Then we can choose a 18 as ρa 18. The coproduct of a 18 is computed as follows: ψa 18 = 1/ψa 9 = 1/(1 a + a 1) 9 a a a 6 + a 6 a + 1 a 18 (mod ). Thus ψa 18 = a a 6 + a 6 a is shown. Consider the inclusion ι : F 4 E 7, we chose a 18 H (ΩE 7 ; Z/) so as to satisfy (Ωι) a 18 = a 18. Because (Ωι) is injective for degrees less than 18, ψa 18 = a a 6 + a 6 a is shown again for this a 18. And in the similar way we put a 0 = 1/a 10, a 4 = 1/a 14 and a 54 = 1/a 7 and obtain the coproduct formulas of the statement. We remark that we can assume that a and a 58 are primitive. Therorem 7. In Proposition 4 we have that P 1 a 18 = ±a. Let G(l) be the connected cover of G(l) and Ω G(l) G(l) (B) G(l) p G(l) i K(Z, ) (C) Ω G(l) Ωp ΩG(l) Ωi K(Z, ) (D) be Serre fibrations. To prove Theorem 7 we have to compute H (Ω G; Z/) and H ( G; Z/). Let ã j be (Ωp) a j, for j 1. Using the Serre spectral sequence associated with the fibration (D), one can easily show that there are generators ã 17 H 17 for l = 4, 6, and ã 5 H 5 for l = 8. We have the following proposition. Let denote E(l) {1} as Ẽ(l). 7

8 Proposition 8. For the dimensions less than n(l) +, the next isomorphism holds : H (Ω G(l); Z/) Z/[ã j j Ẽ(l) {9}] Λ(ã 17), if l = 4, 6, = Z/[ã j j Ẽ(7) {15}]/(ã 10), if l = 7, Z/[ã j j Ẽ(8) {1, 7}]/(ã 14) Λ(ã 5 ), if l = 8. By computing the Serre spectral sequence associated with (B), it is easy to see ã j, (j 15, 1) is universally transgressive. Let x i+1 be τã i. Then we have the following: Proposition 9. For the dimensions less than n(l) +, the next isomorphism holds : H ( G(l); Z/) Λ( x j+1 j Ẽ(l) {9}) Z/[ x 18], if l = 4, 6, = Λ( x j+1 j Ẽ(7)), if l = 7, Λ( x j+1 j Ẽ(8) {7}) Z/[ x 54], if l = 8. Proof of Theorem 7. It is possible to show that P 1 a 18 is not zero as follows. Let σ denotes the cohomology suspension associated to the fibration (C) for l = 4. It is easy to see x 19 = σ βp P 1 u and x = σ (βp 1 u ), where u is the generator of H (K(Z, ); Z/). So we get P 1 x 19 = σ P 1 βp P 1 u = σ P 4 βp 1 u = σ (βp 1 u ) = x, and from this, we have (Ωp) P 1 a 18 = P 1 (Ωp) a 18 = P 1 ã 18 = P 1 σ x 19 = σp 1 x 19 = σ x = ã, where σ is the cohomology suspension associated to (B). Thus P 1 a We fix a as P 1 a Homology groups Therorem 10. The homology ring of ΩG(l) is { H (ΩG(l); Z/) Z/[tj j E(l) {}]/(t ), if l = 4, 6, 7 = Z/[t j j E(8) {, 9}]/(t, t 6 ), if l = 8. (1) where t j = j. The coproduct is given by 0, if j, 9, 11, 9, t φ(t j ) = t t t, if j =, t t 6 t + t t 6 t t 6 t 6 t t 6 t t 6 t t 6 t t 6 t 6 t 6 + t t t 6 + t t t 6, if j = 9. 8

9 Proof. Let t j be the dual element of a j H (ΩG; Z/) as to the monomial basis for j E(l) {9} and t 6, t 18 be the dual element of a, a 9, respectively. It is easy to see t = t 6 = 0 and to show the coproduct formula for t 6 and t 18. Thus we can say that statement (1) is true for < n(l) +. Now it is possible to show that there is no truncation in H (ΩG; Z/) other than the parts generated by t and t 6 and that (1) holds for all dimensions. Since H (ΩG(l); Z/) is the even degree concentrated commutative Hopf algebra, we may suppose H (ΩG(l); Z/) = Z/[u i i I] Z/[v j j J]/(v r j j j J). Consider an Eilenberg - Moore spectral sequence : E = Ext H (ΩG(l):Z/)(Z/, Z/) = E = Gr(H (G(l); Z/)). Since E = Λ(su i i I) Λ(sv j j J) Z/[θv j j J], where deg su i = (1, u i ), deg sv j = (1, v j ), and deg θv j = (, r j v j ), the essential differentials have the forms : d r su i = (θv j ) k j (k j 1) or d r sv j = (θv j ) l j (l j 1). Because H (G(l); Z/) is a finite dimensional vector space, one can easily show E = Λ(su i i I ) Λ(sv j j J ) Z/[θv j j J]/((θv j ) m j j J), (I I, J J) and I + J = I. Here the total dimension of E is I + J j J m j, (m j 1) and the total dimension of H (G(l); Z/) is E(l) f(l) where f(l) = 1 for l = 4, 6, 7 and f(l) = for l = 8. Thus the indices J of the truncation part satisfy that J f(l) and I = E(l). This means that the truncation parts of H (ΩG; Z/) is generated by only t and t 6. Therefore H (ΩG(l) ; Z/) has the form Z/[u i i I] Z/[t ]/(t ) for l = 4, 6, 7 and Z/[u i i I] Z/[t, t 6 ]/(t, t 6 ) for l = 8. Also Theorem 5 means that for j E(l) {9} t j is primitive and indecomposable and t 6, t 18 are indecomposable. Thus {t j j Ẽ(l)} {t 6} {u i i I} for l = 4, 6, 7 and {t j j Ẽ(l)} {t 18} {u i i I} for l = 8. 9

10 Since I = E(l), the theorem is proved. Dualizing the result of Theorem 5 and Theorem 7, we obtain the statement of Theorem 1 except for P 1 t 6, P 1 t 4, P t 4, P 1 t 46, P 1 t 58 and P 9 t 58. To determine these operations, we use the adjoint action of H (G(l); Z/) on H (ΩG(l) ; Z/) which is introduced in the next section. (Remark)The computation of dualizing the result of Theorem 5 and Theorem 7 is not difficult except for P 1 t 18, because P n t is primitive if t is primitive. Moreover, it is easily shown φ(p 1 t 18 ) = P 1 φ(t 18 ) = φ( t t 6 ) and this shows P 1 t 18 = t t 6 modulo primitive elements. By Theorem 5 we can see P 1 a 14 = ɛa 9 and this shows that P 1 t 18 = ɛt 14 t t 6. 5 Adjoint action Put y y = Ad (y y ) and y t = ad (y t) where y, y H (G; Z/) and t H (ΩG ; Z/). Following are the dual result of []. Also see [9]. Therorem 11. For y, y, y H (G; Z/) and t, t H (ΩG ; Z/) (i) 1 y = y, 1 t = t. (ii) y 1 = 0,if y > 0, whether 1 H (G; Z/) or 1 H (ΩG ; Z/). (iii) (yy ) t = y (y t). (iv) y (tt ) = ( 1) y t (y t)(y t ) where y = y y. (v) σ(y t) = y σ(t) where σ is the homology suspension. (vi) P n (y t) = i(p y) i (P n i t). P n (y y ) = i(p y) i (P n i y ). (vii) (y t) = ( y) ( t) = ( 1) y t (y t ) (y t ) where y = y y and t = t t. And (y t) = ( y) ( t). 10

11 (viii) If t is primitive then y t is primitive. Also the result of [] implies the following theorem. See [8]. Therorem 1. We set a submodule A of H (G; Z/) as A = Z/[y 8 ]/(y 8 ) for G = F 4, E 6, E 7 and A = Z/[y 8, y 0 ]/(y 8, y 0 ) for G = E 8 where y i is the dual of x i with respect to the monomial basis. Then there exists a retraction p : H (G; Z/) A and the following diagram commutes. H (G; Z/) H (ΩG ; Z/) ad H (ΩG ; Z/) p 1 A H (ΩG ; Z/) ad (Remark)By Theorem we can see P y 0 = y 8. Since Ad is agreed with the composition µ (1 µ ) (1 1 ι ) (1 T ) ( 1) where µ is the multiplication of G(l) and ι is the inverse map, the next theorem follows. See [9]. Therorem 1. Let y, y H (G). If y is primitive, where [y, y ] = yy ( 1) y y y y. y y = [y, y ] Now we give the proof of Theorem and finish the proof of Theorem 1. Let y i be the dual element of x i H (G(l)) as to the monomial basis. By Theorem and Theorem 1 we see that for j E(l) {, 9} {11, 9} y j+9 for j = 1,, 4, 9, 1, y 8 y j+1 = y j+9 for j = 19, 0 others 11

12 and y j+1 for j =, 7, 9, y 0 y j+1 = y j+1 for j = 1, 0 others. Since σt j = y j+1 for j E(l) {, 9} {11, 9}, Theorem 11 (v) implies σ(y 8 t j ) 0 for j = 1,, 4, 9, 1, 19, σ(y 0 t j ) 0 for j =, 7, 9, 1. () Then the equations y 8 t = t 10, () y 8 t 8 = t 16, (4) y 8 t 6 = t 4, (5) y 8 t 8 = t 46, (6) y 0 t 14 = t 4, (7) y 0 t 6 = t 46 (8) are shown by Theorem 11 (viii). Moreover () implies modulo decomposable elements. Since y 8 t 6 t 14, (9) y 8 t 18 t 6, (10) y 0 t 6 t 6, (11) y 0 t 18 t 8 (1) φ(y 8 t 6 ) = (y 8 t ) t (y 8 t ) t t (y 8 t ) t (y 8 t ) = φ( t 10 t ), one can see that y 8 t 6 t 10 t mod primitive elements. By this and (9), we have y 8 t 6 = t 14 t 10 t. (1) The equations y 8 t 18 = t 6 + t 10 t t 6 t 14 t 6, (14) y 0 t 6 = t 6 (y 0 t )t, (15) y 0 t 18 = t 8 (y 0 t 6 )t 6 1 (16)

13 are shown in the similar way. By the equation (1), we can compute y 8 t 6 as y 8 t 6 = y 8 (t 14 t 10 t ) = y 8 t 14 + t 10. Since y 8 = 0, y 8 t 14 = t 10 and this means y 8 t 14 is a non-zero primitive indecomposable element. We redefine t as Then we have t = y 8 t 14. (17) y 8 t = t 10. By Theorem 7 we can set P t 1 = κt 6 where κ = ±1. Since P t 1 = P (y 1 8 t 14 ) = y 8 t 10, we have y 8 t 10 = κt 6. By the similar manner, we can compute y 8 t 18 and obtain y 8 t 6 = t 14. Therefore y 8 t 4 = y 8 t 6 = t 14. (18) Because t 16 and t 46 are primitive, we can set Operate P to (0) to obtain y 8 t 16 = ρ t 8, (19) y 8 t 46 = ρ t 18. (0) y 8 t 4 = P (y 8 t 46 ) = ρ P (t 18 ) = ρ ɛt 14. Thus by (18), we conclude that ρ = ɛ. y 8 t 58 will be determined after the determination of y 0 t 58. Here we apply P 1 on (5), (6) and (14), P on (5) to see P 1 t 6 = P 1 (y 8 t 18 t 10 t 6 t + t 14 t 6 ) = ɛy 8 t 14 = ɛt, P 1 t 4 = P 1 (y 8 t 6 ) = ɛy 8 t = ɛt 10, P 1 t 46 = P 1 (y 8 t 8 ) = ɛy 8 t 4 = ɛt 14, P t 4 = P (y 8 t 6 ) = y 8 t 14 = t. 1

14 Next we compute y 0 t i. First we apply P 1 to (15) to obtain y 0 t = P 1 (y 0 t 6 ) = P 1 (t 6 (y 0 t )t ) = ɛt. From this, (15) and (16) imply that y 0 t 6 is computed as y 0 t = ɛt, y 0 t 6 = t 6 ɛt t, y 0 t 18 = t 8 + ɛt t 6 t t 6 t 6. 0 = y 0 t 6 = y 0 (y 0 t 6 ) = y 0 (t 6 ɛt t ) = y 0 t 6 + ɛt. Thus y 0 t 46 = y 0 t 6 = ɛt. The similar computation of y 0 t 18 implies y 0 t 8 = t 6. Thus y 0 t 8 is a non zero primitive indecomposable element and we redefine t 58 as y 0 t 8. Hence By applying P to (), we have We obtain also y 0 t 8 = t 58, (1) y 0 t 58 = t 6. () y 8 t 58 = P (y 0 t 58 ) = P (t 6 ) = ɛt. y 0 t = ɛp 1 (y 0 t 6 ) = ɛp 1 t 46 = t 14 by applying P 1 to (8). Since t 4 is primitive, we can set y 0 t 4 = ρ 4 t 18 (ρ 4 Z/). Operating P to the both sides of this equation, ρ 4 ɛt 14 is computed as follows: ρ 4 ɛt 14 = ρ 4 P (t 18 ) = P (y 0 t 4 ) = y 8 t 4 + y 0 t = t

15 So y 0 t 4 = ɛt 18 is shown. Now ad is determined except for y 8 t 16. Finally we operate P 1 to (1) and P 9 to () and see P 1 t 58 = P 1 (y 0 t 8 ) = y 0 (P 1 t 8 ) = ɛy 0 t 4 = t 18, y 0 (P 9 t 58 ) = P 9 (y 0 t 58 ) = P 9 (t 6 ) = t 14. These equations imply that P 1 t 58 = t 18, P 9 t 58 = t. This completes the proof of Theorem 1. Hiroaki HAMANAKA Department of Mathematics, Kyoto University Shin-ichro HARA Nagaoka University of Technology References [1] R. Bott. The space of loops on Lie group. Michigan Math. J.5(1955), [] A. Kono and K. Kozima. The mod homology of the space of loops on the exceptional Lie group. Proceedings of th Royal Society of Edinburgh, 11A(1989), []. The adjoint action of Lie group on the space of loops. Journal of The Mathematical Society of Japan, 45 No.(199), [4] S. Araki. On the non-commutativity of Pontrjagin rings mod of some compact exceptional groups. Nagoya Math.J, 17(1960), [5] A. Kono and M. Mimura. On the cohomology mod of the classifying space of the compact, 1-connected exceptional Lie group E 6. preprint series of Aarhus University, [6] H. Toda. Cohomology of the classifying space of exceptional Lie groups. Conference on manifolds, Tokyo, (197),

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