TRANSFER AND COMPLEX ORIENTED COHOMOLOGY RINGS MALKHAZ BAKURADZE AND STEWART PRIDDY Keywords: transfer, Chern class, classifying space, complex cobor- dism, Morava K-theory 55N22, 55R12. 1. Introduction Let p be a prime and let G p be a subgroup of the symmetric group. In this paper we use the transfer to study homotopy orbit spaces XphG= EGx GXp in complex oriented cohomology. We are particularly interested in computing the ring structure. Thus we are led to consider the relation between cup products and transfer known as Fröbenius reciprocity by analogy with representation theory T r*(x)y = T r*(xæ*(y)) (formula (i) of Section 2) where æ : EGx Xp ! XphG is the covering projection and T r* : E*(Xp) ! E*(XphG) is the associated transfer homomorphism. It is worth noting that the multiplicative structure of the cohomology groups we consider is com- pletely determined by this formula. In case E = K(s) is Morava K-theory, G is cyclic of order p, and X is the classifying space of a finite group, Hopkins-Kuhn-Ravenel [11 ] have studied these cohomology groups as modules over the coefficient ring. Our paper builds on their approach by extending their notion of a good group to spaces. For X = CP 1 we determine the algebra K(s)*(Xph p) for Morava K-theory; for complex cobordism we compute the ring MU*(Xph p) making additional use of the formal group law. This enables us to make explicit computations of the transfer in both cases. In an analogous fashion we compute the algebra BP *(Xph p). The starting point and original motivation for our work comes from Quillen's famous formula for T r*(1), the stable Euler class, for the uni- versal Z=p covering. As explained in Section 2, our results for CP 1 ____________ Date: January 10, 2002. 1 2 MALKHAZ BAKURADZE AND STEWART PRIDDY provide a universal example which enable us to compute the stable Eu- ler classes and the transfer in general for many other cases. For example universal coverings for some nonabelian p-groups, namely those with cyclic subgroups of index p and those which are semi-direct products of elementary abelian p-groups with Z=p. The paper is organized as follows. In Section 2 we recall some pre- liminary results for the transfer. Section 3 is devoted to extending some results of [11 ] in Morava K-theory. The computation of the alge- bra K(s)*(Xph p) for X = CP 1 is given as Theorem 4.5 of Section 4; analogous results are given for X = B(Z=p) along with further compu- tations of the transfer. In Section 5 we turn to the complex cobordism rings of B(ß o U(1)). In Section 6 we derive formulas for transferred Chern classes. The ring BP *(Xph p) is computed in Section 7. In Sec- tion 8 stable Euler classes for the universal coverings associated with p-groups with normal cyclic subgroups of index p are computed. Similar results are given in Section 9 for the semi-direct products (Z=p)soZ=p. Finally in Section 10 we compute coefficients useful in computing the transfer for G = Z=p and G = p. 2. Preliminaries We recall that a multiplicative cohomology theory E* is called com- plex oriented if there exists a Thom class, that is, a class u 2 E2(CP 1) that restricts to a generator of the free one-dimensional E* module E2(CP 1). The universal example is complex cobordism MU*. Then E*(CP 1) = E*[[x]], where x is the Chern class of canonocal complex line bundle , over CP 1 = BU(1) and E*(BU(1)p) = E*[[x1, ..., xp]], xi = c1(,i) and ,iis the pull back bundle over BU(1)p by the projection BU(1)p ! BU(1) on the i-th factor. Much of our paper is written in terms of transfer maps [1, 7] and formal group laws. Let us give a brief review of formal properties of the transfer. For a finite covering æ : X ! X=G there is a stable transfer map T r = T r(æ) : X=G+ ! X+ . 3 For any multiplicative cohomology theory E* the induced map T r* is a map of E*(X=G) modules, i.e., (i) T r*(xæ*(y)) = T r*(x)y, x 2 E*(X), y 2 E*(X=G). For example (ii) T r*(æ*(y)) = T r*(1)y. The element T r*(1) 2 E0(X=G) is called the index or stable Euler class of the covering æ. The following additional properties of the transfer will be used: (iii) The transfer is natural with respect to pullbacks; (iv) T r(æ1 x æ2) = T r(æ1) ^ T r(æ2); (v) If æ = æ2æ1, then T r(æ) = T r(æ2)T r(æ1). More generally for a covering projection æH,G : X=H ! X=G with H G there is a stable transfer map T rH,G : X=G+ ! X=H+ For ease of notation if H = e, as above, we write projection and transfer in several equivalent ways æ = æG = æe,G, T r = T r(æ) = T rG = T re,G. The reverse composition to (ii) is given by (vi) (Double coset formula) If K, H G then X * æ*K,GT r*H,G= T r*K\Hx,K O x-1 O æ*Kx-1\H,H x where the sum is taken over a set of double coset representatives x 2 K\G=H. Here Hx = xHx-1. For a regular covering æH,G, i.e. H E G, X æ*H,GT r*H,G(x) = g*(x). g2G=H In subsequent sections the reduced transfer T rH,G : X=G ! X=H is used. We recall Quillen's formula [13 , 6] mentioned in the Introduction. First, E*(BZ=p) = E*[[z]]=([p](z)) where z is the Chern class of the canonical complex bundle over BZ=p and [p](z) is the p-series or p-fold iterated formal sum. Then (1) T r*Z=p(1) = [p](z)=z, 4 MALKHAZ BAKURADZE AND STEWART PRIDDY where T r*Z=pis the transfer homomorphism for the universal p-covering EZ=p ! BZ=p. The relation [p](z) = 0 is equivalent to the transfer relation zT r*Z=p(1) = T r*Z=p(c1(C)) = T r*Z=p(0) = 0 obtained by applying (ii). Of course since transfer is natural, Quillen's formula enables us to compute the stable Euler class for any regular Z=p covering. In this spirit, let ß p be the subgroup of cyclic permutations of order p. For a given free action of ß on a space Y with a given complex line bundle j ! Y we have an equivariant map ji = (g1, ...gp) : Y ! BU(1)p, where gi classifies the line bundle ti-1j, t 2 ß. So by naturality of transfer, the computation of transferred Chern classes T r*(ci1(j)), i 1 for cyclic coverings can be reduced to the covering æi : Eß x (BU(1))p ! Eß xi (BU(1))p, as the universal example. Similarly for the symmetric group. Note that by transfer property (v), T r(æi)* has the same value on the Chern classes x1, ..., xp in E*(BU(1)p) = E*[[x1, ..., xp]] as above: the group ß permutes the xiand æit = æi, t 2 ß. Thus in computations of the transfer we sometimes write these Chern classes in an equivalent way x, tx, ..., tp-1x. 3. Transfer and K(s)*(Xph p) Fix a prime p and an integer s 0. Let X be a CW complex whose Morava K-theory K(s)*(X) is even dimensional and finitely generated as a module over K(s)* = Fp[vs, vs-1] with |vs| = -2(ps - 1) as usual. We note there is no restriction on p: although K(s) is not a commu- tative ring spectrum for p = 2 our even dimensionality assumption implies the deviation from commutativity is zero in the groups we con- sider by a result of Würgler [16 ]. In this section we study the transfer homomorphism and compute the stable Euler class. We extend some results of Hopkins-Kuhn-Ravenel [11 ] and Hunton [10 ] for the group ß to p. This leads to one of our main results, the computation of the algebra K(s)*(Xph p) for X = 5 CP 1 given as Theorem 4.5 and Corollary 4.6 of the next section. Here, as in the Introduction, Xph p= E p x p (X)p is the homotopy orbit space of p acting on Xp by permuting factors. We recall s K(s)*(Bß) = K(s)*[z]=(zp ) where |z| = 2. We consider the Atiyah-Hirzebruch Serre spectral se- quence: (2) E2 *,*(ß, X) = H*(ß; K(s)*Xp) ) K(s)*(Xphi) By the Künneth isomorphism (3) K(s)*Xp -! (K(s)*X) p Then K(s)*Xp is a ß module where ß acts by permuting factors (see [11 ], Th. 7.3). An element x 2 K(s)*(X) is called good if there is a finite cover Y ! X together with an Euler class y 2 K(s)*(Y ) such that x = T r*(y) where T r* : K(s)*(Y ) ! K(s)*(X) is the transfer. The space X is called good if K(s)*(X) is spanned over K(s)* by good elements. Let fl = '*(z), where ' : Xphi! Bß is the projection and let {xi} be a K(s)* basis for K(s)*(X). The following result is well known in ordinary mod-p cohomology. We follow the K(s)* version for classifying spaces of [11 ] Th.7.3. Proposition 3.1. Let X be a good space. (i) As a K(s)* module K(s)*(Xphi) is free with basis { fli (xj) p | 0 i < ps, 1 j} and X { 1 xi1 xi2 ... xip | I 2 Pp } (i1,i2,...,ip)2I where I = {(i1, i2, ..., ip)} runs over the set Pp of ß-equivalence classes of p-tuples of positive integers at least two of which are not equal. (ii) Xphiis good. Proof: (i) By the Künneth isomorphism, (4) K(s)*(X) p = F T 6 MALKHAZ BAKURADZE AND STEWART PRIDDY as a ß-module, where F is free and T is trivial. Explicitly a K(s)* basis for T is {(xi) p, i 1}, while a K(s)* basis for F is {xi1 xi2 ... xip, ij 1} where not all the factors are equal. Then H*(ß; F ) = F i if * = 0 = 0 if * > 0 and H*(ß; T ) = H*(Bß) T. Thus E2 0,*(ß, X) = K(s)*(Xphi)i = F i T . To continue the proof we consider the covering projection æi : Eß x Xp ! Xphi its associated transfer homorphism (5) T r* = T r*i: K(s)*(Xp) ! K(s)*(Xphi) and induced homomorphism æi* : K(s)*(Xphi) ! K(s)*(Xp). Similarly for the group p. Then æi *T r* = N, where N = Ni is thePnorm or trace map N : K*(Xp) ! K*(Xp) defined by N(x) = g2i gx. Thus we have established the following lemma. Lemma 3.2. If y 2 K(s)*(Xp) is good then there exists a good element z 2 K(s)*(Xphi) such that æ*i(z) = N(y). Lemma 3.3. If x 2 K(s)*(X) is good then there is a good element z 2 K(s)*(Xphi) such that æ*i(z) = x p. Proof: By assumption there is a finite covering f : Y ! X and an Euler class e 2 K(s)*(Y ) such that x = T r*(e). Now consider the covering OE = f x ... x f : Y p! Xp which extends to a covering 1 x OE : Yhpi! Xphi and yields a map of coverings æi p Eß x Y p ______-Yhi | | | | | | OE| 1 x OE| | | | | ?| æi ?| Eß x Xp ______-Xphi 7 The class e p is an Euler class for Y p. Since transfer is natural and commutes with tensor products we have æ*iT r*(1 e p) = T r*æi*(1 e p) = T r*(e p) = T r*(e) ... T r*(e) = x p Corollary 3.4. E2 0,*(ß, X) consists of permanent cycles which are good. Thus as differential graded K(s)* modules, there is an isomorphism of spectral sequences (Er *,*(ß, pt) K(s)*T ) F i- ! Er*,*(ß, X) Thus it follows that as a K(s)* algebra, K(s)*(Xphi) is generated by K(s)*(Bß), T , and F i. (ii) The proof of [11 ] Th. 7.3 carries over. This completes the proof of Proposition 3.1. Lemma 3.5. i) Im(T r*)s. fl = 0. ii) T r*(1) = vsflp -1. iii) If y 2 T then T r*(y) = y . T r*(1). Proof: i) By Fröbenius reciprocity, T r*(w) . fl = T r*(w . æi*(fl)) = T r*(w . 0) = 0. ii) Consider the pull back diagram of coverings '0 Eß x Xp ______-Eß | | | | | | | | | | | | ?| ' ?| Xphi ________-Bß Then T r*(1) = T r*'0*(1) = '*T r*Z=p(1) by naturality of the transfer. s-1 * By Quillen's formula T rZ=p(1) = vszp . Since fl = ' (z), the result follows. iii) If y 2 T then æi*(y) = y. The result follows from transfer property (ii), T r*æi*(y) = y . T r*(1). 8 MALKHAZ BAKURADZE AND STEWART PRIDDY Remarks : 1) From the periodicity of the cohomology of a cyclic group [4] Proposition XII, 11.1, we have isomorphisms Hs(ß; K*(Xp)) -.z!Hs+2(ß; K*(Xp)) for s > 0 and H0(ß; K*(Xp))=Im(N) -.z!H2(ß; K*(Xp)) thus multiplication by z is also injective on T at the E2 term. 2) æi*T r* = N, thus modulo ker(æi *) we have X (6) T r*(xi1 xi2 ... xip) = 1 xff(i1) xff(i2) ... xff(ip). ff2i Note that if the ij in (6) are equal, the right hand side is zero. However T r*(xj p ) = 1 xj p . T r*(1) on the nose by Lemma 3.5 iii). We now turn to K(s)*(Xph p). Lemma 3.6. æi, p : Bß ! B p induces an isomorphism of K(s)* algebras æ*i, p: K(s)*(B p) -! {K(s)*(Bß)}W where W = N p(ß)=ß Z=(p - 1). Computing invariants yields K(s)*(B p) = K(s)*[y]=(yms ) where æ*i, p(y) = zp-1 and ms = [(ps - 1)=(p - 1)] + 1. Proof: The group W has order prime to p and thus acts exactly on the AHS spectral sequence for K(s)*(Bß) producing a new spec- tral sequence of invariants. It is well known that H*(B p; Fp) = H*(Bß; Fp)W = E[a] Fp[b] where |a| = 2(p - 1) - 1 and b = fi(a). Therefore æi, p induces an isomorphism of spectral sequences. E*,*r( p, pt) -! E*,*r(ß, pt)W The corresponding isomorphism on abutments yields the desired result. Let c = '*(y) where ' : E p x p Xp ! B p is projection. 9 Proposition 3.7. As a K(s)* module K(s)*(Xph p) is free with basis { ci (xj) p | 0 i < ms, 1 j } and X { 1 xi1 xi2 ... xip | I 2 Ep } (i1,i2,...,ip)2I where I = {(i1, i2, ..., ip)} runs over the set Ep of p-equivalence classes of p-tuples of positive integers at least two of which are not equal. Proof: The result follows from the AHS spectral sequence since, as in Lemma 3.6, W acts exactly. 4. Applications 1. CP 1. As an application we consider the case X = CP 1 = BU(1). Then Xphi= B(ß o U(1)), Xph p= B( p o U(1)). We compute the Morava K- theory algebras of these spaces as well as the transfer homomorphism (5) of Section 3 and (8) below. We recall K(s)*(Xp) = K(s)*[[z1, z2, ..., zp]] where |zi| = 2. The projection 'G : XphG! BG for G = ß or G = p is induced by the factorisation G o U(1)=U(1)p = G. Proposition 4.1. (i) For X = CP 1 the cup products of K(s)*(Xphi), as an algebra over K(s)*(Bß), are determined by Im(T r*) . fl = 0. (ii) B(ß o U(1)) is good. Proof: (i) These relations hold by Lemma 3.5. By the calculation of a basis in the proof of Proposition 3.1, there are no other relations. (ii) Since BU(1) is good this follows from Proposition 3.1 (ii). Turning to the symmetric group case we denote by ,xp the p-fold product of the canonical line bundle over Xp. This bundle extends to an p-dimensional bundle (7) , p = E p x p ,xp 10 MALKHAZ BAKURADZE AND STEWART PRIDDY on Xph pclassified by the inclusion Xph p= B( p o U(1)) ,! BU(p). Let ci = ci(, p). Then æ p *(ci) = ci(,xp ) = oei, the ith symmetric polynomial in the zj where æ p : E p x Xp ! Xph p is the covering projection. Definition 4.2. We shall also consider the classes ~ci= T r p*(z1z2 . . . zi) for i = 1, ..., p - 1 where (8) T r p*: K(s)*(E p x XP ) ! K(s)*(Xph p) is the transfer homomorphism. Lemma 4.3. æ p *(~ci) = i!(p - i)!oei. Proof: æ p*(~ci) = æ p*T r* p(z1z2...zi) = N p(z1z2...zi). For each sub- set of i integers {j1, j2, ..., ji} with 1 jk p, there are i! bijec- tions {1, 2, ..., i} ! {j1, j2, ..., ji} and (p - i)! bijections {i + 1, i + 2, ..., p} ! {1, 2, ..., p}\{j1, j2, ..., ji}. Thus there are i!(p - i)! sum- mands of zj1zj2...zjiin N p(z1z2...zi). Proposition 4.4. The K(s)* algebra K(s)*(Xph p) is generated by c, c1, ..., cp-1, cp. Similarly with ci replaced by ~cifor 1 i p - 1. Proof: We note that (4) is also a decomposition of p modules. Fur- thermore, H*( p; F ) H*(ß; F ), and T is trivial as a p module. Thus H*( p; F ) = F p if * = 0 = 0 if * > 0 and H*( p; T ) = H*(B p) T. Therefore, as differential graded K(s)* modules, there is an isomor- phism of spectral sequences (Er *,*( p, pt) K(s)*T ) F p- ! Er*,*( p, X) It follows that as a K(s)* algebra, K(s)*(Xph p) is generated by K(s)*(B p), T , and F p. However F p T = (F T ) p and by the fundamental theorem of symmetric functions (F T ) p = K(s)*[oe1, ..., oep] 11 This completes the proof since æ p*(ci) = oei and for 1 i p - 1 æi*(~ci) = fflioei, ffli 6= 0. Theorem 4.5. As a K(s)*(B p) algebra, K(s)*(Xph p) = K(s)*(B p)[[~c1, ..., ~cp-1, cp]]=(c~ci). Proof: Since ~ci2 Im(T r*), the relations follow from Lemma 3.5 i). That there are no other relations follows from Proposition 3.7 and the fact that cp is represented by x p 2 T . Corollary 4.6. Over K(s)*, K(s)*(Xph p) = K(s)*[[c, ~c1, ..., ~cp-1, cp]]=(cms , c~ci). Corollary 4.6 follows immediately from Corollary 3.4 and Lemma 3.5 ii). Proposition 4.7. T r* p(1) = -vscms-1 . Proof: First we prove the formula on the level of classifying spaces, i.e. for X = pt, in which case it reduces to T r p(1) = -vsyms-1 . To do this we analyze æ*i, pT r* p(1). Since the transfer is transitive (v) we can factor Tr*ß * Tr*ß, p * T r* p: K(s)*(Be) -! K(s) (Bß) -! K(s) (B p) Applying the double coset formula (transfer property (vi)) to Tr*ß, p j*ß, p K(s)*(Bß) -! K(s)*(B p -! K(s)*(Bß) we obtain X j*ßg\ß c*g- 1 Tr*ßg\ß,ß K(s)*(Bß) ! K(s)*(Bßg\ß) ! K(s)*(Bßg\ß) ! K(s)*(Bß) g2i\ p=i Since ßg \ ß = e unless g 2 N p(ß)=ß in which case ßg \ ß = ß, æ*ig\i= T r*ig\i,i= id. Thus s-1 X * ps-1 æ*i, pT r* p(1) = æ*i, pT r*i, p(vszp ) = g (vsz ). g2N p(i)=i Now N p(ß)=ß = Z=(p - 1) and g*(z) = gz. Therefore p-1X s-1 ps-1 ps-1 æ*i, pT r* p(1) = [ ip ]vsz = -vsz . i=1 12 MALKHAZ BAKURADZE AND STEWART PRIDDY Since æ*i, pis injective and æ*i, p(y) = zp-1 this proves the special case X = pt. The general case now follows by a pull back argument as in the proof of Lemma 3.5 (ii) using c = '*(y). Remark: In view of Proposition 4.7 we observe that the multiplicative relations of K(s)*(Xph p) are completely determined by the relation Im(T r* p) . c = 0. Let us return to Theorem 4.5 and find the relationship between the two systems of generators ci and ~ci. Corollary 4.8. Let ' p : Xph p! B p be the projection induced by factorisation p o U(1)=U(1)p = p. There exist unique elements ffiipk2 K(s)*(B p) such that in K(s)*(Xph p) eck- k!(p - k)!ck = i 0'* p(ffiipk)cip, k = 1, ..., p - 1. Proof: The existence of ffiipk follows from Lemma 4.3 and Corollary 4.6: the left hand side, as an element of Keræ* p, has the form of the right hand side. As for uniqueness note that multiplication by cp is injective. Definition 4.9. Let t be a generator of ß and let F K(s)*(Xp) be the free ß module of Proposition 3.1 (4). Then elements !k 2 F can be defined modulo KerNi = Im(1 - t), t 2 ß by (9) Ni(!k) = oek, k = 1, ..., p - 1. By transfer property (v) T r*i= T r*it, where T r*i: K(s)*(Eß x BU(1)p) ! K(s)*(Xphi) hence T r*i(!k) is well-defined. Remark 4.10. In order to compute the transfer homomorphism of (8) in terms of Chern classes c, c1, ..., cp we recall [9], p. 44, that we can consider K(s)*(Xp), X = CP 1 as a free K(s)*[[oe1, ..., oep]] mod- ule generated by the elements zi11...zipp2 F K(s)*(BU(1)p), with 0 ij p - j, where F is the free ß module of Definition 4.9. So by Fröbenius reciprocity it suffices to compute thePtransfer on these monomials. Then summed over the symmetric group g*(zi11...zipp) is a symmetric function and hence has the form X X X g*(zi11...zipp) = oe1s1+ ... + oep-1sp-1 = (!1s1+ ... + !p-1sp-1), i p=i i 13 for the elements !k defined in 4.9 and some symmetric functions s1, ...sp-1. Hence we have modulo kerN = Im(1 - t), t 2 ß in F X g*(zi11...zipp) = !1s1 + ... + !p-1sp-1. p=i The left sum consists of (p - 1)! elements each with the same value of transfer. Also !k consists of p-1 pkelements with the value of transfer T r* p(z1...zk). Thus the computation of the elements ~ckand Fröbenius reciprocity is all that is needed for computing T r* p. Relating ß and p we have a lift of æi, p ~æi, p p Xphi ________-X p | | | | | | ' | ' | | | | | ?| æi, p |? Bß ________-B p Lemma 4.11. ~æ*i,(peck) = k!(p - k)!T r*i(!k). Proof: Note that modulo Im(1-t) we have g*(z1z2...zk) = k!(p-k)!!k summed over p=ß. Applying the double coset formula X æ*i, peck= æ*i, pT r* p(z1z2...zk) = T r*i g*(z1z2...zk) = k!(p-k)!T r*i(!* *k). g2 p=i We are grateful to D. Ravenel for supplying us with the proof of the following result which is important for calculations in Section 10. Lemma 4.12. Forsthe formal group law in Morava K-theory K(s) we have mod yp X ` p' s-1 s-1 F (x, y) x + y - vs p-1 (xp )j(yp )p-j. 0 1 we can reduce modulo the ideal ve2s(xp , y ) and get s-1 F (x, y) F (x + y, vsw1(x, y)p ) s-1 ps-1 ps-1 = F (x + y + vsw1(x, y)p , vsw1(x + y, vsw1(x, y) ) , . .). s-1 ps-1 ps-1 ps-1 p2(s-1) F (x + y + vsw1(x, y)p , vsw1(x + y , vs w1(x, y) )* *), s-1 p2(s-1) p2(s-1) and modulo v1+ps (x , y ) we have s-1 F (x, y) x + y + vsw1(x, y)p . Let us write for short oek = oek(x, F (x, z), ..., F (x, (p - 1)z)). Lemma 4.13. 1) The following formula holds in K(s)*(BU(1) x Bß) X ` p' s oek = - ~iikoeip+ p-1 xkvszp -1, 0 i ps k where ~iik= ~iik(zp-1) are polynomials in zp-1. ~i0j= 0, j = 1, ..., p - 2. s+1 2) For 1 k p - 1, equating the coefficients of xp, ..., xp in 1) gives a system of linear equations with invertible determinant. Thus the elements ~iikcan be defined as the solution of this system. Proof. 1). Clearly oep(x, F (x, z), ..., F (x, (p - 1)z)) = xp + zp-1A(x, z) for some polynomial A. Thus in the expression for oek we can substitute xn-p(oep - zp-1A(x, z)) for oek each xn for n p. As a result, high powers of x become replaced either by oep or by terms divisible by zp-1. Repeating this process gives expressions where each occurrence of powers of x exceeding p - 1 is accompanied by higher and higher powers of z, which is nilpotent. By iteration all such powers of x become eliminated, thus giving oek expressed in terms of oep, the only s-1 remaining term containing x being p-1 pkxkvszp . 2). Proof is elementary. 15 We now consider the bundle ,i = ~æ*i,(p, p) over Xphi: (10) ,i = Eß xi ,xp , with the classifying map the inclusion B(ß o U(1)) ! BU(p), where , p ! Xp pis the bundle (7). Then let T r*iand T r* pbe the transfer homomorphisms of (5) and (8) and 'i : Xphi! Bß and ' p : Xp p! B p be projections as before in this section. Let ~iik= ffiiik(z), the polynomials in zp-1 from K(s)*(Bß) = K(s)*[z]=([p]z) defined above Lemma 4.13 and ~ipkbe defined by Corollary 4.8 and ~æ*i,(pffiipk) = k!(p - k)!~iik. Proposition 4.14. There exists unique elements ffiiik2 K(s)*(Bß) such that 1) In K(s)*(Xphi) the following formula holds X ck(,i) = T r*i(!k) - '*i(ffiiik)cip(,i). 0 i ps 2) In K(s)*(Xph p) one has X ck(, p) = T r* p(z1...zk) - '* p(ffiipk)cp(, p). 0 i ps 3) The value of T r*i(c1(,i)) is determined by X s i i-1 c1(,i) = T r*i(c1(,i)) + vs flp -p cpp (,i). 1 i s-1 where fl = '*i(z) Proof: Theorem 5.1 gives the existence and uniqueness of the elements ffiiik. The diagonal map : BU(1) ! BU(1)p induces an inclusion Bß x BU(1) ! Xphiand the commutative diagram 1 x p Eß x BU(1) ______-Eß x BU(1) | | | | | | ß x 1 | æi | | | | | ?| ?| Bß x BU(1) ___________-Xphi 16 MALKHAZ BAKURADZE AND STEWART PRIDDY p Then (1 x )*(!k) = p-1 kxk, x = c1(,). Hence by transfer prop- erties (i) and (iv) we have for the transfer T r = T r(ß x 1): ` ' ` ' p k * -1 p k ps-1 T r*((1 x )*(!k)) = p-1 x T r (1) = p x vsz . k k On the other hand by the existence of the elements ffiiikwe have T r*((1 x )*(!k)) = X oek(x, F (x, z), ..., F (x, (p-1)z))+ ffiiikoeip(x, F (x, z), ..., F (x, (p-* *1)z)), i 0 hence by Lemma 4.13 oek(x, F (x, z), ..., F (x, (p - 1)z)) = X ` p' s - ~iikoeip(x, F (x, z), ..., F (x, (p - 1)z)) + p-1 xkvszp -1. 0 i ps k This proves 1) and shows ffiiik= ~iikfor 0 i ps and is zero otherwise. 2) Combining Corollary 4.8 and Lemma 4.11 this follows from 1). Statement 3) follows from the following explicit formula for oe1: Lemma 4.15. In K(s)*(Bß x BU(1)) one has _ s-1 ! s-1 X ps-pi pi-1 oe1 = vs zp x + z oep . i=1 Proof: First note that for any prime p and any 0 < k < p one has ( 0( mod p), k < p - 1, oek(1, 2, ..., p - 1) -1( mod p), k = p - 1 where oek are the symmetric functions. The polynomial xp - x has p roots 0, 1, ..., p - 1 in Fp, hence it equals x(x - 1)...(x - (p - 1)) in this field. But coefficients of the latter polynomial are exactly the symmetric functions in question. One has s-1 s-1 oe1 = x + F (x, z) + ... + F (x, (p - 1)z) = x + x + z + vsw1(xp , zp ) s-1 ps-1 +... + x + (p - 1)z + vsw1(xp , ((p - 1)z) ) P p-1 ps-1 = px + p(p-1)_2z + vs i=1w1(x, iz) i P P pjs-1 = vs p-1i=1p-1j=1-p-1 pjijxp-jzj i P P psj-1 = vs p-1j=1- p-1i=1ijp-1 pjzjxp-j . 17 P p-1 Now i=1ij is an integral linear combination of oek(1, 2, ..., p - 1) with k i, hence by the above argument it is zero for i < p - 1 and for i = p - 1 it is p - 1 as ip-1 1 mod p for all i < p. Thus (11) ` ` ' 'ps-1 p p-1 p-(p-1) ps-ps-1 ps-1 oe1 = -vs (p - 1)p-1 z x = vsz x . p - 1 Now since F (x, z)p = xp+zp, one has oepp= (x(x + z)...(x + (p - 1)z))p. But again we have x(x+z)...(x+(p-1)z) = xp-xzp-1. Substituting thisione obtains j s-1 P s-1 ps-pi pi-1 vs zp x + i=1 z oep i s s P s i i-1j = vs zp -1x + zp -poep + s-1i=2zp -p (xp - zp-1x)p i s s P s i i i-1 i-1 j = vs zp -1x + zp -poep + s-1i=2zp -p (xp - z(p-1)p xp .) ButP s-1 ps-pi pi (p-1)pi-1pi-1 P s-1 ps-pi pi ps-pi+(p-1)pi-1pi-1 Pi=2z (xsi-zi s ix i )i=-1ii=2-z1P x -zsi i s i-x1 i-1 = s-1i=2zp -p xp -zp -p +p -p xp = s-1i=2zp -p xp -zp -p xp s-ps-1 ps-1 ps-p p = zp x - z x . Henceione has j s-1 P s-1 ps-pi pi-1 vs zp x + i=1 z oep i s s s s-1 s-1 s j (12) = vs zp -1x + zp -poep + zp -p xp - zp -pxp . Nowsone has s s-1 s-1 s zp -pF (x,skz) = zps-p(x+kz +vsw1(xp , (kz)p )) =szp -p(x+kz), hence zp -poep = zp -px(x + z)...(x + (p - 1)z) = zp -p(xp - zp-1x) s-p ps-p p ps-1 as above,ii. e. zp oep = z x -zj x. Substituting this into (12) s-1 P s-1 ps-pi pi-1 ps-ps-1 ps-1 gives vs zp x + i=1 z oep = vsz x , which is oe1 by (11). 2. ß o (Z=pn). We now turn to Gn = ß o (Z=pn) where ß = Z=p. Consider the AHS spectral sequence for ' B(Z=pn)p ! BGn ! Bß Then Ep,q2= H*(Bß; K*(s)B((Z=pn)p) 18 MALKHAZ BAKURADZE AND STEWART PRIDDY where ns p K*(s)B(Z=pn)p = (K(s)*[z]=(zp )) = F T where F and T as above are free (resp. trivial)sß modules. Let c = '*(z) where K(s)*(Bß) = K(s)*[z]=(zp ) as above. Proposition 4.16. As a K(s)* module K(s)*(BGn) is free with basis { ci (zj) p 0 i < ps, 0 j < pns} and X { 1 zi1 zi2 ... zip | I 2 Pp(n) } (i1,i2,...,ip)2I where I = {(i1, i2, ..., ip)} runs over the set Pp(n) of ß-equivalence classes of p-tuples of integers {0 ij < pns} at least two of which are not equal. Proof: This spectral sequence computation is exactly analogous to that of Proposition 3.1. Remarks: (i) Proposition 4.16 gives another derivation of Proposition 3.1. Since CP 1 ^p= [colimn B(Z=pn)]^p, we have K(s)*(CP 1 hi) = limn K(s)*(BGn). (ii) Gn is good for K(s)* by [11 ] Th. 7.3. By analogy with Lemma 3.2 we have Lemma 4.17. i) Im(Tsr*) . c = 0. ii) T r*(1) = vsc2 -1. iii) If y 2 T then T r*(y) = y . T r*(1). Finally we consider the case p = 2. Proposition 4.18. Let p = 2. As a K(s)*(Bß) algebra, K(s)*(BGn) is generatednbysc,n~c1,sc2 subject to the relations ~c1. c = 0 and relations ~c21= c22 = 0 modulo terms divisible by c. Proof: ~c1. c = 0 by Lemma 3.5. The other relations hold in the E1 term of the spectral sequence. The only possible extensions are those on the fiber, involving ~c1, c2. Similar results hold for p o Z=p and p o p. 19 5. Complex cobordism of Xphi We turn to computing the complex cobordism for X = CP 1, Xphi= B(ß o U(1)). Before stating the main result Theorem 5.1 we begin with some preliminary notions. Let , be the canonical complex line bundle over CP 1 = BU(1) and ,i be the pulback bundle over BU(1)p by the projection on the i-th factor as before. Then MU*(BU(1)p) = MU*[[x1, ..., xp]], xi = c1(,i) and x1...xp is Euler class of the bundle ,xp = ,i. Hence for the sphere bundle S(,xp ), one has MU*(S(,xp )) = MU*[[x1, ..., xp]]=(x1...xp). Then for the trace map N we have kerN = Im(1 - t), t 2 ß in MU*BU(1)p and after restricting N to MU*(S(,xp )) we have the exact sequence (13) ... MU*(S(,xp )) N MU*(S(,xp )) 1-tMU*(S(,xp )) N MU*(S(,xp )) ... Then let ,i = Eß xi ,xp be the Atiyah transfer bundle [2] as in (10), (14) S(,i) = Eß xi S(,xp ) be its sphere bundle and (15) D(,i) = Eß xi D(,xp ) be the disk bundle. Then D(,i) is homotopy equivalent to Xphi= B(ß o U(1)). The cofibration D(,i)=S(,i) = (Xphi),ß gives the long exact sequence xcp * p ,ß (16) ... MU*(S(,i)) MU*(Xphi) MU ((Xhi) ) ... where (Xphi),ß is the Thom space of the bundle ,i and the right homo- morphism is multiplication by the Euler class cp = cp(,i). Since the diagonal of BU(1)p is fixed under the permutation action of ß the inclusion Eß ! Eß x BU(1)p; x ! (x, fixpoint) defines the inclusions i : Bß ! Xphiand the inclusion i0 : Bß ! S(,i). The projection ' : Xphi! Bß induced by ß o U(1) ! ß defines the projection (17) '0 : S(,i) ! Bß and the compositions '0i0, 'i are the identity. We can consider '0 as the bundle with fiber S(,xp ). 20 MALKHAZ BAKURADZE AND STEWART PRIDDY Let j be canonical line bundle over Bß and ` = '*(j) ! Xphi be the pull back bundle. Thus i*(`) = j and i*(,i) = C + j + ... + jp-1. Consider the pull back diagram Eß x S(,xp ) ___-Eß x BU(1)p | | | | | | æ0 | æ | | | | | ?| ?| S(,i) ___________-Xphi Let T r = T ri be the transfer of covering æ, and T r0 = S(,i) ! S(,xp ) transfer map of æ0. Theorem 5.1. In MU*(Xphi) a) Annihilator of the Chern class fl = c1(`) coincides with ImT r*; b) Multiplication by cp = cp(,i) is a monomorphism;P c) Any element of Ker(æ*) has the form k 0'*(ffik)ckp, for some * elements ffik 2 M~U (Bß). d) For ß = Z=2, MU*B(ß o U(1)) = MU*[[c, c1, c2]]=(c1 - c*1, c2 - c*2) = MU*(Bß)[[T r*(x), c2]]=(cT r*(x)), where ci = ci(,i), c*i= ci(,i C `), c = c1(`) and x 2 MU*(BU(1) = MU*[[x, tx]]) are Chern characteristic classes. We need the following Lemma 5.2. The left homomorphism in the long sequence (16) is epi- morphism and this gives the short exact sequence 0 MU*(S(,i)) MU*(Xphi) MU*(Xphi),ß 0 Moreover we can construct a stable equivalence of the form '0 _ fi : S(,i) ! Bß _ Xi, with a map fi factoring through the following map S(,i) ! XphiTr!Eß x BU(1)p. and '0 is in (17). 21 Proof: Consider the Serre spectral sequence for the fibration (17) '0 S(,xp ) ! S(,i) ! Bß. Ei,j2= Hi(ß, Hj(S(,xp ); Fq)) with the action of ß on H*(S(,xp ); Fq) by permutations of the cohomological Chern classes. When q = p, E0,j2= Hj(S(,xp ); Fp)i and Ei,02= Hi(Bß; Fp). Then note that H*(S(,xp ); Fq) = Fq[x1, ..., xp]=(x1...xp) in positive dimensions is a permutation representation of ß acting on monomials which have degree zero in at least one indeterminate. Since all the monomials that are fixed under this action have been factored out after taking quotient by the ideal (x1...xp), this representation is a free Fq[ß]-module, hence the cohomology of ß with coefficients in it is trivial in positive dimensions, i.e. Ei,j2= 0 when i, j > 0. Thus the spectral sequence collapses and we have H*(S(,i); Fp) H*(Bß; Fp) ~H*(S(,xp ); Fp)i. Also if q 6= p we have H*(S(,i); Fq) H*(S(,xp ); Fq)i. Let Xi be a stable summand of BU(1)p defined as follows. The action of ß on BU(1)p induces an action of ß on stable decomposition of BU(1)p which is the wedge of all smash products of length 1, ..., p-1, say Yi, and a smash product of length p. Then choose Xi such that NXi = Yi, where N = 1 + t + ... + tp-1. By the stable equivalence S(,xp ) ! BU(1)p ! Yi we can consider Xi as a stable summand of S(,xp ). For any choice of Xi, consider the composition of stable maps fi : S(,i) ! XphiTr!Eß x BU(1)p ! BU(1)p ! Xi. We have to show that the stable map '0 _ fi induces an isomorphism in cohomology for any group of coefficients Fq, q a prime, and hence gives us above stable equivalence by the stable Whitehead lemma. It follows from the above arguments that H~*(S(,i); Fp) = '*0~H*(Bß; Fp) T r*0~H*(S(,xp ); Fp), and H~*(S(,i); Fq) = T r*0~H*(S(,xp ); Fq), when q 6= p. The restriction of T r0 on Xi induces monomorphism on ImT r*0since by the transfer property (iv), æ*0T r*0= N* and the restriction of N* on H~*(Xi; Fq) is monomorphism. Hence '*0_(T r0|Xi)* is isomorphism and so is '*0_f*i by the commutative diagram 22 MALKHAZ BAKURADZE AND STEWART PRIDDY S(,i) _________- Xphi | | | | | | T r0|| T r | | || ?| ?| S(,xp )____-Eß x BU(1)p This proves Lemma 5.2. Now we are ready to prove Theorem 5.1. The statement b) of the Theorem 5.1 follows from the fact that the right homomorphism in the short exact sequence from Lemma 5.2 is multiplication by the Euler class cp(,i). Then taking into account Lemma 5.2 and the fact that ' the composition S(,i) ! Xphi! Bß coincides with '0, we have that any element y 2 MU*(Xphi) has the form y = '*(u) + T r*(v) + y1cp, * * p * p where u 2 M~U (Bß), y1 2 MU (Xhi) and v 2 MU (BU(1) ) is from the direct summand MU*[[x1, ..., xp]]=(x1...xp). Then suppose fly = 0. We know that æ*(`) = C, hence æ * (fl) = 0 and cT r*(v) = T r*(æ*(fl)v) = 0 by the transfer property (i). So we have that fl'*(u) + fly1cp = 0. This implies that both summand are zero: i) fl'*(u) = 0 is the consequence of the transfer relation flT r*(1) = 0 and ii)fly1cp = 0 which implies fly1 = 0 by the statement b). We have to prove i). Applying i* we have 0 = i*(fl'*(u)) + fly1cp = zu in MU*(Bß) = MU*[[z]]=([p])z (i*(cp) = 0 since i*(,i) has a section as above, i*'* is the identity and fl = '*(z)) ). But any relation in MU*(Bß) is the consequence of the relation zT r*Z=p(1) = 0 hence is zu = 0. Then applying '* we have by naturality of transfer fl'*(u) = 0 is the consequence of flT r*i(1) = 0. Since dim(y1) = dim(y) - p repetition of this argument gives us the statement a). For the proof of c) let y 2 Keræ*. Since 'æ = * we have 0 = æ*(T r*(v)) + æ*(y1cp). If T r*(v) 6= 0 then it restricts non-trivially to MU*(S(,i)) by the definition of v. However y1cp restricts to zero by exactness; thus T r*(v) = 0 and so both summands are zero. Further æ*(cp) = x1...xp hence æ*(y1) = 0. So y = '*(u) + y1cp = '*(u) + ('*(u1) + y2cp)cp = '*(u) + '*(u1)cp+ y2c2p. Repetition of this process proves c). 23 The relations d) follow from the bundle relation ,i C ` = (, C æ*(`))i = ,i which in turn follows from transfer property (i). So we have to prove that Chern classes c1, c2 with these relations are a complete system of generators and relations. The fact that c, c1, c2 multiplicatively generate MU*B(ß o U(1)) fol- lows from the above stable splitting of B(ß o U(1)). Let us use the splitting principle and formal group law in complex cobordism and write formally ,i = j1 + j2; u1 = c1(j1) ; u2 = c1(j2) and let X F (x, y) = ffijxiyj be the formal group. Then the above bundle relation after applying Whitney formula for the first and second Chern classes gives two rela- tions of the form: F (u1, c) + F (u2, c) = c1 and (18) F (u1, c)F (u2, c) = c2; or in terms of c, c1 = u1 + u2, c2 = u1u2 X (19) c(2 + fiijkcicj1ck2) = 0 and X (20) c(c1 + flijkcicj1ck2) = 0, for some coefficients fiijk, flijk2 MU*(pt). We claim that relations (19) and (20) are equivalent to the following two obvious transfer relations for T r* : MU*[[x, tx]] ! MU*(B(ß o U(1))) cT r*(1) = 0 and cT r*(x) = 0. For the proof note that if y is annihilated by c and æ*(y) = b + tb then u = T r*(b). This follows from the following arguments: In MU*(BU(1)2) we have Ker(1 + t) = Im(1 - t); also by property (v) of the transfer T r*(a) = T r*(ta). So if z is annihilated by c (hence y = T r*(a) for some a by the first statement of the Theorem 5.1) and æ*(y) = b + tb (hence b + tb = a + ta by the transfer property (iv)) then y = T r*(b) (that is T r*(b) = T r*(a)). Rewrite relations (19) and (20) as follows: 24 MALKHAZ BAKURADZE AND STEWART PRIDDY X 2c + ff11cc1 + ffk1(uk1+ uk2)c + o(c2) = 0; k 2 X cc1 + 2ff11cc2 + ffk1(uk-11+ uk-12)cc2 + o(c2) = 0. k 2 By the above remark this means that X T r*(1) + ff11T r*(x) + ffk1T r*(xk) = (F (u1, c) + F (u2, c) - c1)=c; k 2 X T r*(x) + ff11T r*(1)c2 + ffk1T r*(xk-1)c2 = (F (u1, c)F (u2, c) - c2)=c. k 2 Now since xk = xk-1(x + tx) - uk-2(xtx), transfer property (i) and computation of T r*(x) is sufficient for the computation of T r*(xk), k 2 (see Corollary 6.2). So we have (21) T r*(1)(1 + g0) + T r*(x)h0 = (F (u1, c) + F (u2, c) - c1)=c, and (22) T r*(1)g1 + T r*(x)(1 + h1) = (F (u1, c)F (u2, c) - c2)=c, where g0, h0, g1, h1 2 MU*(Bß o U(1)). This proves d). This completes the proof of Theorem 5.1. 6.Transferred Chern Classes 1. p = 2 Formula (22) for computing T r*(x) is complicated; let us give a simpler form. Consider again (20). Note that the coefficient fl00k 2 MU*(pt) contains a factor 2: the element X c1 + flijkcicj1ck2 as the one annihilating c, belongs to ImT r*. On the other hand æ*T r* = 1 + t; æ*(c) = 0; æ*(c1) = x + tx; æ*(c2) = xtx, hence applying æ* we have that X x + tx + fl0jk(x + tx)j(xtx)k belongs to Im(1 + t). So fl00k(xtx)k = 2flk(xtx)k, that is, fl00k = 2flk for some coefficient flk. 25 Recall that on the other hand F (c, c) = 0 that is 2c = o(c2). So fl00kc = o(c2), hence taking into account the relation F (c, c) = 0 we can rewrite (20) after division by X 1 + fli1kcick2 i,k 0 (the coefficient at cc1) as follows (23) cc1 = d0c + d2cc21+ ... + dnccn1+ ..., where dk = dk(c, c2) 2 MU*[[c, c2]] and d0(0, c2) = 0; the lower index n indicates the coefficient at ccn1. Since æ*(T r*(x) - c1) = 0, it follows from Theorem 5.1c) that there exist some elements * ffij 2 M~U (Bß) such that we have X T r*(x) = c1 + '*(ffij)cj2; j 0 Since for the inclusion i : Bß ! B(ß o U(1)) we have that i*(c1) = i*0(c); i*T r*(x) = 0; i*(c2) = 0, we obtain '*(ffi0) = -c. For the calculation of other elements ffij recall that cT r*(x) = 0, hence (24) ccn1= -cffin; n 1, X ffi = -c + ffijcj2. j 1 Substituting (24) in (23) in place of ccn1, elements ffij, j > 0 can be determined from the recurrence relations which arise from the following formula in MU*(Bß)[[c2]] X ffi = d0 + diffii. i 2 Here we act as in thePproof of the above claim: By the definition the element ffi - d0 - i 2diffii belongs to Keræ* . On the other hand this element is from the annihilator of c hence X " ffi - d0 - diffii 2 ImT r* Ker(æ*) = 0. i 2 So we have 26 MALKHAZ BAKURADZE AND STEWART PRIDDY * Proposition 6.1. For the constructed elements ffij 2 M~U (BZ=2) the following formula holds in MU*B(Z=2 o U(1)) X T r*(x) = c1 - c + '*(ffij)cj2. j 1 Note that Proposition 6.1 is sufficient for the calculation of T r*(xn), n 2. This follows from the following simple formula xn = U(n)x + U[n], where U(n) = U(n)(x + tx, xtx), U[n](x + tx, xtx) are formal series in x + tx and xtx: U(1)= 1; U[n]= 0; U(n+1)= (x + tx)U(n)+ U[n]; U[n+1]= -xtxU(n). Recall that æ*(c1) = x + tx and æ*(c2) = xtx and consider formally U(n)and U[n]as a formal series in c1 and c2. Then transfer property (i) gives the following recurrence formula for T r*(xn). Corollary 6.2. In MU*B(Z=2 o BU(1)), T r*(xn) = U(n)(c1, c2)T r*(x) + U[n](c1, c2)T r*(1). 2. The case of odd prime p Now let us elucidate the meaning of the relations ,i C ` = ,i in the general case of B(ß o U(1)). Again, we can use the splitting principle and write formally ,i = j1 + j2 + ... + jp, um = c1(jm ), m = 1, ..., p. Applying the Whitney formula for the relation j1 C ` + ... + jp C ` = j1 + ... + jp, and taking into account that cm = cm (,i) is the elementary symmetric function oem (u1, ..., up) we have (25) oem (F (u1, fl), ..., F (up, fl)) = cm , m = 1, ..., p, or in terms fl, c1, ..., cp we have X fl(p + fi0i0,i1,...,ipfli0ci11...cipp) = 0; and X (26) fl((p - k)ck + fiki0,i1,...,ipfli0ci11...cipp) = 0; for k = 1, ..., p - 1 and some fi0i0,i1,...,ip, fiki0,i1,...,ip2 MU*(pt). 27 We claim that these relations are equivalent to the following obvious relations flT r*(1) = 0; and flT r*(wk) = 0, for the elements wk 2 MU*(BU(1))p, k = 1, ..., p - 1 defined as follows (see Definition 4.9). The k-th Chern class of the bundle ,xp is the elementary symmetric function oek(x1, ..., xp) in Chern classes xi and is the sum of p(p - 1)...(p - k + 1)=k! number of elementary monomials. The action of ß on the set of these monomials gives us (p - 1)...(p - k + 1)=k! orbits and the transfer homomorphism on the monomials from the same orbit are equal by tranfer property (v). Let wk = wk(u1, ..., up) be the sum of representative monomials of these orbits. The value of T r*(wk) does not depend on the choice of wk since wk is defined modulo Im(1 - t*) and on the elements of Im(1 - t*) transfer homomorphism is zero again by (v). In other words we can take any wk for which Nwk = oek(u1, ..., up) holds. For the proof of our claim multiply the k-th relation from (26) by pk, the inverse of p - k in Fp. Then by Theorem 5.1, Annihilator(c) coincides with ImT r* hence (25) implies that pk(oek+1(F (u1, fl), ..., F (up, fl)) - ck+1)=fl = T r*(ak), for some ak which we have to find. Let us write æ*(pk(oek+1(F (u1, fl), ..., F (up, fl)) - ck+1)=fl) = g(k)(oe1, ..., oep) X (k) = oek(1 + g(k)k(oe1, ..., oep)) + oejgj (oej, oej+1, ..., ~oek, ...* *, oep) j6=k,1 j p X (k) = N(wk)(1+g(k)k(oe1, ..., oep))+ N(wj)gj (oej, oej+1, ..., ~oek, ..., * *oep). j6=k,1 j p Here the symbol o~ekindicates absence of the corresponding term. So we have pk(oek+1(F (u1, fl), ..., F (up, fl)) - ck+1)=fl X (k) = T r*(wk)(1+g(k)k(c1, ..., cp))+ T r*(wj)gj (cj, cj+1, ..., ~ck, ...,* * cp), j6=k,1 j p and (oe1(F (u1, fl), ..., F (up, fl)) - ck+1)=fl X (0) = T r*(1)(1 + g(0)0(c1, ..., cp)) + T r*(wj)gj (cj, cj+1, ..., cp). 1 j p This proves our claim. 28 MALKHAZ BAKURADZE AND STEWART PRIDDY Now let us give a formula analogous to that of Proposition 6.1. It follows from the definition of wk that æ*(ck - T r*(wk)) = 0. So taking into account Theorem 5.1 the difference ck - T r*(wk) P (k) (k) * has the form i 0 '*(ffii )cip, for some elements ffii 2 M~U (Bß). Us- ing this notation we have Proposition 6.3. We can construct explicit elements * ffi(k)i2 M~U (Bß), k = 1, ..., p - 1, such that the value of the transfer of the covering æ : BU(1)p ! Xphiis given by X T r*(wk) = ck + '*(ffi(k)i)cip, i 0 Proof: As mentioned above, existence of elements ffi(k)ifollows from Theorem 5.1. We start with the equations (26) and rewrite as (27) flfk(fl, c1, ..., cp) = 0, k = 1, ..., p - 1, These are equations in a formal series algebra MU*(Bß)[[e]], with e = cp, since we know cck 2 cMU*(Bß)[[cp]]. We now want to find explicitly formal series X (k) (28) ffi(k)(e) = ffii (fl)ek i 0 such that T r*(!k) = ck + ffi(k)(cp) and hence (29) flcjk= -fl(ffi(k)(cp))j, j 1 For this we want to replace the equations (27) by the equations (30) flfek(fl, ffi(1)(cp), ..., ffi(p-1)(cp), cp) = 0, where efk2 Keræ*iand has a form efk= ffi(k)+ flg(k)(fl)(fl, ffi(1), ..., ffi(p-1), cp), where fl(k)(fl) some series. In fact efk= 0 since we know that Ann(fl) = ImT r* and Ker(æ*) \ ImT r* = 0. Then equating each coefficient of the resulting series (31) eefk(cp) = efk(fl, ffi(1)(cp), ..., ffi(p-1)(cp), cp) = 0 29 in the ring MU*(Bß)[[cp]] to zero we will obtain p - 1 infinite strings of equations in MU*(Bß), relating ffi(k)i. We proceed as follows: Let us look at the term cfk(0, 0, ..., 0, cp) in the equations (27). Note that fk(0, 0, ..., 0, cp) is divisible by p: fk 2 Ann(fl) = ImT r* ) æ*ifk 2 ImN ) fk(0, ..., 0, oep) 2 ImN ) fk(0, ..., 0, oep) is divisible by p. Next using the relation [p]F (fl) = 0 we know that pfl is divisible by fl2; hence each occurrence of pfl in these equations can be replaced by terms with higher powers of fl. So flfk(0, 0, ..., 0, cp) can be replaced by the term divisible by fl2. Also the k-th relation from (27) contains the term fl(p - k)ck, and for the condition (31) we have to multiply k th equation from (27) by pk, the inverse of p - k in Fp, so that pk(p - k) 1 (mod p) and as above we can replace fl(p - k)ck by flck + (terms divisible by fl2). Then we use (29) and substitute the series ffi(k)in the resulting equa- tions. Thus obtain (30) and hence we obtain mentioned string of equa- tions relating ffi(k)i: (32) ffi(k)n(1+flf(k,n)(fl))+flg(k,n)(fl)(fl, (ffi(1)i)i n, ..., (ffi(k)i)i