TRANSFERRED CHERN CLASSES IN MORAVA K-THEORY

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 6, Pages S (03) Article electronically published on December 18, 2003 TRANSFERRED CHERN CLASSES IN MORAVA K-THEORY MALKHAZ BAKURADZE AND STEWART PRIDDY (Communicated by Paul Goerss) Abstract. Let η be a complex n-plane bundle over the total space of a cyclic covering of prime index p. We show that for {1, 2,..., np}\{p, 2p,..., np} the -th Chern class of the transferred bundle differs from a certain transferred class ω of η by a polynomial in the Chern classes c p,..., c np of the transferred bundle. The polynomials are defined by the formal group law and certain equalities in K(s) B(Z/p U(n)). 1. Statements Let p be a prime, and let π be the cyclic group of order p. For a given action of π on a space X consider the regular covering ρ : Eπ X Eπ π X. Let us write for short X hπ := Eπ π X. For the permutation action of π on X = Y p we have Y p hπ = Eπ π Y p. Let t be a generator of π and let N π =1+t t p 1 be the trace map. For an n-plane bundle η n, the corresponding classifying map f : X BU(n) induces the classifying map (f, tf,..., t p 1 f) for the bundle N π η n and thereby a map of orbit spaces f η n : X hπ BU(n) p hπ. For the covering ρ π : Eπ BU(n) p BU(n) p hπ and the universal n-plane bundle ξ n BU(n) consider the Atiyah transfer bundle [2] ξπ n BU(n)p hπ, i.e., the np-plane bundle ξπ n = Eπ π (ξ n ) p. Then the map f η n classifies the Atiyah transfer bundle for η n and ρ. So by naturality of transfer [1] we can consider ρ π as the universal example. Received by the editors October 24, 2002 and, in revised form, February 24, Mathematics Subject Classification. Primary 55R12, 55R20; Secondary 55R40. Key words and phrases. Transfer, formal group law, Chern class. The first author was supported by the Max Planc Institute of Mathematics and CRDF grant GM The second author was partially supported by the NSF c 2003 American Mathematical Society

2 1856 MALKHAZ BAKURADZE AND STEWART PRIDDY Let K(s), s 1, be the sth Morava K-theory at p. We recall that by the Künneth isomorphism, K(s) (BU(n) p )=F T as a π module, where F is free and T is trivial. Definition 1.1. Let ω (n) F be defined modulo er N π =Im(1 t) by N π (ω (n) )=c (N π (ξ n )), where the c are Chern classes, {1,..., np}\{p, 2p,..., np}. By naturality of the transfer, Trπ = Tr π t,where Trπ : K(s) (Eπ BU(n) p ) K(s) (BU(n) p hπ ); hence Trπ (ω(n) ) is well defined. We write ω (η n ) for the pullbac by the map f η n of orbit spaces defined above. Recall that K(s) (Bπ) =F p [v s,vs 1 ), where z = c ][z]/(zps 1 (θ) isthechernclass of the canonical complex line bundle over Bπ. Lemma 1.2. We can define a polynomial in n +1 variables, A (n) (zp 1,Z 1,..., Z n ) K(s) [z,z 1,..., Z n ], uniquely by the equation in K(s) B(π U(n)): ( ) C v s z ps 1 p 1 p i 0,i 1,..., i n i 1+2i ni n= i 0+i i n=p c i1 1...cin n = A(n) (zp 1,C p,..., C pn ) where C i = c i (ξ n θ ξ n... θ p 1 ξ n ), c j = c j (ξ n ) are Chern classes, and {1,..., np}\{p,..., np}. For example, in K(s) (Bπ BU(1)) one has ) s 1 C 1 = v s (z ps 1 c 1 + z ps p i Cp pi 1 ; thus i=1 s 1 A (1) 1 (zp 1,Z 1 )=v s z ps p i Z pi 1 1. Then using the polynomials A (n) we evaluate the transferred classes ω (η n )for regular coverings: Theorem 1.3. Let ρ : X X/π be the regular cyclic covering of prime index p defined by a free action of π on X, andlettr = Trρ be the associated transfer homomorphism. Let η n X be a complex n-plane bundle, ηπ n X/π the pnplane bundle defined by Atyiah transfer, and ψ X/π be the complex line bundle associated with ρ. Then c (ηπ) n Tr (ω (η n )) = A (n) (cp 1 1 (ψ),c p (ηπ), n..., c pn (ηπ)), n where {1,..., np}\{p,..., np}. Example 1.4. ω 1 = c 1 (η n ); hence if =1wehave c 1 (ηπ) n Tr (c 1 (η n )) = A (n) 1 (cp 1 1 (ψ),c p (ηπ), n..., c pn (ηπ)). n i=1

3 TRANSFERRED CHERN CLASSES IN MORAVA K-THEORY 1857 If, in addition, n = 1, then by the above example we have for the line bundle η X and the transferred Chern class, s 1 c 1 (η π ) Tr (c 1 (η)) = v s z ps p i c p (η π ) pi Proofs For the proof of Lemma 1.2 we need the following property of formal group laws (FGL) observed in [3]. Lemma 2.1. For the formal group law in Morava K-theory K(s), s>1, we have mod x p2(s 1) (or modulo y p2(s 1) ) ( ) p F (x, y) x + y v s p 1 (x ps 1 ) j (y ps 1 ) p j. j 0<j<p Proof. As D. Ravenel explained, this result can be derived from the recursive formula for the FGL given in [5, 4.3.8]: For the FGL in Morava K-theory it reads F (x, y) =F (x + y, v s w 1 (x, y) ps 1,vs e2 w 2 (x, y) p2(s 1),...) where w i is a certain homogeneous polynomial of degree p i defined by [5, 4.3.5] and e i =(p is 1)/(p s 1). In particular, w 0 = x + y (the first term in the formula above), w 1 = 0<j<p i=1 p 1 ( p j ) x j y p j, and w i / (x p,y p ). For s>1 we can reduce modulo the ideal vs e2,y (xp2(s 1) p2(s 1) )andget F (x, y) F (x + y, v s w 1 (x, y) ps 1 ) = F (x + y + v s w 1 (x, y) ps 1,v s w 1 (x + y, v s w 1 (x, y) ps 1 ) ps 1,...) F (x + y + v s w 1 (x, y) ps 1,v s w 1 (x ps 1 + y ps 1,vs ps 1 w 1 (x, y) p2(s 1) )), and modulo vs 1+ps 1 (x p2(s 1),y p2(s 1) )wehavef(x, y) x + y + v s w 1 (x, y) ps 1. ProofofLemma1.2. We begin by considering the case n =1. Letσ i be the i-th symmetric functions in p variables. Then C i = σ i (x, F (x, z),..., F (x, (p 1)z)), where x = c 1 (ξ 1 ), z = c 1 (θ) are Chern classes in K(s) B(π U(1)) = K(s) [[z,x]]/ (z ps ). Consider the equation C = ( ) p λ i Cp i + p 1 x z ps 1 ; 1 p 1. 0 i p s We want to prove that such λ i exist uniquely as elements in K(s) [[z]]/(z ps ) and to compute these elements as polynomials in z p 1. Then A (1) (zp 1,C p )= 0 i p λ s i Cp i. C is the Chern class of the bundle ξ (1 + θ + θ θ p 1 )andcanbe written as a series in the Chern classes of ξ, thatisx, and the Chern classes of 1+θ + θ θ p 1. But the Chern classes of the latter bundle are elementary

4 1858 MALKHAZ BAKURADZE AND STEWART PRIDDY symmetric functions in z,2z,..., (p 1)z, the Chern classes of θ, θ 2,..., θ p 1,allof which vanish except for the (p 1)-th class, which is z p 1. Hence we can write the classes C as series in x and z p 1. Lemma 2.1 enables us to write C as explicit polynomials in x and z p 1. Now noting that C p = x p mod z p 1 we obtain from the above equation a system of linear equations in variables λ j by equating the coefficients at x i, i 0. Vanishing also implies that λ 0 =0,=1,..., p 2and λ p 10 = c p 1 (1 + θ θ p 1 )= z p 1. Then equating the coefficients at x p,...,x ps+1 in the above equation after rewriting it in terms of x and z p 1 as above, we have a system of p s linear equations in p s variables λ i, i =1,..., p s. The determinant of this system is invertible since the diagonal coefficients are invertible and all other coefficients lie in the (nilpotent) augmentation ideal. Thus the elements λ i are uniquely defined. Of course, equating the coefficients at x i for i p, 2p,..., p s+1 will produce other equations in λ j, j =1,..., p s. But these equations are derived from the old equations above. These additional equations mae the matrix upper triangular. This defines A (1). In the general n case we proceed analogously, noting that C ip = c p i mod zp 1, i =1,..., n. Our additional claim again is that the A (n) are polynomials, that is, only a finite number of elements λ,i1,...,i n are nontrivial. Here we need the splitting principle and Lemma 2.1 to express explicitly the elements C i, i =1,..., np in terms of polynomials in z p 1 and c 1,..., c n.for {1,..., np}\{p,..., np}, let A (n) (zp 1,C p,..., C pn )= λ,i1,...,i n Cp i1...cnp. in 0 i 1,...,i n p s We define λ,0...0 = c (n + nθ nθ p 1 ) again by looing at reductions to K(s) (Bπ). The other n(p s +1) 1elementsλ,i1,...,i n can be defined as the solution of a system of n(p s +1) 1 linear equations with an invertible determinant. This system is obtained from the equation after using Lemma 2.1 to rewrite it in terms of z p 1 and c 1,..., c n and equating coefficients at c pi1 1...cpin n. The solution defines λ,i1,...,i n,0 i j p s. Again the additional equations in these elements arise from the coefficients of other monomials and are not new. The desired polynomials are thus uniquely defined. Proof of Theorem 1.3. Consider the homotopy orbit space BU(n) p hπ = Eπ π BU(n) p as the universal example. The diagonal map BU(n) BU(n) p induces the inclusion i : Bπ BU(n) Eπ π BU(n) p. We use a result from [4] (Proposition 4.2) which implies that since BU(n) isa unitary-lie space (i.e., K(s) BU(n) has no nilpotent elements) the map (i ρ π ) is a monomorphism. Since ρ π Tr = N π, the difference c (ηπ) n Tr (ω (n) ) belongs to er ρ π and hence is detected by i. The result now follows from Lemma 1.2. Note we can replace the cyclic group by the symmetric group Σ p and use the polynomials A (n) to evaluate the disparity or gap between Chern class c (ξσ n p ) and Im TrΣ p,forρ Σp : EΣ p U(n) p BU(n) p hσ p.

5 TRANSFERRED CHERN CLASSES IN MORAVA K-THEORY 1859 Namely, the Euler characteristic of the coset space Σ p U(n)/π U(n) =(p 1)! is prime to p. Hence the inclusion ρ π,σp : π Σ p induces a monomorphism K(s) (B(Σ p U(n))) K(s) (B(π U(n))). Hence Hunton s result above holds for BU(n) p hσ p. Now let ς (n) F be defined modulo er N Σp by N Σp (ς (n) )=c (N π (ξ n )). Again TrΣ p (ς (n) ) is well-defined: as above, er N π =Im(1 t ) and therefore a F er N Σp g a Im(1 t ) TrΣ p ( g a)=0 g Σ p/π (p 1)!Tr Σ p (a) =0 Tr Σ p (a) =0. Then for the np-plane bundle over BU(n) p hσ p, ξ n Σ p = EΣ p Σp (ξ n ) p, g Σ p/π the difference belongs to er ρ Σ p c (ξ n Σ p ) Tr Σ p (ς (n) ) and hence is detected by the polynomials A (n) (y, c p(ξ n Σ p ),..., c np (ξ n Σ p )), where y K(s) (B(Σ p )) = K(s) [[y]]/(y ms ), y =2(p 1), and m s =[(p s 1/(p 1))] + 1. Thus we have: Theorem 2.2. c (ξσ n p ) TrΣ p (ς (n) )=A(n) (y, c p(ξσ n p ),..., c n p(ξσ n p )), for {1,..., np}\{p,..., np}. Acnowledgment We would lie to than Neil Stricland for suggesting an elegant proof of Lemma 1.2 completely within the context of subgroups of formal groups. This material is available from N.P.Stricland@sheffield.ac.u. We have decided not to include it since it uses rather different techniques and is no shorter than our proof for which we now offer some motivation. The lemma says that, modulo image of the transfer Tr : K(s) B(U(n)) K(s) B(π U(n)) for the covering π 1:Eπ BU(n) Bπ BU(n), the class C can be written as a polynomial in z p 1 and C p,..., C np, that this expression vanishes when restricted to K(s) BU(n), and that the indeterminacy is obtained by applying the transfer to p 1 C after restricting to K(s) (BU(n)). This uses Fröbenius reciprocity and the fact that Tr (1) = [p](z)/z = v s z ps 1 in Morava K-theory.

6 1860 MALKHAZ BAKURADZE AND STEWART PRIDDY References 1. J. F. Adams, Infinite Loop Spaces, Annals of Mathematics Studies, Vol. 90, Princeton University Press, Princeton, NJ, MR 80d: M. F. Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. 9 (1961), MR 26: M. Bauradze and S. Priddy, Transfer and complex oriented cohomology rings, Algebr. Geom. Topol. 3 (2003), (electronic). 4. J. R. Hunton, The Morava K-theories of wreath products, Math. Proc. Cambridge Philos. Soc. 107 (1990), MR 91a: D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press, Orlando, FL, MR 87j:55003 Razmadze Mathematical Institute, Tbilisi , Republic of Georgia Current address: Max-Planc-Institut Für Mathemati, Bonn, Germany address: bauradz@mpim-bonn.mpg.de Department of Mathematics, Northwestern University, Evanston, Illinois address: priddy@math.northwestern.edu