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THE BROWN-PETERSON COHOMOLOGY of the CLASSIFYING SPACES OFTHEPROJECTIVEUNITARYGROUPSPU(P) and EXCEPTIONAL LIE GROUPS

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THE BROWN-PETERSON COHOMOLOGY of the CLASSIFYING SPACES OFTHEPROJECTIVEUNITARYGROUPSPU(P) and EXCEPTIONAL LIE GROUPS TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 5, May 2008, Pages 2265–2284 S 0002-9947(07)04425-X Article electronically published on December 20, 2007 THE BROWN-PETERSON COHOMOLOGY OF THE CLASSIFYING SPACES OFTHEPROJECTIVEUNITARYGROUPSPU(p) AND EXCEPTIONAL LIE GROUPS MASAKI KAMEKO AND NOBUAKI YAGITA Abstract. For a fixed prime p, we compute the Brown-Peterson cohomologies of classifying spaces of PU(p) and exceptional Lie groups by using the Adams spectral sequence. In particular, we see that BP ∗(BPU(p)) and K(n)∗(BPU(p)) are even dimensionally generated. 1. Introduction Let p be a fixed odd prime and denote by BP ∗(X)(resp. P (m)∗(X)) the Brown- ∗ Peterson cohomology of a space X with the coefficient ring BP = Z(p)[v1,v2, ···] ∗ k (resp. P (m) = Z/p[vm,vm+1, ···]) where deg vk = −2p +2.Wedenoteby PU(n) the projective unitary group which is the quotient of the unitary group U(n) by its center S1. Recall that the cohomologies of PU(p) and the exceptional Lie groups F4,E6,E7,E8 have odd torsion elements. In this paper, we compute the Brown-Peterson cohomologies of classifying spaces BG of these Lie groups G as BP ∗-modules using the Adams spectral sequence. Let us write H∗(X; Z/p)by simply H∗(X)andletA be the mod p Steenrod algebra. Ourmainresultisasfollows: Theorem 1.1. Let (G, p) be one of cases (G = PU(p),p) for an arbitrary odd prime p and G = F4,E7 for p =3,andG = E8 for p =5. Then the E2-terms of the Adams spectral sequences abutting to BP ∗(BG) and P (m)∗(BG) for m ≥ 1, s,t ∗ ∗ s,t ∗ ∗ ExtA (H (BP),H (BG)), ExtA (H (P (m)),H (BG)), have no odd degree elements. An immediate consequence is as follows: Corollary 1.2. For (G, p) in Theorem 1.1, the Adams spectral sequences abutting ∗ ∗ to BP (BG) and P (m) (BG) in the previous theorem collapse at the E2-level. In particular BP odd(BG)=P (m)odd(BG)=0. ∗ ∼ ∗ ∗ Recall K(m) (X) = K(m) ⊗P (m)∗ P (m) (X)istheMoravaK-theory. From the above theorem and corollary, we see K(m)odd(BPU(p)) = 0. Then we have the following corollary ([Ko-Ya], [Ra-Wi-Ya]). Received by the editors November 2, 2005. 2000 Mathematics Subject Classification. Primary 55P35, 57T25; Secondary 55R35, 57T05. Key words and phrases. BP-theory, PU(p), exceptional Lie groups. c 2007 American Mathematical Society 2265 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2266 MASAKI KAMEKO AND NOBUAKI YAGITA Corollary 1.3. For (G, p) in Theorem 1.1, the following holds: ∗ ∗ ∗ ∗ ∼ (1) BP (BG) is BP -flat for BP (BP)-modules, i.e., BP (BG × X) = ∗ ∗ BP (BG) ⊗BP ∗ BP (X) for all finite complexes X. ∗ ∼ ∗ ∗ (2) K(n) (BG) = K(n) ⊗BP ∗ BP (BG). ∗ ∼ ∗ ∗ (3) P (n) (BG) = P (n) ⊗BP ∗ BP (BG). We give the BP ∗-module structure of BP ∗(BPU(p)) more explicitly in this introduction. For the exceptional Lie group cases, the explicit formulas are given in §5. Theorem 1.4. There exists a BP ∗-algebra exact sequence ∗ ∗ ∗ 0 → BP ⊗M → grBP (BPU(p)) → BP ⊗IN/(f0,f1) → 0 where ∼ (1) M = Z [x ,x ,...,x ] as Z -modules (but not Z -algebras). ∼ (p) 4 6 2p (p) (p) (2) IN =Z(p)[x2p+2,x2p(p−1)]{x2p+2}, the principal ideal of Z(p)[x2p+2,x2p(p−1)] generated by x2p+2. 2 (3) Relations f0,f1 are given with modulo (p, v1,v2, ···) : ≡ − p−1 ··· ≡ − ··· f0 v0 v2x2p+2 + ,f1 v1 v2x2p(p−1) + . Remark. In the above theorem, subscript i of xi means its degree. The alge- ∗ ∗ bra BP (BPU(p)) does not contain the subalgebra BP ⊗Z(p)[x4,...,x2p], but contains a subalgebra which is isomorphic as BP ∗-modules to the above BP ∗- subalgebra. For an algebraic group G over C, Totaro [To] defines its Chow ring and con- ∗ ∼ ∗ jectures that BP (BG) ⊗BP ∗ Z(p) = CH (BG)(p). Recall that PGL(p, C)isthe algebraic group over C corresponding to the Lie group PU(p). Theorem 1.5. There exists an isomorphism ∗ ∼ ∗ BP (BPU(p)) ⊗BP ∗ Z(p) = CH (BGL(p, C))(p). Hence there exists an additive isomorphism ∗ ∼ CH (BGL(p, C))(p) = Z(p)[x4,x6, ··· ,x2p] ⊕ Fp[x2p+2,x2p(p−1)]{x2p+2}. Remark. Vistoli [Vi] also determined the additive structure of the Chow ring and integral cohomology of BPGL(p, C) by using stratified methods of Vezzosi [Ve] (see also [Mo-Vi]). Moreover he shows that for G = PGL(p, C), ∗ ∗ H (G; Z) → H (BT; Z)WG(T ) is epic. ∗,∗ Let MGL (X) be the motivic cobordism ring defined by V. Voevodsky [Vo1] 2∗,∗ 2i,i and MGL (X)= i MGL (X). 2∗,∗ ∼ ∗ Corollary 1.6. MGL (BPGL(p, C))(p) = MU (BPU(p))(p). We prove Theorem 1.1 by using the Adams spectral sequence converging to the Brown-Peterson cohomology. The E1-term of the spectral sequence could be given by ∞ ∗ Fp[v0,v1, ···]⊗H (X)withd1x = vkQkx k=0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BROWN-PETERSON COHOMOLOGY OF BPU(p) 2267 where the Qk’s are Milnor’s operations. (Here we identify Fp[v0 = p, v1, ···]= P (0)∗.) By the change-of-rings isomorphism, the E2-term is ∗ ∗ ∼ ∗ ExtA(H (BP),H (X)) = ExtE (Fp,H (X)) ∗ where E = Λ(Q0,Q1, ···). The E∞-term is given by grBP (X). ∗ To state the cohomology H (BPU(p)), we recall the Dickson algebra. Let An be an elementary abelian p-group of rank n,and ∗ ∼ H (BAn) = Fp[t1, ..., tn] ⊗ Λ(dt1, ..., dtn)withβ(dti)=ti. The Dickson algebra is GL(n,Fp) ∼ Dn = Fp[t1, ..., tn] = Fp[cn,0, ..., cn,n−1] n i with |cn,i| =2(p − p ). The invariant ring under SL(n, Fp) is also given: F − − F SL(n, p) ∼ { p 2} p 1 SDn = p[t1, ..., tn] = Dn 1,en, ..., en with en = cn,0. We also recall Mui’s ([Mu]) result by using Qi according to Kameko and Mimura [Ka-Mi] ∗ SLn(Fp) ∼ grH (BA) = SDn/(en) ⊕ SDn ⊗ Λ(Q0, ..., Qn−1){un} where un = dt1...dtn and en = Q0...Qn−1un. Theorem 1.7. There exists a short exact sequence 0 → M/p → H∗(BPU(p)) → N → 0 where M/p is the trivial E-module given in Theorem 1.4 and ∼ N = SD2 ⊗ Λ(Q0,Q1){u2} = Fp[x2p+2,x2(p2−p)] ⊗ Λ(Q0,Q1){u2} identifying x2p+2 = e2 and x2(p2−p) = c2,1. This theorem is proved by using the following facts. The group G = PU(p) has just two conjugacy classes of maximal elementary abelian p-subgroups, one of ∗ which is toral and the other is nontoral A of rankp = 2. The cohomology H (BG) is detected by these two subgroups. The restriction image to the nontoral subgroup ∗ ∗ ∼ ∗ SL(2,Fp) is iA(H PU(p)) = H (BA) . Similar (but not the same) facts also hold for the exceptional Lie groups given in Theorem 1.1. The algebraic main result in this paper is as follows: ∗ Theorem 1.8. For m ≥ 0, define f , ..., f − in P (m) ⊗SD by 0 n 1 n d1un = vkQk(un)=f0Q0un + ···+ fn−1Qn−1un. k≥m ∗ Then the sequence f0, ..., fn−1 is a regular sequence in P (m) ⊗SDn. With the notation in this theorem, we prove that the complex ∗ C =(P (m) ⊗SDn ⊗ Λ(Q0,Q1,...,Qn−1){un},d1) − with the differential d u = n 1 f Q u is a Koszul complex. This means that 1 n i=0 i i n ∗ P (m) ⊗SDn{en}/(f0,...,fn−1) for i =0 Hi(C, d1)= 0 for i ≥ 1. Thus Theorem 1.1 follows from the above theorem. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2268 MASAKI KAMEKO AND NOBUAKI YAGITA Remark about the convergence of the Adams spectral sequence. By The- orem 15.6 in Boardman’s paper [Bo2], since H∗(BP) is of finite type, the above Adams spectral sequence is conditionally convergent. Moreover, since we prove the above Adams spectral sequence collapses at the E2-level, by the remark after Theorem 7.1 in [Bo1], the above Adams spectral sequence is strongly convergent, so that we know the Brown-Peterson cohomology up to group extension. We now give an outline of the paper. In §2, we note the relation of the Koszul complex and the E1-term of the Adams spectral sequence for Λ(Q0, ..., Qn−1)-free ∗ H (X). Section 3 is stated about the Dickson invariant rings and the Milnor Qi- operation on the cohomology of classifying spaces of elementary abelian p-groups. In §4, we compute H∗(BG), MU∗(BG), CH∗(BG)forG = PU(p). The similar computations are done in §5 for exceptional groups. The last section is devoted to the proof of Theorem 1.8. 2. The Koszul complex and the Adams spectral sequence For spaces X, Y , the Adams spectral sequence converging to the stable homotopy {X, Y } has the E2-term ∗,∗ ∼ ∗ ∗ ⇒{ } E2 = ExtA(H (Y ),H (X)) = X, Y . We want to consider the case Y = P (m). ∼ ∗ ∗ ∼ Recall that A = H (BP) ⊗E with E =Λ(Q0,Q1, ...). Similarly H (P (m)) = ∗ H (BP) ⊗ Λ(Q0, ..., Qm−1). Hence the E2-term of the Adams spectral sequence abutting to P (m)∗(X)isgivenby ∗ ∗ ∼ ∗ ∗ ExtA(H (P (m)),H (X)) = ExtH∗(BP)⊗E (H (BP) ⊗ Λ(Q0, ..., Qm−1),H (X)) ∼ F ∗ E ··· = ExtEm ( p,H (X)) where m =Λ(Qm,Qm+1, ).
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