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
Rallowable 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: BrownPeterson cohomology, Infinite loop spaces, Generalized cohomo*
*logy
operations, Unstable Algebras, DyerLashof operations, Nishida relations,
LandweberNovikov algebra, Steenrod algebra, Hopf ring
2000 MSC: Primary 55N22, 55P47, 55S10, 55S12, 55S25, Secondary 55R40, 55R45
_______________________________________________________________________________*
*__
1. Introduction
It is wellknown that the mod p ordinary homology of an infinite loop space *
*has the
structure of a so called ARallowable Hopf algebra [13], i.e., an Hopf algebra*
* on which
both the Steenrod algebra and the DyerLashof 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 (pcomplete) Ktheory 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*In an unpublished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 Enhomology of QX when Enhomology 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 AR 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.ujfgrenoble.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 BrownPeterson 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*
*ellknown
(c. f. [1] ) that E*(X) is a module over the ring of stable cohomology operat*
*ions E*E.
Less wellknown is the fact that, if X is a space, under certain conditions on *
*E, E*(X) has
a richer set of structures called unstable Ealgebra (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*BPI
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 ARallowable 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 Landweberflat, 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 wellknown 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 BPalgebra, 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 BPalgebra. 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*
* Apalgebra.
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 ARallowable 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*
*Rallowable
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 ARallowable 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 DyerLas*
*hof
algebra from the composition pairing and the action of the Steenrod algebra, we*
* can make
the DyerLashof 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 socalled 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. EilenbergMaclane spaces
The BrownPeterson cohomology of the EilenbergMaclane 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+...+pn1)
which induces the surjection BP *(BP__2(1+p+...+pn1)) ! 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+...+pn1))_____BP *(K(Z=p, n))
 
 
 
 
,BP*(BP_2(1+p+...+pn1)) ,BP*(K(Z=p,n))
 
 
 
? ?
DI(BP *(BP__2(1+p+...+pn1)))__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 torsionfree 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 nconnected. 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*
*sionfree
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 ARallowable 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 (p1)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 bOp11O bOnp.
9
(ii)Let n 1. In H*(BP__2n) we have
Q2('2n) = [v1] O bOp11O 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 bOp11O=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 DyerLashof 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 bOp11O 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 pth 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 bOp11O bOnp)
p1 factors n f*
*actors
z____"_____ z____*
*"_____
= `*((v1s1x1. .x.p1xp. .x.p+n1)*(oe1 fi1 . . .fi1 fip *
* . . .fip))
p1 factors n f*
*actors
z____"_____ z____*
*"_____
= (`(v1s1x1. .x.p1xp. .x.p+n1))*(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 bOp1+jp1O bOnjpwith[ff] 2 BP 2(1+j)(p1).
10
Consider e1O y = [ff] O bOp+jp1O bOnjp. Suppose vidivides ff. Then 1 + j 1 +*
* p + . .p.i1, 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 DyerLashof 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)p2(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)p1(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 2p1*
*(BP__1) be
any element with < ae(`2), '1 >= 1. Then we have < ae(`1), '1 >= 1, but `2*
* O `1 = 0 as
BP 2p1(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)pff*
*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(nk) 1and that the following diagram commutes
rn 1
BP ________BP
 
 
 
ae ae
 
 
? Pn ?
HZ=p ______HZ=p1
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 bOp11O bOnp) = v1'2n+1.
Proof. We see easily that r0n 1(e1 O [v1] O bOp11) = 0 so by the Cartan formul*
*a we get
r0n 1(e1 O [v1] O bOp11O bOnp) = e1 O [v1] O bOp11O 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*
*p11. Denote by C
the sets of BP*linear combinations of the elements of the form e1O [v1] O bp1*
*1O [ff] O bOj01O bOj1p
in QBP(2n+1)p1(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(aeC)= 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 bOp11O=bOnp))BP*(`)(r0n 1(e1 O [v1] O bOp11O *
*bOnp))
= BP*(`)(v1'2n+1)
= v1'2n+1
we deduce that
r0n 1(BP*(`)(e1 O [v1] O bOp11O bOnp)) = e1 O [v1] O bOp11O 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.m1). 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 *(sks1X)). 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 drawbacks, 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(p1)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(p1)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 nonconvergence 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 01(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 *(sks1X)).
(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 AtiyahHirzebruch 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*
*hHirzebruch
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 AtiyaHirze*
*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 AtiyahHirzebruch 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 Hspace
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 (plocal, 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(BPji).
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 ptypical 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 pseries 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 bOppn1+ . .[.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
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