BP infinite loop algebras Takuji Kashiwabara Institut Fourier, Universit'e de Grenoble I UMR 5582, CNRS BP74, 38402, St. Martin d'H`eres, France _______________________________________________________________________________* *__ Abstract In this paper we define the notion of BP infinite loop algebras, a sort of BP -* *analogue of A- R-allowable Hopf algebras, and show that under some conditions, BP -cohomology * *of infinite loop spaces has such a structure. Furthermore we show that the BP infinite loop* * algebra structure gives a serious restriction on underlying unstable BP -algebra struct* *ure. Key words: Brown-Peterson cohomology, Infinite loop spaces, Generalized cohomo* *logy operations, Unstable Algebras, Dyer-Lashof operations, Nishida relations, Landweber-Novikov algebra, Steenrod algebra, Hopf ring 2000 MSC: Primary 55N22, 55P47, 55S10, 55S12, 55S25, Secondary 55R40, 55R45 _______________________________________________________________________________* *__ 1. Introduction It is well-known that the mod p ordinary homology of an infinite loop space * *has the structure of a so called A-R-allowable Hopf algebra [13], i.e., an Hopf algebra* * on which both the Steenrod algebra and the Dyer-Lashof algebra act satisfying certain co* *mpatibility conditions. What about generalized (co)homology? In the case of the complex K* *-theory, the question was more or less solved by McClure in [14]. The rather complicate* *d answer, however, is due to the presence of torsion elements, and the situation appears * *to simplify when dealing with spaces whose integral (p-complete) K-theory has no torsion el* *ements (c. f. [5, 8]. See also [7] on the relationship between the results of [5] and [14].) * *In an unpublished work, N. Strickland deals with En-homology of QX when En-homology of X is free.* * What about connective theories? In [10] the author computed BP -cohomology of QX wh* *en X satisfies certain conditions. In this paper we use this computation to give a d* *efinition of a sort of BP - analogue of the category of A-R allowable Hopf algebras, and we wi* *ll discuss some properties. Another purpose of this paper is to exhibit that the unstable BP -cohomology* * operations are rather accessible, contrary to a common belief. For this purpose we rely ra* *ther heavily on known results on Steenrod operations and use the relationship between ordinary * *cohomology and BP -cohomology. ________________________________ Email address: Takuji.Kashiwabara@fourier.ujf-grenoble.fr (Takuji Kashiwabar* *a) Preprint submitted to Elsevier Novem* *ber 26, 2009 The paper is organised as follows. In section 2, we define the category of * *BP infinite loop algebras and show that the BP -cohomology of an infinite loop space become* *s naturally a BP infinite loop algebra under some conditions. In section 3 we describe expl* *icitly BP infinite loop algebra structure of BP -cohomology of some infinite loop spaces.* * In section 4, we show that the structure of BP infinite loop algebra imposes a serious restri* *ction on the structure of the underlying unstable BP -algebra, and show that there is no non* *trivial BP infinite loop algebra with a spherical class in dimension greater than 2 and th* *at is finitely generated as BP *-module. In appendix A we deal with some issues concerning the* * topology of BP -cohomology of spaces, and in appendix B we recall some standard facts on* * BP*(BP__*) for readers who are not familiar with. The following convention will be used throughout the paper. BP will denote * *the p- completed version of the Brown-Peterson spectrum unless otherwise specified, wh* *ere p will be a fixed odd prime. For spaces or spectra X, BP *(X) will be equipped with t* *he BP - skeletal topology as will be defined in appendix A unless otherwise specified. * *However, the use of BP -skeletal topology instead of the usual skeletal topology is not cruc* *ial. In fact it is only used in Theorem 2.11 ii), and in case of interests the two topologies agre* *e anyway. We denote H*(-) and H*(-) the mod p ordinary cohomology and homology. 2. The definition of BP infinite loop algebras 2.1. BP *(BP )-modules, unstable BP - algebras, and the destabilization Let X be a spectrum, E a generalized cohomology theory with product. It is w* *ell-known (c. f. [1] ) that E*(X) is a module over the ring of stable cohomology operat* *ions E*E. Less well-known is the fact that, if X is a space, under certain conditions on * *E, E*(X) has a richer set of structures called unstable E-algebra (c. f. [3]). BP is known t* *o verify the required condition (loc. cit. ) by results in [25, 19]. In view of recent works* * by Andrews and Whitehouse [24], it is not clear if the definition given there is the best one.* * In any event, we need following properties of the category of unstable BP -algebras (which we* * will denote KBP ) : i Let X be a space. Then BP *(X) is naturally an unstable BP -algebra. ii The following diagram of functors is commutative P ointed spaces_____Spectra- 1 | | |BP*(-) |BP*(-) | | |? |? KBP __________MBP*BP-I where I denotes the augmentation ideal functor, and MBP*BP the category of* * BP *BP - modules. 2 iiiThere is a natural isomorphism Hom KBP(BP *(BP__n), A) ~=An for unstable BP -algebras A. iv In KBP the categorical sum is represented by a completed tensor product ove* *r BP *. v Cokernels exist in KBP . All of the properties follow easily from the definition of unstable BP -algebra* *s in [3]. These properties allow us to define Definition 2.1. The destabilization functor MBP*BP ! KBP is the left adjoint to* * the aug- mentation ideal functor I : KBP ! MBP*BP . (here we use the augmentation ideal and not the forgetful functor because we* * are using the unreduced theory for spaces). Note that such a functor transforms the direct sum into the tensor product, is * *right exact, and sends nBP *(BP ) to BP *(BP__n). These properties characterize the destabiliza* *tion functor. As we know completely algebraically BP *(BP__n)'s and the induced map among the* *m, our functor is determined in a completely algebraic way (including the topology/fil* *tration). 2.2. BP infinite loop algebras In this section we define the category of BP infinite loop algebras, which c* *an be considered as a some sort of BP counterpart of the category of A-R-allowable Hopf algebras* *, and show that indeed when X is a nice infinite loop space, its BP -cohomology has a stru* *cture of a BP infinite loop algebra. First we recall from [10] the following Theorem 2.2. Let X be a space satisfying the following conditions. (H1) BP *(X) ^BP*Z=p H*(X). (H2) BP *(X) is Landweber-flat, namely the sequence (p, v1, v2, . .).is regula* *r on BP *(X). * * Then the natural map DB"P (X) ! BP (QX) is an isomorphism. Now suppose X itself is an infinite loop space, i.e., X = 1 Y for a spectrum * *Y . Then by adjunction we get a map of infinite loop spaces QX ! X. Thus we get a map of unstable BP -algebras BP *(X) ! BP *(QX). One can see that this map satisfies * *several compatibility conditions coming from the formal properties of adjunction, notab* *ly it is a section of the obvious map BP *(QX) ! BP *(X). This motivates the following def* *inition : 3 Definition 2.3. An BP infinite loop algebrais a coalgebra over the comonad ([12* *]) associ- ated to the adjunction pair (D, I), that is, an unstable BP -algebra A equipped* * with a map of unstable BP -algebras ,A : A ! DI(A) making the following diagrams commutative A _________DI(A)-,A A | | Q | | | Q ,A| |D(OEIA) ,A| Q | | | iQd |? |? |? Q QQs DI(A) ______DIDI(A)-DI(, DI(A) _________-A A) 'A where OE : id ! ID and ' : DI ! id are the adjunction maps. Often by abuse of l* *anguage we say simply that A is a BP infinite loop algebra. Thus the above discussion leads to : Theorem 2.4. Let X be an infinite loop space satisfying the properties (H1) a* *nd (H2). Then BP *(X) is a BP infinite loop algebra. Before we study the structure of BP infinite loop algebras, we need to know * *more on the structure of unstable BP -algebras obtained by the destabilization. The fir* *st results are more or less formal : Proposition 2.5. Let M be a BP *(BP )-module. Then (i)D(M) is naturally a completed Hopf algebra. (ii)There is a composition copairing D(M) ! D(M) ^BP*D(BP *(S0)) which agrees with the map induced by the composition pairing map Proof. The diagonal map M ! M M induces a map D(M) ! D(M) ^BP*D(M) which makes D(M) a completed coalgebra. One easily sees that the coproduct is compati* *ble with product. This proves i). The assertion ii) is a special case of the following : Lemma 2.6. Let L, M, and N be BP *(BP )-modules with a map L ! M ^BP*N. Then we have a natural map D(L) ! D(M) ^BP*D(N). Proof. By adjunction we have natural maps M ! ID(M) and N ! ID(N), which give rise to a map M ^BP*N ! ID(M) ^BP*ID(N) ~=I(D(M) ^BP*D(N)). By composing with the map L ! M ^BP*N we get a map L ! I(D(M) ^BP*D(N)). By adjunction we get the desired map. * * 2 We can also relate them to ordinary homology, we start with : Proposition 2.7. Let A be an object in KBP . Then A ^BP*Z=p has a natural struc* *ture of unstable algebra over Ap, the mod p Steenrod algebra with trivial action of the* * Bockstein. 4 Proof. Let P ibe the Steenrod power operation. It is well-known that there is* * a stable BP -operation i which covers P i([29]). Let M be an object in MBP*BP . Since t* *he ideal I(1) = (p, v1, . .,.vn) is invariant under the action of BP *BP ([11]) one sees* * that_one_can let the subalgebra of Ap generated by P i's on M ^BP*Z=p via the formula P i(__* *x) = i(x). Now let A be an unstable BP -algebra. We have already seen that Ap acts on A ^B* *P*Z=p (with trivial action of Bockstein). Let x be an element of degree k in A ^BP*Z=* *p. We will show that if k < 2i then P i(x) = 0. Consider i('k) where 'k 2 BP k(BP__k)_is* * the class corresponding to the identity map. As P i(__'k) = 0 in HZ=p*(BP__k) BP *(BP__* *k) ^BP*Z=p, and since BP *(BP__k) is generated by 'k as an unstable BP-algebra, we see that* * there is a series of element ij2 BP *(BP__k) such that i('k) = p i0('k) + . .+.vn in('k) . ... By the universality of 'k we get i(x) = p i0(x) + . .+.vn in(x) . . . for any x in degree k part of an unstable BP-algebra. Therefore we have _____ P i(__x) = i(x)= 0. Other conditions can be verified in a similar way. * * 2 As a corollary, we can recover the following result originally due to Quille* *n ([18]). Corollary 2.8. Let A be an unstable BP -algebra. Then A ^BP*Z=p is trivial in * *negative degrees. Proof. It suffices to note that id = P 0= 0 in negative degrees for an unstable* * Ap-algebra. 2 Now we can prove Proposition 2.9. Let M be a BP *(BP )-module. Then D(M)# ~=Hom Z=p(D(M) ^BP*Z=p* *, Z=p) has a natural structure of A-R-allowable Hopf algebra. Proof. All of the structure has been already shown to exist except the action o* *f the Dyer- Lashof algebra ([13]). To see this, let M P0 P1 be a BP *(BP )-module prese* *ntation of M. Thus we have a coexact sequence of algebras (which also is an exact sequ* *ence of completed Hopf algebras in view of results above) BP * D(M) D(P0) D(P1). A* *s the tensor product is right exact we see that D(M) ^BP*Z=p is the cokernel (as alge* *bras/Hopf algebras) of the map D(P0) ^BP*Z=p D(P1) ^BP*Z=p. By dualizing we see that D(* *M)# is the Hopf kernel of the map D(P0)# ! D(P1)# . Since D(Pi)# 's are just mod p hom* *ology of the infinite loop spaces associated to a free BP -module spectra, and the maps * *between them are induced by a spectra map, we see that the kernel has the strucutre of an A-* *R-allowable Hopf algebra. * * 2 Remark 2.10. Note that one can identify N# with the set of continuous BP *-li* *near maps from N to Z=p. 5 With these preparations we are now ready to study properties of BP infinite * *loop alge- bras. As a matter of fact we get all the properties of D(M)'s that we have seen* *, namely Theorem 2.11. Let A be an BP infinite loop algebra. (i)A is a completed Hopf algebra, and the structure map ,A is a map of comple* *ted Hopf algebras. (ii)The category of BP infinite loop algebras is abelian. (iii)There is a natural "composition copairing" A ! A ^BP*D(BP *(S0)) and ,A commutes with the copairing. (iv)A ^BP*Z=p is an A-R-allowable Hopf algebra. Proof. Basic idea is to use the fact that these properties hold for D O I(A), a* *nd the fact that A embeds naturally into D O I(A). One can use the composition A ! D O I(A) ! D O I(A) ^BP*D O I(A) ! A ^BP*A to define the coproduct. Unfortunately with such a definition of the coproduct* *, it is not obvious that the structure map commutes with the coproduct. To see this, first * *consider the case when A = D(M). We show that the coproduct defined earlier agrees with the * *new one, which proves the compatibility of the coproduct with the structure map in the g* *eneral case by naturality. To prove that the two coproducts agree, we need to show the comm* *utativity of the following square. D(M) _______________-DID(M) _________D(ID(M)- ID(M)) ___DID(M)-^BP*D* *ID(M) | ,, , ,,: | | ,, , | | ,, , | | , , , | | , , , | |? , , , |? D(M M) _________________________________________________________D(M)-^BP*D* *(M) The top left triangle commutes by naturality and the bottom pentagon commutes b* *y gener- alities of adjunctions. The part ii) follows immediately in view of the appendi* *x A. Note that without using the properties of the BP -skeletal topology, we still get all of * *the properties of the abelian category except the equality between the image and the coimage. * *The part iii) can be proved by a method similar to the part i). To prove the part iv), * *notice that the functor ^ BP*Z=p transforms our composition copairing into the composition * *pairing H*(QS0) A ^BP*Z=p ! A ^BP*Z=p. As one can recover the action of the Dyer-Las* *hof algebra from the composition pairing and the action of the Steenrod algebra, we* * can make the Dyer-Lashof algebra act on A ^BP*Z=p. All the compatibility conditions are* * satisfied because they are satisfied on DI(A) ^BP*Z=p which surjects to A ^BP*Z=p. * * 2 6 3. Examples of BP infinite loop algebras 3.1. The adjunction maps In this section we will exhibit explicitly the structure of BP infinite loop* * algebra for some known BP -cohomology of infinite loop spaces. We start with the simplest c* *ase when the structure maps can be obtained from the adjunction maps, especially for BP * **(BP__n) ~= D( nBP *(BP )) or BP *(QX) ~=DI(BP *(X)) with BP *(X) satisfying the conditions* * (H1) and (H2) of Theorem 2.2. So we explain how to describe algebraically the adjunc* *tion map. Let's first consider the adjunction idMBP*BP ! I O D. By the construction of D,* * the general cases can be reduced to the case when M is free and monogenic, i.e., M is of th* *e form M = nBP *(BP ), in which case the adjunction map nBP *(BP ) ~=BP *( nBP ) ! BP *(* *BP__n) is obtained by considering a stable operation as an unstable operation, in othe* *r words by evaluating a stable operation on the class 'n 2 BP *(BP__n). Next let's consider the adjunction D O I ! idKBP. For the ease of the descri* *ption we only deal with the case of BP -cohomology of a space, the general case being left to* * the reader. Consider a free BP *BP -resolution of IBP *(X) realized by maps X ! P0 ! P1 * *! . . . where Pi's are wedge of suspensions of BP . Thus BP *(X) is a quotient of BP ** *(P0) and DI(BP *(X)) that of D(BP *(P0)) ~=BP *( 1 P0). As we are supposed to know BP *(* *X) as an object of KBP , we know how BP -unstable operations act on it, and thus we c* *an extend the map BP *(P0) ! IBP *(X) to D(BP *(P0)) ! BP *(X). One verifies easily that* * by passing to the quotient, we get the desired map DI(BP *(X)) ! BP *(X). 3.2. Wilson spaces Let's examine now the case of so-called Wilson spaces, namely the spaces of * *the form BP__jwith j 2(1 + p + . .p.i), which splits off BP__j([26, 4, 3]), that is* * we have a map of spaces ` : .BP__j! BP__jsuch that the composition with the usual map aei_j: * *BP__j! BP__j is homotopic to the identity. We shall prove Proposition 3.1. Let j 2(1 + p + . .p.i). Then the structure map ,BP*(BP_j): BP *(BP__j) ! DI(BP *(BP__j)) is given by the composition BP*(`) * ,BP*(BP_j) * DI(BP*(aei_j) * BP *(BP__j) ! BP (BP__j) ! DI(BP (BP__j)) ! DI(BP (BP__* *j)). Proof. It suffices to take the adjoint of the commutative diagram BP *(BP__j)________________oBPe*(BP__j)id HHY H H ss BP *(BP__j) Note that each of theses arrows can be described completely algebraically (c. f* *. [3, 4] for BP *(`)), which means that we have a completely algebraic description of ,BP*(B* *P_j). 7 3.3. Eilenberg-Maclane spaces The Brown-Peterson cohomology of the Eilenberg-Maclane spaces were calculate* *d in [20]. For concreteness' sake, we will deal with the case of K(Z=p, n)'s, but the othe* *r cases are similar. According to [20], there are maps of infinite loop spaces K(Z=p, n) ! K(Z, n + 1) ! BP_<1>_n+2p! . .!.BP__2(1+p+...+pn-1) which induces the surjection BP *(BP__2(1+p+...+pn-1)) ! BP *(K(Z=p, n))* *. As this is a map of BP infinite loop algebra, we deduce Proposition 3.2. ,BP*(K(Z=p,n)is determined by the commutative diagram BP *(BP__2(1+p+...+pn-1))_____-BP *(K(Z=p, n)) | || | | | | | | ,BP*(BP_2(1+p+.|..+pn-1)) ,BP*(K(Z=p|,n)) | | | | | | |? |? DI(BP *(BP__2(1+p+...+pn-1)))__DI(BP-*(K(Z=p, n))) 4. Finitely generated BP infinite loop algebras In this section we will see how the structure of BP infinite loop algebra gi* *ves restrictions on the structure of the underlying unstable BP -algebra structure. Let's first* * see what happens if X is an infinite loop space with torsion-free cohomology. (It happen* *s that by a work of Slack [21] we know all of them, but let's forget if for the time being.* *) Suppose also X is n - 1 connected, but not n-connected. Thus we have the following commutative* * diagram. Sn ________QSn- | | | | | | | | | | | | |? |? X ________-QX ________-X ______K(Z=p,-n) | | ` | | | | |? BP__n Here the composition of maps all the way from Sn to K(Z=p, n) gives a genera* *tor of Z=p. Such a space X satisfies clearly the conditions (H1) and (H2), thus by taking B* *P -cohomology 8 we get the following diagram BP *(Sn) ______DI(BPo*(Sn))e |6 |6 | | | | | | | | | | BP *(X) ______oDI(BPe*(X)) ______oBPe*(X), BP*(X) where the composition from BP *(X) to BP *(Sn) is surjective. This motivates th* *e following definition. Definition 4.1. Let A be an unstable BP -algebra. We say that A has a spherical* * class in degree n if There is a surjective map of unstable BP -algebras from A to BP *(S* *n). Remark 4.2. By the above, we see that if X is an infinite loop space with tor* *sion-free cohomology then BP *(X) has a spherical bottom in degree n. Now we are ready to state our main result. Theorem 4.3. Let X be an unstable BP -algebra with a spherical class in dimen* *sion n. If n 3 then A ^BP*Z=p is not bounded above. In particular A is not finitely gen* *erated as BP *-module. Proof. The key point is the following Proposition 4.4. Let A be as above. Then 9a 2 HomZ=p(A ^BP*Z=p, Z=p)n such that ae Q1(a) 6= 0if n is odd Q2(a) 6= 0if n is even Granted Proposition, we can conclude the proof of Theorem 4.3 using the followi* *ng Lemma 4.5. Let M be an A-R-allowable module ([13], Definition 2.8), and x 2 M* * be an 0 j element such that P i(x) = 0 for 8i > 0, 9j > 0 such that Q (x) 6= 0. Then we * *have 0 i* * * * Qpj(x) 6= 0. We use the notation P irather than the more usual P to keep the * *notation ` for the maps induced by `. 0pj Proof. By Nishida relation ([13, 16]), we have P (p-1)jQ (x) = Qj(x). * * 2 Now we go back to the proof of Proposition. First we need to know the action* * of Q1 and Q2 on the bottom class of H*(BP__n). Denote 'n the element of Hn(BP__n) which i* *s the image of the unit map in ssn(BP__n) by the Hurewicz map. Lemma 4.6. (i)Let n 0. In H*(BP__2n+1) we have Q1('2n+1) = -e1 O [v1] O bOp-11O bOnp. 9 (ii)Let n 1. In H*(BP__2n) we have Q2('2n) = -[v1] O bOp-11O bOnp. Proof. Basically in [19], Theorem 6.1, the action of Qi's on 'n was determined* *, and their result was extended by Turner ([23]) for the action of Qi's on any element of H* **(BP__*). However here we present a simpler proof in the spirit of [25]. First of all we * *have -e1 O (e1 O [v1] O bOp-11O=bOnp)-[v1] O bOp1O bOnp(as e1 O e1 = b1) = (b?p1) O bOnp(by main relation, see appendix B ) = (b1 O bOn1)?p(the distributivity, see B ) = '?p2n+1 = e1 O Q1('2n+1) (property of Dyer-Lashof operati* *ons) We also know that the map e1O :H*(BP__2n+1) ! H*+1(BP__2n+2) factors through QH* **(BP__2n+1), and by [25] we have P H*(BP__2n+1) ~=QH*(BP__2n+1) ,! H*+1(BP__2n+2). Thus we g* *et i) since both Q1('2n+1) and -e1 O [v1] O bOp-11O bOnpmap to '?p2n+1. To prove ii), note* * that the map e1 O : QH*(BP__2n+1) ! P H*+1(BP__2n+2) is also injective ([25]). Furthermore, * *the kernel of the map P H*(BP__2n) ! QH*(BP__2n) consists of the p-th powers. Since QH*(BP__2* *n) is trivial if * 2n + 2(p - 1) unless * = 0 or 2n, we see that in degree |Q2('2n)| = 2np * *+ 4(p - 1), the map P H*(BP__2n) ! QH*(BP__2n) is bijective. Thus we derive ii) from i). * * 2 These elements have an interesting property. Let ` be an element of BP 2n+ff* *l(BP__2n+ffl), ffl = 1, 2. If ` comes from a stable map, then we clearly have `*(Qffl'2n+ffl) = Qffl(`*'2n+ffl) = aQffl'2n+ffl for some a 2 Z=p. It turns out that this still is the case with unstable maps `* *, namely Proposition 4.7. Let ` 2 BP 2n+ffl(BP__2n+ffl), ffl = 1, 2, 2n+ffl 1. Then in* * Hp(2n+ffl)-ffl(BP__2n+ffl) we have 9a 2 Z=p, such that`*(Qffl'2n+ffl) = aQffl'2n+fflmodulo decomposable* *s. Proof. We only deal with the case ffl = 1, the other case being similar. Denote* * by s1 and oe1 the fundamental classes in BP 1(S1) and BP1(S1) respectively. We then have `*(-Q1'2n+1) = `*(e1 O [v1] O bOp-11O bOnp) p-1 factors n f* *actors z____"_____- z____* *"_____- = `*((v1s1x1. .x.p-1xp. .x.p+n-1)*(oe1 fi1 . . .fi1 fip * * . . .fip)) p-1 factors n f* *actors z____"_____- z____* *"_____- = (`(v1s1x1. .x.p-1xp. .x.p+n-1))*(oe1 fi1 . . .fi1 fip * * . . .fip) So we can conclude that `*(Q1'2n+1) lies in the coalgebraic subring of H*(BP__** *) generated by e1, b1 and bp over Z=p[BP *]. Denote B this coalgebraic subring. Let y 2 P Hp(2* *n+1)-1(BP__*)\ B. Then for degree reasons y is a linear combination of elements of the form [ff] O e1 O bOp-1+jp1O bOn-jpwith[ff] 2 BP 2(1+j)(p-1). 10 Consider e1O y = [ff] O bOp+jp1O bOn-jp. Suppose vidivides ff. Then 1 + j 1 +* * p + . .p.i-1, so by Lemma B.4, e1 O y is decomposable. Since ff has to be divided by some vi, we* * see that e1O y is decomposable. As in the proof of Lemma 4.6, we see that y is a multipl* *e of Q1'2n+1. Combining everything together, we get the desired result. * * 2 We also need to know some particularity of unstable BP -operations that are * *"dual" to these Dyer-Lashof operations. Denote ae the obvious map BP ! HZ=p. Note that * *ae can also be considered as an element of Hn(BP__n) that is dual to 'n. Then we have Proposition 4.8. (i)Let n 2, `1 2 BP 2n(BP__2n), `2 2 BP (2n+2)p-2(BP__2n) * *such that < ae(`2),Q2('2n) > = 1 < ae(`1), '2n > = 1. Then we have < ae(`2 O `1), Q2('2n) >= 1. (ii)Let n 1, `1 2 BP 2n+1(BP__2n+1), `2 2 BP (2n+2)p-1(BP__2n+1) such that < ae(`2),Q1('2n+1) > = 1 < ae(`1), '2n+1> = 1. Then we have < ae(`2 O `1), Q1('2n+1) >= 1. Remark 4.9. (i) An analogous fact holds for any stable operations, since BP * **(BP ) is generated as a topological algebra by operations that cover Steenrod reduc* *ed powers. (ii)Let `1 2 BP 1(BP__1) be the splitting map BP__1! S1 ! BP__1, `2 2 BP 2p-1* *(BP__1) be any element with < ae(`2), '1 >= 1. Then we have < ae(`1), '1 >= 1, but `2* * O `1 = 0 as BP 2p-1(S1) = 0, so an analogous statement doesn't hold for BP__1. Proof. We have < ae(`), 'k >=< `*(ae), 'k >=< ae, `*('k) > which implies that < ae(`), 'k >= 1 () `*('k) = 'k. Similarly we have < ae(`), Qffl('2n+ffl) >= 1 () `*(Qffl('2n+ffl)) = ('(2n+2)p-ff* *l) so it suffices to show that `*('2n+ffl) = '2n+fflimplies `*(Qffl('2n+ffl)) = Qf* *fl('2n+ffl) for ffl = 1, 2. To prove this, we consider the induced map in BP -homology. As it is a map of BP * **(BP )- modules, we can use the action of BP *(BP ). Recall HZ=p*(HZ=p) ~= (q1, q2, .* * .). P [,1, ,2, . .]., and denote Pn 1 the element of dual polynomial basis (the Mil* *nor basis [15]) dual to ,n1. Let rn 1 2 BP *(BP ) be as in [29]. We know that its coproduct i* *s given by (rn 1) = krk 1 r(n-k) 1and that the following diagram commutes rn 1 BP ________BP- | | | | | | |ae |ae | | | | |? Pn |? HZ=p ______HZ=p-1 11 (According to [29], Proof of Lemma 3.7, we get O(Pn 1) instead of Pn 1, but as * *O(,1) = ,1, O being an algebra map we have O(,n1) = ,n1so O(Pn 1) = Pn 1. ) The operation Pn * *1is known to coincide with the Steenrod reduced power P n, so for spaces X, we have P0n 1* *(x) = 0 if x 2 H*(X) with * pn and P0n 1(x) = V (x) where V is the Verschiebung (and 0de* *notes the action in homology). Now we have Lemma 4.10. (i)Let n 1. In BP*(BP__2n+1)=p we have r0n 1(e1 O [v1] O bOp-* *11O bOnp) = v1'2n+1. (ii)Let n 2. In BP*(BP__2n)=p we have r0n 1([v1] O bOp-11O bOnp) = v1'2n+1. Proof. We see easily that r0n 1(e1 O [v1] O bOp-11) = 0 so by the Cartan formul* *a we get r0n 1(e1 O [v1] O bOp-11O bOnp) = e1 O [v1] O bOp-11O r0n 1(bOn* *p). We also see that r01 1(bp) = b1 modulo [p], p, and ?-decomposables. Since [p] O* * e1 = pe1, and e1 O (-) kills the decomposables, we get the equality i) using the lemma B.5. T* *he proof of ii) is similar and omitted. * * 2 Now we can finish the proof of Proposition 4.8. Note that by the naturality* * of all structures of coalgebraic rings, all named elements in H*(BP__*) is the image o* *f the ele- ments with same name in BP*(BP__*). Let ` be an element of BP 2n+1(BP__2n+1) su* *ch that H*(`)('2n+1) = '2n+1(from now on, we use notations H*(`) and BP*(`) instead of * *`* to avoid confusion). Then we have also BP*(`)('2n+1) = '2n+1modulo p. Note that as in th* *e proof of Proposition 4.7 we see that BP*(`)('2n+1) lies in the coalgebraic subring gener* *ated by e1, b1 and bp over BP*[BP *]. With a little more careful analysis, we see also that mo* *dulo decom- posables it has to lie in the image of the circle multiplication with e1O[v1]Ob* *p-11. Denote by C the sets of BP*-linear combinations of the elements of the form e1O [v1] O bp-1* *1O [ff] O bOj01O bOj1p in QBP(2n+1)p-1(BP__2n+1). For degree reasons |ff| 6= 0| unless j0 = p - 1 and * *j1 = n. The proof of Proposition 4.7 shows us that the elements of [I2] O BP*(BP__*) \ C is* * in the kernel of ae. Thus we have Ker(ae|C)= C \ ([I2] O BP*(BP__*) + p) C \ ([I2] O BP*(BP__*) + I[I] O BP*(BP__*) + I2BP*(BP__*)) where denotes the submodule generated by a. However, [I2] O BP*(BP__*) + I* *[I] O BP*(BP__*) + I2BP*(BP__*) is invariant under r0n 1, so from r0n 1(BP*(`)(e1 O [v1] O bOp-11O=bOnp))BP*(`)(r0n 1(e1 O [v1] O bOp-11O * *bOnp)) = BP*(`)(v1'2n+1) = v1'2n+1 we deduce that r0n 1(BP*(`)(e1 O [v1] O bOp-11O bOnp)) = e1 O [v1] O bOp-11O bOnpmodu* *lo Kerae. Thus we have the desired result. * * 2 12 Remark 4.11. The case of BP 2(BP__2) deserves a little discussion. Lemma 4.10* * ii) almost holds, with an extra term '?p2on the right hand side. As ` doesn't have to be a* *dditive,BP*(`) doesn't commute with the ? product, so BP*(`)('?p2) is not necessarily a decomp* *osable element, and as a matter of fact it can be equal to v1'2. The proof of Proposition 4.4 can be concluded as follows. We treat the case* * when n is odd, the case n is even is left to the reader. Take an element ff 2 A that * *maps to a generator BP n(Sn). Consider the element D(u)(,A(ff)) 2 D(BP *(Sn)) ~=BP *(QSn* *). By the hypothesis on ff it maps to a generator in BP n(Sn), so it also maps to a g* *enerator in Hn(QSn). Therefore, 9`1 2 BP n(BP__n) such that (i)D(u)(,A(ff)) = o*(`1) = `1(o) where o : QSn ! BP__nis the unit map. (ii)ae(`1) 2 Hn(BP__n) is a generator. We can suppose that < ae(`1), 'n >= 1, multiplying ff with a unit if necessary.* * Now choose an element `2 2 BP *(BP__n) as in Proposition 4.8. Given an unstable BP -algebr* *a B, denote by <, > the pairing B x HomZ=p(B ^BP*Z=p; Z=p) ! Z=p. Then we have < `2`1(o), Q1(oen) >= < o*(`2`1), Q1(oen) > = < (`2`1), o*Q1(oen) > = < (`2`1), Q1('n) > as o is a map of infinite loop sp* *ace = 1 by Proposition 4.8 But we also have < `2`1(o), Q1(oen) >= < `2(D(u)(,A(ff))), Q1(oen) > = < `2(ff), ,A# D(u)# (Q1(oen)) > = < `2(ff), Q1(,A# D(u)# (oen)) > as , and D(u) are maps of BP infinite loop algebras. This concludes the proof o* *f Proposition 4.4. * * 2 Remark 4.12. Here we see some of the difficulties of dealing with the general* *ized cohomol- ogy theories. First of all unlike the case of BP -homology (which is connective* *), the existence of spherical class is not guaranteed at all in BP -cohomology of a space. Secon* *dly, it is tempt- ing to consider our arguments above as some sort of refinement of Nishida relat* *ions, and to state an equality like r0n 1Q1 = -v1id. Unfortunately it is not clear at all on* * what kind of algebraic structure this equality makes sense. To conclude, we show that one can use Proposition 4.4 to get some restrictions * *on the homotopy type of infinite loop spaces. For example, Corollary 4.13. Let X be a (n - 1)-connected space with H*(X; Z(p)) is free ove* *r Z(p), Hn(X; Z(p)) 6= 0 and Hp(n+ffl)-ffl(X; Z(p)) ~=0 where ffl = 1 if n is odd and 2* * if n is even. Let G* be a torsion abelian graded group of finite type such that Gm ~=0 unless 2(1* *+p+. .p.m) (p - 1)(n + ffl). Then the product space X x K(G*, *) doesn't have the homotopy* * type of an infinite loop space. 13 Proof. By [20] K(G*, *) satisfies the conditions (H1) and (H2), and we have BP* * *(X x K(G*, *)) ~=BP *(X) ^BP*BP *(K(G*, *)), thus X x K(G*, *) also satisfies the co* *nditions (H1) and (H2). On the other hand according to [20], BP *(K(G*, *)) ^BP*Z=p ~=0 * *in degrees less than 2(1 + p + . .p.m-1). Thus BP *(X x K(G*, *)) ^BP*Z=p is trivial in t* *he degree p(n + ffl) - ffl. Thus by Proposition 4.4 it can't have a structure of a BP inf* *inite loop algebra. 2 A. Topologies on BP *(X) One of the technical difficulties concerning the BP -cohomology is the issue* * of its topol- ogy. That is, quite often while dealing with the BP -cohomology of an infinite * *dimensional complex, one would like to consider infinite sums, which means that we need a t* *opology. The traditional solution is tu use the "classical" skeletal topology, that is the t* *opology associ- ated to the filtration given by F s(BP *(X)) ~=Ker(BP *(X) ! BP *(sks-1X)). It * *turns out that this topology is nice enough so that it has become the default topology to* * work with. Unfortunately it also has several draw-backs, notably the lack of the rigidity.* * That is, for example, if f : BP *(X) ! BP *(Y ) is a continuous homomorphism of BP *-modules* *, then it is not clear whether the topology on Im(f) induced by that of the topology o* *f BP *(Y ) agrees with the quotient topology. In [10] one approach to settle this was atte* *mpted, unfor- tunately it requires the ordinary cohomology as a part of initial data, and we * *certainly don't want to use such an approach to deal with general problems involving unstable B* *P -algebras, even though in practice we are only interested in BP -cohomology of spaces or s* *pectra whose ordinary cohomology is known. Tamanoi, on the other hand, used another natural topology called BP -topolog* *y in [22], and showed that it has some nice properties. Unfortunately his topology is too* * fine for our purpose. For example, a sum of the form i(v1)ix(p-1)iwith x 2 BP 2(CP 1) * *doesn't converge in this topology. Consequently BP *(CP 1) ^BP*Z=p where the completed * *tensor product is taken with respect to the BP -topology doesn't inject to H*(CP 1; Z=* *p). There also are several other natural topology on BP *(X), arising from its a* *lgebraic structure. In [28], Yamaguchi mentions the "skeletal topology", which we will * *refer to as "algebraic skeletal topology" to distinguish from the classical skeletal topolo* *gy. With this topology i(v1)ix(p-1)iconverges. However, a homogeneous sum of the form iviyi* * doesn't converge. Now we notice that the problems of convergence we have with the BP -t* *opology and with the algebraic skeletal topology are complementary. That is, with the B* *P -topology, the non-convergence comes from the high powers of the ideal (v1, . .,.vn) where* *as with the algebraic skeletal topology the problem comes from the presence of vn's with in* *finitely many n's. This motivates us to consider the intersection of the two topologies, whic* *h we will call the BP -skeletal topology. It turns out that it agrees with the classical skele* *tal topology in many cases of interest, and it also has a good rigidity. We will discuss the de* *tails in the rest of this appendix. We start with some definitions. Definition A.1. Let X be a space or spectrum. 14 (i)The BP -topology on BP *(X) is the topology defined by the decreasing filt* *ration BP k(X) = F 0-1(X) F 00(X) . . .F 0n(X) = Ker(BP k(X) ! BP k(X)) * *. ... Note that we complete BP at p, so we will do the same with BP . (ii)The algebraic skeletal topology on BP *(X) is defined by the decreasing f* *iltration F 00n(BP k(X)) is the submodule generated by[i nBP k+i(X). (iii)The classical skeletal topology on BP *(X) is defined by the filtration F s(BP *(X)) ~=Ker(BP *(X) ! BP *(sks-1X)). (iv)The BP -skeletal topology on BP *(X) is the intersection of the BP -topolo* *gy and the algebraic skeletal topology, in other words it is the topology defined by * *the fundamental system of neighbourhood of 0 {F 0n(BP *(X)) + F 00m(BP *(X))}. Now, according to [22] Proposition 2.8, the BP -topology is finer than the clas* *sical skeletal topology. It is clear that the algebraic skeletal topology is finer than the cl* *assical skeletal topology. Thus the BP -skeletal topology is finer than the classical skeletal * *topology, too. We prove a partial inverse, namely Theorem A.2. Let X be a space satisfying the conditions (H1) and (H2) of Theo* *rem 2.2. Then the BP -skeletal topology on BP *(X) agrees with the classical skeletal to* *pology. Proof. We start with the simplest case, when BP *(X) is topologically free. In * *this case we don't need X to be a space, the proof will be valid when X is a spectrum as wel* *l. Let {xi} be a topological basis of BP *(X) with respect to the classical skeletal topolo* *gy. Thus all elements of BP *(X) can be written uniquely as X x = ffixi, with |ffi| + |xi| = |x|, xi2 F |xi|(BP *(X)). i Now fix n. Note that BP * ~= ^Zp[v1, . .v.n] is a Noetherian ring, so the i* *deal {f 2 BP *, |f| l} is finitely generated. Call the generators f1, . .,.fm . Now,* * one can rewrite the sum as X X x = ff0ixi+ ff00ixiwhere ff0i2 BP *, ff00i2 Ker(BP *! BP ** *). i i Suppose x 2 F |x|+l(BP *(X)). Then we have |ff0i| l, so each ff0ican be rewri* *tten as linear combination of f1, . .,.fm . Thus the first sum is contained in F 00l(BP *(X)).* * Obviously the second sum is in F 0n(BP *(X)), so we get F d+l(BP d(X)) F 0n(BP d(X)) + F 00* *l(BP d(X)). Thus the BP -skeletal topology is coarser than the classical skeletal topology * *as desired. Now we will deal with the general case. According to the proof of Theorem 1* *.20 of [20], the minimum set of generators of BP *(X) also generates the E1 -term of t* *he Atiyah- Hirzebruch spectral sequence H*(X, BP *) ! BP *(X). Thus any element of BP *(X)* * can 15 be representedPin the E1 -term of the Atiyah-Hirzebruch spectral sequence by an* * element of the form x = iffixi, with |ffi| + |xi| = |x|, and xi's are in E*,01. The rest* * of the argument is similar. The prototype case of the second situation is as follows. Consider the Atiya* *h-Hirzebruch spectral sequence H*(BZ=p, BP *) ! BP *(BZ=p). We have E2 ~=E1 ~=BP *=p[[x]],* * and BP *(BZ=p) ~=BP *[[x]]=([p](x)) where [p]x = px + v1x + . ...In BP *(BZ=p), the* * element x has filtration 2, however px = -v1x + . .h.as filtration 2p. In the Atiya-Hirze* *bruch spectral sequence, px = 0 and it is represented by -v1x+. .,.which, indeed, has the corr* *ect filtration. 2 Remark A.3. It is not clear if the condition (H2) is really necessary here, a* *s we know from [20] that the Atiyah-Hirzebruch spectral sequence for BP *(X) behaves more* * or less reasonably. Of course, the interest of defining a new topology is not that it agrees wit* *h an old one, but that it has something new to offer, in our case the rigidity. Note that the alg* *ebraic skeletal topology is completely algebraic. We show that for a space X, if we take into * *account unstable operations, the BP -topology on BP *(X) is determined by its algebraic* * structure. Fix d. Let n be a positive integer such that 2(1 + . .p.n) > d. Then using th* *e H-space splitting BP__d! BP__d([25, 4]), we get an operation `d,n: BP__d! BP__dsuch * *that x 2 Ker(BP d(X) ! BP d(X)) if and only if`d,n(x) = x. Thus the the BP -topology on BP *(X) is determined by the action of `d,n's and * *the abelian group structure. As the algebraic skeletal topology on BP *(X) is determined by* * the (dis- crete) BP *-module structure, we see that the BP -skeletal topology on BP *(X) * *is deter- mined by its underlying (discrete) algebraic structure. Furthermore our argumen* *t apply to any unstable BP -algebra. Putting them altogether, we have proven : Theorem A.4. An unstable BP -algebra admits a natural inherent topology TBP s* *uch that (i)TBP is finer than the "classical skeletal topology". (ii)TBPagrees with the classical skeletal topology on BP *(X) where X is a sp* *ace satisfying the conditions (H1) and (H2) in Theorem 2.2. (iii)TBPon an unstable BP -algebra A depends only on the "algebraic structures* *" on A. More precisely, let A and A0 be unstable BP -algebras, and f : A ! A0 a h* *omo- morphism of BP *-modules that commutes with all unstable operations. Then* * f is a homeomorphism (with respect to TBP). Thus, for example, we have Corollary A.5. The category of completed unstable BP -Hopf algebras equipped w* *ith TBP (instead of the default topology) is abelian. Here a completed unstable BP -Ho* *pf algebra means an unstable BP -algebra A equipped with the diagonal A ! A ^BP*A which is* * a map of unstable BP -algebras. Proof. It is easy to see that the standard proof of the fact that the category * *of Hopf algebras is abelian applies. The only issue would be the uniqueness of the topology on t* *he image, but in view of iii) above, we see that this doesn't cause a problem. * * 2 16 B. The Coalgebraic ring BP*(BP__*) In this appendix we gather some facts on E*(BP__*) where E = H or BP . Most * *of the material presented here is taken from [19], [27] and [9]. Another good referen* *ce on the subject is [3]. The results of this section hold for the usual (p-local, witho* *ut completion) BP -theory. The first computation of H*(BP__*) was done in [25]. It was shown Theorem B.1 ([25 ]). H*(BP__i) is a polynomial algebra concentrated in even deg* *rees for i even, and an exterior algebra generated by odd degree elements for i odd. Fu* *rthermore, dim(QHj(BP__i)) = rank(BPj-i). This is somewhat surpassed by later works that we shall describe. However, for * *low degree computations, this dimension formula is quite handy. Besides we used it implici* *tely in the proof of Lemma 4.6. Note that by the space BP__irepresent the degree i part of the BP -cohomolog* *y. Thus the ring structure of the BP -cohomology is represented by maps ~+ : BP__ix BP__i! BP__i ~x : BP__ix BP__j! BP__i+j which induce in homology the following maps. ? = H*(~+) : H*(BP__i) H*(BP__i)! H*(BP__i) O = H*(~x) : H*(BP__i) H*(BP__j)! H*(BP__i+j) Since BP__i's are spaces, they have the diagonal : BP__i! BP__ix BP__iwhich mak* *es H*(BP__i) a coalgebra. Clearly the two products ? and O are maps of coalgebras, and they * *are related to each other via the "distributivity law". More precisely, there is a relation* * of the form a O (b ? c) = (a0O b) ? (a00O c) where (a) = a0 a00 "up to sign". That is according to the bidegrees of elements concerned, there a* *re the mul- tiplication by -1 and/or the conjugation that appear. However we only need to * *use the distributivity law for elements in H2*(BP__2i) 's so the reader can forget abou* *t the signs. All these make H*(BP__*) a ring object in the category of coalgebras, which were ca* *lled Hopf rings in [19]. However we follow [6] and call them coalgebraic ring. The theory of coalgebraic rings have its own interest, especially in connect* *ion with that of coalgebraic modules (c. f. [6]). Here, we are interested in them because tha* *nks to these products, we can produce lots of elements starting from a few elements, which w* *e are going to define now. First of all for a 2 BP *~= [S0, BP__-*] we have [a] = a*(1) 2 * *H0(BP__-*). Note that we have ([a]) = [a] [a], [a] ? [a0] = [a + a0] and [a] O [a0] = [a* *a0]. Second, let x 2 BP 2(CP 1) be the orientation class, fii2 BP2i(CP 1) to be dual to xi. Then* * we have bi = x*(fii) 2 H2i(BP__2). Note that we have (fii) = j+k=i(fij fik). So far* * all elements we have defined live in even degrees, and to remedy this we define e1 2 H1(BP__* *1) to be 17 the image under the suspension map of 1 2 H0(BP__0). It turns out that these e* *lements "generate" H*(BP__*) under our two products, that is , every element of H*(BP__* **) can be written as a linear combination of ? products of O products of theses elements.* * However we don't have the uniqueness, i.e., there are some relations that we describe now. Consider the composition CP 1 x CP 1 ! CP 1 x! BP__2. This is just the forma* *l sum x1 +BP x2 where +BP denotes the universal p-typical formal group law ([17]). * *Thus in homology it induces the map fi(x1) +[BP]fi(x2) 2 Hom(H*(CP 1 x CP 1), H*(BP__2)) ~=H*(BP__2)[[x1, x2* *]] where we denote fi(X) = fiiXi, and +[BP]means the formal sum with x and + repl* *aced with O and ?. However, the map CP 1x CP 1 ! CP 1 induces in cohomology the ring* * map that sends x to x1 + x2 so we see that the above map is equal to fi(x1 + x2). T* *hus we have fi(x1) +[BP]fi(x2) = fi(x1 + x2) (The main relation). Then the main result of [19] is Theorem B.2 ([19 ], Theorem 4.2). H*(BP__*) is the quotient of the free coalge* *braic ring generated by the elements [a]'s for a 2 BP *, bi's and e1 by the main relation * *and the relation e1 e1 = b1. Among other things, this implies that everything comes from a product of S1's a* *nd CP 1's. Denote CP Sthe full subcategory of the homotopy category of spaces whose object* *s are finite products of S1's and CP 1's, and CP S=BP__*the category whose objects are maps * *from an object of CP S to BP__*and whose morphisms are commutative triangles. Then we h* *ave Theorem B.3 ([9]). The natural map colimCPS=BP_*H*(source(-)) ! H*(BP__*) is a* *n iso- morphism. Now, although the main relation gives a complete set of relations in purely alg* *ebraic way, it is not quite practical to work with. Fortunately there are simpler versions.* * Consider the p 1 x maps CP 1 ! CP ! BP__2and the induced maps in homology. Then as in above, we * *get b(px) = [p[BP]](b(x)) where [p[BP]](X) is the p-series for BP ([pBP ](X) )with the sum and product re* *placed by the star and circle products. It turns out that bi's with i equal to a power of* * p is necessary to generate H*(BP__*), and if one uses only these b's instead of all of them, t* *hen only the simplified form of the main relation is necessary ([19]). Now, let's take a lo* *ok at these relations. Modulo ([v1], [v2] . .[.vn] . .).O ([p], [v1], [v2] . .[.vn] . .).we* * get [v1] O bOp1+ b?p1=0 2 ?p [v1]O bOpp+ [v2] O bOp1+=bp0 .. . 2 Opn ?p [v1]O bOppn+ [v2] O bOppn-1+ . .[.vn] O=b10 + bpn from which we derive 18 n Lemma B.4 ([2]). We have [vn] O bOp1+...+p= (-1)n(bO1)?p As we know that BP*(BP__*) is free, the above arguments apply to BP*(BP__*) * *with a slight modification (especially concerning the main relation). For example, th* *e simplified form of the main relation becomes b([pBP ]x) = [p[BP]](b(x)) and we get Lemma B.5. In BP*(BP__*)=p we have [v1] O bOp1+ b?p1= v1b1. Note also that the freeness implies that BP *(BP__*) is just its dual, which sh* *ould mean that it is enough to know BP*(BP__*) to understand BP *(BP__*) . However, as BP j(B* *P__i) ~= [BP__i, BP__j], there is a composition BP j(BP__i) x BP k(BP__j) ! BP k(BP__i),* * and there is no structure in BP*(BP__*) which is dual to this. Recent works in [24] suggest a n* *ew approach to deal with this problem, but here we will stick to the traditional approach (c. * *f. [27]). Given ` 2 BP *(BP__*) ~= [BP__*, BP__*], consider `* 2 HomBP*(BP*(BP__*), BP*(BP__*))* *. Of course we know the most concretely HomBP*(BP*(BP__*), BP*). So if denote by ^`the ele* *ment corresponding to ` in HomBP*(BP*(BP__*), BP*) what is really interesting to kno* *w is the formula expressing `* in terms of ^`. Although such a formula exists ([27], [3]* *) as we don't need it in the current version of our paper, we content ourselves to express `** * in terms of `, and we just consider the induced map in ordinary homology (although the c* *ase for BP -homology is just as simple). This is extremely simple using the colimit mod* *el, and we have Lemma B.6. Let ` 2 BP j(BP__i) ~=[BP__i, BP__j]. In H*(BP__j) we have `*(f*(f* *i)) = (`(f))*(fi) where f 2 BP i(X), fi 2 H*(X) with X a finite product of S1's and CP 1's. Proof. Juste note that `(f) = ` O f. * * 2 The last formula we need is the action of BP *(BP ) on BP*(BP__*). Again, th* *e colimit model makes things easy. Let r 2 BP *(BP ) and denote by r0 its right action o* *n BP - homology. Then we have Lemma B.7. In BP*(BP__*) we have r0(f*(fi)) = f*(r0(fi)) where f 2 BP i(X), f* *i 2 BP*(X) with X a finite product of S1's and CP 1's. Proof. By the definition of the homology operations f* commutes with r0. * * 2 References [1]J. F. Adams. 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