SPECTRA OF BP-LINEAR RELATIONS, vn -SERIES, AND
BP COHOMOLOGY OF EILENBERG-MAC LANE SPACES
Hirotaka Tamanoi
University of California at Santa Cruz
Abstract. On Brown-Peterson cohomology groups of a space, we introduce a *
*natu-
ral inherent topology, BP topology, which is always complete Hausdorff fo*
*r any space.
We then construct a spectra map which calculates infinite BP-linear sums *
*conver-
gent with respect to the BP topology, and a spectrum which describes infi*
*nite sum
BP-linear relations in BP cohomology. The mod p cohomology of this spectr*
*um is a
cyclic module over the Steenrod algebra with relations generated by produ*
*cts of ex-
actly two Milnor primitives. We show a close relationship between BP-line*
*ar relations
in BP cohomology and the action of the Milnor primitives on mod p cohomol*
*ogy. We
prove main relations in the BP cohomology of Eilenberg-Mac Lane spaces. T*
*hese are
infinite sum BP-linear relations convergent with respect to the BP topolo*
*gy. Using
BP fundamental classes, we define vn-series which are vn-analogues of the*
* p-series.
Finally we show that the above main relations come from the vn-series.
Contents
1.Introduction and summary of results 1
2.BP topology and a spectra map which calculates infinite sums 8
3.Spectrum L of infinite sum BP-linear relations 15
4.Spectrum L of finite sum BP-linear relations and its relation
to L 21
5.BP-linear relations and Milnor primitives 29
6.Main BP-linear relations in BP cohomology of Eilenberg-Mac
Lane spaces: vn-series 33
References 40
x1. Introduction and summary of results
The Brown-Peterson (BP) cohomology theory generally has more interesting
structures and yet simpler descriptions than the BP homology theory.
In generalized cohomology theories we need to deal with infinite sums of ele-
ments, and to discuss convergences we need a topology on these cohomology group*
*s.
For infinite dimensional CW complexes, skeletal filtration topologies are commo*
*nly
______________
1991 Mathematics Subject Classification. 55.
Key words and phrases. Brown-Peterson (co)homology theory, BP fundamental cl*
*ass, BP
topology, Eilenberg-Mac Lane spaces, Milnor primitives, -spectrum, Steenrod alg*
*ebra, Sullivan
exact sequence, vn-series.
The research is partially supported by a Faculty Research Grant, UC Santa Cr*
*uz
Typeset by AM S-T*
*EX
1
2 HIROTAKA TAMANOI
used. However, these topologies often fail to be complete Hausdorff, and there *
*can
exist elements of infinite filtration, so-called phantom maps, which vanish when
restricted to any finite skeleton. For a generalized cohomology theory satisfy*
*ing
the Milnor's additivity axiom [M3], being Hausdorff and being complete Hausdorff
are equivalent. When a topology on cohomology groups is not complete Hausdorff,
convergence problems are tricky.
However, the good news is that for the BP cohomology theory the situation is
very good. We show that there is a very natural and inherent topology on BP
cohomology groups of any spectrum (which is not necessarily a CW-spectrum)
derived from the global structure of BP theory, namely the existence of the BP-
tower. This is the following sequence of BP-module spectra and BP-module spectra
maps:
(1-1) BP -! . .-.!BP -! BP -! . .-.!BP <0> = HZ(p)-! HZp;
where ss*(BP ) = Z(p)[v1; v2; : :;:vn] with |vi| = 2(pi-1) for 1 i n, and *
*HZ(p)
and HZp are Eilenberg-Mac Lane spectra for the ring Z(p)of localized integers at
p and the ring Zp of mod p integers.
For any spectrum X and k 2 Z, we consider a decreasing filtration
BP k(X) F -1 F 0 F 1 . . .F n . .;.
(1-2)
where F n= Ker {ae*: BP k(X) -! BP k(X)}:
The BP topology is defined to be the topology defined by this filtration. Altho*
*ugh
the BP topology on BP *(X) can be defined for any spectrum X, it is inherently *
*an
unstable notion and it only exhibits nice properties when X is a space.
Proposition 1-1 [Proposition 2-1]. For any k 2 Z, the BP-topology on BP k(X)
is always complete Hausdorff for any space X.
Thus any Cauchy sequence in BP k(X) always converges to a unique limit when
X is a space. Although many results in this paper are valid for any spectrum X,
we must assume that X is a space (not necessarily a CW complex) when we need
convergences.
We are interested in infinite BP-linear sums in BP *(X) of the following for*
*m:
X1
(1-3) vnbn = pb0 + v1b1 + . .+.vnbn + . .;.
n=0
where bn 2 BP *+2pn-1 (X) for n 0.
Corollary 1-2 [Corollary 2-3]. Any infinite sum of the form (1-3) always con-
verges to a unique element in BP k+1 (X) with respect tonthe BP topology for any
space X and for any collection of elements bn 2 BP k+2p -1 (X) for n 0.
We want to calculate the limit of the infinite sum (1-3). We consider the fo*
*llowing
composition of BP-module maps:
1Y i ~ _1 i W v 1_
(1-4) : 2(p -1)BP --= 2(p -1)BP - -!i BP folding----!BP:
i=0 i=0 i=0
SPECTRA OF BP-LINEAR RELATIONS AND Vn-SERIES 3
Here, k denotes the k-fold formal suspension of spectra, and the first arrow is*
* a
homotopy equivalence [Lemma 2-4]. For any spectrum X, the induced map
1Y i
* : BP k+2p -1(X) --! BP k+1 (X)
i=0
provides us with a well-defined element *(b0; b1; : :;:bn; : :):for any sequenc*
*e of
elements "b= (b0; b1; : :;:bn; : :):of appropriate degrees. Note that *("b) is *
*a well-
defined element in BP cohomology of any spectrum X. However, when X is a
space, we can identify this element as the limit of (1-3).
Theorem 1-3 [Theorem 2-7]. For any elements bi 2 BP k+2pi-1(X) for i 0,
where X is a space, the limit of (1-3) is given by *. Namely,
X
(1-5) *(b0; b1; : :;:bi; : :):= vibi in BP k+1(X);
i0
where the convergence on the right hand side is with respect to the BP-topology.
When X is an infinite dimensional CW complex, we can also consider the skele*
*tal
filtration topology on BP *(X). Although these two topologies have very differe*
*nt
origin, it turns out that the BP topology is finer than the skeletal filtration*
* topology
[Proposition 2-8]. So any convergent sequence with respect to the BP topology a*
*lso
converges with respect to the skeletal filtration topology.
Let L be the cofibre spectrum of the spectra map . We have the following
cofibre sequence:
Q q 1Y i
(1-6) -1 L ---!i 2(p -1)BP --! BP --! L:
i=0
Since is a BP-module map, the spectrum L is a BP-module spectrum. This
spectrum L turns out to have very interesting properties.
Theorem 1-4 [Theorem 3-1]. Let X be a space and let k 2 Z.
(I) For any element z 2 Lk(X), let bi = qi*(z) 2 BP k+2pi-1(X) for i 0. Then
pb0 + v1b1 + . .+.vnbn + . .=.0 in BP k+1(X);
where the convergence is with respect to the BP-topology.
(II) There exists a spectra map j :L -! HZp such that the following diagram str*
*ictly
commutes (not up to an unknown nonzero constant in Zp) for any i 0:
Lk(X) --qi*--!BP k+2pi-1(X) ---*-! Lk+2pi-1(X)
? ? ?
(1-7) j*?y ?yae* ?yj*
HZkp(X) --Qi--! HZk+2pi-1p(X) ________HZk+2pi-1p(X);
where Qi is the ith Milnor primitive in the Steenrod algebra.
4 HIROTAKA TAMANOI
(III) The mod p cohomology of the spectrum L is the following cyclic module over
the Steenrod algebra A(p) generated by j:
h . X i
(1-8) HZ*p(L) ~= A(p) A(p)QiQj . j:
i;j0
Part (I) says that each element in L*(X) can be thought of as an infinite sum
BP-linear relation in BP cohomology. Thus we call L the spectrum of BP-linear
relations. Part (II) shows that in L-theory, there exist Milnor operations bqi,*
* namely
O qi for i 0. But any product among them is zero, since qj O = 0 for any j
by exactness. Part (III) is a reflection of this fact, and it shows that there *
*are no
other relations in the mod p cohomology of L.
We can also consider finite BP-linear sums in BP *(X) of the form
(1-9) pb0 + v1b1 + . .+.vnbn:
The spectra map which calculates this summation is the following composition of
BP-module maps:
(1-10) Yn n W n
i-1) ~= _ 2(pi-1) vi_ folding
: 2(p BP -- BP - -! BP ----! BP :
i=0 i=0 i=0
Let L be the cofibre of . Then L is a BP-module spectrum with properti*
*es
corresponding to a finite version of Theorem 1-4 [Theorem 4-1]. These BP-module
spectra L fit into the following tower [Proposition 4-5]:
j
(1-11) L -! . .-.!L ----! L -! . .-.!L<1> -! L<0> = HZp:
This tower can be used to construct infinite sum BP-linear relations in BP *(X)
from finite sum BP-linear relations in BP *(X) [Theorem 4-6].
The BP cohomology theory and mod p cohomology theory are closely related
by the Thom map ae* : BP *(X) -! HZ*p(X). Through ae*, BP-linear relations in
BP *(X) translate into a certain property of the action of Milnor primitives on*
* the
mod p cohomology of X.
Proposition 1-5 [Proposition 5-1]. Let X be a space and let k be a positive
integer. Suppose we have
pb0 + v1b1 + . .+.vnbn + . .=.0 in BP k+1(X)
for some elements bn 2 BP k+2pn-1 (X) for n 0. Then there exists an element
x 2 HZkp(X) such that
(1-12) ae*(bn) = Qn(x) for n 0:
Proposition 1-5 for finite sum BP-linear relations was first proved in [Y1] *
*when
X is a finite complex using a geometric method of manifolds with singularities.*
* Our
general result is proved in the stable category of spectra.
We remark that we can easily write down the corresponding BP homology version
of the above proposition.
We consider a converse problem of constructing (infinite sum) BP-linear rela*
*tions
in BP cohomology from information on the action of the Milnor primitives on mod
p cohomology.
SPECTRA OF BP-LINEAR RELATIONS AND Vn-SERIES 5
Theorem 1-6 [Theorem 5-6]. Let X be a space and let k; n be non-negative
integers such that k 2(pn-1 + . .+.p + 1). Then for any element
(1-13) x 2 Im {ae*: BP k(X) --! HZkp(X)};
there exist elements bn+j 2 BP k+2pn+j-1(X) for j 0 such that in BP k+1(X),
vnbn + vn+1 bn+1 + . .+.vn+j bn+j + . .=.0;
(1-14) and ae
*(bn+j ) = Qn+j (x) for allj 0:
We apply our results to study the BP cohomology of Eilenberg-Mac Lane spaces.
In this introductory summary, we describe our results for the integral Eilenber*
*g-
Mac Lane spaces localized at p, K(Z(p); n + 2) with n 1. Let
(1-15) S+n = {(s1; s2; : :;:sn) 2 Zn | 0 < s1 < . .<.sn}
be the set of strictly increasing sequences of n positive integers. In [T1], we*
* produced
nontrivial elements bS 2 BP * K(Z(p); n + 2) for each S 2 S+n with the property
(1-16) ae*(bS ) = QS (on+2 ) = Qs1Qs2. .Q.sn(on+2 ) 6= 0
in HZ*pK(Z(p); n + 2) . Here on+2 2 HZn+2p K(Z(p); n + 2) is the mod p funda-
mental class. Our main result in [T1] was that for X = K(Z(p); n + 2),
(1-17) Im {ae* : BP *(X) --! HZ*p(X)} = Zp[ QS (on+2 ) | S 2 S+n]:
Here, the right hand side is a polynomial subalgebra of HZ*pK(Z(p); n + 2) . In
[RWY], they proved that (quoting results in [RW1]) these elements bS actually
generate the entire BP cohomology of K(Z(p); n + 2). As the next step, we want *
*to
study infinite sum BP-linear relations in the BP cohomology of Eilenberg-Mac La*
*ne
spaces. Repeatedly applying the connecting homomorphisms
m -1
m : BP r(X) --! BP r+2p (X)
in Sullivan exact sequences to a Z(p)-lift bon+2of the mod p fundamental class *
*on+2 ,
we can produce an element z 2 BP 2(pn-1+...+p+1)(X) such that
(1-18) ae*(z) = Qn-1 . .Q.1(on+2 ) 6= 0:
For a systematic study of the mod p cohomology of Eilenberg-Mac Lane spaces
in terms of the Milnor basis, see [T2]. Applying Theorem 1-6, we then obtain an
infinite sum BP-linear relation in the BP cohomology of K(Z(p); n + 2).
Theorem 1-7 [Theorem 6-3]. Let n 1. There exist nontrivial elements
n+j+pn-1+...+p+1)
(1-19) bn+j 2 BP 2(p K(Z(p); n + 2)
for j 0 such that in BP 2(pn-1+...+p+1)+2K(Z(p); n + 2) we have
vnbn + vn+1 bn+1 + . .+.vn+j bn+j + . .=.0;
(1-20) and ae
*(bn+j ) = Qn+j Qn-1 . .Q.1(on+2 ) 6= 0 for j 0:
6 HIROTAKA TAMANOI
Here bn+j = b(1;2;:::;n-1;n+j)in our previous notation bS for S 2 S+n.
We have corresponding statements for the mod pj Eilenberg-Mac Lane spaces.
It is well known that the BP cohomology of K(Zp; 1) is given by
*
(1-21) BP * K(Zp; 1) ~=BP [ [ x ] ]= [ p ](x) ;
where [ p ](x) is the p-series with respect to the formal group law in BP theor*
*y. In
analogy, we would like to think of the BP-linear relation in (1-20) as a vn-ana*
*logue
of p-series. But then we must ask ourselves "what is a vn-series?" Is there suc*
*h a
thing at all? If it exists, where does it live? We answer these questions by ex*
*plicitly
presenting a vn-series with right properties.
Let {BP__*} be the -spectrum of BP . By Wilson's Splitting Theorem [W],
BP__2(pn+...+p+1)+tis a factor space of BP__2(pn+...+p+1)+twhen t 0. Let
(1-22) 2(pn+...+p+1)+t: BP__2(pn+...+p+1)+t--! BP__2(pn+...+p+1)+t; *
*t 0;
be the inclusion map afforded by Splitting Theorem. The map above can be
thought of as a BP cohomology class which we call a BP fundamental class of the
space BP__2(pn+...+p+1)+tfor t 0. When t > 0, BP fundamental classes do not
exist.
Definition 1-8 [Definition 6-4]. Let 2(pn-1+...+p+1)+2be the BP fundamen-
tal class for the space BP__2(pn-1+...+p+1)+2. The pull-back of this class *
*by vn-
multiplication map
(1-23) vn : BP__2(pn+...+p+1)--! BP__2(pn-1+...+p+1)+2
is defined to be the vn-series denoted by [vn]. Namely,
2(pn-1+...+p+1)+2
(1-24) [vn] = v*n2(pn-1+...+p+1)+2 2 BP BP__2(pn+...+p+1):
Observe that when n = 0, we have BP__<0>2 = K(Z(p); 2) = CP 1(p)and [v0] 2
BP 2 CP 1(p)is the following map:
<0>2
(1-25) [ p ] = [v0] : CP 1(p)p--!CP 1(p)= BP__<0>2- -! BP__2:
Thus, the element [ p ] 2 BP 2(CP 1(p)) defined by (1-24) coincides with the or*
*dinary p-
series [ p ](x) when we choose <0>2= x 2 BP 2(CP 1(p)) to be the usual BP-orien*
*tation.
To study the vn-series (1-24) in detail, we introduce the following maps for j *
* 1:
vn
(1-26) - n+j: BP__2(pn+...+p+1)--! BP__2(pn+...+p+1)---!BP__2(pn-1+...+*
*p+1)+2
proj------!BP
____2(pn+j+pn-1+...+p+1)----! BP__2(pn+j+pn-1+...+p+*
*1):
Here the map proj is the projection map afforded by Wilson's Splitting
Theorem which, in this case, says
(1-27) Y
BP__2(pn-1+...+p+1)+2~=BP_2(pn-1+...+p+1)+2x BP__2(pn+j+pn-1+...+p*
*+1):
j1
Note that -n+j2 BP 2(pn+j+pn-1+...+p+1)BP__2(pn+...+p+1) for j 1. Let n
be the BP-fundamental class 2(pn+...+p+1)of BP__2(pn+...+p+1).
SPECTRA OF BP-LINEAR RELATIONS AND Vn-SERIES 7
Theorem 1-9 [Proposition 6-5, Proposition 6-6]. (I) The vn-series [vn] 2
BP 2(pn-1+...+p+1)+2BP__2(pn+...+p+1) is of the following form:
(1-28) [vn] = vnn+ vn+1 n+1+ . .+.vn+j n+j+ . .;.
where the convergence is with respect to the BP topology.
(II) The pull-back of the vn-series [vn] by the map
n O . .O.1 : K(Z(p); n + 2) -1-!BP__<1>2p+n+1 -2-! . .-.n-!BP__2(pn+...+p+*
*1)
to the BP cohomology of Eilenberg-Mac Lane space K(Z(p); n + 2) is equal to zer*
*o,
and in BP 2(pn-1+...+p+1)+2K(Z(p); n + 2) the vn-series induces the following *
*in-
finite sum BP-linear relation:
(1-29) vnbn + vn+1 bn+1 + . .+.vn+j bn+j + . .=.0;
where bn+j = (n O . .O.1)*(n+j) 2 BP 2(pn+j+pn-1+...+p+1)K(Z(p); n + 2) for
j 0 has the property
(1-30) ae*(bn+j ) = Qn+j Qn-1 . .Q.1(on+2 ) 6= 0:
Thus, our BP-linear relation (1-20) discovered by our general theory actually
comes from the vn-series, which in turn comes from the BP fundamental class
of the space BP__2(pn+...+p+1). Thus the relation (1-29) can be appropriatel*
*y called
the main relation in the BP cohomology of Eilenberg-Mac Lane space K(Z(p); n+2).
In [RWY], they give a description of BP * K(Z(p); n + 2) as a certain quoti*
*ent.
More precisely, they prove that
* * ffi *
(1-31) BcPp K(Z(p); n + 2) ~=BcPp BP__2(pn+...+p+1) (vn);
where (v*n) is the ideal generated by the image of the following map induced fr*
*om
the vn-multiplication map (1-23):
* *
(1-32) v*n: cBPp BP__2(pn-1+...+p+1)+2 --! BcPp BP__2(pn+...+p+1):
Here, BcPp is the p-adic completion of the BP-spectrum. Our definition of the
vn-series (1-24) was motivated by their result. Unfortunately, [RWY] is not ve*
*ry
explicit about the ideal (v*n) which gives the entire relations in the BP cohom*
*ology
of Eilenberg-Mac Lane spaces. Our results on main relations and the vn-series go
some distance towards clarifying the ideal (v*n) of relations in the BP cohomol*
*ogy.
The organization of this paper is as follows. In x2, we introduce the BP top*
*ology
on BP cohomology groups of any spectrum X, and we show that this topology is
complete Hausdorff when X is a space. In x3 nd x4, we introduce spectra L and L*
*
of BP-linear relations and we show that these theories have Milnor primitives, *
*and
we calculate their mod p cohomology as modules over the Steenrod algebra. In x5,
we demonstrate a close connection between BP-linear relations in BP cohomology
and actions of Milnor primitives in mod p cohomology. Finally, in x6 we prove o*
*ur
main BP-linear relations in the BP cohomology of Eilenberg-Mac Lane spaces and
show that these relations come from vn-series.
8 HIROTAKA TAMANOI
x2. BP topology and a spectra map which calculates infinite sums
For a generalized cohomology theory h*(X) of an infinite dimensional CW com-
plex X, the skeletal filtration topology is commonly used. If the generalized c*
*oho-
mology theory h*( . ) further satisfies the additivity axiom of Milnor [M3], th*
*en we
have the Milnor's exact sequence:
0 --! lim-1h*-1 (X(m) ) --! h*(X) --! lim-h*(X(m) ) --! 0;
m m
where {h*(X(m) )}m2Z is the inverse system formed by the skeletal filtration o*
*n X.
In this exact sequence, lim-1m-term describes the set of elements of infinite f*
*iltration.
If elements of infinite filtration, also called phantom maps, do not exist in h*
**(X),
then the topology is complete Hausdorff and Cauchy sequences converge to unique
limits. Since in generalized cohomology theories, we must routinely consider in*
*finite
sums of elements, a non-Hausdorff topology on cohomology groups causes a serious
problem in studying relations among generators of cohomology groups.
Fortunately, on BP cohomology groups there exists a very natural and inherent
topology which is always complete Hausdorff for any space X which may not be ev*
*en
a CW complex. We call this topology BP-topology which we now define. Recall
that there exists a tower of BP-module spectra BP for n 0 and BP-module
maps [JW1]:
ae ae ae
(2-1) BP ----! BP - ---! BP --! BP <-1> = HZp:
Using the BP-module map ae: BP -! BP , we let
k k k
(2-2) F n BP (X) = Ker ae*: BP (X) --! BP (X) ;
for any spectrum X and for any n -1, k 2 Z. This defines a decreasing filtrati*
*on
on the BP cohomology of X:
T
BP k(X) F -1 F 0 F 1 . . .F n . . . nF n:
The topology on BP k(X) for k 2 Z defined by this filtration is the BP topology.
That is, the base for the neighborhood system of an element x 2 BP k(X) is {x +
F n}n. Thus a sequence of elements {xi} in BP k(X) converges to an element x if*
* for
any integer n there exists an integer N such that ae*(x - xm ) = 0 for all m*
* N.
The BP topology is inherently an unstable notion.
Proposition 2-1. Let X be any topological space (which may not be even a CW
complex ). Then the BP-topology on BP *(X) is complete Hausdorff. Namely, we
have the followings:
"
(2-3) F n BP *(X) = {0};
n-1
~= k
(2-4) abe*= lim-ae*: BP k(X) --! lim-BP (X):
n n
For the proof of this proposition, we need Wilson's Splitting Theorem.
SPECTRA OF BP-LINEAR RELATIONS AND Vn-SERIES 9
Theorem 2-2. [W] (i) Let k 2(pn + . .+.p + 1). Then we have the following
homotopy equivalence among spaces of the -spectra of BP and BP's:
Y
(2-5) BP__k~=BP__k x BP__k+2(pj-1):
jn+1
If k < 2(pn + . .+.p + 1), then this equivalence is as H-spaces.
(ii) Let m n and k 2(pn + . .+.p + 1). Then we have
Ym
(2-6) BP__k ~=BP__k x BP__k+2(pj-1):
j=n+1
If k < 2(pn + . .+.p + 1), then this equivalence is as H-spaces.
(Proof of Proposition 2-1). Wilson's Splitting Theorem shows that for a fixed k,
BP k(X) is a direct summand of BP k(X) for n satisfying k 2(pn + . .+.p + 1*
*),
and for such an n, the induced map ae*: BP *(X) -! BP *(X) is surjective.
We fix one such n0. Then for any m n0, we have
Y j
(*) BP k(X) ~= BP k(X) x BP k+2(p -1)(X);
jn0+1
Ym j
(**) BP k(X) ~= BP k(X) x BP k+2(p -1)(X):
j=n0+1
To show injectivity of bae*, suppose bae*(x) = 0 for an element x 2 BP k(X).*
* By
definition, this means that ae*(x) = 0 in BP k(X) for any m. In the decom-
position (*), let the element in the right hand side corresponding to x 2 BP k(*
*X) be
(xn0; xn0+1 ; : :;:xj; : :):, where xj 2 BP k+2(pj-1)(X). Since the decompos*
*itions
(*) and (**) are compatible, ae*(x) = 0 implies that xn0 = . .=.xm = 0. Sin*
*ce
m n0 is arbitrary, we have xj = 0 for all j n0. Thus,Tfrom (*), thisTimplies *
*that
x = 0. This proves that bae*is injective. Since nF n BP k(X) = nKer ae*=
Ker bae*, (i) follows also.
Next we prove surjectivity of bae*. Let {yn}n 2 lim-nBP k(X) be any eleme*
*nt
in the inverse limit. This means that elements yn are compatible in the sense t*
*hat
for any m n, we have ae*(ym ) = yn. Thus, we only have to consider eleme*
*nts
from the nth0term on. Let m n0. In (**), let (xn0; : :;:xm ) be the element in*
* the
right hand side corresponding to ym 2 BP k(X). If we use different m0such t*
*hat
m0 m, ym0 defines the same element xj 2 BP k+2(pj-1)(X) for n0 j m,
because ym0 and ym are compatible. Since m is arbitrary, we obtain an infinite
sequence of elements
Y j
(xn0; xn0+1 ; : :;:xj; : :):2 BP k(X) x BP k+2(p -1)(X):
jn0+1
Let x 2 BP k(X) be the element in the left hand side of (*) corresponding to the
above sequence. We then have ae*(x) = ym for any m n0, since both elements
correspond to (xn0; : :;:xm ) in the decomposition (**). Hence bae*(x) = {ym }m*
* and
bae*is surjective. This completes the proof that bae*is an isomorphism.
10 HIROTAKA TAMANOI
Corollary 2-3. Let X be any topological space.P Let bi 2 BP k+2(pi-1)(X) be any
element for i 0. Then the infinite sum 1i=0vibi always converges to a unique
element in BP k(X) with respect to the BP-topology.
P n
Proof. Let xn = i=0 vibi be a finite sum, and let yn = ae*(xn) 2 BP k(X)
for any n 0. We claim that elements yn define an element {yn} in the inverse
limit lim-nBP k(X). To see this, observe that
ae*(yn+1 ) = ae*O ae*(xn+1 ) = ae*(xn + vn+1 bn+1 )
= ae*(xn) + ae*(vn+1 ) . ae*(bn+1 ) = ae*(xn)*
* = yn;
where the last two equalities hold because the BP-module map ae*has the prop-
erty ae*(vn+1 ) = 0. Thus, the sequence {yn}n defines an element in the inve*
*rse
limit and by Proposition 2-1, it defines a unique element x 2 BP k(X) such that
ae*(x) = yn for all n.
We now show that the sequence of finite sums x0; x1; : :;:xn; : :c:onverges *
*to x
in BP k(X) with respect to the BP topology. We observe that for any m n,
ae*(x - xm ) = ae*(x) - ae*(xm ) = yn - yn = 0 2 BP k(X);
since ae*(vk) = 0 for k > n. This means that for any given n, we have x - xm*
* 2
F n BP k(X) for all m n. Hence the sequence {xn} converges to x in the BP
topology.
Next, we construct a spectraPmap which automatically calculates infinite sums
of elements of the form i0 vibi in BP *(X), where the convergence is with re*
*spect
to the BP topology.
For this, we recall a few facts about a family of spectra. For details, see*
* Part
III, x3 of Adams [Ad]. Let {Xff}ff2A be a family of CW spectra indexed by ff 2 *
*A.
Then by E.QH. Brown's Representability Theorem, we can consider the product
spectrum ffXffdefined by the property
Q Q
[Y; Xff] = [Y; Xff]
ff ff
W
for any CW spectrum Y . The coproduct ffXffis defined by the property
W Q
[ Xff; Y ] = [Xff; Y ]:
ff ff
The coproduct can be taken to be the one point union spectrum. From the above
two properties, we have a canonical map
_ Y
Xff- -! Xff;
ff ff
whose component Xff-! Xfiis the identity map if ff = fi, and 0 if ff 6= fi. T*
*he
following lemma is well known.
SPECTRA OF BP-LINEAR RELATIONS AND Vn-SERIES 11
Lemma 2-4 [Ad, p157]. Suppose for each n, we have ssn(Xff) = 0 for all but fin*
*itely
many ff. Then the following canonical map is an equivalence:
_ Y
Xff- -! Xff:
ff ff
We go back to constructing a spectra map calculating infinite sums. We consi*
*der
the following composition of BP-module maps:
Y i ~= _ i W vi _ folding
(2-7) : 2(p -1)BP -- 2(p -1)BP - -! BP ----! BP :
i0 i0 i0
The equivalence of the first map is due to Lemma 2-4. Components of the second
map is induced by the vi-multiplication for i 0:
i-1) 2(pi-1) vi^1
(2-8) vi : 2(p BP ~= S ^ BP - --! BP ^ BP - -! BP ;
where is the multiplication map in BP , and S2(pi-1)is a suspension of the sph*
*ere
spectrum. The folding map is the map whose restriction to each component of the
coproduct is the identity map. Namely,
W Q
[ BP ; BP ] = [BP ; BP ]
(2-9) i0 i0Q
folding ! i0 IdBP
We also consider the following related spectra maps:
Yn i ~ _n i W v _n
(2-10) n : 2(p -1)BP --= 2(p -1)BP - -!i BP folding----!BP;
i=0 i=0 i=0
(2-11)
Yn i ~ _n i Wv n_
: 2(p -1)BP -=- 2(p -1)BP --!i BP -folding---!BP:
i=0 i=0 i=0
Of course, the corresponding induced maps on the BP cohomology of a spectrum
X are finite BP-linear sums:
Yn i
n * : BP k+2(p -1)(X)--! BP k(X);
(2-12) i=0 Pn
(b0; b1; : :;:bn)7- ! vibi
i=0
Yn i
*: BP k+2(p -1)(X)- -! BP k(X);
(2-13) i=0 Pn
(b00; b01; : :;:b0n)7- ! vib0i
i=0
We are most interested in the cohomology map induced by :
Y i
* : BP k+2(p -1)(X) --! BP k(X)
(2-14) i0
(b0; b1; : :;:bi; :7:):-! *(b0; b1; : :;:bi; : :)::
12 HIROTAKA TAMANOI
When X is a general spectrum, we do not know whether BP *(X) is a completeP
Hausdorff topological space or not, and an infinite sum of the form 1i=0vibi *
*may
or may not make sense as an element of BP *(X). However, even for a general
spectrum X, the element *(b0; b1; : :;:bi; : :):is always well defined in the BP
cohomology. P
We want to identify the element *(b0; : :;:bi; : :):with the infinite sum *
*i0 vibi
which is convergent with respect to the BP topology by Corollary 2-3 when X is a
space.
We first consider the behavior of * when almost all bi's are zero.
Lemma 2-5. For any spectrum X, let bi 2 BP k+2(pi-1)(X) for 0 i n be any
elements. Then * reduces to a finite BP-linear sum map. Namely,
Xn
(2-15) *(b0; b1; : :;:bn; 0; : :):= vibi:
i=0
Proof. We consider the following commutative disgram of spectra and spectra map*
*s:
1Q i ~= 1W i Wvi 1W folding
: 2(p -1)BP ---- 2(p -1)BP ----! BP ----! BP
i=0 x i=0 x i=0x fl
inclusion?? inclusion?? inclusion?? flfl
Qn i ~= nW i Wvi Wn folding
n : 2(p -1)BP ---- 2(p -1)BP ----! BP ----! BP :
i=0 i=0 i=0
Induced cohomology maps give the following commutative diagram:
1Q i *
(b0; b1; : :;:bn; 0; :2:): BP k+2(p -1)(X) ----! BP k (X)
x? i=0 x? flfl
? ? fl
Qn i n *
(b0; b1; : :;:bn)2 BP k+2(p -1)(X) ----! BP k (X)
i=0
P n
The bottom row is a finite sum map (2-12) and we have n *(b0; : :;:bn) = i=0 *
*vibi.
Hence the commutativity of the above diagram proves (2-15).
Thus, when there are only finitely many non-trivial elements, the map * is
really the summation map with vi-coefficients. We cannot just let n tend to 1
because we must deal with the convergence with respect to the BP topology. Since
the BP topology is defined using the spectra map ae: BP -! BP , we exami*
*ne
the behavior of * with respect to ae*.
Lemma 2-6. Let X be any spectrum. For arbitrary elements bi 2 BP k+2(pi-1)(X)
for i 0, let x = *(b0; b1; : :;:bi; : :):2 BP k(X). Then letting bi= ae**
*(bi) for
i 0, we have
Xn
(2-16) ae*(x) = vi. bi in BP k(X):
i=0
SPECTRA OF BP-LINEAR RELATIONS AND Vn-SERIES 13
Proof. We consider the following commutative diagram:
1Q i ~= 1W i W vi 1W folding
2(p -1)BP ---- 2(p -1)BP ----! BP ----! BP
i=0 ??1Q i i=0 ??1W i i=0??1W ??
y i=02(p -1)ae yi=02(p -1)ae yi=0ae y ae<*
*n>
1Q i ~= 1W i W vi W1 folding
2(p -1)BP ---- 2(p -1)BP ----! BP ----! BP
i=0 ? i=0 ? i=0 x fl
proj?y proj?y inclusion?? flfl
nQ i ~= nW i W vi Wn folding
2(p -1)BP ---- 2(p -1)BP ----! BP ----! BP
i=0 i=0 i=0
The commutativity of squares are obvious, possibly exceptithe lower middle one.
This one commutes because the multiplication map vi: 2(p -1)BP -! BP is
a zero map for i n. The induced cohomology diagram of a spectrum X is
"b= (b0; : :;:bn; bn+1 ; :2:):1QBPk+2(pi-1)(X) --*--! BP k(X) 3 x = *("b)
?? i=0 ?? ??
y y ae*y
Qn i * Pn
(b0; : :;:bn; 0;2: :):BPk+2(p -1)(X)----! BP k(X) 3 vibi :
i=0 i=0
The commutativity of this diagram proves the result.
Now we show that the spectra map does calculate infinite sums with respect
to the BP topology.
Theorem 2-7. ForPany space X, let bi 2 BP k+2(pi-1)(X) for i 0 be any ele-
ments. Let xn = ni=0vibi be a finite sum. Then the sequence {xn} converges to
the element *(b0; : :;:bi; : :):in the BP-topology. That is, in BP k(X) we have
X1
(2-18) *(b0; b1; : :;:bi; : :):= vibi:
i=0
Proof. Let x = *(b0; : :;:bn; : :):2 BP k(X). By Proposition 2-1, we know that
the sequence {xn} converges to a unique element. We must show that this element
is x. For any n, by Lemma 2-6 we have
Xn
ae*(x) = ae**(b0; : :;:bn; bn+1 ; : :): = vi. ae*(bi):
i=0
LetP"bm= (b0; : :;:bn; : :;:bm ; 0; : :):for m n. From Lemma 2-5, we have *("b*
*m) =
m "
i=0vibi = xm . Applying Lemma 2-6 to bm , we have
Xn
ae*(xm ) = ae**(b0; : :;:bn; : :;:bm ; 0; : :): = vi. ae*(b*
*i):
i=0
Hence ae*(x) = ae*(xm ) for all m n. In other words, x - xm 2 F n BP k(*
*X)
for all m n. This means that the sequence {xn} converges to x.
14 HIROTAKA TAMANOI
When X is an infinite dimensional CW complex, we can consider two different
topologies on BP *(X): the BP topology and the skeletal filtration topology. *
*We
compare these two topologies.
Recall that the skeletal filtration on BP k(X) is a decreasing filtration
BP k(X) = G0 G1 . . .Gn . . .
(2-19)
where Gn = Gn BP k(X) = Ker {r*n-1: BP k(X) --! BP k(X(n-1))}:
Here rn : X(n) -! X is the inclusion map of the n-skeleton of X. The skeletal
filtration topology may not be complete Hausdorff due to existence of phantom
maps.
On the other hand, the BP topology is always complete Hausdorff for any infi*
*nite
dimensional CW complex by Proposition 2-1. Although the definitions of these two
topologies are very different, we show that we can compare these two topologies
and in fact the BP topology is finer than the skeletal filtration topology. For
convenience, let FB*P and Fs*keletondenote the BP-filtration (2-2) and the skel*
*etal
filtration (2-19), respectively.
Proposition 2-8. Let X be an infinite dimensional CW complex. Then the BP-
topology is finer than the skeletal filtration topology. More precisely, for a *
*given k,
let m0 be any integer such that k 2(pm0 + . .+.p + 1). Then for any m m0,
k k+2(pm+1 -1) k
(2-20) FBmP BP (X) Fskeleton BP (X) :
Thus, any sequence convergent in the BP-topology is also convergent in the skel*
*etal
filtration topology.
Proof. We consider the following diagram:
ae*
BP k(X) ----! BP k(X)
?? ?
y * ?y*
m+1 ae* m+1
BP k X(k+2p -3) ----! BP k X(k+2p -3) ;
where : X(k+2pm+1 -3)-! X is the inclusion map. For the upper horizontal map,
Ker ae*= FBmP BP k(X) . Since k 2(pm + . .+.p + 1), by Wilson's Splitting
Theorem we have
Y j
BP k(X) ~=BP k(X) x BP k+2(p -1)(X):
jm+1
The above condition on k; m is satisfied if m m0. For the k + 2(pm+1 - 1) - *
*1 -
skeleton of X, for j m + 1 we have
j-1) (k+2pm+1 -3) (k+2pm+1 -3)
BP k+2(p X = [X ; BP__k+2(pj-1)] = 0;
since BP__k+2(pj-1) is at least (k + 2pm+1 - 3)-connected for j m + 1. Thus
(k+2pm+1 -3) ae* k (k+2pm+1 -3)
BP k X - --!~ BP X :
=
Then the above commutative diagram implies that
k * k+2(pm+1 -1) k
FBmP BP (X) = Ker ae* Ker = Fskeleton BP (X) ;
for any m m0. This completes the proof.
SPECTRA OF BP-LINEAR RELATIONS AND Vn-SERIES 15
x3. Spectrum L of infinite sum BP-linear relations
We consider a spectrum which is closely related to the infinite BP-linear sum
map of (2-7). Namely, let L be the cofibre spectrum of the spectra map . The
resulting cofibre sequence is
Q q 1Y i
(3-1) -1 L ---!i 2(p -1)BP --! BP --! L;
i=0
where qi : L --! 2pi-1BP is the map to the ith factor. We study the mod p
cohomology of L and properties of qi. The cofibre sequence (3-1) induces the
following cohomology exact sequence for any spectrum X:
(3-2)
Q q Y1 i
. .-.! L*-1 (X) ---!i* BP *+2(p -1)(X) -*--!BP *(X) -*-! L*(X) --! . . .:
i=0
The mod p cohomology exact sequence of the cofibre sequence (3-1) is of the form
Y1 i *
(3-3) . .-. HZ*-2(pp-1)(BP ) --- HZ*p(BP )
i=0 * P qi* Y1 i
-- HZ*p(L) ----- HZ*-2pp+1(BP ) - . . .:
i=0
When * = 0, the map * : HZ0p(L) -! HZ0p(BP ) is an isomorphism by dimensional
reason and by the fact that p* = 0 on mod p cohomology. Let j : L -! HZp be an
element in HZ0p(L) corresponding to the Thom map ae : BP -! HZp in HZ0p(BP )
under the isomorphism *. That is, ae = *(j) = j O .
Let Qi be the ith Milnor primitive in the mod p Steenrod algebra A(p) [M1, M*
*2].
These elements are defined by
(
Q0 = Bockstein operator;
(3-4) pn pn
Qn+1 = P Qn - QnP ; n 0:
The operation Qi raises the cohomology degree by 2pi- 1.
The purpose of this section is to prove the following theorem.
Theorem 3-1. The spectrum L is a BP-module spectrum with the following prop-
erties. i
(I) Let X be a space. For any z 2 Lk(X), let bi = qi*(z) 2 BP k+2p -1(X) for i *
* 0.
Then we have
pb0 + v1b1 + . .+.vnbn + . .=.0 in BP k+1(X):
Here the convergence is with respect to the BP topology.
(II) For any i 0, the following diagram commutes:
2pi-1 i
L --qi--! 2pi-1BP ------! 2p -1L
? ? ?
(3-5) ?yj ?y2pi-1ae ?y2pi-1j
HZp --Qi--! 2pi-1HZp ________ 2pi-1HZp:
16 HIROTAKA TAMANOI
The Milnor operation bqiin L-theory can be defined by bqi= 2pi-1 O qi for i 0,
and they satisfy bqiO bqj= 0 for any i; j 0.
(III) The mod p cohomology of the spectrum L is the following cyclic module over
the mod p Steenrod algebra A(p) generated by j:
h . X i
(3-6) HZ*p(L) ~= A(p) A(p) . QiQj . j:
i;j0
(IV) The coefficient group of L-theory is such that L* = 0 when * > 0 and L0 ~=*
*Zp
spanned by j. When * < 0, the group L* is torsion free and we have the following
exact sequence:
Q q Y i
0 -! L* ---!i* BP *+2p -1 -*--!BP *+1 -! 0; * < 0;
i0
P
where *(ff0; ff1; : :;:ffi; : :):= i0 viffi is a finite sum map.
(Proof of (I) and (IV)). (I) For a given z 2 Lk(X), by exactness ofP(3-2), we h*
*ave
*(b0; : :;:bn; : :):= 0 in BP k+1 (X). By Theorem 2-7, this means i0 vibi =*
* 0.
This proves (I).
(IV) The homotopy exact sequence of the cofibre sequence (3-1) is
Q q Y i
. .-.! L*-1 - --i*! BP *+2(p -1)--*-! BP * -*-! L* -! . .:.
i0
We observe that Im (*) is the ideal I1 = (p; v1; : :;:vn; : :): BP * and BP *=*
*I1 ~=
Zp concentrated in degree 0. Thus, when * < 0, * is surjective and we obtain the
short exact sequence in (IV).
Part (I) shows that any element z 2 L*(X) gives rise to an infinite sum BP-l*
*inear
relation in BP *(X). This is why we call the spectrum L the spectrum of BP-line*
*ar
relations.
Part (II) is proved by a sequence of lemmas. Note that the commutativity of
the right square in (3-5) follows from the definition of the map j. We prove t*
*he
commutativity of the left square.
To examine the spectra map , we compare it with the vi-multiplication map on
BP. We examine the following cofibre sequence:
i-1) vi ji
(3-7) -1 BP (vi) -fii-!2(p BP - -! BP - -! BP (vi); i 0:
Here BP (vi) is the cofibre of the vi-multiplication map. Its homotopy group is
ss* BP (vi) ~= BP *=(vi). Taking the mod p cohomology of the cofibre sequence
(3-7), we get
* i v*
(3-8) . .-. HZ*+1pBP (vi) -fii--HZ*-2(pp-1)BP -i- HZ*pBP
-j*i-HZ* fi*i *-2pi+1
pBP (vi) --- HZp BP - . .:.
Observe that when * = 0, j*iis an isomorphism. This is clear when i > 0 by
dimensional reason. When i = 0, we get the same conclusion since p* = 0 in mod
p cohomology. Let aei:BP (vi) -! HZp be the map corresponding to the Thom map
ae : BP -! HZp through the isomorphism j*i. Thus, ae = j*i(aei) = aeiO ji for *
*i 0.
The mod p cohomology modules of the spectra BP and BP (vi) are known. Let
P(p) = A(p)=(Q0) be the algebra of Steenrod reduced powers, where (Q0) is the
two-sided ideal generated by Q0.
SPECTRA OF BP-LINEAR RELATIONS AND Vn-SERIES 17
Lemma 3-2 [BM]. As modules over the Steenrod algebra, the mod p cohomologies
of BP and BP (vi) are the following cyclic modules generated by ae and aei:
(3-9) h . i
HZ*p(BP ) = A(p) A(p)(Q0; Q1; : :;:Qn; : :):ae ~=P(p)ae;
h . i
HZ*pBP (vi) = A(p) A(p)(Q0; : :;:bQi; Qi+1; : :):aei ~=P(p)aei P(p)Qi(aei*
*):
Here A(p)(Qj's) is the left ideal generated by Qj's, and bQimeans that Qi is om*
*itted.
It is known that the left ideal (Q0; Q1; : :;:Qi; : :):coincides with the tw*
*o-sided
ideal (Q0). Observe that HZ*p(BP ) is even dimensional and HZ2pi-1pBP (vi) ~=Zp
is spanned by Qi(aei).
We examine the exact sequence (3-8) in the light of Lemma 3-2. We have seen
that p* = 0 by a trivial reason in mod p cohomology. It turns out that for all *
*i 0,
we have v*i= 0 in (3-8).
Lemma 3-3. The induced map v*iin (3-8) on mod p cohomology is trivial and we
have the following short exact sequence:
* i j* i
(3-10) 0 -! HZ*p(BP ) -fii--!HZ*+2pp-1 BP (vi) --i! HZ*+2pp-1(BP ) -! 0:
Here, both fi*iand j*iare A(p)-module maps such that
(3-11) fi*i(ae) = i. Qi(aei); j*i(aei) = ae and
for some nonzero constant i 2 Zp which may depend on i. In (3-10), fi*imaps
P(p) . ae isomorphically onto a summand P(p) . Qi(aei), and j*imaps P(p) . aei *
*iso-
morphically onto P(p) . ae.
Proof. By Lemma 3-2, both HZ*p(BP ) and HZ*pBP (vi) are cyclic A(p)-modules
and we know that j*imaps the module generator aei of HZ*pBP (vi) to the module
generator ae of HZ*p(BP ). Hence j*iin (3-8) is surjective and, by exactness, v*
**iis a
zero map. Thus, we have the short exact sequence (3-10). When * = 0, we have
~= i
an isomorphism fi*i: HZ0p(BP ) --! HZ2pp-1 BP (vi) ~=ZpQi(aei), since the mod*
* p
cohomology of BP is even dimensional. This proves the first formula in (3-11).*
* The
second one is the definition of aei. Since both fi*iand j*iare A(p)-linear, we*
* have
the last statement.
We want to show that the constants i in (3-11) are independent of i, and in
fact they are all equal to 1.
We combine the cofibre sequences (3-7) for all i, and compare it with the co*
*fibre
sequence (3-1). Consider the following diagram:
1W W fii W i Wvi W Wji W
-1 BP (vi) ----! 2(p -1)BP ----! BP ----! BP (vi)
i=0 ?? i0 ?? i0?? i0 ??
(3-12) y -1o ~=yh.e. yfolding yo
Q q Q i
-1 L ----i! 2(p -1)BP ----! BP ----! L:
i0
The commutativity of the middle square comesWfrom the definition of . The map
o is a spectra map between cofibres of vi and induced from the commutative
middle square. With this definition of o , the above diagram commutes.
We know the behavior of the mod p cohomology of the top row by Lemma 3-3.
This implies the following result for the mod p cohomology of the bottom row.
18 HIROTAKA TAMANOI
Lemma 3-4. The spectra map induces a zero map in mod p cohomology, and
we have the following short exact sequence:
Y i (Q qi)*=P q*i *
(3-13) 0 -! HZ*-2pp+1(BP ) ---------! HZ*p(L) --! HZ*p(BP ) -! 0:
i0
With the same nonzero constant i 2 Zp as in (3-11), we have
(3-14) q*i(ae) = i. Qi(j) for i 0:
Proof. Considering the mod p cohomology of the middle square of (3-12), we have
the following commutative diagram:
Q *-2(pi-1) Q v*i Q
HZp (BP ) ---- HZ*p(BP )
i0 flfl i0 x?
fl ? diagonal
Q *-2(pi-1) *
HZp (BP ) ---- HZ*p(BP ):
i0
Note that the cohomology map induced from the folding map is the diagonal map.
Since v*i= 0 by Lemma 3-3, it follows that * = 0. QThus we obtain the short
exactPsequence (3-13). Note that the induced map ( iqi)* is actually a finite *
*sum
map i q*iby dimensional reason. From (3-12), we obtain the following diagram
in which both rows are short exact:
Q Q j*i Q Q fi*i Q i
0 HZ*p(BP ) ---- HZ*pBP (vi) ---- HZ*-2pp+1(BP ) 0
i0 x? i0 x? i0 flfl
?diagonal ?o* fl
* P q*i Q i
0 HZ*p(BP ) ---- HZ*p(L) ---- HZ*-2pp+1(BP ) 0:
i0
First we let * = 0 in this diagram. Then the rightQend groups are both zero bec*
*ause
HZ*p(BP ) is even dimensional. Thus, both * and j*iare isomorphisms in this
degree. Since *(j) = ae and j*i(aei) = ae for i 0 by definition, it follows th*
*at
(3-15) o *(j) = (ae0; ae1; : :;:aei; : :)::
Next, we letQ* = 2p` -P1 for some ` 0. This time, left end groups are zero and
both maps fi*iand q*iare isomorphisms in this case, and consequently o *is
also an isomorphism in this degree. Let ae 2 HZ0p(BP ) be in the `th factor of*
* the
right end group. We examine the behavior of this element in this diagram.
* Q * *
o * q`(ae)= ( fii)(0; : :;:0; ae; 0; : :):= (0; : :;:0; fi`(ae); 0; *
*: :):
= (0; : :;:0; `Q`(ae`); 0; : :):;
where the last equality is due to (3-11). On the other hand Q`(j) 2 HZ2p`-1p(L),
and by naturality of cohomology operations, we have
*
o * Q`(j) = Q` o (j) = Q`(ae0; ae1; : :;:ae`; : :)::
SPECTRA OF BP-LINEAR RELATIONS AND Vn-SERIES 19
Since Q`aei = 0 when i 6= ` by lemma 3-2, by derivation property of Q`, the
above is further equal to (0; : :;:0; Q`ae`; 0; : :):. Comparing this with the *
*previous
calculation, we see that o * q*`(ae) = o * `Q`(j) . Since o *is an isomorphis*
*m in
the degree we are working, we finally have q*`(ae) = ` . Q`(j). Since ` 0 is
arbitrary, we get (3-14). This completes the proof.
This proves Part (II) of Theorem 3-1 up to nonzero constant multiples ` 2 Zp.
Our next task is to show that all the constants ` are equal to 1. For this, *
*we
need a preparation. We consider the following cofibre sequence:
(3-16) . .-.!-1 HZp -fi-!HZ(p)--p! HZ(p)--j! HZp -! . .:.
The Bockstein operator Q0 is then defined by
(3-17) Q0 = fi*(j) = j O fi 2 HZ1p(HZp):
We first show that 0 in (3-11) and (3-14) is equal to 1.
Lemma 3-5. Let q0 : -1 L -! BP and fi0 : -1 BP (vi) -! BP be as in (3-1) and
(3-7). Then
(3-18) fi*0(ae) = Q0(ae0); q*0(ae) = Q0(j):
Proof. By Lemma 3-4, the first identity implies the second identity. To see the
first identity, we consider the following commutative diagram between the cofib*
*re
sequence (3-16) and (3-7) with i = 0:
. . .----! -1 BP (v0) --fi0--! BP ---p-! BP --j0--! BP (v0) ----! . .*
* .
?? ? ? ?
y -1ae0 ?yae<0> ?yae<0> ?yae0
. . .----! -1 HZp --fi--!HZ(p) ---p-! HZ(p) ---j-! HZp ----! . . *
*.:
Here, v0 = p. In the associated commutative diagram of mod p cohomologies, we
have p* = 0 and we obtain the following diagram:
fi*0 p*=0
. . .---- HZ1pBP (v0) ---- HZ0p(BP ) ---- 0
x? x
? ae*0 ??ae*<0>
* p*=0
. . .---- HZ1p(HZp) --fi-- HZ0p(HZ(p)) ---- 0:
Since fi*(j) = Q0 by (3-17) and ae*<0>(j) = ae, we have
* * * * *
fi*0(ae) = fi*0ae<0>(j) = ae0fi (j) = ae0 Q0(1) = Q0 ae0(1) = Q0(ae0*
*):
Here we used the naturality of Q0. This proves the first identity of (3-18), an*
*d the
second one follows from this.
Next we show that ` = 1 for all ` 0. Since these constants are "universal"
constants, we only have to prove this for a particular example. As such an exam*
*ple,
20 HIROTAKA TAMANOI
we use the infinite dimensional lens space Lp. Recall that the mod p cohomology
of the lens space is given by
V
HZ*p(Lp) ~=Zp[ x ] Zp(ff); |x| = 2; |ff| = 1;
(3-19) i
where Qi(ff) = xp; i 0:
The BP cohomology of Lp was calculated in [L] and it is given by
2
BP *(Lp) = BP *[ [ x ] ] = [ p ]BP(x) ; x 2 BP (Lp)
(3-20)
where [ p ]BP(x) = x +BP x +BP . .+.BPx
is the p-series for the BP-formal group law [Ar, H]. One can show easily that
X i
(3-21) [ p ]BP(x) = expBP(px) +BP BP vixp :
i0
Let I1 = (p; v1; : :;:vn; : :):be the maximal ideal of BP *. From (3-21), it f*
*ollows
that there exist elements yi 2 BP *[ [ x ] ] for i 0 such that
[ p ]BP(x)= py0 + v1y1 + . .+.viyi+ . .;.
(3-22) i
yi xp mod I1 for i 0:
Note that we may take y0 = x exactly. From the cofibre sequence (3-1), we have
the following induced cohomology exact sequence for the lens space Lp:
Q q Y i
. . .-!L1(Lp) ---!i* BP 2p(Lp) -*--!BP 2(Lp) -! . .:.
i0
P
Since *(y0; y1; : :;:yi; : :):= i0 viyi = 0 in BP *(Lp) by (3-20) and (3-22)*
*, from
the exactness of the above sequence, there exits an element z 2 L1(Lp) such that
yi = qi*(z) for all i 0. Formula (3-14) implies commutativity of the following
diagram for each ` 0:
L1(Lp) --q`*--!BP 2p`(Lp)
? ?
(3-23) ?yj* ?y ae*
HZ1p(Lp) -`Q`---!HZ2p`p(Lp):
To show that ` = 1 for all ` 0, it is necessary that all elements are chosen in
a coherent way. Thus, we fix x = xBP 2 BP 2(Lp) and define xH 2 HZ2p(Lp) by
xH = ae*(xBP ). Then we fix ff 2 HZ1p(Lp) by Q0(ff) = xH .
Lemma 3-6. The diagram (3-23) commutes with ` = 1 2 Zp for all ` 0. Thus
(3-24) q*`(ae) = Q`(j) in HZ*p(L) for all` 0:
Proof. First we examine (3-23) with ` = 0. From Lemma 3-5 we know that the
diagram (3-23) commutes with 0 = 1. Hence
x = xH = ae*(xBP ) = ae* q0*(z) = Q0 j*(z) :
SPECTRA OF BP-LINEAR RELATIONS AND Vn-SERIES 21
Since HZ1p(Lp) ~=Zpff and Q0(ff) = x, we must have j*(z) = ff exactly. But then
the commutativity of the diagram (3-23) for ` 1 implies
` p`
xp = ae*(y`) = ae* q`*(z) = `Q` j*(z) = `Q`(ff) = `x :
Thus, we must have ` = 1 for all ` 1. This completes the proof.
This completes the proof of Part (II) of Theorem 3-1.
(Proof of Part (III) of Theorem 3-1). First we prove the relations QiQj(j) = 0 *
*for
any i; j 0. Part (II) gives q*j(ae) = Qj(j) in HZ*p(L). By naturality of cohom*
*ology
operations,
QiQj(j) = Qi q*j(ae) = q*jQi(ae) = 0
since Qi(ae) = 0 for any i 0 by Lemma 3-2. In the proof of Lemma 3-4, we had
the following exact sequence:
Y i P q*i *
0 -! HZ*-2pp+1(BP ) ---! HZ*p(L) --! HZ*p(BP ) -! 0:
i0
Let R be a sequence of non-negative integers almost all zero, and let PR be the
corresponding Milnor's Steenrod reduced power operation [M1]. By naturality
q*iPR (ae) = PR q*i(ae) = PR Qi(j) for any sequence R and for i 0. Since
*(j) = ae, as Zp-vector spaces (not as A(p)-modules) we have
M hL i hL i
HZ*p(L) = ZpPR Qi(j) ZpPR (j) ;
i0 R R
P
where the first summand is the monomorphic image of q*iand the second sum-
mand maps isomorphically onto HZ*p(BP ) by *. This shows that QiQj(j) = 0 for
i; j 0 are the only relations in HZ*p(L). This completes the proof of Part (II*
*I) of
Theorem 3-1.
x4. Spectrum L of finite sum BP-linear
relations and its relation to L
In x2, we introduced the following spectra map:
(4-1) Yn n W n
i-1) ~= _ 2(pi-1) vi _ folding
: 2(p BP -- BP --! BP ----! BP :
i=0 i=0 i=0
The induced map on the cohomology of a spectrum X is a finite BP-linear sum of
(2-13). In x3, we constructed the spectrum L of BP-linear relations as the cofi*
*bre
of . Here, we define a spectrum L as the cofibre of the spectra map . We
have the following cofibre sequence:
Q q Yn i
(4-2) -1 L ----i-! 2(p -1)BP ---! BP --! L:
i=0
Note that when n = 0, we have BP = HZ(p), and the cofibre sequence (4-2)
reduces to the cofibre sequence (3-16) for the p-multiplication map HZ(p)-p!HZ(*
*p),
and consequently we have L<0> = HZp.
22 HIROTAKA TAMANOI
We examine the following portion of the mod p cohomology of the cofibre se-
quence (4-2):
Yn i P q
. .-.! HZ-2pp+1 (BP ) ----i--!*HZ0p(L)
i=0 * *Yn i
--! HZ0p(BP ) ---! HZ-2(pp-1)(BP ) -! . . .
i=0
The left end group is zero by dimensional reason. The right end group is trivi*
*al
except the i = 0 case, and the map *reduces to p* which is zero in mod p
cohomology. Hence *is an isomorphism. Let j: L -! HZp be the map
corresponding to the generator ae: BP -! HZp of the group HZ0p(BP ) ~*
*=Zp
under *. That is, ae = *(j) = jO .
We prove results for L which correspond to Theorem 3-1 for L.
Theorem 4-1. (I) Let X be a spectrum, and let z 2 Lk(X) be any element.
Let bi = qi*(z) 2 BP k+2pi-1(X) for 1 i n. Then we have
pb0 + v1b1 + . .+.vnbn = 0 in BP k+1 (X):
(II) There exists a map j:L -! L for n 0 such that j = jO j, and t*
*he
following diagram commutes for each 0 i n:
2pi-1 i
L --qi--! 2pi-1BP - ----! 2p -1L
?? ? ?
yj ?y2pi-1ae ?y2pi-1j
(4-3) L -qi---!2pi-1BP -2pi-1------!2pi-1L
?? ? ?
yj ?y2pi-1ae ?y2pi-1j
HZp --Qi--! 2pi-1HZp ________ 2pi-1HZp:
Milnor operations bqi in L-theory can be defined by bqi = 2pi-1O qi*
*
for 0 i n, and they satisfy bqi O bqj = 0 for 0 i; j n.
(III) The mod p cohomology of the spectrum L is the following cyclic module
over the mod p Steenrod algebra generated by j:
h . X i
(4-4) HZ*p(L) ~= A(p) A(p)QiQj . j:
0i;jn
(IV) The coefficient group of L cohomology theory is described as follows:
L* = 0 if* > 0; L0 ~=Zp; L