The multiplicative structure of K(n)*(Ba4)

Publicacions Matem`atiques, Vol 41 (1997), 605–612. THE MULTIPLICATIVE STRUCTURE OF K(n)∗(BA4) Maurizio Brunetti Abstract Let K(n)∗(−)beaMoravaK-theory at the prime 2. Invariant theory is used to identify K(n)∗(BA4) as a summand of K(n)∗(BZ/2×BZ/2). Similarities with H∗(BA4;Z/2) are also discussed. Introduction Let Gbe a finite group, and let Nand Cdenote respectively the normalizer and the centralizer of a p-Sylow subgroup Hof G. For a large family of cohomology theories including the Brown-Peterson cohomology BP∗(−) and Morava K-theories K(n)∗(−), the author described h∗(BG) when His cyclic [3], and discussed the case “p-rank (H)<3” in [5], proving in particular that h∗(BG) is generated as h∗-module by at most two elements if |N:C|divides p−1. Results in this paper show that the condition above is really necessary, in fact we have Theorem 0.1. Let K(n)∗(−)be a Morava K-theory at the prime 2. K(n)∗(BA4)restricts to those elements in K(n)∗(BZ/2×BZ/2) ∼ =K(n)∗[x, y]/(x2n,y2n) which algebraically depend on ¯σ=x2+y2+xy +νn(x2n−1+1y2n−1+x2n−1y2n−1+1), ¯τ1=x3+y3+x2y+νn(x2n−1y2n−1+2), ¯τ2=x3+y3+xy2+νn(x2n−1+2y2n−1). 1991 Mathematics subject classifications: 55N20, 55N22. 606 M. Brunetti This paper has several motivations. The knowledge of K(n)∗(BA4) could help to have explicit formulæ for the K(n)∗-Dickson classes. Furthermore, similarities among H∗(BA4;Z/2) and K(n)∗(BA4) suggest to study K(n)∗(BAm) —whose rank as K(n)∗-module can be calculated [6]— to get information on H∗(BAm;Z/2) which is not entirely known for m≥16 (see [1] for the cohomology of several alternating groups). The author would like to thank the anonymous referee, who drew attention to certain inaccuracies contained in the first version. 1. Preliminaries. H∗(BA4) From now on Vwill denote the group Z/2×Z/2, and H∗(−) ordinary cohomology with coefficients in Z/2. In [7], the authors describe H∗(PSL2Fq) for any odd q: they first calculate the cohomology of the generalized quaternion group Q2n+1 of order 2n+1, and then use the diagram Z/2−−−−→SL2Fq−−−−→PSL2Fq      i  j Z/2−−−−→Q2n+1 −−−−→Dn where Dnis the dihedral group of order 2n, rows are fibrations, and i and jare inclusions of 2-Sylow subgroups. Nevertheless, we show in this section that the special case PSL2F3∼ =A4 can be approached in a more direct way. The alternating group A4is the central term of the short exact sequence of groups 0−→ V−→ A4−→ Z/3−→ 0, therefore for any mod 2 cohomology theory h∗(−), h∗(BA4) is isomorphic to the ring of invariants [h∗(BV )]Z/3under the action determined by the map h∗(Bφ) induced by an automorphism φof order 3 in Aut(V). On H∗(BV )∼ =F2[x, y] the action of a generator of Z/3≤GL(V)is xαH −−→ yand yαH −−→ x+y. Consider now the map Φ from F2[x, y] to itself which maps any element cto the sum Φ(c)=c+αH(c)+α2 H(c); The Morava K-theories of BA4607 Φ is commonly known as norm map. It is easy to see that ImΦ=[H∗(BV )]Z/3; furthermore the image of Φ restricted to the set of monomials generates [H∗(BV )]Z/3regarded as graded F2-vector space. The invariant of lowest positive degree in F2[x, y]is σ=Φ(xy)=x2+xy +y2. This element is actually the Dickson class known in literature as Q2,1 (see [9]). The reader will find the relevant invariant theoretic computation in [2] to prove the algebraic dependence of every invariant on Φ(xy), Φ(x2y), Φ(xy2). In fact we have the following proposition. Proposition 1.1. As a graded ring, H∗(BA4)is isomorphic to F2[σ, τ1,τ 2]/R, where deg σ=2,deg τ1= deg τ2=3, and Ris the ideal generated by σ3+τ2 1+τ1τ2+τ2 2. The proposition above can be restated in terms of pure invariant theory. Corollary 1.2. Suppose that a Z/3-action on F2[x, y]is given by x−→ yand y−→ x+y. The ring of the invariants is a polynomial ring generated by σ=x2+y2+xy, τ1=x3+y3+x2y, τ2=x3+y3+xy2, quotiented by R=(σ3+τ2 1+τ1τ2+τ2 2). 608 M. Brunetti 2. The Morava K-theory of BA4 We recall that Morava K-theory at the prime 2 is a complex oriented cohomology theory with coefficients K(n)∗({pt})=F2[νn,ν−1 n] where deg νn=−2(2n−1), and we have K(n)∗(BV )∼ =K(n)∗[x, y]/(x2n,y2n) where deg x= deg y= 2. As noticed in section 1, K(n)∗(BA4) is isomorphic to [K(n)∗(BV )]Z/3 where the Z/3-module structure is defined by the map K(n)∗(Bφ), being φa generator of Z/3≤Aut(V). The following lemma helps to give a concrete description of the K(n)-invariants. Lemma 2.1. One of the two generators φof Z/3≤Aut(V)acts as follows on K(n)∗(BV ): αK def =K(n)∗(Bφ):x−→ yand αK:y−→ x+y+νnx2n−1y2n−1. Proof: See [4]. The element αK(y) is actually the formal sum of xand ywith respect to the formal group law of mod 2 Morava K-theory FK(n)(x, y)mod(x2n,y2n). Consider now the norm map Ψ defined as follows: Ψ:c∈K(n)∗(BV )−→ c+αK(c)+α2 K(c)∈[K(n)∗(BV )]Z/3. The map Ψ is obviously the analogue of Φ defined in section 1: it is surjective, and the invariants regarded as F2-vector space are spanned by the image of Ψ restricted to monomials. Notice also that we can equip K(n)∗(BV )∼ =K(n)∗[x, y]/(x2n,y2n) with a different Z/3-module structure just by posing αH(x)=yand αH(y)=x+y. Abusing notation, we shall use again Φ to denote the endomorphism defined on the generic element of K(n)∗(BV ) as follows: c−→ c+αH(c)+α2 H(c). We are ready now to prove our main result. The Morava K-theories of BA4609 Theorem 2.2. K(n)∗(BA4)restricts to those elements in K(n)∗(BV ) which algebraically depend on Ψ(xy)=¯σ, Ψ(x2y)=¯τ1and Ψ(xy2)=¯τ2. Proof: Since K(n)∗(−)is2(2 n−1)-periodic we can look at classes in K(n)∗(BV ) whose degree is between 2 and 2(2n−1). In this range, elements of type ν2 nxhyk are necessarily zero, since either hor kis greater than 2n. It follows that for any monomial xhyk∈K(n)∗(BV )wehave (1) Ψ(νnxhyk)=νnΦ(xhyk). An element c∈K(n)∗(BV ) is invariant under αKif and only if Ψ(c)=c, and supposing 2≤t≤2(2n−1), we have c=p(x, y)+νnq(x, y), where p(x, y) and q(x, y) are homogeneous polynomials of F2[x, y]of degree tand t+ 2(2n−1) respectively. If Ψ(c)=c, it follows from the considerations above that Φ(p(x, y)) = p(x, y), and by Corollary 1.2 there exists a polynomial r1in three indeterminates such that r1(σ, τ1,τ 2)=p(x, y). Define now Ψ(xy)=¯σ, Ψ(x2y)=¯τ1and Ψ(xy2)=¯τ2. The element c−r1(¯σ, ¯τ1,¯τ2)=νns(x, y) is invariant under αK. Notice now that s(x, y) can be regarded as a polynomial in F2[x, y]; it follows by (1) that s(x, y) is invariant under αH, and again by Corollary 1.2 there exists a polynomial r2in three indeterminates such that r2(σ, τ1,τ 2)=s(x, y). 610 M. Brunetti We finally get c=r1(¯σ, ¯τ1,¯τ2)−νnr2(¯σ, ¯τ1,¯τ2) as we claimed. Theorem 2.2 also gives some information on K(n)∗(BA5). Notice in fact that 2-Sylow subgroups in A5are abelian, and a 2-Sylow normalizer in A5is isomorphic to A4. It follows by a theorem in [8] that BA4 and BA5are stably 2-homotopy equivalent. Hence the map induced by inclusion K(n)∗(BA5)−→ K(n)∗(BA4) is an isomorphism. Remark 2.3. The element ¯σ3+¯τ2 1+¯τ1¯τ2+¯τ2 2 is zero in K(n)∗(BA4), as the analogous algebraic expression in σ,τ1,τ2 for ordinary cohomology. The relation above is not however of minimal positive degree: the element ν2 n¯σ2n is zero and has degree four. It is known that the subring of Heven(BA4) generated by Chern classes is proper (see, for example [10, p. 100]), and the reader could ask if ¯σ, ¯τ1,¯τ2are K(n)-Chern classes of suitable representations. We recall that up to equivalence the group A4has just four distinct complex irreducible representations. Three of them are one-dimensional, and their restriction to Vis trivial. The fourth one has instead non-trivial total Chern class in K(n)∗(BA4), as the next proposition shows. Proposition 2.4. Let ξbea3-dimensional irreducible representation of A4. The restriction ξ|Vto the 2-Sylow subgroup Vhas Chern classes c1(ξ|V)=νn¯σ2n−1,c 2(ξ|V)=¯σ, c3(ξ|V)=¯τ1+¯τ2+νn¯σ2n−1+1 in K(n)∗(BV ). Proof: Let g1and g2be two generators in V. Consider two onedimensional representations ρ1and ρ2defined as follows ρi:gi−→ −1ρi:g3−i−→ 1 The Morava K-theories of BA4611 for i=1,2. The transfer ξof ρ1to A4represents the equivalence class of the 3-dimensional irreducible representations of A4; its restriction to Vis given by ρ1⊕ρ2⊕(ρ1⊗ρ2). It follows that the total Chern class c.(ξ|V) is equal to (1 + x)(1 + y)(1 + x+y+νnx2n−1y2n−1). Hence the proposition follows. References 1. A. Adem, J. Maginnis and R. J. Milgram, Symmetric invariants and cohomology of groups, Math. Ann. 287 (1990), 391–411. 2. D. Benson,“Polynomial invariants of finite groups,” London Math. Soc., Lecture Notes 190, 1993. 3. M. Brunetti, A family of 2(p−1)-sparse cohomology theories and some actions on h∗(BCpn), Math. Proc. Cambridge Philos. Soc. 116 (1994), 223–228. 4. M. Brunetti, On the canonical GL2(F2)-module structure of K(n)∗(BZ/2×BZ/2), in “Algebraic Topology: New Trends in Localization and Periodicity,” (C. Broto, C. Casacuberta, G. Mislin, eds.), Barcelona Conference on Algebraic Topology 1994, Birkh¨auser Verlag, 1996, pp. 51–59. 5. M. Brunetti, On groups of order p2qand some complex oriented cohomology theories, Preprint. 6. M. J. hopkins, N. J. Kuhn and D. G. Ravenel, Morava K-theory of classifying spaces and generalized characters of finite groups, in “Algebraic Topology: Homotopy and Group Cohomology,” (J. Aguad´e, M. Castellet, F. R. Cohen, eds.), Proceedings of the 1990 Barcelona Conference on Algebraic Topology, Springer LNM 1509, 1992, pp. 186–209. 7. S. A. Mitchell and S. Priddy, Symmetric product spectra and splittings of classifying spaces, Amer. J. Math. 106 (1984), 219–233. 8. G. Nishida, Stable homotopy types of classifying spaces of finite groups, in “Algebraic and Topological Theories,” to the memory of T. Miyata, Kinokuniya Comp. Ltd., Tokyo, 1986, pp. 391–404. 9. W. Singer, Invariant theory and the Lambda Algebra, Trans. Amer. Math. Soc. 280 (1981), 673–693. 612 M. Brunetti 10. C. B. Thomas,“Characteristic classes and the cohomology of finite groups,” Cambridge University Press, 1986. Dipartimento di Matematica e Applicazioni Universit`a di Napoli Via Claudio 21 I-80125 Napoli ITALY Primera versi´o rebuda el 3 de Setembre de 1996, darrera versi´o rebuda el 17 de Mar¸c de 1997