Geometry & Topology Monographs 10 (2007) 213 226 213 On the Rothenberg Steenrod spectral sequence for the mod 3 cohomology of the classifying space of the exceptional Lie group E 8 MASAKI KAMEKO MAMORU MIMURA We show that the Rothenberg Steenrod spectral sequence converging to the mod 3 cohomology of the classifying space of the exceptional Lie group E 8 does not collapse at the E 2 level. 55R40; 55T99 1 Introduction One of the most powerful tools in the study of the mod p cohomology of classifying spaces of connected compact Lie groups is the Rothenberg Steenrod spectral sequence. For a connected compact Lie group G, there is a strongly convergent first quadrant spectral sequence of graded p algebras fer p;q ; d r W Er p;q! E pcr;q rc1 r such that E p;q 2 D Cotor p;q H G. p; p / and E 1 D gr H BG. The Rothenberg Steenrod spectral sequence, sometimes mentioned as the Eilenberg Moore spectral sequence, has been successful in computing the mod p cohomology of classifying spaces of connected compact Lie groups. In all known cases, for an odd prime p, the Rothenberg Steenrod spectral sequence converging to the mod p cohomology of the classifying space of a connected compact Lie group collapses at the E 2 level. So, one might expect that this collapse should always occur. In this paper, however, we show that this is not the case for the mod 3 cohomology of the classifying space of the exceptional Lie group E 8. Theorem 1.1 The Rothenberg Steenrod spectral sequence converging to the mod 3 cohomology of the classifying space of the exceptional Lie group E 8 does not collapse at the E 2 level. g Published: 29 January 2007 DOI: 10.2140/gtm.2007.10.213
214 Masaki Kameko and Mamoru Mimura We prove Theorem 1.1 by computing the ring of invariants of the mod 3 cohomology of a nontoral elementary abelian 3 subgroup of E 8. According to Andersen, Grodal, Møller and Viruel [1], up to conjugates, there are exactly two maximal nontoral elementary abelian 3 subgroups, which they call EE 5a 8 and EE 5b 8. They described the action of Weyl groups on these nontoral elementary abelian 3 subgroups explicitly. In this paper, we compute the ring of invariants of the polynomial part of the mod 3 cohomology of BE 5a E 8. By comparing degrees of algebra generators of the above ring of invariants with those of algebra generators of the cotorsion product Cotor H E 8. 3 ; 3 / computed by Mimura and Sambe [2], we prove Theorem 1.1. In Section 2, we set up a tool, Theorem 2.5, for the computation of certain rings of invariants. In Section 3, we recall some facts on maximal nontoral elementary abelian p subgroups of simply connected compact simple Lie groups and their Weyl groups. Then using Theorem 2.5, we compute some of rings of invariants of the above Weyl groups. In Section 4, we complete the proof of Theorem 1.1. The first named author was partially supported by the Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C) 13640090. 2 Invariant theory In this section, we consider the invariant theory over the finite field q of q elements where q D p k with k 1 and p is a prime number. For a finite set fv 1 ; : : : ; v n g, we denote by q fv 1 ; : : : ; v n g the n dimensional vector space over q spanned by fv 1 ; : : : ; v n g. Let us write GL n. q / for the set of invertible n n matrices whose entries are in q. We also write M m;n. q / for the set of m n matrices whose entries are in q. Let G be a subgroup of GL n. q /. The group G acts on the n dimensional vector space V D q fv 1 ; : : : ; v n g as follows: for g in G, gv i D nx a j;i.g/v j ; jd1 where a i;j.g/ is the.i; j / entry of the matrix g. We denote by fx 1 ; : : : ; x n g the dual basis of fv 1 ; : : : ; v n g and write V for the dual of V, that is, V D Hom q.v; q / D q fx 1 ; : : : ; x n g: We denote by q ŒV D q Œx 1 ; : : : ; x n
The Rothenberg Steenrod spectral sequence 215 the polynomial algebra over q in n variables x 1,..., x n. Then the group G acts on both V and q ŒV as follows: for g in G,.gx/.v/ D x.g 1 v/ g.y z/ D.gy/.gz/ for x in V ; v in V I for y; z in q ŒV : Using entries of a matrix g 2 G, we may describe the action of g as follows: Proposition 2.1 Proof gx i D nx a i;j.g 1 /x j : jd1 nx.gx i /.v j / D x i.g 1 v j / D x i a k;j.g 1 /v k D a i;j.g 1 /: In order to prove Theorem 2.5 below, we recall a strategy of Wilkerson [4, Section 3]. It can be stated in the following form. Theorem 2.2 Let G be a subgroup of GL n. q / acting on V as above. Let f 1 ; : : : ; f n be homogeneous polynomials in q ŒV. We have q ŒV G D q Œf 1 ; : : : ; f n if and only if the following three conditions hold: (1) f 1 ; : : : ; f n are G invariant; (2) q ŒV is integral over the subalgebra R generated by f 1 ; : : : ; f n ; (3) deg f 1 deg f n D jgj. In the statement of Theorem 2.2, deg f is the homogeneous degree of f, that is, we define the degree deg x i of indeterminate x i to be 1. For the proof of this theorem, we refer the reader to Corollaries 2.3.2 and 5.5.4 and Proposition 5.5.5 in Smith s book [3] and Wilkerson s paper [4, Section 3]. To state Theorem 2.5, we need to set up the notation. Let G 1 GL m. q / and G 2 GL n m. q /. Let V 1 D q fv 1 ; : : : ; v m g and V 2 D q fv mc1 ; : : : ; v n g. Let G 1 and G 2 act on V 1 and V 2 by mx nx m g 1 v i D a k;i.g 1 /v k and g 2 v j D a k;j m.g 2 /v mck ; respectively, where i D 1; : : : ; m and j D m C 1; : : : ; n. The following proposition is immediate from the definition and Proposition 2.1.
216 Masaki Kameko and Mamoru Mimura Proposition 2.3 The following hold: (1) If f.x 1 ; : : : ; x m / 2 q ŒV 1 is G 1 invariant, then for all g 1 2 G 1, we have X m f a 1;k.g 1 1 /x k; : : : ; mx a m;k.g1 1 /x k D f.x 1 ; : : : ; x m /I (2) If f.x mc1 ; : : : ; x n / 2 q ŒV 2 is G 2 invariant, then for all g 2 2 G 2, we have f n Xm n a 1;k.g2 1 /x Xm mck; : : : ; a n m;k.g2 1 /x mck D f.x mc1 ; : : : ; x n /: Suppose that G consists of the matrices of the form 0 1 @ g 1 m 0 A ; 0 g 2 where g 1 2 G 1 GL m. q /, g 2 2 G 2 GL n m. q / and m 0 2 M m;n m. q /. We denote respectively by xg 0, xg 1, xg 2 the subgroups of G consisting of matrices of the form 0 1 0 1 0 1 @ 1 m m 0 A ; @ g 1 0 A ; @ 1 m 0 A ; 0 1 n m 0 1 n m 0 g 2 where g 1 2 G 1, g 2 2 G 2, m 0 2 M m;n m. q / and 1 k is the identity matrix in M k;k. q /. We denote by xg 1, xg 2 the elements in xg 1, xg 2 corresponding to g 1, g 2, respectively. Considering V2 as a subspace of V, let us define OX in q ŒV ŒX by OX D Y.X C x/: x2v 2 The following proposition is well-known (see Wilkerson [4, Section 1]). Proposition 2.4 There are c n where c n m;n m D 1. m;k 2 q ŒV 2 such that nx m OX D. 1/ n m k c n m;k X qk ; kd0 Now, we state our main theorem of this section.
The Rothenberg Steenrod spectral sequence 217 Theorem 2.5 With the above assumption on G, suppose that rings of invariants q ŒV 1 G 1 and q ŒV 2 G 2 are polynomial algebras q Œf 1 ; : : : ; f m and q Œf mc1 ; : : : ; f n ; respectively, where f 1 ;: : :;f m are homogeneous polynomials in m variables x 1 ;: : :; x m and f mc1 ; : : : ; f n are homogeneous polynomials in.n m/ variables x mc1 ; : : : ; x n. Then the ring of invariants q ŒV G is also a polynomial algebra q Œ x f 1 ; : : : ; x f m ; f mc1 ; : : : ; f n ; where for i D 1,..., m, x f i D f i.ox 1 ; : : : ; Ox m /: To prove Theorem 2.5, we verify that the conditions (1), (2) and (3) in Theorem 2.2 hold for the polynomials f x 1 ; : : : ; f x m, f mc1 ; : : : ; f n in Theorem 2.5. Step 1 To prove that f1 x ; : : : ; f x m ; f mc1 ; : : : ; f n are G invariant, it suffices to prove the following propositions. Proposition 2.6 Suppose f.x 1 ; : : : ; x m / 2 q ŒV 1 G 1. Then f.ox 1 ; : : : ; Ox m / is G invariant in q ŒV. Proposition 2.7 Suppose f.x mc1 ; : : : ; x n / 2 q ŒV 2 G 2. Then f.x mc1 ; : : : ; x n / is G invariant in q ŒV. To simplify the argument, we use the following. Lemma 2.8 f 2 q ŒV is G invariant if f is xg 0 invariant, xg 1 invariant and xg 2 invariant. Proof We may express each g in G as a product of elements xg 0, xg 1, xg 2 in xg 0, xg 1, xg 2 respectively, say g D xg 0 xg 1 xg 2 as follows: 0 1 0 1 0 1 0 1 @ g 1 m 0 A D @ 1 m m 0 g2 1 A @ g 1 0 A @ 1 m 0 A : 0 g 2 0 1 n m 0 1 n m 0 g 2 Firstly, we prove Proposition 2.6. Lemma 2.9 O is an q linear homomorphism from V to q ŒV.
218 Masaki Kameko and Mamoru Mimura Proof For, ˇ 2 q and for x, y 2 V, we have. x C ˇy/ qk D x qk C ˇy qk. By Proposition 2.4, we have nx m O. x C ˇy/ D. 1/ n m k c n m;k. x C ˇy/ qk kd0 n m X D. 1/ n m k c n m;k. x qk C ˇy qk / kd0 D Ox C ˇOy: Lemma 2.10 The following hold for k D 1; : : : ; m: (1) xg 0 Ox k D Ox k ; mx (2) xg 1 Ox k D a k;`.g1 1 /Ox` ; `D1 (3) xg 2 Ox k D Ox k. Proof (1) We have xg 0 x k D x k C y for some y in V2, and xg 0x D x for any x in V2. Then y C x ranges over V2 as x ranges over V2. Hence, we have xg 0 Ox k D Y.x k C y C x/ D Ox k : x2v 2 (2) We have xg 1 x k D P m `D1 a k;`.g1 1/x`, and xg 1 x D x for any x in V2. Hence, by Lemma 2.9, we have xg 1 Ox k D Y x2v 2 X m D O D mx `D1 m X `D1 `D1 a k;`.g1 1 /x` C x a k;`.g 1 1 /x` a k;`.g 1 1 /Ox`: (3) We have xg 2 x k D x k. Then xg 2 x ranges over V2 xg 2 Ox k D Y x2v 2.x k C xg 2 x/ D Ox k : as x ranges over V 2. Hence,
The Rothenberg Steenrod spectral sequence 219 Proof of Proposition 2.6 By Lemma 2.8, it suffices to show that f.ox 1 ; : : : ; Ox m / is xg i invariant for i D 0, 1, 2. By Lemma 2.10 (1) and (3), it is clear that the above element is invariant with respect to the action of xg i for i D 0, 2. By Lemma 2.10 (2) and by Proposition 2.3 (1), we have X m xg 1 f.ox 1 ; : : : ; Ox m / D f Secondly, we prove Proposition 2.7. a 1;k.g 1 1 /Ox k; : : : ; D f.ox 1 ; : : : ; Ox m /: Lemma 2.11 The following hold for k D m C 1; : : : ; n: (1) xg 0 x k D x k ; (2) xg 1 x k D x k ; (3) nx m xg 2 x k D a k m;`.g2 1 mc`. `D1 mx a m;k.g1 1 /Ox k Proof (1) and (2) are immediate from the definitions of xg 0 and xg 1. (3) follows immediately from the fact that a k;`.xg 1 2 / D a k m;` m.g 1 2 / for ` m C 1 and that a k;`.xg 2 1 / D 0 for ` m. Proof of Proposition 2.7 As in the proof of Proposition 2.6, it suffices to show that f.x mc1 ; : : : ; x n / is xg i invariant for i D 0, 1, 2. It is clear from Lemma 2.11 (1) (2) that the above element is xg i invariant for i D 0, 1. By Lemma 2.11 (3) and by Proposition 2.3 (2), xg 2 f.x mc1 ; : : : ; x n / D f n Xm n a 1;k.g2 1 /x Xm mck; : : : ; a n m;k.g2 1 /x mck D f.x mc1 ; : : : ; x n /: Step 2 We prove that the inclusion R! q ŒV is an integral extension, for R the subalgebra of q ŒV generated by f x 1 ; : : : ; f x m, f mc1 ; : : : ; f n. Let S be the subalgebra of q ŒV generated by f x 1 ; : : : ; f x m, c n m;0 ; : : : ; c n m;n m 1. Since G 2 GL n m. q /, we see that c n m;k 2 R. So, S is a subalgebra of R. Therefore, it suffices to prove the following proposition.
220 Masaki Kameko and Mamoru Mimura Proposition 2.12 For k D 1; : : : ; n, the element x k is integral over S. Proof Firstly, we prove that x k is integral over S for k D 1; : : : ; m. By Theorem 2.2, x k is integral over q ŒV 1 G 1. Hence, there exists a monic polynomial F.X / and polynomials ' j s over q in m variables for j D 0; : : : ; r 1 such that rx 1 F.X / D X r C ' j.f 1.x 1 ; : : : ; x m /; : : : ; f m.x 1 ; : : : ; x m //X j jd0 and that F.x k / D 0 in q Œx 1 ; : : : ; x m. Replacing x i in the equality F.x k / D 0 above by Ox i for i D 1; : : : ; m, we have the following equality in q ŒV : Let rx 1.Ox k / r C ' j.f 1.Ox 1 ; : : : ; Ox m /; : : : ; f m.ox 1 ; : : : ; Ox m //.Ox k / j D 0: jd0 rx 1 F 0.X / D.OX / r C ' j. f x 1 ; : : : ; f x m /.OX / j : jd0 By Proposition 2.4, F 0.X / is a monic polynomial in SŒX. Since, by definition, x f i D f i.ox 1 ; : : : ; Ox m /, it is clear that F 0.x k / D 0 in q ŒV. Hence x k is integral over S. Secondly, we verify that x k is integral over S for k D m C 1; : : : ; n. By Proposition 2.4, OX is a monic polynomial in SŒX. It is immediate from the definition that Ox D 0 for x 2 V 2. Therefore, x k is integral over S. Step 3 Finally, we compute the product of degrees of f1 x ; : : : ; f x m, f mc1 ; : : : ; f n. Since deg Ox is of degree q n m for x 2 V, we have By Theorem 2.2, we have deg x f i D deg f i q n m : and Therefore deg f 1 deg f m D jg 1 j deg f mc1 deg f n DjG 2 j: deg f x 1 deg f x m.n m/ m deg f mc1 deg f n D deg f 1 deg f n q m.n m/ D jg 1 j jg 2 j q D jgj: This completes the proof of Theorem 2.5.
The Rothenberg Steenrod spectral sequence 221 3 Rings of invariants of Weyl groups Let p be an odd prime. Let G be a compact Lie group. We write H BG and H BG for the mod p homology and cohomology of the classifying space BG of G. We write A for an elementary abelian p subgroup of the compact Lie group G. Let H BG D H BG= p 0; where p 0 is the ideal of nilpotent elements in H BG. It is clear that H BA is a polynomial algebra H BA D p Œt 1 ; : : : ; t n ; where the cohomological degree of each t i is 2 and n is the rank of A. We called it the polynomial part of H BA in Section 1. Choosing a basis for A, we may consider the action of GL n. p / on A. We recall the relation between the action of GL n. p / on A and the one on H BA. For the sake of notational simplicity, let V D H 1 BA. On the one hand, V is identified with A as a GL n. p / module, where g 2 GL n. p / acts on V as the induced homomorphism Bg. As an p algebra, H BA is isomorphic to p ŒV. As in the previous section, we may consider the GL n. p / module structure on p ŒV. On the other hand, GL n. p / acts on H BA by gx D B.g 1 / x, where x 2 H BA and g 2 GL n. p /. The relation between these actions is given by the following proposition. Proposition 3.1 As a GL n. p / module, H BA D p ŒV. Proof The Bockstein homomorphism induces an isomorphism of GL n. p / modules ˇW H 1 BA! H 2 BA: Since, for x 2 V D H 1 BA, v 2 V D H 1 BA, we have.gx/.v/ D.x/.g 1 v/ D.x/.B.g 1 / v/ D.B.g 1 / x/.v/; we see that H 2 BA D H 1 BA D V as GL n. p / modules. Hence, we may conclude that H BA D p ŒV as GL n. p / modules. The Weyl group W.A/ D N G.A/=C G.A/ acts on A as inner automorphisms. So, we have the action of W.A/ on H BA. Choosing a basis for A, we consider the Weyl group W.A/ as a subgroup of GL n. p /. In this section, we compute rings of invariants of Weyl groups of the polynomial part of the mod p cohomology of the classifying spaces of maximal nontoral elementary abelian p subgroups of simply connected compact simple Lie groups.
222 Masaki Kameko and Mamoru Mimura It is well-known that for an odd prime p, a simply connected compact simple Lie group G does not have nontoral elementary abelian p subgroups except for the cases p D 5, G D E 8, and p D 3, G D F 4, E 6, E 7, E 8. Andersen, Grodal, Møller and Viruel [1] described Weyl groups of maximal nontoral elementary abelian p subgroups and their action on the underlying elementary abelian p subgroup explicitly for p D 3, G D E 6, E 7, E 8. Up to conjugate, there are only 6 maximal nontoral elementary abelian p subgroups of simply connected compact simple Lie groups. For p D 5, G D E 8 and for p D 3, G D F 4, E 6, E 7, there is one maximal nontoral elementary abelian p subgroup for each G. We call them EE 3 8, EF 3 4, E3E 4 6, E2E 4 7, following the notation in [1]. For p D 3, G D E 8, there are two maximal nontoral elementary abelian p subgroups, say EE 5a 8 elementary abelian p subgroup. For a detailed account on nontoral elementary abelian p subgroups, we refer the reader to [1, Section 8] and its references. In this section, we compute and E 5b E 8, where the superscript indicates the rank of. H BA/ W.A/ for A D E 4 3E 6, E 4 2E 7, E 5a E 8 using Theorem 2.5. Proposition 3.2 We have the following isomorphisms of graded p algebras: (1) For p D 5, G D E 8, A D E 3 E 8,. H BA/ W.A/ D 5 Œx 62 ; x 200 ; x 240 I (2) For p D 3, G D F 4, A D E 3 F 4,. H BA/ W.A/ D 3 Œx 26 ; x 36 ; x 48 I (3) For p D 3, G D E 6, A D E 4 3E 6,. H BA/ W.A/ D 3 Œx 26 ; x 36 ; x 48 ; x 54 I (4) For p D 3, G D E 7, A D E 4 2E 7,. H BA/ W.A/ D 3 Œx 26 ; x 36 ; x 48 ; x 108 I (5) For p D 3, G D E 8, A D E 5a E 8,. H BA/ W.A/ D 3 Œx 4 ; x 26 ; x 36 ; x 48 ; x 324 ; where the subscript of x indicates its cohomological degree. Proof We use Theorem 2.5 for (3), (4) and (5). In these cases, we described G 1, G 2, V1, V 2, pœv 1 G 1, p ŒV 2 G 2 in Theorem 2.5. (1) The case p D 5, G D E 8. A D E 3 E 8. The Weyl group W.A/ is the special linear group SL 3. 5 /. The ring of invariants of the special linear group is well-known as Dickson invariants. Then we have 5 Œt 1 ; t 2 ; t 3 W.A/ D 5 Œx 62 ; x 200 ; x 240 ; where x 4 62 D c 3;0, x 200 D c 3;2, x 240 D c 3;1 and c 3;k s are Dickson invariants described in Proposition 2.4.
The Rothenberg Steenrod spectral sequence 223 (2) The case p D 3, G D F 4, A D E 3 F 4. The Weyl group W.A/ is the special linear group SL 3. 3 /. The ring of invariants are known as Dickson invariants as before: 3 Œt 1 ; t 2 ; t 3 W.A/ D 3 Œx 26 ; x 36 ; x 48 ; where x 2 26 D c 3;0, x 36 D c 3;2, x 48 D c 3;1 as above. (3) The case p D 3, G D E 6, A D E3E 4 6. The Weyl group W.A/ is the subgroup of GL 4. 3 / consisting of matrices of the form 0 1 @ g 1 m 0 A ; 0 g 2 where g 1 2G 1 Df1g, g 2 2G 2 DSL 3. 3 /, m 0 2M 1;3. 3 /. Consider V1 D 3ft 1 g and V2 D 3ft 2 ; t 3 ; t 4 g. Then we have 3 ŒV 1 G 1 D 3 Œt 1 and 3 ŒV 2 G 2 D 3 Œx 26 ; x 36 ; x 48, where x26 2 D c 3;0, x 36 D c 3;2, x 48 D c 3;1 and c 3;k s are Dickson invariants in 3 ŒV 2. By Theorem 2.5, we have where x 54 D Q t2v 2.t 1 C t/. 3 Œt 1 ; t 2 ; t 3 ; t 4 W.A/ D 3 Œx 26 ; x 36 ; x 48 ; x 54 ; (4) The case p D 3, G D E 7, A D E2E 4 7. The Weyl group W.A/ is a subgroup of GL 4. 3 / consisting of matrices of the form 0 1 @ g 1 m 0 A ; 0 g 2 where g 1 2 G 1 D GL 1. 3 /, g 2 2 G 2 D SL 3. 3 / and m 0 2 M 1;3. 3 /. Consider V1 D 3ft 1 g and V2 D ft 2; t 3 ; t 4 g. Then we have 3 ŒV 1 G 1 D 3 Œt1 2 and 3ŒV 2 G 2 D 3 Œx 26 ; x 36 ; x 48 as in (3). By Theorem 2.5, we have where x 108 D Q t2v 2.t 1 C t/ 2. 3 Œt 1 ; t 2 ; t 3 ; t 4 W.A/ D 3 Œx 26 ; x 36 ; x 48 ; x 108 ; (5) The case p D 3, G D E 8, A D EE 5a 8. The Weyl group W.A/ is a subgroup of GL 5. 3 / consisting of matrices of the form 0 1 0 1 @ g g 1 m 0 1 m 0 0 m00 o A D B 0 g 0 g @ 2 0 0 C A ; 2 0 0
224 Masaki Kameko and Mamoru Mimura where g 1 2 G 1 D GL 1. 3 /, g 2 D.g2 0 ; / 2 G 2 D SL 3. 3 / GL 1. 3 / GL 4. 3 / and m 0 D.m 0 0 ; m00 0 / 2 M 1;4. 3 / D M 1;3. 3 / M 1;1. 3 /. Consider V1 D 3ft 1 g and V2 D 3ft 2 ; t 3 ; t 4 ; t 5 g. Then we have 3 ŒV 1 G 1 D 3 Œt1 2 and 3 ŒV 2 G 2 D 3 Œx 26 ; x 36 ; x 48 ; t 2 5 ; where x 26, x 36, x 48 are Dickson invariants in 3 Œt 2 ; t 3 ; t 4 as in (3). By Theorem 2.5, we have 3 Œt 1 ; t 2 ; t 3 ; t 4 ; t 5 W.A/ D 3 Œx 4 ; x 26 ; x 36 ; x 48 ; x 324 ; where x 4 D t 2 5 and x 324 D Q t2v 2.t 1 C t/ 2 : 4 Proof of Theorem 1.1 Let p be a prime, including p D 2. As in the previous section, let G be a compact Lie group and A an elementary abelian p subgroup of G. We denote by i A;G W A! G the inclusion. Then the induced homomorphism Bi A;G W H BG! H BA factors through the ring of invariants of the Weyl group W.A/. Proposition 4.1 The inclusion of the image is an integral extension. Im Bi A;G!. H BA/ W.A/ Proof By the Peter Weyl theorem, for a sufficiently large n, there exists an embedding of a compact Lie group G into a unitary group U.n/, say i G;U.n/ W G! U.n/: Through the induced homomorphism Bi G;U.n/ W H BU.n/! H BG, the mod p cohomology H BG is an H BU.n/ module. Recall here that H BU.n/ D p Œc 1 ; : : : ; c n ; where each c i is a Chern class and deg c i D 2i. So, H BU.n/ is a Noetherian ring. It is well-known that H BG is a finitely generated H BU.n/ module, so that H BG is a Noetherian H BU.n/ module. We defined H BG as a quotient module of H BG. Therefore, H BG is also a Noetherian H BU.n/ module. Considering the case G D A, we may conclude that H BA is also a Noetherian H BU.n/ module. Since the ring of invariants. H BA/ W.A/ is an H BU.n/ submodule of a
The Rothenberg Steenrod spectral sequence 225 Noetherian H BU.n/ module H BA, it is also a Noetherian H BU.n/ module. Hence, the ring of invariants. H BA/ W.A/ is a finitely generated H BU.n/ module. Thus, the inclusion Im Bi A;G!. H BA/ W.A/ is an integral extension. In the case p D 3, G D E 8, A D E 5a E 8, the ring of invariants of the Weyl group W.A/ is computed in the previous section and it is as a graded 3 algebra. 3 Œx 4 ; x 26 ; x 36 ; x 48 ; x 324 Now, we recall the computation of the cotorsion product Cotor H E 8. 3 ; 3 / due to Mimura and Sambe in [2]. From this, we need just an upper bound for the degree of algebra generators of the cotorsion product. Namely, the following result which is immediate from the computation of Mimura and Sambe suffices to prove the noncollapsing of the Rothenberg Steenrod spectral sequence. Proposition 4.2 As a graded 3 algebra, the cotorsion product Cotor H E 8. 3 ; 3 / is generated by elements of degree less than or equal to 168. As a consequence of Proposition 4.2, if the Rothenberg Steenrod spectral sequence collapsed at the E 2 level, then H BE 8 and H BE 8 would be generated by elements of degree less than or equal to 168 as graded 3 algebras. The image of the induced homomorphism BiA;E 8 would also be generated by elements of degree less than or equal to 168. Therefore, Im BiA;E 8 would be a subalgebra of 3 Œx 4 ; x 26 ; x 36 ; x 48 : It is clear that x 324 is not integral over 3 Œx 4 ; x 26 ; x 36 ; x 48, and so the inclusion Im Bi A;E 8!. H BA/ W.A/ D 3 Œx 4 ; x 26 ; x 36 ; x 48 ; x 324 would not be an integral extension. This contradicts Proposition 4.1. Hence, the Rothenberg Steenrod spectral sequence does not collapse at the E 2 level.
226 Masaki Kameko and Mamoru Mimura References [1] K Andersen, J Grodal, J Møller, A Viruel, The classification of p compact groups for p odd arxiv:math.at/0302346 [2] M Mimura, Y Sambe, On the cohomology mod p of the classifying spaces of the exceptional Lie groups II, III, J. Math. Kyoto Univ. 20 (1980) 327 379 MR582171 [3] L Smith, Polynomial invariants of finite groups, Research Notes in Mathematics 6, A K Peters Ltd., Wellesley, MA (1995) MR1328644 [4] C Wilkerson, A primer on the Dickson invariants, from: Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982), Contemp. Math. 19, Amer. Math. Soc., Providence, RI (1983) 421 434 MR711066 Toyama University of International Studies Toyama 930-1292, Japan Department of Mathematics, Okayama University Okayama 700-8530, Japan kameko@tuins.ac.jp, mimura@math.okayama-u.ac.jp Received: 31 July 2004 Revised: 26 October 2005