* = Z(p)[v1; : :;:vq]: There are spectra E(k; n) with coefficient rings E(k; n)* ' v-1nBP*=Ik with similar long exact sequences. A special case, when k = n > 0, is the nth Morava K-theory, K(n)*(X), with K(n)* ' Fp[vn; v-1n]. Before we state our main theorem we have a result which makes the statements easier to make. Throughout this paper we assume all of our spaces to be of the homotopy type of CW complexes with H*(X; Z(p)) of finite type. We will say that a graded object (such as the generalized cohomology of a space) is even dimensional if it is concentrated in even degrees. Theorem 1.2 If K(n)*(X), X a space, is even dimensional for an infinite number of n, then K(n)*(X) is even dimensional for all n > 0. We say X has even Morava K-theory if K(n)*(X) is even dimensional for all n > 0. We use the weaker sounding assumption to prove our results, 3 but when all is said and done, the proofs show it is equivalent to having even Morava K-theory. We get simple, but interesting, corollaries: Corollary 1.3 If K(q)*(X) has a non-zero odd degree element for some q > 0 then K(n)*(X) has an odd degree element for all but a finite number of n. This does not apply to the usual extension of Morava K-theories to in- clude K(0)*(X) = H*(X; Q) and there are examples (X = K(Z; 2n + 1), [RW80 ]) where this is non-zero in odd degrees but X has even Morava K- theory. Originally, this led us to worry a lot about the possibility of bad low Morava K-theories with K-theory "stabilizing" to even degrees. We thought that our results or proofs would need exotic types of completion. However, such examples cannot exist. Because K(0) does not fit the pattern we must sometimes go to the p-adic completion of BP for our results. Proposition 1.4 If X and Y have even Morava K-theory, then so does X x Y . This follows from the K"unneth isomorphism for Morava K-theories and shows that the class of spaces for which our main results hold is closed under finite products. We can now state our main theorem. To avoid unnecessary repetition we have: Definition 1.5 Let P (0) be BP if lim1 BP *(Xm ) = 0 (for each space X under discussion), and the p-adic completion of BP , BPp^, otherwise. Theorem 1.6 If a space X has even Morava K-theory then P (0)*(X) is even dimensional and is a flat BP *-module for the category of finitely pre- sented BP *(BP )-modules. 4 Note that our results are strictly unstable. There are counter-examples if X is a spectra and not a space. Flatness in the sense of our theorem has been explored by Peter Landwe- ber in [Lan76 ] where he proves his exact functor theorem. He shows that flat means that vn-multiplication on M=InM is always injective. So, we get flatness from him by proving the following: Theorem 1.7 Let a space X have even Morava K-theory. For k 0 we have short exact sequences (where v0 = p): vk * * 0 -! P (k)*(X) -! P (k) (X) -! P (k + 1) (X) -! 0 and P (k)*(X) is even dimensional. Remark 1.8 Flatness was further studied by Zen-ichi Yosimura in [Yos76 ] and Nobuaki Yagita in [Yag76 ]. Their papers show that the above result implies that P (k)*(X) is also flat for the analogous category. This follows from Landweber's results once it is known that P (k)*(P (k)) is a BP *(BP )- module from [Yag76 ]. A K"unneth isomorphism follows if one space is nice. Corollary 1.9 Let X be a space with even Morava K-theory. For k 0 we have K"unneth isomorphisms: P (k)*(X x Y ) ' P (k)*(X) bP(k)*P (k)*(Y ): Recall that our spaces are all CW complexes of finite type and that P (0) is chosen according to Definition 1.5. There are similar isomorphisms for the theories E(k; n)*(-) if K(n)*(X) is even. Remark 1.10 By this K"unneth isomorphism, if X is an H-space with even Morava K-theory then P (k)*(X) has all the structure of a Hopf algebra. 5 Although it is reasonable to ask for even Morava K-theory if you want all of these theories to be even dimensional, flatness is the really interest- ing property and it should have nothing to do with even Morava K-theory. This is just the first nontrivial place the general phenomenon of a class of spaces having flat Brown-Peterson cohomology has shown up. Having ob- served it here one would expect to see it frequently in the future in a more general setting. For example, it seems unlikely that P (0)*(QS2n) is flat but P (0)*(QS2n+1) is not. Likewise, the flatness of the evenly indexed spaces in the -spectrum for BP fits into our scheme of things but the known flatness for the odd spaces does not. In fact, it seems some sort of fluke that there are so many examples of spaces with even Morava K-theory around. Pos- sibly such spaces have a significantly deeper reason for having even Morava K-theory than their association with flatness. Our results are a simultaneous generalization of previous observations on these two rather different concepts: even Morava K-theory and flatness. First, if X is a finite complex then the Atiyah-Hirzebruch spectral sequence must collapse for K(n)*(X) when the dimension of the space is less than 2(pn - 1), i.e., for all big n. If K(n)*(X) is even dimensional for such an n and X, the mod p cohomology must also be even dimensional, which implies that there is no torsion and the integral cohomology is even dimensional. It then follows that BP *(X) is free over BP *and is even dimensional. Our theorem generalizes this to infinite complexes. Second, over twenty years ago Peter Landweber [Lan70 ] computed the Brown-Peterson cohomology (at the time he worked with complex cobordism) of BG where G is a finitely generated abelian group and showed it was flat and even dimensional. This is a special case of our result applied to the first Eilenberg-Mac Lane space, K(G; 1). The K(G; n) have many similar properties. Our results have nothing new to say about finite complexes. Infinite complexes can have many a subtle unpleasant property in cohomology. This, and other factors, motivated J. Frank Adams to steer people in the direc- tion of homology rather than cohomology [Ada74 ] [Ada69 ]. What we are 6 observing is that things are not as bad as they seemed and that looking at cohomology can be rewarding. In particular it is turning out to be easier to compute and describe the cohomology than the homology in several exam- ples. Landweber's example for BG where G is abelian should have showed the way. It was much later that BP*(BG), for G an elementary p-group, was computed ([JW82 ] ) and little progress has been made on more complicated abelian groups. Likewise, BP *(BO) was computed in a reasonable fashion ([Wil84 ]) before BP*(BO) was properly understood ([Yan ]). We can now add all Eilenberg-Mac Lane spaces to the list of spaces whose Brown-Peterson co- homology is completely described but whose Brown-Peterson homology is still a mystery. Although for many spaces that fit our hypothesis we do not have more de- tailed descriptions of the cohomology, our result is still way ahead of anything we can produce for homology. A brief description of our proof is now in order. First we show Propo- sition 4.12 that any given nontrivial element of P (k)*(X) maps nontrivially to E(k; n)*(X) if n is big enough. (P (k)* and E(k; n)* were defined in the opening paragraph.) Second, we show Lemma 5.1 that if K(n)*(X) is even dimensional then E(k; n)*(X) is also. (This allows us to "compute" E(k; n)*(X) for all spaces with K(n)*(X) even degree.) Thus, if X has even Morava K-theory, then P (k)*(X) is also even dimensional. This is proved using the Atiyah-Hirzebruch spectral sequence. Because all of our spaces are infinite complexes, there are technicalities to worry about. For example, we must show that for k > 0 there are no phantom maps in E(k; n)*(X). This, and more, is achieved using a generalization of Quillen's theorem saying that P (k)*(X) has only non-negative dimensional generators and a generaliza- tion of the Landweber exact functor theorem, which says that tensoring with E(k; n)* is exact in the category of finitely presented P (k)*(P (k))-modules. In the process of proof some subtle differences between cohomology and homology for infinite complexes come to the surface. The Morava structure theorem for complex cobordism (see [JW75 ]) allows one to use the Morava 7 K-theory, K(n)*(X), to compute the vn-torsion free part of P (n)*(X). Not so in the cohomology of infinite complexes. In fact, in all of our examples, all elements of P (n)*(X) are vn-torsion free, but only some show up in the Morava K-theory. This partial failure of the Morava structure theorem is compensated for by the lack of infinite divisibility by vk in E(k; n)*(X), k < n, whereas in E(k; n)*(X) it is commonplace and shows up in the proof of the Conner-Floyd conjecture of [RW80 ]. In that proof it was important that the Morava structure theorem detected all of the vn torsion free part of homology and that one could have infinite divisibility as well. In the present work we can live without the Morava K-theory detecting all of the vn torsion free part but we must be able to eliminate infinite divisibility. The Morava structure theorem still tells us that we can recover K(n)*(X) from P (n)*(X) K(n)*(X) ' K(n)* P(n)*P (n)*(X): for infinite complexes, except that now this doesn't pick up all of the vn torsion free part. Since P (n)*(X) is determined by P (0)*(X) for our special spaces with even Morava K-theory, we have a result which was first suggested in papers of Tezuka and Yagita, [TY89 ] and [TY90 ], and later in a paper of A. Kono and N. Yagita, [KY93 ]: K(n)*(X) ' K(n)* P(0)*P (0)*(X): In fact we can replace K(n) with E(k; n). We started this project with the belief that the time had come to seriously attack the Brown-Peterson cohomology of Eilenberg-Mac Lane spaces. We tried many approaches, including the Adams spectral sequence, before we found the present one. Calculations led us to believe that it was possible everything was even dimensional; motivating our study even more. Since we began with Eilenberg-Mac Lane spaces we have a measure of satisfaction that these spaces all satisfy our conditions ([RW80 ]), and we are even more pleased that we can describe their BP-cohomology completely. We will give an algebraic construction of the Brown-Peterson cohomology of Eilenberg- Mac Lane spaces. 8 There is a BP -module spectrum BP , [JW73 ] and [Wil75 ], with ss*(BP) = Z(p)[v1; : :;:vq] and for each q > 0 there is a stable cofibre sequence q-1) vn 2pq-1 2(p BP-! BP-! BP-! BP: (1.11) This gives rise to corresponding fibrations in the -spectra for the BP, {BP_ _*}. The following is also true using P (n). Theorem 1.12 Let g(q) = 2(pq+1 - 1)=(p - 1). Then BPp^*(K(Z(p); q + 2)) is isomorphic to: BPp^*(BP_ _g(q))=(v*1; : :;:v*q) ' BPp^*(BP__g(q))=(v*q) and BP *(K(Z=(pi); q + 1)) is isomorphic to: * * * * i* * BP *(BP__g(q))=(pi ; v1; : :;:vq) ' BP (BP__g(q))=(p ; vq) for q > 0. For q = 0 delete the v*qfrom the ideal. (The notation v*nmeans the image in Brown-Peterson cohomology of the map vn of (1.11).) We need the p-adic completion for K(Z(p); n), n > 2, because there are phantom maps for these spaces. However, we don't need it for finite groups. Remark 1.13 Because all of this comes from spaces and maps of spaces we have much more here than just the BP *module structure. In fact, these things are as good as Hopf algebras and the structure maps are included in what is known. Furthermore, everything is completely understood as unstable modules over BP *(BP__*) (or BPp^*(BPp^_*)) from [BJW95 ]. Later we will give a set of algebra generators. 9 The q = 0 version of the theorem was known to Stong and presumably others, in the 1960s. Landweber, in [Lan70 ], showed these q = 0 cases were flat and then calculated the result for products of these spaces. Some explanation is called for. The ideal is generated by the images of the maps in Brown-Peterson cohomology induced from the maps of spaces in the -spectrum which come from the stable maps described above in 1.11. There is a map which induces this isomorphism. It comes from the iterated boundary maps of 1.11. Unstably, the boundary map is: BP__j ! BP_ _j+2pk-1: and the iteration is: K(Z=(pi); q + 1) ! K(Z(p); q + 2) ! BP_<1>_q+2p+1! . .!.BP_ _g(q): This is the same map used by Hopkins-Ravenel [HR92 ] to prove that suspen- sion spectra are harmonic. The reason this is a satisfactory answer for us is that everything is "known" about BP *(BP__g(q)), BP *(BP__g(q)-2(pq-1)), and the map v*q between them. This is because BP__g(q)splits off of BP__g(q)and all of the spaces BP__* are well understood from [RW77 ]. In particular, in that paper we give an algebraic construction for BP*(BP__k). BP*(BP__k) is a well defined quotient of this construction for k g(q) (just set all [vi] = 0 for i > q where [vi] is defined in BP0(BP__-2(pi-1)) using vi 2 ss2(pi-1)(BP ) ' [pt; BP__-2(pi-1)]). Since the spaces BP__k, k g(q) are all torsion free for regular homology, we know that they are BP* free and the Brown-Peterson cohomology is just the BP *dual. Likewise, the maps are just the dual maps. This theorem gives insights into H. Tamanoi's results, [Tam83b ], [Tam83a ](an announcement with no proofs), (see [Yag86 , Theorem 3.3]), and vice versa. H. Tamanoi has only recently written up his work in [Tam95 ]. Not only do we know the BP -homology of the above spaces and use it to describe the BP -cohomology of the Eilenberg-Mac Lane spaces, but the same can be done for Morava K-theory. Although in principle the maps 10 are all known, in practice it can be difficult to compute them. Our main technical proposition about these spaces which allows us to go up to our BP cohomology answer is (see [HRW ] for the category of K(n)* Hopf algebras): Proposition 1.14 Let g(q) = 2(pq+1-1)=(p-1). There is an exact sequence in the category of K(n)* Hopf algebras: K(n)* ! K(n)*(K(Z(p); q + 2)) ! vq K(n)*(BP__g(q)) -------! K(n)*(BP__g(q)-2(pq-1)): The K"unneth isomorphism, Corollary 1.9, gives us the Brown-Peterson cohomology for all (abelian) Eilenberg-Mac Lane spaces. In order to translate this Morava K-theory information into information about Brown-Peterson cohomology we have to have some general results about exactness for Morava K-theory implying exactness for BP . We have theorems about injectivity, surjectivity and just enough exactness for our purposes: Theorem 1.15 Let spaces Xi, i = 1; 2, have even Morava K-theory. If f : X1 -! X2 has f* : K(n)*(X2) -! K(n)*(X1) surjective (injective) for all n > 0, then f* : P (k)*(X2) -! P (k)*(X1) is also surjective (injective), for k 0. Theorem 1.16 Let spaces Xi, i = 1; 2; 3, have even Morava K-theory. If f1 f2 X1 -! X2 -! X3 has f2 O f1 ' 0 and gives rise to an exact sequence (as K(n)* modules) f*1 * f*2 * 0 - K(n)*(X1) - K(n) (X2) - K(n) (X3) for all n > 0 then for all n 0 we get another exact sequence: f*1 * f*2 * 0 - P (n)*(X1) - P (n) (X2) - P (n) (X3): 11 Theorem 1.17 Let spaces Xi, i = 1; 2; 3, have even Morava K-theory. As- sume that f1 f2 X1 -! X2 -! X3 has f2 O f1 ' 0 and all spaces are H-spaces and all maps are H-space maps. Assume also that we have an exact sequence of bicommutative Hopf algebras for all n > 0: f1* f2* K(n)* -! K(n)*(X1) -------! K(n)*(X2) -------! K(n)*(X3): Then, P (n)*(X1) ' P (n)*(X2)=(f*2) for all n 0. The above theorem will be used repeated for our examples. What is sur- prising is that, more often than not, our spaces do not come from a fibration. Although we cannot give the level of detail for most of our examples that we give for Eilenberg-Mac Lane spaces, there are some general statements which we can make about generators and relations. Theorem 1.18 Let a space X have even Morava K-theory. For n 0, one can pick a set Tn in P (n)*(X) such that the elements Tn satisfy (a)-(d). (a) They generate P (n)*(X) topologically as a P (n)*-module, (b) are all essential to generate, and (c) are almost all in F s, the sth skeletal filtration of the Atiyah-Hirzebruch spectral sequence. (d) These elements, Tn, reduce to a set, Tq, in P (q)*(X), q > n, with the same properties. Let Rn be the set of relations on the elements Tn in P (n)*(X). Then 12 (e) all relations must be infinite sums, in particular, the elements of Tn are linearly independent over P (n)*, (f) any relation, in Rq, q > n, on the reduced set Tq in P (q)*(X) comes from Rn, and (g) any relation whose coefficients all map to zero in P (q)* can be written P q-1 i=nviri, with ri in Rn. The last statement is a nice generalization of "regular" in Landweber's paper [Lan70 ]. It is clear that the image of the set Tn in K(q)*(X) generates. From the next result we see that every element in Tn must show up in some Morava K-theory (or else it would be unnecessary). In fact, it follows that every generator must be detected by an infinite number of the Morava K- theories. It seems reasonable, but we were unable to prove, that if a generator shows up K(q)*(X), then it also shows up in K(q + 1)*(X). Such is the case for Eilenberg-Mac Lane spaces. Our next result says that if elements generate the Morava K-theories then they actually generate everything. This is a strong result which allows us to prove our exactness theorems and go on to attack Eilenberg-Mac Lane spaces. Theorem 1.19 Let a space X have even Morava K-theory. Let n 0 and let Tn P (n)*(X) be such that (a) the elements of Tn are almost all in F s, the sth skeletal filtration of t* *he Atiyah-Hirzebruch spectral sequence and (b) for each q n (q > 0), K(q)*(X) is generated topologically as a K(q)*- module by the image of Tn. Then Tn generates P (n)*(X) topologically as a P (n)*-module. Remark 1.20 If Tn is multiplicatively generated by a finite subset Gn of elements of positive skeletal filtration, then Tn satisfies (a). 13 Corollary 1.21 Let X and Tn be as in Theorem 1.19 with all of the elements of Tn essential. Then Tn satisfies the conditions of Theorem 1.18. Corollary 1.22 Let X and Tn be as in Theorem 1.18. Every t 2 Tn maps non-trivially to K(q)*(X) for an infinite number of q > n. We see structure in many examples of Brown-Peterson cohomology where there was not known to be structure before. We consider what we have done as just a start. The problem of computing these examples more completely is a problem that remains, but now with more than a glimmer of hope that the answers we will find will be nice. We hope this work will inspire others to tackle these explicit computations. We leave people with the question: If these things are so nice, what are they? Section 2 elaborates on our examples. In Section 3 we organize the pre- liminaries needed in the rest of the paper. After that we have a section on the Atiyah-Hirzebruch spectral sequence for our theories. Then we move on to assume even Morava K-theory and deduce the main result. We then do our work with generators and relations followed by our work with exactness. Our final section works out the details of the Eilenberg-Mac lane example. We would like to thank the Japan-United States Mathematics Institute (JAMI) at The Johns Hopkins University for making it possible for the au- thors to get together during the fall of 1991 for this work. The second author wishes to thank the Johns Hopkins University for leave support during the time of this work and John Sheppard and the University of Zimbabwe for computer support during the first writing of the paper in the spring of 1992. The authors would like to thank Michael Boardman, David Johnson, Jack Morava, Hal Sadofsky, and Katsumi Shimomura for various conversations about the material. We would also like to acknowledge the influence of the work of H. Tamanoi, which led us to believe this project was possible and which we should have paid more attention to years ago. 14 2 Examples Before listing our examples, we need the following easy result. Proposition 2.0.1 Let f F - i!E -! B be a fibration in which K(n)*(F ) and H*(B) (= H*(B; Z=(p))) are concen- trated in even dimensions. Then K(n)*(i) is one-to-one, K(n)*(f) is onto, and K(n)*(E) is even degree. Moreover, if the fibration is one of loop spaces, then f* K(n)* -! K(n)*(F ) -i*!K(n)*(E) -! K(n)*(B) -! K(n)* is an short exact sequence of Hopf algebras. Corollary 2.0.2 If we have a fibration of double loop spaces as in Proposi- tion 2.0.1, F has even Morava K-theory and H*(B) is even, then P (n)*(F ) ' P (n)*(E)=(f*): Proof of Proposition. The Atiyah-Hirzebruch-Serre spectral sequence con- verging to K(n)*(E) with E2 = H*(B; K(n)*(F )) collapses since it is concentrated in even dimensions. The result follows. 2 Proof of Corollary. The double loops implies bicommutative Hopf algebras so this follows from Proposition 2.0.1 using Theorem 1.17. 2 15 2.1 Finite Postnikov systems All Eilenberg-Mac Lane spaces not having the circle S1 as a homotopy factor, and all products of such spaces satisfy the conditions of our theorems. The first from [RW80 ] and the products from Proposition 1.4. Furthermore, we can describe the Brown-Peterson cohomology of all of these spaces explicitly. See the introduction and the last section of the paper. This is our main example. In Hopkins-Ravenel-Wilson, [HRW ], we show that the loop space of a finite Postnikov system has even Morava K-theory, provided that it does not have an S1 as a factor. If F is such a space, but with double loops replacing loops, then K(n)*(F ) is isomorphic as a Hopf algebra to K(n)*(E), where E is the product of Eilenberg-Mac Lane spaces having the same homotopy as F . In other words, k-invariants are not seen by Morava K-theory. 2.2 Classifying spaces of compact Lie groups In [HKR92 ] it was conjectured that the Morava K-theory of a finite group, i.e., K(n)*(BG), should be even dimensional. If this conjecture is true, (and it is presently in doubt, see [Kri]) then our result would apply to all finite groups. The Brown-Peterson cohomology is not given explicitly in these cases and we have no conjecture for this analogous to our results for Eilenberg-Mac Lane spaces. The information we do have should inspire someone to come up with an explicit conjecture though. As it is, our result applies only to those groups which have had their Morava K-theory com- puted. That list starts with finite abelian groups. Their Brown-Peterson cohomology was known, in detail, to Landweber [Lan70 ] who also knew of their flatness. Perhaps next on the list, in terms of interest, are the sym- metric groups. Hopkins-Kuhn-Ravenel, [HKR92 ] and [HKR ], and Hunton, [Hun90 ], independently proved that the Morava K-theory of these groups is even. These would be good examples to understand explicitly. Hopkins, Kuhn, and Ravenel give other examples where the result is known. The re- 16 sult is known for groups G with rankpG 2. For groups with rankpG = 2 all but one case is done by Tezuka and Yagita in [TY89 ] and [TY90 ]. All cases, including the missing one, are done in [Yag93 ]. M. Tanabe also has an interesting class of examples in [Tan95 ]. The result about symmetric groups mentioned above is a consequence of the following statement. If the conjecture is true for a finite group G, then it is also true for the wreath product Z=(p)o G. By this we mean the evident semidirect product in the split group extension Gp -! Z=(p) o G -! Z=(p) in which Z=(p) acts on Gp by permuting the factors cyclically. The proofs given by Hunton and Hopkins-Kuhn-Ravenel differ slightly in the assumption made about K(n)*(BG). The latter assume that it is additively generated by images under the transfer map of Euler classes of representations of subgroups H G, while Hunton assumes that there is a map f : BG ! Y with K(n)*(f) onto and K(n)*(Y ) a finitely generated power series ring. He calls such a map a `unitary like embedding' because in the case where G is abelian, Y can be taken to be BU(m) for a suitable unitary group U(m). We can improve on the wreath product result of [HKR92 ] as follows. For a group G let TreK(n)(G) denote the subalgebra of K(n)*(BG) generated by transferred Euler classes of irreducible representations of subgroups of G, and similarly for TreBP (G) and TreP(n)(G). (In [HKR92 , Cor. 8.3] it is shown that the module generated by transferred Euler classes of all representations of subgroups of G is the same as this algebra.) We let Tr* denote the transfer and e(ae) the Euler class of a representation ae. Then Tr*(e(ae)) stands for the transferred Euler class of a representation. We will say that a group G is good if K(n)*(BG) = TreK(n)(G): for all n. We know that finite abelian groups and groups G with rankpG 2 are good. A group G is good if its p-Sylow subgroup is. In [HKR92 ] it 17 was shown that W = Z=(p) o G is good if G is. The following result is a consequence of Theorem 1.19. Corollary 2.2.1 Let G be a finite group which is good in the sense above, and let W = Z=(p) o G. Then BP *(BG) = TreBP(G); P (n)*(BG) = TreP(n)(G); BP *(BW ) = TreBP(W ) and P (n)*(BW ) = TreP(n)(W ): Proof. First note that for X = BG or BW , we know from [BM68 ] and [Lan72 ] that lim1BP *(Xm ) = 0, so P (0) is BP as in Definition 1.5, so the statements about BP -cohomology are special cases of those about P (n)- cohomology. Now let Tn P (n)*(BG) be the subalgebra generated by the set of transferred Euler classes of irreducible representations of subgroups of G. There are finitely many such classes, so Tn is multiplicatively generated by a finite set as required by Remark 1.20 so the statements about the cohomology of BG follow from Theorem 1.19. Let Tn0 P (n)*(BW ) be similarly defined. Since W is good, Tn0also satisfies the hypotheses of Theorem 1.19 because of Remark 1.20 and the statements about the cohomology of BW follow. 2 We want to give a more detailed description. Theorem 2.2.2 Let G be good and BP *(BG)BP* Z=(p) ' Z=(p){b }, that is, the b are BP *generators. Then BP *(B(Z=(p) o G)) BP* Z=(p) ' 0 0 Z=(p){P (); oe(1; : :;:p)ys; ys | s 0; s > 0; 9i 6= j} 18 where oe(1; : :;:p) = Tr*(b1 . . .bp ); with Tr* : BP *(BGp) ! BP *(B(Z=(p) o G)); y = ss*("y) with ss* : BP *(BZ=(p)) ' BP *[["y]]=[p]("y) ! BP *(B(Z=(p) o G)); P () = Tr*(e(a^e)) if b = Tr*(e(ae )) for some representation ae of H G, and a^eis the representation of Z=(p) o H with a^e|Hp = ae . . .ae . Proof. The exact sequence 1 -! Gp -! Z=(p) o G -! Z=(p) -! 1 induces the spectral sequence H*(BZ=(p); K(n)*(Gp)) =) K(n)*(B(Z=(p) o G)): In [HKR92 , between 8.3 and 8.6], the differentials are computed and they get a similar theorem as ours for K(n) except that there are some restrictions. One must use a subset of the b's, s pn - 1 and s0 pn - 1. All of the elements in the statement of the theorem can be defined for BP cohomology and we now see that their reductions generate all of the Morava K-theories. Our result follows from Theorem 1.19 and we see that all of these elements are necessary as well. 2 Now let X = BG, where G is a compact Lie group. From Buhstaber-Mischenko, [BM68 ], and Landweber, [Lan72 ], it is known that lim1BP *(Xm ) = 0. In [KY93 ], Kono-Yagita conjecture that BP *(X) is even degree and flat in our sense. They go on to prove this for O(n), SO(2n + 1), P U(3) and F4. The Brown-Peterson cohomology of BO, BO(n) and MO(n) was computed in [Wil84 ]. Remark 2.2.3 Our results show that the Hopkins-Kuhn-Ravenel conjecture about finite groups is equivalent to the Kono-Yagita conjecture (for finite groups). 19 2.3 The sphere spectrum The evenly indexed spaces in the -spectrum for the sphere, QS2n, have even Morava K-theory as they are the limit of spaces which have even Morava K- theory. (This follows from Hunton's theorem about the Morava K- theory of wreath products [Hun90 ].) We want to thank Takuji Kashiwabara for bringing this example to our attention. Kashiwabara has, since we proved our basic theorems, pushed this example to its ideal conclusion in [Kas94 ]. There, he shows that if E is a bouquet of BP spectra and there is a map, f, from BP to E such that f* * 0 - BP *(S0) - BP *(BP ) - BP (E); is an exact sequence, then there is an exact sequence of K(n)* Hopf algebras (f2k)* K(n)* -! K(n)*(QS2k) -! K(n)*(BP__2k) - ! K(n)*(E_2k) which, by Theorem 1.17, gives BP *(QS2k) as a quotient, BP *(BP__2k)=(f*2k): Note that these maps and spaces do not form a fibration. It is easy to come up with a spectrum E and a map from BP . A minimal i-1) one, for example, is just to have E be the wedge of 2(p BP for each i > 0. The maps just cover the generators for BP *(BP ). In principle, this gives complete information not just for BP *(QS2k) but for K(n)*(QS2k). Everything about the map from BP to E is known and one can use the techniques developed in [BJW95 ]. Kashiwabara has recently pushed his work even further, see [Kas ]. 2.4 Image of J and related spaces We thank Stewart Priddy and Fred Cohen for tutorials which allowed us to include this example. 20 We will outline a computation of K(n)*(J) for an odd prime p, where J is the fibre of k-1 BU(p)-------! BU(p) for a suitable choice of the integer k, namely it must be congruent to a primitive (p - 1)th root of unity mod p but not mod p2. It is also known that if k is a power of some prime other than p, then the fibre of the map above is the p-localization of algebraic K-theory of the field Fk [Qui72 ]. In any case this space is known to be a direct limit of the classifying spaces of finite groups studied by Tanabe in [Tan95 ] ([Qui72 ]). He shows that each of them has even Morava K-theory. In particular J has even Morava K-theory, so the theorems of this paper apply to it. Moreover, the fibration k-1 J -------! BU(p)-------! BU(p) gives a short exact sequence of Hopf algebras in Morava K-theory by Propo- sition 2.0.1. Corollary 2.0.2 then gives us the result that BP *(J) ' BP *(BU(p))=(( k - 1 )*) because there are no lim1 problems (J is torsion and BU(p)has no torsion). This discussion could be made self contained, and thus not dependent on Tanabe, by showing that ( k - 1)*is surjective as a map from K(n)*(BU(p)) to itself. A simple argument then shows that K(n)*(J) is even. As it is, Tanabe, with the collapsing of the spectral sequence, gives us this surjectivit* *y. The effect of the map k - 1 in BP -cohomology can be computed with the help of the formal group law and the splitting principle. We have BP *(BU) = BP *[[c1; c2; . .].]; the power series ring on the Chern classes. Consider the formal expression X c(t) = citi where c0 = 1: i0 21 Under the splitting principle we can write Y c(t) = (1 + xjt); j which should be understood to mean that ci is the ith elementary symmetric function in the xj. Then we have Y ( k)*(c(t)) = (1 + [k]BP* (xjt)); j where [k]BP* (x) denotes the k-series for the formal group law associated with BP *and this gives the action of ( k)* on our generator for BP *(CP 1). The expression on the right is symmetric in the xj, so the coefficient of ti is a certain symmetric polynomial (with coefficients in BP *) in the xj, so it can be written in terms of the elementary symmetric functions. Then ( k)*(ci) is the corresponding polynomial in the Chern classes. In a similar sense we have ! Y 1 + [k]BP* (xjt) ( k - 1)*(c(t)) = _______________ : j 1 + xjt 2.5 BO The object of this section is to recover the second author's computation of BP *(BO) [Wil84 ] using the results of this paper. It is shown there that BP *(BO) is a certain quotient of BP *(BU) = BP *[[c1; c2; . .].] (the power series ring on the Chern classes of the universal complex vector bundle) under the map Bi : BO ! BU induced by the complexification map i : O ! U. Let c*idenote the ithChern class of the conjugate of the universal bundle [MS74 , page 167]. Then the result of [Wil84 ] that we want to reprove is BP *(BO) = BP *(BU)=(ci- c*i: i > 0): 22 We do not know if similar methods can be used to recover BP *(BO(m)). For more on BO(m) the reader should see [Kri, Section 5]. We first observe that BO does not have a lim1 problem so that we really can use BP and not BPp^. In general this is done by Landweber in [Lan72 ] but in this case it it quite easy to see because the rational cohomology of BU surjects to BO. The only way there can be an infinite number of differentials in the Atiyah-Hirzebruch spectral sequence, giving a phantom class, is if there are an infinite number of differentials on one of the integral classes of BO. Some multiple is in the image of the spectral sequence from BU and that spectral sequence collapses, so this cannot happen. Next we observe that BO has no odd prime torsion so we have nothing to prove to get the result at odd primes. For p = 2 we begin by showing that BO has even Morava K-theory. K(n)*(BO) can be computed as follows. We know that (the mod 2 homology) H*(BO) = P (b1; b2; . .). where bi 2 Hi(BO) is the image of the generator of Hi(RP 1). The action of the Milnor primitive Qn is given by ( bi+1-2n+1 if i is even and i 2n+1 Qn(bi) = 0 otherwise. It follows that in the Atiyah-Hirzebruch spectral sequence for K(n)*(BO) we have ( vnbi+1-2n+1 if i is even and i 2n+1 d2n+1-1(bi) = 0 otherwise, so we have E2n+1= K(n)*[b2; b4; . .b.2n+1-2] K(n)*[b22i: i 2n]: It follows that K(n)*(BO) and K(n)*(BO) are even dimensional. Recall that we are trying to show that the map Bi : BO ! BU induces a surjection in BP -cohomology. Bott periodicity ([Bot59 ]; see also Milnor's 23 treatment in [Mil63 , x24]) gives us a fibre sequence j Z x BO -Bi!Z x BU -! Sp=U = Sp: Delooping this gives j 2Sp -! U -! Sp where j is the usual inclusion map. Delooping twice more gives f 2 Sp -! U -! O where f is the usual inclusion map. Now Sp has even dimensional homology. To see this, recall [Whi78 , xVII.4] that H*(Sp; Z) = E(x3; x7; . .;.x4m+3 ; . .). with |x4m+3 | = 4m + 3: The Eilenberg-Moore spectral sequence for the homology of its loop space collapses, giving H*(Sp; Z) = P (x2; x6; . .;.x4m+2 ; . .). with |x4m+2 | = 4m + 2: Likewise, the bar spectral sequence collapses giving E0H*(BSp; Z) = (x4; x8; . .;.x4m ; . .). with |x4m | = 4m: Hence Proposition 2.0.1 and Corollary 2.0.2 apply to the fibration j BO -Bi!BU -! Sp and we have BP *(BO) ' BP *(BU)=((j)*) which is not quite what we want yet. It came from a short exact sequence of Hopf algebras from Proposition 2.0.1 K(n)* ! K(n)*(BO) ! K(n)*(BU) ! K(n)*(Sp) ! K(n)* 24 Now consider the fibration Bj Sp=U ______BUw _______BSpw ||||||||||||||||||||||||| | |||||||||||||||||||||||||||| |||||||||||||||||||||||||||| |||||||||||||||||||||||||||| |||||||||||||||||||||||||||| |||||||||||||||||||||||||||| |||| f |||||||||||| Sp ______wSU All three spaces have even dimensional homology, so by Proposition 2.0.1 we have another short exact sequence of Hopf algebras K(n)* ! K(n)*(Sp) ! K(n)*(BU) ! K(n)*(BSp) ! K(n)* Thus we get a diagram Bi (fj) Z x BO ______Zwx BU _____________________wZ x BU ||||||||||||| | |||||||| ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | | | j f |||||||| U __________Spw __________Uw and we can splice together the two exact sequences to get an exact sequence K(n)* -! K(n)*(BO) -! K(n)*(BU) -! K(n)*(BU): This no longer comes from a fibration but the result of [Wil84 ] can be recovered by using Theorem 1.17 after we have identified the self-map (fj) on BU as the one inducing the difference between the universal complex bundle and its conjugate. To do this, consider the composite j f U(2m) -! Sp(2m) -! U(4m): To study this we suppose that we have inclusions R2m C2m H2m = C2m + jC2m where the quaternion j 2 H has its usual meaning. Then for M 2 U(2m) Sp(2m) and a; b 2 C2m we have ___ M(a + jb) = Ma + jM b; 25 note here that conjugation in U(2m) is well defined since we have chosen a real subspace of C2m . It follows that the map fj sends M to ! M _0_ 2 U(4m): 0 M It follows that (fj) = 1 c where c : U ! U is the conjugation map. All that remains is to show c = -Bc. Restricting to U(1) ' Z, we see that c induces multiplication by -1 in ss0. To evaluate c on the 0-component BU ' SU, recall [Mil63 , Theorem 23.3] that this equivalence is derived from a certain map g Gm (C2m ) -! SU(2m) where Gm (C2m ) is the Grassmannian of complex m-planes in C2m . (Bott proves the complex case of his theorem by showing that the map g is an equivalence through a range of dimensions that increases with m.) The map is defined by associating to each point in Gm (C2m ) a path in SU(2m) from I to -I as follows. Choose a basis of C2m such that the m-dimensional subspace in question is spanned by the first m basis elements. We parametrize the path by 2 [0; ss] with ! ei I 0 7! -i 2 SU(2m); 0 e I where I here denotes the identity element in U(m). In other words (indepen- dently of the choice of basis) we send to a unitary transformation having eigenvalue ei on the given subspace and e-i on its complement. Again we note that there is a well defined conjugation map c on SU(2m), given our choice of a real subspace of C2m . Applying it does two things. First it conjugates the basis, replacing each subspace by its conjugate. It also conjugates the coefficient ei , so that with respect to the conjugated basis the map above becomes ! e-i I 0 7! i 2 SU(2m): 0 e I 26 Thus the direction of the path gets reversed, which effectively replaces the conjugated subspace by its unitary complement. It follows that the conjugation map c on SU(2m) restricts on the subspace Gm (C2m ) to the map which sends each complex m-plane through the origin in C2m to the complement of its conjugate. Passing to the limit as m ! 1, we see that Z x BU -c! Z x BU is the map inducing the Whitney inverse of the conjugate universal bundle as required. 2.6 Connective covers of BU and related spaces and spectra Let BU<2m> denote the (2m - 1)-connected cover of BU for m 2 and consider the fibration F - ! BU<2m> -! BSU: Then K(n)*(F ) is even dimensional for n > 0 by [HRW ], as is H*(BSU). Thus Proposition 2.0.1 applies and we conclude that K(n)*(BU<2m>) is even dimensional. We also know that K(0)*(BU<2m>) (the rational homology of BU<2m>) is even dimensional. Thus the results of this paper give informa- tion about BPp^*(BU<2m>), and similarly for MU<2m>, the associated Thom spectrum. Localizing at an odd prime, we can say the same about the fibration F 0-! BO<4m> -! BSO; so we can say a lot about BP *(BO<4m>) and BP *(MO<4m>). 27 3 Some generalized cohomology theories We need a large selection of theories to state and prove our results. First, there is the Brown-Peterson cohomology, BP *(-), associated with a prime, p. Some basic references for BP are Brown-Peterson, [BP66 ], Adams, [Ada74 ], Quillen, [Qui69 ], Ravenel, [Rav86 ], and Wilson, [Wil82 ]. Next, we need the p-adic completion of BP , BPp^, defined by BPp^= inv limBP ^ M(pi) (3.1) where M(pi) is the mod pi Moore spectrum. The coefficient ring for BP , BP *, is Z(p)[v1; v2; : :]:where the degree of vn is -2(pn - 1). The coefficient ring for BPp^is just the p-adic completion of this. Either of these theories can be labeled P (0). Next, we need the theories introduced by Morava, P (n). Their coefficient ring is BP *=In where In = (p; v1; : :;:vn-1). For references, see Johnson-Wilson, [JW75 ], W"urgler, [W"ur77 ], and Yagita, [Yag77 ]. Thanks to their construction using Baas-Sullivan singularities, [Baa73 ], [BM71 ], th* *ey come equipped with stable cofibrations: n-1) vn 2(p P (n) -------! P (n) -------! P (n + 1); (3.2) which give us long exact sequences in cohomology. Note that P (0) can be either BP or BPp^in this cofibration. Letting BP* be Z(p)[v1; : :;:vn], theories, E(k; n) can be constructed, using Baas-Sullivan singularities and localization, which have coefficients v-1nBP *=Ik, and similar stable cofi- brations. These spectra are discussed by Baker-W"urgler in [BW89 , page 523] and in [Hun92 ]. The earliest reference to these theories is probably in [Yos7* *6 , Prop. 4.6] (where they go by a different name). The theories without local- ization play a prominent role in [Yos76 ], [Yag76 ] and [BW ]. For k = 0 these theories are usually denoted by E(n); they have been studied in [JW73 ], [Lan76 ], [Rav84 ], and others. It is proven in [JW73 , Remark 5.13, p. 347], and later follows from the Landweber exact functor theorem of [Lan76 ], that E(n)*(X) = E(n)* BP* BP*(X): 28 A similar result for k > 0 (with BP* replaced by P (k)*) was proved in [Yag76 ]. As a special case, when k = n > 0, we have the nth Morava K-theory, K(n), see [JW75 ], [Hop87 ], [Rav86 ], [Rav92 ], [W"ur91 ] and [Yag80 ]. One of our main tools is the Atiyah-Hirzebruch spectral sequence E*;*r) G*(X) (3.3) where Es;t2' Hs(X; Gt) (3.4) which we will denote E*;*r, E*;*r(X), or E*;*r(G*(X)), depending on the con- text. The differential, dr, has bidegree (r; 1 - r). When G is one of our con- nected spectra this is a fourth quadrant spectral sequence. If not, it is a fir* *st and fourth quadrants spectral sequence. Let F s= ker(G*(X) ! G*(Xs-1)) where Xs-1 is the s - 1 skeleton of X. Then we have F s=F s+1' Es;*1and F 1 gives the phantom maps. The spectral sequence really converges to G*(X)=F 1 so it will be impor- tant for us to be able to show that F 1 is zero in our cases. By Milnor's theorem, [Mil62 ]: Theorem 3.5 (Milnor) There is a short exact sequence 0 ! lim1G*(Xm ) ! G*(X) ! limG*(Xm ) ! 0: Since the term on the right of Milnor's theorem is what the Atiyah- Hirzebruch spectral sequence converges to, the triviality of F 1 is equivalent to the lim1 term being zero. Remark 3.6 One way to show the lim1 term is zero is by using the Mittag- Leffler condition. In our case, we have a sequence of subgroups Im{Gn(Xm+i ) ! Gn(Xm )}: 29 If they stabilize for big i and all n we say the Mittag-Leffler condition is satisfied. In this case, the lim1 term in Milnor's theorem is zero. See [Ada74 ] for more details. Several assumptions on the G or X can give us the Mittag- Leffler condition: (i) If Es;t2is always finite. This can happen if one of Gs or Ht(X) is always finite and the other is finitely generated. This is the case for several of our theories; e.g. P (n) and K(n), n > 0. (ii)If Es;t2is always a compact group. This happens for us when we use the p-adic completion of BP , BPp^, for P (0). (iii)If Es;t*always has only a finite number of nontrivial differentials on it. We show that this is the case for E(k; n) when 0 < k < n, which is a bit surprising and our most difficult technical lemma. The skeletal filtration of G*(X) associated with the Atiyah-Hirzebruch spectral sequence also gives a topology on G*(X) which is nontrivial if X is an infinite complex. Since all of our spaces are infinite complexes this topology is always there. For a detail reference on such things, see [Boa95 ]. We will n* *ot need much about the topology here except that when we look at subgroups generated by a set of elements, we will mean topologically generated, i.e., the closure of the literal subgroup. We need two theorems which generalize known results. First we need a generalization of Quillen's theorem. Theorem 3.7 ([Qui71 ] for n = 0 and [Yag84 ] for n > 0) For X a finite complex, P (n)*(X), n 0, is generated by non-negative degree elements. Quillen proved the n = 0 version of this in [Qui71 ]. A second proof for Quillen's result, n = 0 of this, was given in [Wil75 ]. More recently, Quillen's result follows from abstract information about unstable BP oper- ations, [BJW95 ]. In [Yag84 ] the result is only stated for odd primes but the 30 argument follows Quillen exactly and goes through for p = 2 as well. The result, for all primes, was also proven in [Yag ] using much simpler techniques of Shibata, [Shi75]. We sketch a proof based on this approach here. In ad- dition, a new proof for odd primes, analogous to the one in [Wil75 ], follows from the splitting theorem of [BW ] and is included there. Proof. Recall that the normalizer of Z=(p) in the symmetric group Sp is the meta-cyclic group, Zp;p-1, defined by 0 ! Z=(p) ! Zp;p-1! Z=(p - 1) ! 0: Since |Z=(p-1)| is prime to p we know BP *(BZp;p-1) ' BP *(BZ=(p))Z=(p-1), the invariant subring under the action of Z=(p - 1). The action on u in BP *(BZ=(p)) ' BP *[[u]]=([p](u)) by i 2 Z=(p - 1) is given by [i](u). Hence we can prove that the BP -cohomology of BZp;p-1 is the BP *-subalgebra Q p-1 generated as a BP *-algebra by the element w = i=0[i](u), [Shi75] and [Shi74]. We easily show that BP *(BZp;p-1) satisfies the conditions of the Landweber exact functor theorem, P (n)*(BZp;p-1) ' P (n)* BP*BP *(BZp;p-1) and P (n)*(X ^ BZp;p-1) ' P (n)*(X) bBP*BP *(BZp;p-1): By [Yag84 , Definitions 1.2-1.6], the Gysin map f* is also defined for P (n)- theory. For each element x 2 P (n)*(X), x = f*(1) for some complex oriented map f : ^Z! X from a (p; v1; v2; : :;:vn-1)-manifold ^Zto X. For ease of argument, assume that x is even degree in P (n)*(X). Since Zp;p-1 Sp, we get a map * Q : P (n)2*(X) -P!P (n)2p*(Xp xZp;p-1EZp;p-1) -! P (n)2p*(X x BZp;p-1) where P (x) = fp ^ 1 and is the diagonal map. Let I = (vn; : :): P (n)*. Since ([p](u)) \ P (n)*[[w]] I[[w]] P (n)*([[u]]); we get P (n)*(BZp;p-1) P(n)*Z=(p) ' Z=(p)[w]: 31 Thus we can define Rk : P (n)2s(X) -! P (n)2s+2k(p-1)(X) P(n)*Z=(p) by Xs Q Z=(p) = R2k(x) ws-k: k0 Then Rk is a cohomology operation, in a sense, which is natural. This oper- ation is stable by the usual diagram chasing and P (n)*(S2) ' P (n)*[u]=(u2). The Cartan formula and R|x|=2(x) = xp 1 also hold. By the definition it is obvious that Rk(x) = 0 if dimension x is less than 2k. On the other hand, the operation R0 induces R0 : P (n)*(P (n)) -! P (n)*(P (n)) P*(n)*Z=(p)' Z=(p){rff} ^(Q0; : :;:Qn-1) where rffare the Quillen operations. For dimensional reasons, R0(1) = 1 for 1 2 P (n)0(P (n)). This means R0(x) = x mod I, by naturality. Let the dimension of x be negative, then 0 = R0(x) = x mod I. This means that x is not a P (n)*-generator. Thus we get the Quillen theorem for P (n)-theory. (When p = 2 we identify Z2;1with Z=(2).) 2 All of our proofs go through for p = 2. Normally, there can be problems with this case because most of our theories that are mod 2 do not have a com- mutative multiplication on them. However, in our case they are always even dimensional. In [W"ur77 ], W"urgler computes the obstruction to commutativ- ity and shows that it factors through odd degrees and is thus of no concern to us. Where it could bother us because we do have odd degree elements, is in the Atiyah-Hirzebruch spectral sequence, but that is commutative by itself so it is no problem. The rest of the arguments are no problem. The next result that we need is a generalization of the Landweber exact functor theorem, [Lan76 ]. 32 Theorem 3.8 ([Lan76 ] for k = 0 and [Yag76 ] for k > 0) Let In;kbe the ideal (vk; : :;:vn-1) in P (k)*. M is flat for the category of finitely present* *ed P (k)*(P (k))-modules if vn multiplication is injective on M=In;kM for all n k. Note that this result is not a cohomological version of the Landweber exact functor theorem, but merely an algebraic statement about finitely presented P (k)*(P (k))-modules. Proof of Corollary 1.9. The theorem is true for all of our theories, P (k) or E(k; n), where if k = 0 we may have to use p-adic completion. If Y is a finite complex we already have the result by Theorem 1.6. By [Lan70 , Lemma 6] we do not have to use p-adic completion for the product of two spaces unless we have to for one of the spaces. We have E*(Y ) ' inv limE*(Y N) ' inv limE*(Y )=F N(Y ): For each N there is an nN such that Im (E*(Y nN) ! E*(Y N)) = Im (E*(Y ) ! E*(Y N)) ' E*(Y )=F N(Y ): We can get a sequence E*(XN1 ) E*(X)=F N1(X) E*(XN2 ) E*(X)=F N2(X) . .:. (3.9) We now have E*(X x Y ) ' inv limE*(X x Y N) ' inv limE*(X) E* E*(Y N): E*(X) bE*E*(Y ) is defined by i j inv lim (E*(X) E* E*(Y ))=(F N(X) E* E*(Y ) + E*(X) E* F N(Y )) which is i j ' inv lim (E*(X) E* E*(Y ))=(E*(X) E* F N(Y )) ' inv lim(E*(X) E* (E*(Y )=F N(Y ))) which, by 3.9, is the same as the above. 2 33 4 The Atiyah-Hirzebruch spectral sequence In this section we develop the Atiyah-Hirzebruch spectral sequence for the things that we need. In particular, we accomplish two main goals. First, we show that there are no phantom maps in G*(X) for all of the theories that we are concerned with except BP . This simplifies our life considerably. Originally it seemed that some sort of exotic completions would be necessary to state our theorem, but because of this lack of phantom maps in general, the only place we have to go to completion is occasionally with BP where we have to resort only to p-adic completion. This lack of phantom maps is just the same as having all elements of G*(X) represented in the Atiyah- Hirzebruch spectra sequence; i.e., having the infinite filtration, F 1, equal to zero. We must eliminate the phantom maps so that in the next section we can show K(n)*(X) even implies E(k; n)*(X) is also even. With the advantage of hindsight, an alternative route to these results might be to use Yosimura's work [Yos88 ]. Second, we show that for any given element of P (k)*(X), k > 0, there is some N such that the element maps nontrivially to E(k; n)*(X) for every n > N. Our proof is somewhat technical and uses the Atiyah-Hirzebruch spectral sequence extensively. However, to see what is going on is not so difficult. Consider the k = 0 case where we are working with BP and E(0; n) = E(n), the localization of BP . From [Wil75 ] we know that BP_ _ksplits off of BP__kfor k 2(pn+1 - 1)=(p - 1). It is easy to see that it splits off of E(n)_k as well. Any 0 6= x 2 BP k(X) must reduce to a non-trivial element in BP k(X) for some large n. Thus we can see, quite geometrically by looking at the classifying spaces, the result we want. No such splitting was around for k > 0 when we wanted to generalize this. Our proof depends heavily on the theorem that P (n)*(X) is generated by non- negative degree elements for finite complexes. Since this proof, a splitting theorem has been found, [BW ], which would allow us to prove the result by looking at the representing spaces for the cohomology theories. This gives an alternative approach to this part of the proof as well. Unless otherwise stated, let E*;*r(X) ) P (n)*(X), n > 0, be the Atiyah- 34 Hirzebruch spectral sequence with X a space. Although we do not need X to be a space for the first few lemmas we will assume it anyway. X could just as well be a -1-connected spectra. We will point out when having X a space becomes necessary. Lemma 4.1 Let E*;*r) P (n)*(X), X a space, with n > 0. For each r and s there is a number N = N(r; s) such that there is a finitely generated Z=(p)[vn; : :;:vN-1 ]-module, Asr= As;*r, generated in nonpositive degrees and satisfying Es;*r' Asr P (N)*: Remark 4.2 Here and in the future for such results we are considering Z=(p)[vn; : :;:vN-1 ] and P (N)* both as subrings of P (n)*. Proof. The proof is by induction on r. For r = 2 we can take As2= Hs(X; Z=(p)) and N = n. Assume the case r. Then we have integers N(s; r) as in the lemma for all s. Let {yi(s)} be a (finite) set of generators of Asr with degrees |yi(s)|. Now fix an s. Write M = max {N(k; r): 0 k s + r} S = Z=(p)[vn; : :;:vM-1 ] d = min{|yi(k)|: 0 k s + r; i > 0}: Then there is a bigraded S-module Bqr= Bq;*rgenerated by {yi(q)} for 0 q s + r, such that Eq;*r' Bqr P (M)*; 0 q s + r; with 0 |yi(q)| d for all 0 q s + r. Consider the differential X dr(yi(s0)) = ci;j(s0+ r)yj(s0+ r); ci;j(s0+ r) 2 P (n)* j for 0 s0 s. Then, 1 - r + |yi(s0)| = |ci;j(s0+ r)| + |yj(s0+ r)|; 35 so ci;j(s0+ r) 2 Z=(p)[vn; : :;:vM0-1 ] 0-1 where M0 is the smallest number such that -2(pM - 1) 1 - r + d. Take Sr+1 = Z=(p)[vn; : :;:vM0-1 ]. Then dr induces an Sr+1 map dr : Bsr Z=(p)[vM ; : :;:vM0-1 ] -! Bs+rr Z=(p)[vM ; : :;:vM0-1 ]: ker(dr : Es;*r! Es+r;*r) = ker(dr|Bsr Z=(p)[vM ; : :;:vM0-1 ]) P (M0)*: Similarly, consider dr : Es-r;*r! Es;*r. Then Es;*r+1' As P (M0)* for some Sr+1-module As = As;*. As is a subquotient of the finitely generated Ss+1- module Bsr Z=(p)[vM ; : :;:vM0-1 ]. Hence As is finitely generated as an Sr+1-module since Sr+1 is Noetherian. Take N = N(s; r + 1) as M0 and Asr+1 as As. This completes the induction. 2 Recall Xm is the m-skeleton of X and i : Xm ! X is the inclusion. Lemma 4.3 Let E*;*r(X) ) P (n)*(X), X a space, n > 0. For each r, Es;*r(X) ' Es;*r(Xm ) for all 0 s m - r(r - 1)=2: Proof. Since Es;*2(X) ' Es;*2(Xm ) for s m - 1 and |dr| = (r; 1 - r) this follows by an easy induction. 2 Lemma 4.4 Let E*;*r(X) ) P (n)*(X), X a space, n > 0. For all m (s + 1)s=2 + s and all r, i* : Es;*r(X) -! Es;*r(Xm ) is injective. Proof. By Lemma 4.3 we have an isomorphism for r s + 1 when 0 s m - (s + 1)s=2, i.e., m (s + 1)s=2 + s. Then, because |dr| = (r; 1 - r), Es;tr is not in the image of dr for r > s so we have 36 Es;*s+1(X)_____ Es;*s+1(Xm ) u| u| | | | | | | |y |y * Es;*r(X) _______Es;*r(Xmw)i u| u| | | | | | | |y |y * Es;*1(X) _______Es;*1(Xmw)i for r > s. This implies the result. 2 The next lemma uses Yagita's version (Theorem 3.7) of Quillen's theo- rem for P (n)*(-), and is the main technical lemma which makes the Atiyah- Hirzebruch spectral sequence approach work. At this stage it becomes essen- tial that we are working with spaces and not spectra. Lemma 4.5 Let E*;*r(X) ) P (n)*(X), X a space, n > 0. For each s there is an m such that i* : Es;*1(X) ' Es;*1(Xm ) and dr(Es;*r(X)) = 0 for r > m i.e., Es;*r(X) ' Es;*1(X). Proof. Since each group Es;tris finite we can find an R s such that dr 0;t00 0 0 restricted to {Esr |s s; t -s } is zero for all r R. From Lemma 4.3 0;t s0;t m and Lemma 4.4 we can find m s such that EsR (X) ' ER (X ) for all 0;t s0;t m 0 s0 s and i* : Esr (X) ! Er (X ) injects for all r and s s. Certainly Es;*m-s+1(X) ,! Es;*m-s+1(Xm ) ' Es;*1(Xm ): We want to show that Es;*m-s+1(X) ' Es;*1(X). Assume there is some 0 6= "x2 Es;tr(X), and r m - s + 1 with dr("x) 6= 0. We have i*("x) = "xm2 Es;tr(Xm ) ' Es;t1(Xm ): 37 This is non-zero by our injectivity. Furthermore, t < -s because our original choice says dr = 0 if t -s. Thus the total degree of "xis negative. By Yagita's version of Quillen's Theorem for P (n)*(Xm ), Theorem 3.7, we know there are non-negative degree P (n)* generators, gmi, for P (n)*(Xm )=F s+1in 0;t0m 0 0 0 Es1 (X ) with s s and t -s , which our starting assumption says is 0;t0 isomorphic to Es1 (X). (The fact that X is a space rather than a connective spectrum is crucial at this point of the proof. In the latter case, Theorem 3.7 would not give us the precise control we need on the dimensions of these generators.) Thus P (n)*(X)=F s+1surjects to P (n)*(Xm )=F s+1in non-negative de- grees. So we can choose generators {gi 2 P (n)*(X)} which reduce to the 0;t0 0 generators {gmi2 P (n)*(Xm )=F s+1}. This is a finite set because Es2 , s s and t0 -s0, is finite. If we have our dr("x) 6= 0, then "xm= i*("x) 2 Es;tr(Xm ) = Es;t1(Xm ) is not in the image of i* : Es;t1(X) ! Es;t1(Xm ). Let xm 2 P (n)*(Xm )=F s+1be P m P an element represented by "xm. Then xm = v(i)gi . Define z = v(i)gi 2 0;t 0 P (n)*(X). Then i*(z) = xm so "zmust be in Es1 with s < s. This is so because "xm is not in the image but the element it represents is; therefore 0;t 0 the element representing the element that hits it must be in Es1 with s < s. But, that means "zmust go to zero in order to do its duty of changing filtrations to hit "xm. This contradicts the injectivity of Lemma 4.4 and we have Es;*m-s+1(X) ' Es;*1(X). The argument just given also shows that Es;*r(X) maps surjectively to Es;*r(Xm ) for r m - s + 1. 2 We have some quick corollaries now. Corollary 4.6 Let X be a space and let n > 0. For each r, there is an m such that P (n)*(X)=F r' P (n)*(Xm )=F r 38 where F ris the rth filtration of the Atiyah-Hirzebruch spectral sequence. Corollary 4.7 Let X be a space and let n > 0. P (n)*(X) is (topologically) generated by non-negative degree elements. Corollary 4.8 Let X be a space. Let k > 0. Let G be P (k), E(k; n), v-1nP (k), or K(k). Let E*;*r(G*(X)) ) G*(X) be the Atiyah-Hirzebruch spectral sequence. Then F 1 = 0, every element in G*(X) is represented in E*;*1, G*(X) ' inv limG*(Xm ), and lim1G*(Xm ) = 0. Furthermore E*;*r(E(k; n)*(X)) ' E*;*r(P (k)*(X)) P(k)*E(k; n)* ' E*;*r(v-1nP (k)*(X)) v-1nP(k)*E(k; n)* and E*;*r(v-1nP (k)*(X)) ' E*;*r(P (k)*(X)) P(k)*v-1nP (k)*; which is just localization. Remark 4.9 This is not true for spectra. For example, P (n)*(k(n)) = 0 but K(n)*(k(n)) is not, so the first line for k = n cannot hold. Proof. The first four statements are equivalent so it is enough to show any one of them. For G = P (k) we have that P (k)s is finite so Es;tris finite. By the Mittag-Leffler condition, Remark 3.6, we are done. The other G will follow from the displayed tensor products and Lemma 4.5. Keep in mind that K(k) is just a special case of E(k; n), with n = k. All of the tensor products are true for E*;*2. Since the last tensor product is just localization and localization preserves exactness we see that it is true. Es;*rfor P (k) is always a finitely generated P (k)*(P (k)) module so tensoring with E(k; n)* is exact by Yagita's version of the Landweber exact functor theorem for P (k)*(-) which says that E(k; n)* is P (k)* flat in this situation, Theorem 3.8. This proves the first equivalence (F 1 = 0) for E(k; n) by Remark 3.6(iii). The second isomorphism follows from the first and third. 2 39 Remark 4.10 Although this is a very technical result, it is an exciting one because it removes most of our lim1 problems. Since all of our theorems are about infinite complexes, worrying about phantom maps was a major concern which this corollary eliminates. For example, certain types of elements cannot exist. In the spectral sequence for P (n)*(X) it is quite possible, as we shall see in the next section (see Remark 5.5), to have an element whose filtration is raised every time you multiply by vn, and which is never zero. Such an element, by the corollary, can never give rise to an element in K(n)*(X) even though it is a torsion free element. Corollary 4.11 Let X be a space. Let 0 < k < n. If x is infinitely divisible by vk in E(k; n)*(X), then it is zero. We see from the proof that any such element would have to be a phantom map, and there are none. Proof. By Lemma 4.8 we have E(k; n)*(X) ' inv limE(k; n)*(Xm ) so if x is non-zero it maps nontrivially to some E(k; n)*(Xm ) and is still infinitely divisible by vk there; but this contradicts the finite generation that comes with finite complexes. 2 Proposition 4.12 Let X be a space. Let k > 0. Given 0 6= x 2 P (k)*(X), there exists an N such that x maps nontrivially to E(k; n)*(X) for all n N. Proof. The element x is represented in the Atiyah-Hirzebruch spectral se- quence by an element "xin Es;*1, which, by Lemma 4.5 is isomorphic to Es;*r for some big r. For degree reasons we can pick an N such that "xis a P (N)* generator. Using Lemma 4.1 we can assure that N is big enough so that "xsurvives the tensor product of Corollary 4.8 to the spectral sequence for E(k; n)*(X) for n > N. In this spectral sequence we still have Es;*r' Es;*1 so our element x maps nontrivially. 2 40 5 Even Morava K-theory We complete the proof of the main Theorem 1.6 in this section. Lemma 5.1 Let X be a space. Let 0 < k n. If K(n)*(X) is even dimen- sional, then E(k; n)*(X) is even dimensional and has no vk torsion. Remark 5.2 This lifts up to E(n)*(X) or its p-adic completion if neces- sary. It follows from the proof that E(n)*(X) maps onto K(n)*(X). This was proven by Hunton in [Hun92 , Theorem 11], for finite complexes. This * is improved in [BW91 ], p. 559, to give a surjection of Ed(n) (X), the In- adic completion. As we only need p-adic completion our result is slightly stronger, but not useful for them because other things force them to use their completion. Proof. We prove this by downward induction on k. Since E(n; n) = K(n) our induction is grounded by our assumption. By induction, assume that E(k + 1; n)*(X) is even dimensional and has no vk+1 torsion. We have a long exact sequence from the cofibration analogous to 3.2: vk E(k; n)*(X) ________________________E(k;wn)*(X) 4 4 ffi 4447ae E(k + 1; n)*(X) Since E(k + 1; n)*(X) is even dimensional and ffi is an odd degree map, there are two possible types of odd degree elements in E(k; n)*(X): (i)an element that never shows itself in E(k + 1; n)*(X) because it is infinitely divisible by vk and not vk torsion; (ii)an element that is infinitely divisible by vk but is vk torsion (the eleme* *nt that vk kills comes by way of ffi). 41 Either way the element is infinitely divisible by vk, which cannot happen by Corollary 4.11. Thus ffi is zero and all elements are even degree. If any element were vk torsion, then it would have to be hit by ffi coming from an odd degree element, which doesn't exist by our induction assumption. Thus we get a short exact sequence and all elements are vk torsion free. 2 We can now prove Theorem 1.2. Lemma 5.3 Let X be a space. If K(n)*(X) is even dimensional for an infinite number of n, then P (k)*(X) and K(k)*(X) are even dimensional for all k > 0. Proof. If 0 6= x 2 P (k)*(X) pick N as in Lemma 4.12 so x maps nontrivially into E(k; n)*(X) for n > N. Find some n > N for which K(n)*(X) is even dimensional. By Lemma 5.1, E(k; n)*(X) is even dimensional so x must be even dimensional as well. This concludes the proof for P (k)*(X). By Lemma 4.8, all elements of P (k)*(X) are represented in E*;*1(P (k)*(X)) which is even dimensional. Furthermore, E*;*1(K(k)*(X)) is just the tensor product with K(k)* so it too is even dimensional, and it also represents elements. 2 Corollary 5.4 Let X be a space with even Morava K-theory. For k > 0 we have the short exact sequence: vk * * 0 -! P (k)*(X) -! P (k) (X) -! P (k + 1) (X) -! 0: Proof. The three terms fit into a long exact sequence with odd degree con- necting term, by 3.2. By Lemma 5.3 all terms are even dimensional so the boundary homomorphism must be zero. 2 This just about finishes the proof of Theorem 1.7. Remark 5.5 This is a good time to insert a fundamental example which illustrates the phenomenon described in Remark 4.10. This is an old, well 42 known example but it supplies useful guidance. Let X = BZ=(p). The mod p cohomology is E(e1) P (x2), so E2 of the spectral sequence is E(e1) P (x2) P (n)*: n The only nontrivial differential takes e1 to vnxp2 leaving E1 to be a copy of P (n)* for each xi2for i < pn and a copy of P (n + 1)* for each xi2for i pn. Tensoring this with K(n)* we get the correct answer for K(n)*(BZ=(p)); free on generators xi2for i < pn. However, we know from our corollary that there is no vn-torsion. If you take an element in P (n)*(BZ=(p)) which is represented by xi2for i pn and you multiply by vn, then it is represented n+pn+1 by vn+1xi-p2 . So, iterating the multiplication by vn continues to raise filtration and give a nontrivial element. However, it does not give rise to an element of K(n)*(BZ=(p)). Looking briefly at P (n)*(BZ=(p)) we see that E1 is a free P (n + 1)* on elements ffi in degree 2i - 1 for i > 0 and free P (n)* on fii in degree 2i for 0 < i < pn. The relations on the ff come from the p-sequence. In particular, we have vn+1ffi+ vnffi+pn-pn+1mod (vn; vn+1)2. We see that all of the ffi are infinitely divisible by vn in E(n; n + 1)*(X). Corollary 5.6 If X is a space with even Morava K-theory and k > 0 then P (k)*(X) is even degree and is a flat P (k)*-module for the category of finite* *ly presented P (k)*(P (k))-modules. Proof. P (k)*(X) is even dimensional by Lemma 5.3. To prove flatness we need only invoke Yagita's version of the Landweber exact functor theorem for P (k), Theorem 3.8, and Corollary 5.4. 2 We can now complete our proof of our main theorem. Proof of Theorem 1.6. Consider the long exact sequence from 3.2. p P (0)*(X) _________________wP (0)*(X) flflfli A A fffli AADae P (1)*(X) 43 Since P (1)*(X) is even dimensional by Lemma 5.3, any odd degree element in P (0)*(X) must be infinitely p-divisible. By our assumption, every element restricts to a finite subcomplex (by our choice of P (0)), and elements in a finite complex cannot be infinitely p-divisible. To prove flatness and finish the proof Theorem 1.7 of we need the original Landweber exact functor theorem, see Theorem 3.8, and Corollary 5.4. 2 Remark 5.7 If Hk(X; Z(p)) is finite for all k then the Mittag-Leffler condi- tion, Remark 3.6, is satisfied and the Atiyah-Hirzebruch spectral sequence for BP *(X) converges giving lim1BP *(Xm ) = 0, so the results of Theorem 1.6 hold for BP *(X). As we shall see later, this is the case for X = K(Z=(pi); n). When Hk(X) is not finite we may have to resort to the p-adic completion of BP , such as with X = K(Z(p); n), n > 2, which is known to have phantom maps. 6 Generators and relations In this section we will prove Theorems 1.18 and 1.19, and Corollaries 1.21 and 1.22. Proof of Theorem 1.18. We know that the Atiyah-Hirzebruch spectral se- quence converges and Es;tr(X) is a finitely generated group. By Yagita's and Quillen's Theorem 3.7 the generators must be represented by elements with s + t 0. Since t 0, there can only be a finite number of generators for P (n)*(X) represented in Es;*1. Assume inductively that we have chosen a minimal number of generators for P (n)*(X)=F s. Then pick a few more, if necessary, that are represented in Es;*1in order to get minimal generators of P (n)*(X)=F s+1. The construction of Tn with properties (a), (b) and (c) is now proven. We now show (d), that Tn reduces to a set Tq with the same properties. We do this inductively. Because we know, Theorem 1.7, that P (n)*(X) 44 surjects to P (n + 1)*(X) we get part (a) that Tn+1 generates. The map is a filtered map so part (c) follows. Part (b), that all elements remain essential, is really the only thing left to prove. If some proper subset of Tn could be used to generate P (n + 1)*(X) then we could write some t 2 Tn+1 in terms of the t's: X t = citi where ci 2 P (n + 1)* can all be lifted to ci 2 P (n)* and we use the same no- tation for elements in P (n)*(X) and P (n + 1)*(X). From the exact sequence of Theorem 1.7 we can lift this to X X t = citi+ vn diti in P (n)*(X), contradicting (b). The result follows for Tq by induction. Proof of part (e). Let F Rn be the set of finite linear relations among P the elements of Tn in P (n)*(X). A typical relation looks like iciti where ti 2 Tn and ci 2 P (n)*. We can write the ci in terms of monomials in the vk (where we let v0 = p for P (0)*). We can define the length of a monomial as P the sum of the powers of v's, i.e., for vI = vinnvin+1n+1:w:e:define l(vI) = * * ik. We now extend this definition to the elements of F Rn. We take the length of a relation to be the maximal length of a monomial occurring in any of its coefficients, ci. There is an obvious map from F Rn to F Rn+1. Because there are a finite number of coefficients, every element goes to zero after enough of these maps have been applied. Let us find a relation, r, which has the minimal length as defined above. Let us assume that it is in F Rn and maps to zero in F Rn+1. We can do this because the length of a relation can never increase under these maps. Recall from Theorem 1.7 that we have a short exact sequence. Since each coefficient, ci, maps to zero in P (n + 1)* it must be divisible by vn. Thus we can divide the sum r by vn to get r=vn. This is a finite sum with a smaller length than our minimal one so it must be a non- zero element. This cannot be true as it is a vn-torsion element in P (n)*(X) which by Theorem 1.7 is known to have no such torsion. Thus there are no finite relations anywhere. 45 Proof of part (f). We prove this by downward induction. Let r 2 Rq be P * * written citi with ci 2 P (q) . Lift each ci to P (q - 1) (and note that vq-1 P * does not divide them). Then the element citi 2 P (q - 1) (X) reduces to zero in P (q)*(X). If it is not zero in P (q - 1)*(X), then it is divisible by P 0 0 vq-1 and we can write 0 = citi+ vq-1r where r can be written in terms of the ti. Thus we have a relation which reduces to our r. Proof of part (g). This follows from (f). 2 We need a couple of lemmas to prove Theorem 1.19. Lemma 6.1 Let X be a space and let q > 0, and K(q)*(X) be even di- mensional. Let J2 = (vq+1; vq+2; : :):. Let T be a set in v-1qP (q)*(X) such that (a) all but a finite number of the elements of T are in F s, the s filtration for the Atiyah-Hirzebruch spectral sequence and (b) the image of T in K(q)*(X) generates (topologically). Then the image of T generates v-1qP (q)*(X)=(F s+1+ JN2) for all s and all N. Proof. P (q)*(X)=F s+1is a finitely presented P (q)*(P (q))-module and, as such, it has a Landweber filtration; see [Yag76 ] or [Yos76 ]. When you lo- calize at vq such a filtration becomes a finitely generated free module over v-1qP (q)*. It is then easy to see that K(q)*(X)=F s+1 ' K(q)* v-1qP(q)* v-1qP (q)*(X)=F s+1. Pick a set of generators xi for v-1qP (q)*(X)=F s+1. The image of these xi must generate K(q)*(X)=F s+1; as does the image of T . Thus, modulo J2, the xi must be in the submodule generated by T . We have P si;k P (finite) sums xi = vq tk + ci;jxj where ci;j2 J2. (Note that si;k2 Z.) Now, to show that the xi are in the image modulo J22we just substitute the equations for the xj into this. Iterate to get the theorem modulo JN2. 2 46 Lemma 6.2 Let q > n 0. Let X be a space with even Morava K-theory. Let J2 = (vq+1; vq+2; : :): and J1 = (vn; : :;:vq-1): Let Tn be a set in v-1qP (n)*(X) such that (a) all but a finite number of the elements of Tn are in F s, the s filtration for the Atiyah-Hirzebruch spectral sequence and (b) the image of Tn in K(q)*(X) generates (topologically). Then the image of Tn generates v-1qP (n)*(X)=(F s+1+JN1+JN2) for all s and N. Proof. Recall, Lemma 5.3, that P (k)*(X) is even dimensional for all k from n to q. From Corollary 5.4 we have short exact sequences which we can localize with respect to vq. We then see that v-1qP (q)*(X) ' v-1qP (q)*b v-1qP(n)*v-1qP (n)*(X) Thus, if we have x; y 2 v-1qP (n)*(X) which reduce to the same element in P v-1qP (q)*(X), then x = y + eiri where ei 2 J1 and the sum is possibly infinite. Fix the N and s of the Lemma. Let T be the image of Tn in v-1qP (q)*(X). We can pick generators, {yi}, for P (n)*(X) with property (a) above by 0+1 picking (a finite number of) generators for P (n)*(X)=F s , lifting them to P (n)*(X) and extending this choice by enlarging s0. See the first part of the proof of Theorem 1.18 above for more detail. Map these gener- ators to a set of generators, {xi}, for v-1qP (n)*(X). Reduce these ele- ments further to zi 2 v-1qP (q)*(X). By Lemma 6.1, we can write zi, in 0+1 N v-1qP (q)*(X)=(F s + J2 ), in terms of the reduction of T . Taking the limit, we can write each X X zi = di;ktk + ci;jzj 47 P where di;k2 v-1qP (q)* and ci;j2 JN2. The two elements, xi and di;ktk + P ci;jxj both reduce to the same element and we see from the above that X X X xi = di;ktk + ci;jxj + ei;mxm where ei;m 2 J1 and the sums are possibly infinite. Reduce this to v-1qP (n)*(X)=F s+1and the sums are now finite. As in the proof of Lemma 6.1, substitute this formula in for the xi and iterate in order to show that the t's generate modulo JN1+ JN2 2 Remark 6.3 Note that to use Theorem 1.19 we can weaken the hypothesis somewhat. We do not require X to have even Morava K-theory. It is enough that the Tn generates an infinite number of Morava K-theories that are even. By Theorem 1.2 of the Introduction we have X has even Morava K-theory. In our proof, we will point out where it is enough that we generate not all, but infinitely many, of the Morava K-theories. Proof of Theorem 1.19. We prove our theorem by induction on s. Assume that we have it for smaller s. Now, if we have an x 2 P (n)*(X)=F s+1which is not in the submodule generated by Tn then we will derive a contradiction. Since there are only a finite number of the Tn which are non-zero, the quotient of P (n)*(X)=F s+1by the submodule generated by Tn must be coherent, [Smi69 ]. We will show that is not the case. Our x must be represented in Es;*1. Pick an N such that we can see that our x is not in JN1+ JN2 for strictly dimensional reasons. (For the case n = 0 we have to modify this a little. Put a weight on p to act as a non-trivial degree so that the previous statement still holds. Otherwise, we can just prove the result for n > 0 first and then lift it to n = 0 easily afterwards.) Pick N0 such that P (N0)* acts freely on P (n)*(X)=F s+1by Lemmas 4.1 and 4.5. Thus P (n)*(X)=F s+1injects to v-1qP (n)*(X)=F s+1for q N0. In the last group, by the previous lemma, we can write x in terms of t's modulo 48 JN1 + JN2. (We don't need to do this for every q N0, only for an infinite number of such q.) However, we may use negative powers of vq to do so. Since all sums are finite, we can multiply by some power of vq, say sq, so that vsqqx is in the image of the submodule generated by Tn modulo JN1+ JN2. This is true for all q N0. Thus we see that there are an infinite number of relations; one each with a term vsqqx in it, q > N0. Thus it is not coherent and we have our contradiction. x must therefore be in the submodule generated by Tn. 2 Proof of Corollary 1.21. This is immediate. 2 Proof of Corollary 1.22. If for some t 2 Tn, t goes to zero in K(q)*(X) for q N for some large N, then t is not essential to generate P (N)*(X) by Theorem 1.19. However, Theorem 1.18 says the reduction of Tn to Tq retains property (b) of Theorem 1.18. Contradiction. 2 Remark 6.4 For some of the most interesting examples which we "under- stand" completely, all of the generators reduce to mod p cohomology where they are still independent. This is the case for QS2k and Eilenberg-Mac Lane spaces and it probably contributes a great deal to our being able to under- stand them. This is not always the case though. When all generators are of this sort, then they never change filtration when we map from the spec- tral sequence for P (k)*(X) to that for P (n)*(X), n > k. The filtration can change when we map from P (k)*(X) to K(n)*(X) though. Generators that do not map to mod p cohomology must behave quite differently. They must change filtration when we map from the spectral sequence for P (k)*(X) to that for P (n)*(X) if n is large enough because the location in the spectral sequence, x 2 Es;t2, t < 0, is zero when t > -2(pn - 1). So, as n grows, the filtration of such a generator must keep changing and it never shows up in mod p cohomology. An example of this behavior was pointed out to us by Takuji Kashiwabara. The example is BSO(4) which was computed in [KY93 , Theorem 5.5]. 49 7 Exactness Once again we want properties of Morava K-theories to imply similar prop- erties for Brown-Peterson cohomology. We have four theorems to prove in this section: one for surjectivity, one for injectivity, and two for the exactn* *ess that we need in our applications. Although we state our theorems with the assumption of even Morava K-theory and injectivity or surjectivity for all of the Morava K-theories, we can get by with only assuming these things for an infinite number of the Morava K-theories. The proofs are unchanged. The statements of the theorems are much cleaner this way and there are no examples that need our greater generality. In this section we give the proofs for Theorems 1.15, 1.16, and 1.17 from the introduction. Proof of surjectivity in Theorem 1.15. By Theorem 1.18 we can pick a set T0 which generates P (0)*(X2). We know that it reduces to generators for each P (n)*(X2) and thus also for all K(n)*(X2). Map these generators to P (n)*(X1). By naturality and the fact that the Morava K-theories surject, we have that the image of T0 in the P (n)*(X1) satisfies the conditions of Theorem 1.19 and so we see that the image generates. 2 To prove the theorem on injectivity we need a lemma. Lemma 7.1 Let X1 and X2 be spaces with even Morava K-theory. Let f : X1 -! X2. If f* : K(n)*(X2) -! K(n)*(X1) is injective, then so is f* : E(k; n)*(X2) -! E(k; n)*(X1), 0 < k n. Proof. The proof is by downward induction on k. By Lemma 5.1 we have short exact sequences: vk * * 0 -! E(k; n)*(Xi) -! E(k; n) (Xi) -! E(k + 1; n) (Xi) -! 0: Given 0 6= x 2 E(k; n)*(X2), we know it cannot be infinitely divisible by vk by Corollary 4.11. Find a y 2 E(k; n)*(X2) and a j such that x = vjky and y 50 maps non-trivially to E(k + 1; n)*(X2). By our induction, E(k + 1; n)*(X2) injects to E(k+1; n)*(X1) so y must map non-trivially to E(k; n)*(X1). Since this group has no vk torsion, x = vjky must map non-trivially. 2 Proof of injectivity in Theorem 1.15. For k > 0 and x 2 P (k)*(X2) we use Proposition 4.12 to see that x maps non-trivially to some E(k; n)*(X2). By the injectivity of K(n)*(-) and Lemma 7.1 we have that this group injects to E(k; n)*(X1). By the naturality of maps between all of the cohomology theories involved, we must have x mapping non-trivially to P (k)*(X1). When k = 0 things are just a little bit more complicated. We do not have infinite p- divisibility in P (0)*(X2) so write 0 6= x 2 P (0)*(X2) as x = pjy where y maps non-trivially to z 2 P (1)*(X2). We just showed that the map on P (1)*(-) injected, so, by naturality, we see that y maps non-trivially to P (0)*(X1). Since P (0)*(X1) has no p torsion, x must map non-trivially. 2 f1 r Proof of Theorem 1.16. Take the cofibre X1 ! X2 ! C(f1). This gives rise to a long exact sequence in any cohomology theory. By Theorem 1.15 we have the surjectivity of f*1. By this surjectivity of f*1, we have a short exact sequence f*1 * r* * 0 G*(X1) G (X2) G (C(f1)) 0: for all G = P (k). Since f2 O f1 ' 0 , f2 factors through C(f1). By the assumption of exactness for all K(n) the map C(f1) X3 is surjective for all of the Morava K-theories. Thus, by Theorem 1.15 we have surjectivity for the P (k). We then just patch up our surjectivity with the short exact sequence to get the result. 2 Proof of Theorem 1.17. For bicommutative Hopf algebras we have that the cokernel can be constructed using the tensor product and the kernel from the cotensor product, see [HRW , Section 4], so that although this is a theorem about Hopf algebras, algebras play the main role here. We want to reduce 51 this theorem to Theorem 1.16. Define a map F by X2 ! (X2x X3)=(X2x *) by diag (I;f2) X2 -------! X2 x X2 -------! X2 x X3 ! (X2 x X3)=(X2 x *): We have F O f1 ' 0. Our exact sequence of Hopf algebras implies an exact sequence of K(n)* modules (this is from the cotensor product model for the kernel): K(n)*(X1) ! K(n)*(X2) -F*!K(n)*((X2 x X3)=(X2 x *)) which dualizes to (the tensor product model for cokernel): * * K(n)*(X1) K(n)*(X2) -F K(n) ((X2 x X3)=(X2 x *)): The lim1 condition of Theorem 1.16 is satisfied for the product by Landwe- ber, [Lan70 , Lemma 6], because X2 and X3 both satisfy the condition. By Theorem 1.16 we now have an exact sequence: * * P (n)*(X1) P (n)*(X2) -F P (n) ((X2 x X3)=(X2 x *)): Let I(-) be the augmentation ideal. Then I(P (n)*(X2)) ^P (n)*(X3) maps to the last module. We claim that this map is surjective. To see this, pick sequences of generators, {ti} and {si} for I(P (n)*(X2)) a* *nd P (n)*(X3) respectively. The elements {ti sj} map to generators of K(n)*((X2 x X3)=(X2 x *)) because K(n)*(-) has a K"unneth isomorphism (and our X and Y are very nice spaces of the sort we are studying). Mapping this elements over to P (n)*((X2xX3)=(X2x*)) we see that since they gener- ate all of the Morava K-theories then they must, by Theorem 1.19, generate. Since the tensor product maps onto generators, it must be surjective. The result follows. 2 52 8 Eilenberg-Mac Lane spaces In this section we give a purely algebraic construction for the P (0)* algebra which is isomorphic to the P (0) cohomology of an Eilenberg-Mac Lane space, and then, of course, we go on to show the isomorphism of Theorem 1.12. 8.1 Preliminaries From [RW77 ] we have a completely algebraic construction for the Hopf ring E*(BP__*) whenever E is a complex orientable generalized homology theory. Because the answer is a free E* module we have duality and have also given a construction for E*(BP__*). In particular, we can use K(n), BP , and BPp^ for E. The nice properties all come from the fact that H*(BP__*; Z(p)) has no torsion, [Wil73 ]. For the evenly indexed spaces this is even dimensional and is a bi-polynomial Hopf algebra (i.e. both it and its dual are polynomial algebras) and for the odd spaces it is an exterior algebra. The same is true for the cohomology. These properties lift to E*(BP__*), and, by duality, give completed exterior algebras (for odd spaces) and power series algebras (for evenly indexed spaces) for cohomology. We really need information about the BP_ _*and we can derive it from the above using: Theorem 8.1.1 ([Wil75 ]) Let g(q) = 2(pq+1 - 1)=(p - 1). For k g(q), the standard maps, BP__k! BP__kand BP__k! BP_ _k, split. For k < g(q) this splitting is as H-spaces. The second splitting splits the fibrati* *on: vq+1 BP__k+2(pq+1-1)-------! BP_ _k- ! BP_ _k to give a homotopy equivalence: BP__k' BP_ _kx BP__k+2(pq+1-1): It now follows that K(n)*(BP_ _2k) is a polynomial algebra for 2k < g(q) and is even dimensional for 2k = g(q). This is also true for BP*(-). 53 For our computations we need the bar spectral sequence. In our cases all of our maps are of infinite loop spaces and we only need it for Morava K-theory. Theorem 8.1.2 (Bar spectral sequence) Let F ! E ! B be a fibration of infinite loop spaces, then we have a spectral sequence of Hopf algebras, converging to K(n)*(B), with E2 term: Tor K(n)*(F)(K(n)*(E); K(n)*): Next we need to know how this behaves in a special case that has already been computed. The following was proved in [RW80 , Theorem 12.3, p. 743]. Theorem 8.1.3 For the path space fibration: K(Z(p); q + 1) ! P K(Z(p); q + 1) ! K(Z(p); q + 2) the bar spectral sequence is even dimensional and collapses. Looking at the statement of Theorem 1.12 we see that for each type of Eilenberg-Mac Lane space we really have two statements. For them to both be true we must have that the ideals (v*q) and (v*1; v*2; : :;:v*q) are equal. * *We will prove the theorem for the ideal (v*q). Since this is contained in the "bigger" ideal, it is enough to show that our map of P (0)*(K(Z(p); q + 2)) P (0)*(BP__g(q))=(v*q) factors through P (0)*(BP__g(q))=(v*1; v*2; : :;:v*q): The stable cofibration sequence 1.11 is one of BP module spectra, [JW73 ]. Thus all of the boundary maps used to define our map K(Z(p); q + 2) ! BP__g(q) 54 commute with multiplication by vj. Since the map vj j K(Z(p); q + 2) -! K(Z(p); q + 2 - 2(p - 1)) is homotopically trivial, we have what we need. We admit that the equality of the two ideals was quite a surprise to us which we did not notice until late in the game. 8.2 Construction In [RW77 ], a completely algebraic construction for the Hopf ring, E*(BP__*) is given. By the splitting above, we know that the algebraic construction of E*(BP__k) maps surjectively to E*(BP__k) for k g(q) and that it factors through the quotient given by setting all [vi] = 0 for i > q (see [RW77 ]). There is a minor concern that maybe there could be some other relation in order to get injectivity. However, this is not the case. Lemma 8.2.1 Let ER*(BP__*) be the algebraic construction for E*(BP__*) from [RW77 ]. If we mod out by I(q) = ([vq+1]; [vq+2]; : :):we have ER*(BP__k)=I(q)* * ' E*(BP__k) for 0 < k g(q). We should point out that neither the statement nor proof of Theorem 1.12 depends on this lemma. The theorem is given strictly in terms of spaces and we do need the splitting 8.1.1. The attraction of the theorem to us is this lemma because it gives us a purely algebraic construction for everything in the theorem. [vq] Proof. The map E*(BP__i) -! E*(BP__i-2(pq-1)) is just the induced algebraic map coming from multiplication by vq. Each of the spectra BPis a BP module spectra ([JW73 ]) so the maps between spectra commute with the maps of vq. We prove our lemma with a multiple induction. It is enough to prove our lemma for mod p homology because all of our spaces are torsion free and everything is therefore E* free. Our main induction is on j - k 55 in Hj(BP_~~_k). Our second induction is downward induction on s. To ground our first induction, there is nothing to prove if j = k (and k g(s)). To ground our second induction, we see that Hj(BP__i) ' Hj(BP_~~~~_i) for j - i < 2(ps+1 - 1) because these spaces are homotopy equivalent in this range. For a fixed j - k we must pick s such that j - k < 2(ps+1 - 1). Then we can start the second induction to prove our result for this degree. To do the second induction we need only observe that the split fibration in 8.1.1 must give rise to a short exact sequence of Hopf algebras where the first map is just [vq+1]O multiplication. 2 One can go further with this and write E*(BP_~~_2k) as a power series ring on generators dual to the primitives E*(BP__2k) for 2k g(q), which can be written down directly from [RW77 ] as was done in [Sin76]. We will discuss this more after the proof. In principle, [RW77 ] tells you how to compute the map [vq]O BP*(BP__g(q)) -------! BP*(BP__g(q)-2(pq-1)): What that really means is that you can train a computer to do it, because, in practice, it is very difficult although complete information is available. Because everything is BP* free you can take the duals and the dual map and again, everything is, in principle, computable. It is certain that BP *(BP__g(q))=(v*q) is a well-defined algebraic construct, as is its p-adic completion. Likewise for * * BP *(BP__g(q))=(pi ; vq) which comes from the product map (pi; vq) BP__g(q)! BP__g(q)x BP__g(q)-2(pq-1): 56 8.3 Proof for K(Z(p); q + 2). We can now prove Theorem 1.12 for the integral spaces. It is known that BP *(K(Z(p); q)), q > 2, has phantom maps. This follows from [AH68 ] where they show this for complex K-theory and [Lan72 ] where it is shown that the situation for complex cobordism is the same as that for complex K-theory. We must work, therefore, with P (0) = BPp^. In fact, because of Theorem 1.17, all we must prove is Proposition 1.14. We just let K(Z(p); q + 2) = X1, BP__g(q)= X2, f1 the iterated boundary map given in the introduction, BP__g(q)-2(pq-1)= X3 and f2 the map coming from vq. Observe that the composition of the two maps is indeed null homotopic because the first map factors through the boundary map BP__g(q-1)+1-! BP_ _g(q) which is just the inclusion of the fibre of the map f2. Note that this is an example where the spaces do not form a fibration. The proof breaks up into two pieces. First, we need to show injectivity of the map K(n)*(K(Z(p); q + 2)) -! K(n)*(BP__g(q)): which we do using the Steenrod algebra and a bit of Hopf algebra machinery. Second we need to show that the coker of this map injects to the third Hopf algebra: K(n)*(BP__g(q))==K(n)*(K(Z(p); q + 2)) -! K(n)*(BP__g(q)-2(pq-1)): We will do this using the bar spectral sequence a few times. 8.3.1 Proof of injectivity. This is going to reduce to a calculation over the Steenrod algebra. With apologies to the reader, understanding this proof will also require an intimacy 57 with the Morava K-theory of Eilenberg-Mac Lane spaces from [RW80 ]. From that paper we know [RW80 , Corollary 12.2, p. 742] that lim-!K(n)*(K(Z=(pi); q + 1)) ' K(n)*(K(Z(p); q + 2)): i Furthermore, we know ([RW80 , Theorem 11.1(b), p. 734]) that the very first space in this limit, K(Z=(p); q +1) picks up all of the Hopf algebra primitives for K(n)*(K(Z(p); q + 2)). To get our injection for this last space we want to just show that the primitives P K(n)*(K(Z=(p); q + 1)); and thus also for K(n)*(K(Z(p); q +2)), inject to those for K(n)*(BP__g(q)). An injection on primitives automatically gives an injection on K(n)*(K(Z(p); q + 2)), see, for example, [HRW , Lemma 4.2]. This calculation is probably contained in H. Tamanoi's Master's Thesis, [Tam83b ], and should have been deduced by us from [Yag86 ]. It is certainly contained in [Tam95 ]. Those proofs are in cohomology and we work in ho- mology but the results are the same. H. Tamanoi computes the image of the map: BP *(K(Z(p); q + 2)) ! H*(K(Z(p); q + 2); Z=(p)): In the proof one sees that these elements all come from BP *(BP__g(q)). As it turns out, these elements generate. At any rate, it is Tamanoi who first made the connection between BP *(K(Z(p); q +2)) and BP *(BP__g(q)) more than ten years ago! We now assume a working knowledge of [RW80 ]. Let A* be the dual of the Steenrod algebra. We have the usual map from H*(K(Z=(p); q + 1); Z=(p)) to A* which is an A* comodule map. K(n)*(K(Z=(p); 1)) has elements a(i), 0 i < n, in degrees 2pi which are represented in the Atiyah-Hirzebruch spectral sequence by the elements used to define the oiof the Steenrod algebra ([RW80 , Theorem 5.7]). Under the usual map q+1O K(n)*(K(Z=(p); 1)) -! K(n)*(K(Z=(p); q + 1)) 58 all elements a(i0)O a(i1)O . .O.a(iq) with 0 i0 < i1 < i2 . .<.iq < n are nontrivial ([RW80 , Theorem 9.2]). They are therefore represented by elements which map to oi0oi1oi2. .o.iq in the Steenrod algebra. The elements which are primitive are those with i0 = 0, ([RW80 , Theorem 9.2]). It is not important, but note that there are only a finite number of these elements. Define a subvector space, E(q; n), of A* with basis oi1oi2. .o.ij with 0 i1 < i2 . .<.ij < n with j q in A*. This is clearly a subcomodule of A* over A* and we can take its quotient, A*=E(q; n) which is now a comod- ule over A*. Note that the above set of elements of H*(K(Z=(p); q + 1); Z=(p)) which survive to primitives in the Atiyah-Hirzebruch spectral sequence maps injectively to a subcomodule (over A*) of A*=E(q; n); call it E(q). We have our map, K(Z=(p); q + 1) ! BP__g(q)which induces a map of A* comod- ules in mod p homology. All of the elements in H*(BP__g(q); Z=(p)) sur- vive in the Atiyah-Hirzebruch spectral sequence to the Morava K-theory because the space has no torsion. Thus, it is enough to show that our el- ements which represent primitives map nontrivially and independently to H*(BP__g(q); Z=(p)). In cohomology, the iterated boundary map, K(Z=(p); q + 1) ! BP__g(q); takes the fundamental class in H*(BP__g(q); Z=(p)) to Q0Q1 . .Q.qtimes the fundamental class in H*(K(Z=(p); q + 1); Z=(p)), see [Wil75 ]. This tells us two things we need to know. First, it says our map is trivial on E(q; n) because Qi is dual to oi and there are q or fewer o but q + 1 Q. So, we get a map of A* comodules, E(q) ! H*(BP__g(q); Z=(p)). Second, it says our map is non-zero on the lowest dimensional element o0o1 . .o.q: 59 All we have to do now is show that this element forces an injection of E(q). Recall that the coproduct on oi is X pj oi 1 + i-j oj: 0ji We can ignore the first term in computing the comodule expansion on oI because it will lead to a product of the o on the right in E(q; n). Because oI always has o0 in it (recall these are the primitives), there is only one term we can use from its coproduct, 1 o0. Recall also that o2j= 0, so we cannot use the o0 term of any of the other o's. Thus, modulo E(q; n) we have X pJ (oI) = I-J o0oJ; 0_g(q-1)) ' K(n)*(K(Z(p); q + 1)) P Aq-1 where P Aq-1 is a polynomial algebra. This induction is trivial to ground; just use K(n)*(BP_<0>_2) = K(n)*(K(Z(p); 2)) which we know because we know K(n)*(CP 1). Here the polynomial part is vacuous. The first step in our induction is to compute K(n)*(BP_ _g(q-1)+1). To do this we use the bar spectral sequence 8.1.2 with E2 term: K(n)*(BP _ ) Tor g(q-1)(K(n)*; K(n)*): By induction and the K"unneth isomorphism, we see that this breaks into two parts: TorK(n)*(K(Z(p);q+1))(K(n)*; K(n)*) TorPAq-1(K(n)*; K(n)*) where TorPAq-1(K(n)*; K(n)*) ' EAq-1 where EAq-1 is an exterior algebra on the homology suspension of the gen- erators of the polynomial algebra P Aq-1. These generators lie in the first filtration in the spectral sequence so all differentials on them are trivial. We also know how to compute TorK(n)*(K(Z(p);q+1))(K(n)*; K(n)*) 61 from Theorem 8.1.3. Furthermore, we have maps of fibrations: K(Z(p); q + 1)________wpt: ________wK(Z(p); q + 2) ||||||||| | | ||||||||||||||||| || | ||||||||||||| | | | |||||||||||||||| | | |||||||||||||||| | |u ||||||||||||| | |u BP_ _g(q-1)______wpt: ______BP_ _g(q-1)+1:w By naturality, we have no differentials on this part of the bar spectral sequence we are using to compute K(n)*(BP_ _g(q-1)+1). Since there can be no differentials on the exterior part, we see that the spectral sequence collapses and, as algebras, we have: K(n)*(BP_ _g(q-1)+1) ' K(n)*(K(Z(p); q + 2)) EAq-1: All of the algebra extension problems in the K(n)*(K(Z(p); q+2)) part having been solved by naturality. In case there is any question about this algebra splitting as a tensor product recall that EAq-1 is a free commutative algebra (if our prime is odd) on odd degree elements. We certainly have a short exact sequence with EAq-1 the quotient. Because it is free we can split it. If p = 2 we must observe that the generators of P Aq-1 come from BP*(BP__g(q-1)) and thus, so do the generators of EAq-1 come from BP*(BP__g(q-1)+1). Since they are exterior generators in BP there can be no extension problems where we are working. That ends the proof of the first step of the induction and we can move on the next (and final) step. We will study the bar spectral sequence for the fibration: BP_ _g(q-1)+1! BP_ _g(q)! BP__g(q)-2(pq-1): We know quite a lot about things already. (i)We know that K(n)*(BP__g(q)-2(pq-1)) is a polynomial algebra. 62 (ii)We know that K(n)*(BP__g(q)) is even dimensional. (iii)We have just "computed" K(n)*(BP__g(q-1)+1): (iv) We know that we have the injection part of our desired exact sequence K(n)* ! K(n)*(K(Z(p); q + 2)) ! K(n)*(BP_ _g(q)): Because of (ii) we see that the map K(n)*(BP__g(q-1)+1) ! K(n)*(BP_ _g(q)) must take EAq-1 to zero. All of this allows us to simplify our computation of the E2 term of the bar spectral sequence converging to K(n)*(BP__g(q)-2(pq-1)): The E2 term starts out as K(n)*(BP_ ) Tor g(q-1)+1(K(n)*(BP_ _g(q)); K(n)*) and simplifies, by [Smi70 , Theorem 2.4, p. 67], to K(n)*(BP__g(q))==K(n)*(K(Z(p); q + 2)) TorEAq-1(K(n)*; K(n)*) where the Tor is just a divided power Hopf algebra. In particular, it is even dimensional, as is the first part; thus this spectral sequence collapses. We can now just read off our answers. The quotient Hopf algebra is just the coker which we wanted to inject into K(n)*(BP__g(q)-2(pq-1)) and the map is just the edge homomorphism. This gives us the desired injection. However, to complete our induction we must show that this cokernel is poly- nomial. It is a sub-Hopf algebra of a polynomial Hopf algebra and so it must be polynomial as well. Now we have a short exact sequence of Hopf algebras K(n)*(K(Z(p); q + 2)) ! K(n)*(BP__g(q)) ! P Aq: Because P Aq is free we see that this splits as algebras and we have completed our induction. 63 8.4 Proof for K(Z=(pi); q + 1). This proof is only a slight modification of the previous proof. Our sequence of spaces is now: (pi;vq) K(Z=(pi); q + 1) -! BP__g(q)-------! BP__g(q)x BP__g(q)-2(pq-1) so we need an exact sequence of Hopf algebras: K(n)* | |u K(n)*(K(Z=(pi); q + 1)) | |u K(n)*(BP__g(q)) (pi;[v|q]) * |u K(n)*(BP__g(q)) K(n)*(BP__g(q)-2(pq-1)): The last tensor product is just the product in the category of Hopf algebras and we have already computed the kernel of the map [vq] to the right side. It was just K(n)*(K(Z(p); q + 2)). All we have to do now is worry about the kernel of the map pi*restricted to this part. From [RW80 , Corollary 13.1, p. 745], we have an extension of Hopf algebras which solves that problem: K(n)* | |u K(n)*(K(Z=(pi); q + 1)) | |u K(n)*(K(Z(p); q + 2)) pi| *|u K(n)*(K(Z(p); q + 2)); and we are almost done with this case. We want to use BP as opposed to BPp^in this case. Our assumptions in Theorem 1.17 require us to have 64 lim1BP *(Xm ) = 0, which we have by Remark 5.7. Because the other spaces have no torsion we know that the Atiyah-Hirzebruch spectral sequence col- lapses and the lim1 for them is zero. 8.5 Generators and relations If one really wants to use the construction given above to describe BPp^*(Z(p); q + 2) there are some simplifications which an intimacy with [RW77 ] can give you quite quickly but which the reader has been spared the necessity of knowing so far. We will just briefly describe here what can be done. Both BP *(BP__g(q)) and BP *(BP__g(g)-2(pq-1)) are power series rings on generators dual to the primitives in the BP homology. In [RW77 ], a basis for the primitives is written down explicitly and one can see that most of them are mapped to basis elements for primitives in the second space. The consequence in the dual is that we do not need to worry about those primitives at all. The remaining primitives for BP*(BP__g(q)) are identified in [RW77 ] as 2 Opq b(0)O bOp(j1)O bOp(j2)O . .O.b(jq) where 0 j1 . . . jq. This element is in degree 2(1 + pj1+1+ pj2+2+ . .+.pjq+q). Note that as q goes up, the degree of the generators goes up much faster. (Recall that the Brown-Peterson cohomology of the Eilenberg- Mac Lane spectra is trivial.) When reduced to K(n)*(BP__g(q)) we can see that each of these elements is in the image of K(n)*(K(Z(p); q + 2)) for some n large enough. If we take a bunch of dual generators, say cJ, we can see that BPp^*(K(Z(p); q + 2)) is a quotient of the power series algebra on the cJ. To see what the relations would be requires a good deal more work. For a slight check on reality, there is only one case here that is degenerate enough to be familiar. Let q = 0 and we are talking about BPp^*(K(Z(p); 2)) and there is only one generator in degree 2. It may or may not be an interesting 65 exercise to try to say more about the relations. These generators are those found by Tamanoi. He just did not know that he had found them all. Because of the splitting of Theorem 8.1.1 the map K(Z=(pi); q + 1) -------! BP__g(q) is really a map to BP__g(q). Tamanoi ([Tam95 ]) calls this the fundamental cla* *ss and from the above it is easy to see that using stable BP operations, the algebra structure, and topological completion, this class generates everything in BP *(KZ=(pi); q + 1). References [Ada69] J. F. Adams. Lectures on generalized cohomology, volume 99 of Lecture Notes in Mathematics, pages 1-138. Springer-Verlag, 1969. [Ada74] J. F. Adams. Stable Homotopy and Generalised Homology. University of Chicago Press, Chicago, 1974. [AH68] D. W. Anderson and L. Hodkin. The K-theory of Eilenberg-Mac Lane complexes. Topology, 7(3):317-330, 1968. [Baa73] N. A. Baas. On bordism theory of manifolds with singularities. Math. Scand., 33:279-302, 1973. [BJW95] J. M. Boardman, D. C. Johnson, and W. S. Wilson. 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