MULTIPLICATIVE INDECOMPOSABLE SPLITTINGS OF MSp[2]
Hirotaka Tamanoi
Institut des Hautes Etudes Scientifiques
Abstract. When 2 is inverted, the symplectic cobordism cohomology theory *
*be-
comes complex oriented with respect to the Buhstaber orientation. We stud*
*y multi-
plicative idempotents in this theory in detail. Such multiplicative idemp*
*otents can
only annihilate polynomial generators in degree -4n, where 2n + 1 is not *
*a prime
power. We split off a multiplicatively indecomposable smallest possible *
*nontrivial
cohomology theory from the above theory. The coefficient ring of this th*
*eory has
generators in the same degrees as the odd primary Brown-Peterson theories*
* for ar-
bitrary odd primes.
Contents
1.Introduction and summary of results. 1
2.Complex and symplectic orientations: ring maps for the Thom
spectra MU and MSp. 6
3.Buhstaber splitting and ring maps of MU[2]factoring through
MSp[2]. 11
4.Constructing multiplicative idempotents in MSp[2]. 17
5.General multiplicative idempotents in MSp[2]. 25
6.Multiplicative indecomposable splittings of MSp[2]. 31
References. 33
x1. Introduction and Summary of Results
Let MSp be the symplectic cobordism Thom spectrum. It is a ring spectrum
whose homotopy groups are isomorphic to the symplectic cobordism ring, MSp* =
Sp*. It is well-known that the symplectic cobordism ring has 2-torsion and when
2 is inverted, it becomes isomorphic to the oriented cobordism ring SO* with 2
inverted, that is, in terms of spectra we have MSp[2]' MSO[2] . The connect*
*ion
aeSpU
between these two spectra is that there are natural ring spectra maps MSp --!
U
MU -aeSO-!MSO corresponding to the forgetful functors to the complex cobordism
theory, and then to the oriented cobordism theory.
______________
1991 Mathematics Subject Classification. 55.
Key words and phrases. Complex cobordism cohomology, formal group law, multi*
*plicative
idempotent, symplectic cobordism cohomology.
Typeset by AM S-T*
*EX
1
2 HIROTAKA TAMANOI
It is well known that the MU spectrum localized at a prime p splits into a w*
*edge
sum of infinitely many suspension copies of the BP spectrum for the prime p, and
that the BP spectrum is the smallest ring spectrum which MU(p) can contain.
In this paper, we decompose the symplectic cobordism Thom spectrum MSp[2]
localized away from 2 in terms of the "smallest" ring spectrum in MSp[2], after
describing multiplicative idempotents in MSp[2]-theory.
Our method to split off the smallest ring spectrum from MSp[2]is rather dif-
ferent from the method used in [Q] to split off the BP spectrum from the local-
ized complex cobordism spectrum MU(p). Quillen controlled logarithms of mul-
tiplicative idempotents on MU(p), whereas we control maps on the cohomology
group MSp*[2](HP1 ) = MSp*[2][ [ w ] ] induced by multiplicative idempotents, w*
*here
w 2 MSp4[2](HP1 ) is the symplectic orientation given in (2-4) below. Our basic
tool is the following result.
Proposition 1 [Proposition 2-3]. Ring spectra maps o : MSp[2]-! MSp[2]are
in 1 : 1 correspondence with power series of the form
X
(1-1) g(w) = w + (-1)n2flnwn+1 2 MSp4[2][ [ w ] ]:
n1
Furthermore, the ring spectra map o is a multiplicative idempotent if and only *
*if
the induced map o* : MSp*[2]-! MSp*[2]annihilates all the higher coefficients of
g(w), that is, o*(fln) = 0 for all n 1.
This is completely analogous to the corresponding fact for the complex cobor*
*dism
Thom spectrum MU [Proposition 2-1]. The above proposition can be generalized
to a slightly more general context [Proposition 2-4, Lemma 2-5].
Since all torsion elements in MSp* are 2-primary and the forgetful map aeSpU*
**is
an injection modulo torsion, we can regard MSp*[2] = MSp* Z[ 1_2] as a subring
of MU[2] *. Buhstaber constructed a multiplicative idempotent : MU[2]-! MU[2]
such that for z 2 MU*[2], we have *(z) = z if and only if z 2 MSp*[2] [B1, B2].
Thus the spectrum MSp[2]can be thought of as a subring spectrum of MU[2]. Let
MU[2]-ss!MSp[2]-j!MU[2]be the projection and the inclusion map between these
ring spectra. For more on the Buhstaber splitting, see Theorem 3-2.
To any ring spectra map o : MSp[2]-! MSp[2], there corresponds a ring spectra
map ^o: MU[2]-ss!MSp[2]-o! MSp[2]-j! MU[2]factoring through MSp[2]. By
Proposition 1, there corresponds a degree 4 element g(w) 2 MSp4[2](HP1 ) to o .*
* On
the other hand, to ^othere corresponds a degree 2 element ^o*(x) in MU*[2](CP1 *
*) =
MU*[2][ [ x ] ] by Proposition 2-1 in x2, where x 2 MU2(CP1 ) is the standard M*
*U-
orientation. Since MUQ * = Q[m1; m2; : :;:mn; : :]:where mn = [CPn]=(n + 1), *
*it
is convenient to do calculations in MUQ * to study properties of ring spectra *
*maps
o between MSp[2]. To describe the element ^o*(x), let __x= expMU - logMU(x) be
the inverse power series with respect to the MU-formal group law.
Proposition 2 [Propositions 3-4, 3-5]. Let o : MSp[2]-! MSp[2]be a ring
spectra map, and let ^o : MU[2]-! MU[2]be the associated ring map factoring
through MSp[2]. Then ^o*(x) is an odd power series given by
r _________
(1-2) o^*(x) = x -g(xx_)_x22 MU2[2][ [ x ] ];
MULTIPLICATIVE IDEMPOTENTS OF MSp
[2] 3
P
where g(w) = o*(w) = w + n1 (-1)n2flnwn+1 2 MSp*[2][ [ w ] ] is the power ser*
*ies
associated to o . Furthermore, in MUQ *, we have
ae ^o(m ) = 0; for allk 1;
(1-3) * 2k-1
^o*(m2k) m2k - flk; mod (decomposables ) for allk 1:
We are primarily interested in multiplicative idempotents on MSp[2]among gen-
eral ring spectra maps. We construct such multiplicative idempotents as success*
*ive
compositions of certain multiplicative idempotents. Let
^L= {` 2 N | 2` + 1 is not a prime power};
(1-4) ^
P = {k 2 N | 2k + 1 is a prime power}:
Obviously, ^L[ ^P= N. Note that the smallest integer in the set ^Lis 7. By Miln*
*or's
criterion, for any ` 2 ^Lthere exists an indecomposable element ` 2 MSp-4`[2]
such that s`(`) = 1, where s`( . ) is a Pontryagin number which vanishes on
decomposable elements. For k 2 ^P, there exists an indecomposable element k 2
MSp-4k[2]such that sk(k) = p if 2k + 1 = pj for some odd prime p and j 1. The
choice of these elements ` and k is not unique.
First we describe the simplest nontrivial multiplicative idempotents in MSp[*
*2]-
theory. For ` 2 ^L, let ` 2 MSp-4`[2]be an indecomposable element such that
s`(`) = 1. Let o = o [ `] : MSp[2]-! MSp[2]be a ring spectra map whose
associated power series is of the form
(1-5) go(w) def=o*(w) w + (-1)`2`w`+1 mod (`w`+2);
that is, the first nontrivial higher coefficient is (-1)`2` and the rest of the*
* higher
coefficients are in the ideal (`) MSp*[2]. We know that such a ring map exists
by Proposition 1.
Proposition 3 [Proposition 4-1]. For ` 2 ^L, let ` 2 MSp-4`[2]be any indecom-
posable element such that s`(`) = 1. Then any ring spectra map o : MSp[2]-!
MSp[2]whose associated power series is of the form (1-5) is a multiplicative id*
*em-
potent such that Ker o* = (`) MSp*[2].
Let us call idempotents described in Proposition 3 basic idempotents. Two ba*
*sic
idempotents do not commute in general under compositions. However, by succes-
sively composing such multiplicative idempotents on the right, we can construct*
* a
multiplicative idempotent with specified kernel. When the kernel is finitely g*
*en-
erated, such multiplicative idempotents are constructed in Theorem 4-2. For the
general case, a limiting argument in Proposition 4-4 proves the following theor*
*em.
Theorem 4 [Theorem 4-5]. For any subset L ^L and for any choice of inde-
composable elements ` 2 MSp-4`[2]with s`(`) = 1 for each ` 2 L, there exists a
multiplicative idempotent o : MSp[2]-! MSp[2]such that the kernel of the induced
map o* : MSp*[2]-! MSp*[2]is the ideal generated by the elements ` for ` 2 L.
Furthermore, the associated power series o*(w) 2 MSp*[2][ [ w ] ] is of the form
X
(1-6) o*(w) w + (-1)`2`w`+1 mod (` | ` 2 L) \ (decomposables ):
`2L
4 HIROTAKA TAMANOI
On the other hand, if we compose basic idempotents successively on the left,
the resulting ring map is not an idempotent in general. However, we can mod-
ify idempotents inductively so that the composition gives rise to a multiplicat*
*ive
idempotent [Theorem 4-6]. By considering the limiting case, we obtain an altern*
*ate
proof of Theorem 4 above.
The kernels of the multiplicative idempotents in Theorem 4 are generated by
indecomposable elements in degree -4`, where ` is in the set ^L. It turns out
that no indecomposable elements in degree -4k for k 2 ^Pcan be annihilated by
multiplicative idempotents on MSp[2]in which only 2 is inverted. To be more
precise,Pfor a given multiplicative idempotent o : MSp[2]-! MSp[2], let o*(w) =
w + n1 (-1)n2flnwn+1 2 MSp*[2][ [ w ] ] be the associated power series. No*
*t all
coefficients in o*(w) are indecomposable. But those which are indecomposable pl*
*ay
an important role in the description of o* : MSp*[2]-!MSp*[2]. We let
(1-7) Lo = {n 2 N | sn(fln) 6= 0};
where fln's are higher coefficients of o*(w). We show that this subset of N can*
*not
be arbitrarily large. In fact, the set Lo is always contained in ^L.
Theorem 5 [Theorem 5-3]. Let o : MSp[2]-! MSp[2]be a multiplicative idem-
potent, and let the associated power series o*(w) and the set of integers Lo N*
* be
as above. Then the following statements hold.
(i) We have Lo ^L. Furthermore, for each ` 2 Lo, we have in fact s`(fl`) = 1.
(ii)For any integer k 2 N \ Lo, there exists an indecomposable element k 2
MSp-4k[2]such that (a) MSp*[2]= Z[ 1_2][ fl`; k | ` 2 Lo; k 2 N \ Lo ] as*
* a poly-
nomial algebra, and (b) the induced multiplicative idempotent o* : MSp*[2*
*]-!
MSp*[2]on the homotopy groups is given by
aeo (fl ) = 0; ` 2 L
(1-8) * ` o
o*(k) = k; k 2 N \ Lo:
The kernel Ker o* MSp*[2]is the ideal generated by the indecomposable el*
*e-
ments fl` for ` 2 Lo.
(iii)For any k 2 N \ Lo, the corresponding coefficient flk of wk+1 in o*(w) i*
*s a
decomposable element in the ideal (fl` | ` 2 Lo).
As a simple consequence of Theorem 4, we have the following description of t*
*he
kernel of o* : MSp*[2]-!MSp*[2].
Corollary 6 [Corollary 5-4]. Let o : MSp[2] -! MSp[2]be a multiplicative
idempotent. Then the kernel of the induced map o* : MSp*[2]-! MSp*[2]is the
idealPgenerated by the higher coefficients of the associated power series o*(w)*
* =
w + n1 (-1)n2flnwn+1 MSp4[2][ [ w ] ].
In Theorem 5 and Corollary 6, we assumed that a multiplicative idempotent
o : MSp[2] -! MSp[2]is given, and we studied its properties through its as-
sociatedPpower series. Now suppose we are given a power series g(w) = w +
k1 (-1)k2flkwk+1 2 MSp*[2][ [ w ] ]. We consider the problem of determining *
*the
MULTIPLICATIVE IDEMPOTENTS OF MSp
[2] 5
necessary and sufficient condition on the power series g(w) so that the corresp*
*ond-
ing ring map og : MSp[2]-! MSp[2]is actually a multiplicative idempotent. We
give an inductive criterion on g(w). For n 1, let
Xn
(1-9) g(n)(w) = w + (-1)k2flkwk+1
k=1
be a truncated polynomial of g(w), and let o (n): MSp[2]-! MSp[2]be the corre-
sponding ring map. In Lemma 5-6, we observe that o is a multiplicative idempote*
*nt
if and only if o (n)is a multiplicative idempotent for all n 1. By examining t*
*he
transition from o (n)to o (n+1)[Lemma 5-7, Lemma 5-8], we obtain the following
inductive characterization of those power series corresponding to idempotents.
Theorem 7 [Theorem 5-9]. The ring spectra map og : MSp[2]-! MSp[2]cor-
P
responding to the power series g(w) = w + k1 (-1)k2flkwk+1 is an idempotent
if and only if o*(n-1)(fln) = sn(fln)fln for all n 1, where o (n)is the ring m*
*ap
corresponding to the degree n + 1 truncated polynomial of g(w). Equivalently a*
*nd
more concretely, og is a multiplicative idempotent if and only if for all n 1,*
* we
have
(1) fln 2 Ker o*(n-1)and fln is decomposable, if 2n + 1 is a prime power,
(2) fln 2 Ker o*(n-1), if 2n + 1 is not a prime power and fln is decomposab*
*le,
(3) fln 2 Im o*(n-1)and sn(fln) = 1, if 2n + 1 is not a prime power and fln*
* is
indecomposable.
From (i) in Theorem 5 above, the set Lo is always contained in ^L, and from
Theorem 4 there are multiplicative idempotents o on MSp[2]for which Lo = ^L.
Let LSpo be the ring spectrum split off from MSp[2]as the image of a multiplica*
*tive
idempotent o for which Lo = ^L. From Theorem 5 (ii), its homotopy group is a
polynomial algebra given by
(1-10) LSpo *= Z[ 1_2][ k | k 2 ^P];
where k 2 MSp-4k[2]is an indecomposable element such that sk(k) = p if 2k + 1 is
a power of the odd prime p. Although the way LSpo sits inside of MSp[2]depends
on the multiplicative idempotent o , it turns out that the homotopy type of the
spectrum LSpo is independent of o satisfying the property Lo = ^L.
Theorem 8 [Theorem 6-2, Proposition 6-3]. Let o : MSp[2]-! MSp[2]be a
multiplicative idempotent such that Lo = ^L. Then the ring spectrum LSpo split *
*off
from MSp[2]as the image of o is multiplicatively indecomposable in the sense th*
*at
any multiplicative idempotent on LSpo is the identity map. For any multiplicati*
*ve
idempotents o1 and o2 on MSp[2]such that Loi = ^Lfor i = 1; 2, the associated r*
*ing
spectra LSpo1 and LSpo2 are homotopy equivalent.
Since the homotopy type of LSpo is independent of o , we simply denote this
spectrum LSp without any reference to the multiplicative idempotent used. It
follows that MSp[2]can be decomposed as a wedge sum of many suspension copies
of LSp as follows [Corollary 6-4] :
_
(1-11) MSp[2]' I2|I|LSp;
6 HIROTAKA TAMANOI
where I ranges over all finite (possibly empty) sequences of integers from 2 ^L:
I = (i1 i2 . . .ir | ik 2 2 ^Lfor1 k r; r 0);
P
and |I| = j ij.
The organization of this paper is as follows. In x2, we describe how complex
orientations and symplectic orientations control ring spectra maps for complex *
*or
symplectic cobordism Thom spectra. Here, we describe the symplectic case in de-
tail. We also characterize multiplicative idempotents among ring spectra maps. *
*In
x3, we first describe the relationship between the standard symplectic orientat*
*ion
and the standard complex orientation. Using the Buhstaber splitting, we describe
the ring spectra map on the symplectic cobordism Thom spectra MSp[2]localized
away from 2 in terms of the complex orientation and we derive various useful fo*
*rmu-
lae. In x4, we construct various multiplicative idempotents on MSp[2]as success*
*ive
compositions of basic multiplicative idempotents on the right or on the left. *
*In
x5, we discuss general properties of multiplicative idempotents in MSp[2], and *
*in
particular we show a close relationship between kernel ideals of idempotents and
coefficients of the associated power series. We also give an inductive characte*
*riza-
tion of those power series corresponding to multiplicative idempotents on MSp[2*
*].
In x6, we split off multiplicatively indecomposable ring spectra from MSp[2]and
we decompose MSp[2]into a wedge sum in terms of these indecomposable spectra.
Acknowledgement. The author thanks the referee for various comments which lead
to improvement of the exposition.
x2. Complex and Symplectic Orientations:
Ring Maps for the Thom Spectra MU and MSp
In this section, we describe how orientations in MU-theory and MSp-theory can
be used to control ring spectra maps. Since the literature is readily availabl*
*e for
relevant facts on MU-theory (see [A2], for example), we only recall the basic f*
*acts
for MU-theory. Here we describe the MSp-theory case in some detail.
Notation. Let P be a set of primes. For a spectrum E, let E[P] denote the
spectrum obtained from E by inverting primes in P . When P is the set of all
primes, then E[P]is also denoted by EQ because its homotopy group is a Q-algebr*
*a.
If P contains all primes except one prime p, then E[P]is the spectrum E(p)local*
*ized
at p. In particular, E[2]denotes the spectrum obtained by inverting only 2.
Let j -! CP1 = BU(1) be the universal (tautological) complex line bundle on
CP1 . The inclusion map of CP1 as a zero section into j induces a homotopy
equivalence with the associated Thom complex: CP1 --'! Mj = MU(1). The
standard complex orientation is defined by h:e:
(2-1) x = xMU : CP1 -'-! Mj = MU(1) -! 2MU:
h:e:
This defines a nontrivial element x 2 MU2(CP1 ). If we apply the Thom map ae :
MU -! HZ from MU to the integral Eilenberg-Mac Lane spectrum, the resulting
element ae*(x) 2 HZ2(CP1 ) is the Euler class of the universal bundle j -! CP1
generating the second integral cohomology group of CP1 .
MULTIPLICATIVE IDEMPOTENTS OF MSp
[2] 7
A spectral sequence argument implies that MU* (CPn) = MU* [ x ]= xn+1 for
all n 0, where x is the restriction of the standard complex orientation xMU t*
*o CPn.
The relation xn+1 = 0 comes from the fact that xn+1 : CPn -! MU(n + 1) is null
homotopic, since MU(n + 1) is 2n + 1-connected and CPn is 2n dimensional. From
this calculation, it follows that the restriction map MU* (CPn+1 ) -! MU* (CPn)
is surjective for all n 0. Then from the Milnor exact sequence [M1], we have
MU* (CP1 ) = MU* [ [ xMU ] ], a formal power series ring with the usual filtrat*
*ion
topology. There are no nonzero elements of infinite filtration because the surj*
*ectiv-
ity of each map in the inverse system implies the vanishing of the lim-1term in*
* the
Milnor exact sequence. As usual, let mk = [ CPk ]=(k + 1) 2 MUQ 2k = MU-2kQ for
k 1. We let m0 = 1. The logarithm series logMU T is defined by
X
(2-2) logMU (T ) = T + mkT k+1 2 MU*Q[ [ T ] ];
k1
where T is a formal variable. We remark that the set of all the primitive el-
ements in MU*Q(CP1 ) is MU*Q-free generated by logMU (x). The power series
inverse to logMU (T ) is denoted by exp MU(T ), that is, exp MU logMU(T ) = T*
* =
logMU expMU (T ) .
Now let o : MU[P] -! MU[P] be a ring spectra map for some set of primes
P . Let o* : MU*[P](CP1 ) -! MU*[P](CP1 ) be the induced algebra map. Then
o*(x) = g(x) 2 MU*[P][ [ x ] ] is a formal power series in x. The following fa*
*ct is
well-known. For example see Lemma 4.6 and Proposition 15.3 in [A2], or Lemma
1.53 and the formula (3.8) in [W].
Proposition 2-1. (I)The above correspondence induces a bijection between the
set of ring spectra maps o : MU[P] -! MU[P] and the set of formal power series
g(x) 2 MU*[P][ [ x ] ] of homogeneous degree 2 such that g(x) x mod (x2).
(II) Given a power series g(x) as above, the effect on MU*[P]of the correspo*
*nding
ring map o = og : MU[P]-! MU[P] is given by the following formula:
X
(2-3) T + o*(mk)T k+1 = logMU g-1 (T ) ;
k1
where T is a formal variable.P
(III) Let g(x) = x+ k1 akxk+1 2 MU* [ [ x ] ] be the power series correspo*
*nding
to a ring map o : MU[P]-! MU[P]. Then o is a multiplicative idempotent if and
only if o*(ak) = 0 for all k 1.
We call power series g(x) as in (I) general complex orientations. The part (*
*III)
of Proposition 2-1 is not usually found in the literature, although the proof is
straightforward. See the proof of Proposition 2-3 below.
The identity (2-3) follows from the fact that logMU is primitive. That is, *
*when
o is a ring map, the induced algebra map o* : MU*[P](CP1 ) -! MU*[P](CP1 ) is
also a coalgebra map preserving primitives.PBy checking the leading coefficient*
*, we
have logMU x = o*(logMU x) = g(x) + k1 o*(mk)g(x)k+1 . If we let g(x) = T or
x = g-1 (T ) in this formula, we obtain (2-3).
We show that there is a corresponding statement for the symplectic cobordism
Thom spectrum MSp. First we define the standard MSp-orientation. Let i -!
8 HIROTAKA TAMANOI
HP1 be the universal (tautological) symplectic line bundle. Its Thom complex
is Mi = MSp(1). The inclusion map of HP1 as a zero section into i induces a
homotopy equivalence with the Thom complex HP1 --'! MSp(1). The symplectic
orientation w 2 MSp4 1 h:e:
[2](HP ) is defined by
(2-4) w : HP1 -'-! MSp(1) -! 4MSp:
h:e:
By an argument using spectral sequences and the Milnor exact sequences similar
to the one used for MU*[P](CP1 ) for any set of primes P , we have
(2-5) MSp*[P](HP1 ) = lim-MSp*[P](HPn) = MSp*[P][ [ w ] ]:
n
We note that the spectral sequence H*(HP1 ; MSp[P] *) =) MSp[P] *(HP1 ) col-
lapses since the image of the element [ HPn ] 2 MSp4n(HP1 ) in H4n(HP1 ; Z)
under the Thom map ae : MSp -! HZ is the generator of that group. Thus, by the
Universal Coefficient Theorem [A1,A2], we have
1
(2-6) MSp*[P](HP1 ) ~=Hom MSp[P] MSp[P] (HP ); MSp[P]
* * *
For any i 0, let qi 2 MSp[P] 4i(HP1 ) be defined by = ffiij for i; j*
* 0.
We have q0 = 1. Then MSp[P] *(HP1 ) is a MSp[P] *-free module generated by
qi's. Let Qi 2 MSp[P] 4i(MSp[P]) be the image of qi+1 under the induced map
w* : MSp[P] *+4(HP1 ) -! MSp[P] *(MSp[P]). Note that MSp[P] *(MSp[P]) is a
MSp[P] *-algebra. Using Thom isomorphisms and spectral sequences, we have
(2-7) MSp[P] *(MSp[P]) = MSp[P] *[ Q1; Q2; : :;:Qn; : :]::
Since the spectral sequence for (2-7) collapses, again by the Universal Coeffic*
*ient
Theorem, the Kronecker pairing induces the following isomorphism:
(2-8) MSp*[P](MSp[P]) ~=Hom MSp[P] MSp[P] (MSp[P]); MSp[P] :
* * *
The above isomorphism is as MSp[P] *-modules. Given a spectra map o : MSp[P]-!
MSp[P], the corresponding MSp[P] *-module map o under (2-8) is given by
o : MSp[P] *(MSp[P]) -(1^o)*---!MSp[P]*(MSp[P]) -*! MSp[P] *;
where : MSp[P]^ MSp[P]-! MSp[P]is the product map in the Thom spectrum
MSp. The next lemma is a straightforward generalization of the MU case and the
proof is essentially the same. But we give its proof for the sake of completene*
*ss.
Lemma 2-2. Under the isomorphism (2-8), there is a bijective correspondence
between the set of ring spectra maps o : MSp[P]-! MSp[P] and the set of algebra
maps : MSp[P] *(MSp[P]) -! MSp[P] *.
Proof. Suppose a spectra map o : MSp[P] -! MSp[P] is a ring map making the
following diagram commutative:
MSp[P]^ MSp[P] --o^o--!MSp[P]^ MSp[P]
? ?
(2-9) ?y ?y
MSp[P] ---o-! MSp[P]:
MULTIPLICATIVE IDEMPOTENTS OF MSp
[2] 9
Applying MSp[P] * ( . ), we obtain the following commutative diagram:
* * * (1^o)*(1^o)*----------!* * *-**---! * * *
? ? fl
(2-10) ?y* ?y* flfl
* ----(1^o)*------! * --*--! *:
Here, for simplicity, we have let = MSp[P] . Using the definition of the map o,
we see that the commutativity of (2-10) shows that o is an algebra map.
Conversely, let : MSp[P] *(MSp[P]) -! MSp[P] * be an MSp[P] *-algebra map
and let o : MSp[P] -! MSp[P] be the corresponding spectra map under (2-8).
Since = * O (1 ^ o )*, the condition that is an algebra map implies that we
have a commutative diagram as in (2-10) with o replaced by o . This implies that
we have the following commutative diagram:
(o ^o )*
MSp[P] * MSp[P]^ MSp[P] - ----! MSp[P] * MSp[P]^ MSp[P]
?? ?
y* ?y*
MSp[P] *(MSp[P]) - o-*--! MSp[P] *(MSp[P]) -*! MSp[P] *
The commutativity of this diagram means that under the isomorphism
MSp*[P]MSp[P]^ MSp[P] ~= Hom MSp[P] MSp[P] (MSp[P]^ MSp[P]); MSp[P] ;
* * *
the MSp[2] *-module maps corresponding to the two maps O(o ^o ) and o O from
MSp[P]^ MSp[P] to MSp[P] are the same. Hence, these two spectra maps must
be the same. But this means that the diagram (2-9) commutes, with o replaced by
o . Hence o is a ring spectra map.
Given a ring spectra map o : MSp[P] -! MSp[P], let o* : MSp*[P](HP1 ) -!
MSp*[P](HP1 ) be the induced map. By (2-5), o*(w) 2 MSp*[P](HP1 ) is a power
series in w with coefficients in MSp*[P]. Note that the coefficient of w0 = 1 o*
*f this
power series is an element in MSp4[P], hence it must be zero.
Proposition 2-3. The above correspondence gives rise to a bijection between the
set of ring spectra maps o : MSp[P]-! MSp[P] and the set of formal power series
g(w) 2 MSp*[P][ [ w ] ] of homogeneous degree 4 such that g(w) w mod (w2).
P -4k
Let g(w) = w + k1 akwk+1 with ak 2 MSp[P] be a power series. The corre-
sponding ring map og : MSp[P]-! MSp[P]is an idempotent, that is og O og = og, if
and only if og*(ak) = 0 for all k 1.
Proof. Given a ring map o : MSp[P]-! MSp[P], let o*(w) = go(w) 2 MSp*[P][ [ w ]*
* ]
be the induced power series, and let o : MSp[P] *(MSp[P]) -! MSp[P] * be the
algebra map correspondingPto o given in Lemma 2-2.
Let go(w) = k0 akwk+1 , and o(Qk) = dk 2 MSp[P] 4k for k 0. Since
Q0 = 1 and o is an algebra map, we must have d0 = 1. Now, from the definition
of Qk's and the Kronecker pairings, we have dk = o(Qk) = o w*(qk+1 ) =
10 HIROTAKA TAMANOI
= . Here, w*(o ) = o Ow = o*(w) : HP1 -w! 4MSp[P]*
*-o!
P
4MSp[P]. Since o*(w) = k0 akwk+1 and = ffiij for i; j 0, continuing
P
the above calculation, we have dk = < k0 akwk+1 ; qk+1 > = ak for all k 0. S*
*ince
d0 = 1, we must have a0 = 1. Thus the leading term of go(w) is w.
Let o1; o2 : MSp[P]-! MSp[P]be two ring maps, and suppose that their induced
power series are the same, that is, go1(w) = go2(w). From the above calculatio*
*n,
this implies that the corresponding algebra maps o1 and o2 must be the same. But
then, by Lemma 2-2, we must have o1 = o2. This shows that the correspondence
o -! go(w) is injective.
Now let g(w) 2 MSp[P] *[ [ w ] ] be an arbitrary power series of the form g(*
*w) =
P k+1
k0 akw with a0 = 1. Let : MSp[P] *(MSp[P]) -! MSp[P] * be an algebra
map given by (Qk) = ak for all k 0. If o : MSp[P] -! MSp[P] is the unique
ring map corresponding to the algebra map by Lemma 2-2, then we must have
g(w) = go(w) by the choice of o . Hence our correspondence o -! go(w) is surjec*
*tive.
This proves the first part of PropositionP2-3.
Next, given a power series g(w) = w + k1 akwk+1 2 MSp*[P][ [ w ] ] whose
leading term is w, let og : MSp[P]-! MSp[P]be the corresponding ring map. This
map is idempotent, that is, og O og = og, if and only if og*O og*(w) = og*(w) b*
*ecause
the induced power series uniquely determine ring spectra maps by the first part
of Proposition 2-3. By construction of og, we havePog*(w) = g(w). Hence, the
above identity means that g(w) = og* g(w) = g(w)+ k1 og*(ak)g(w)k+1 . Since
g(w) = w + (higher powers of w ), by examining the coefficients of wk inductive*
*ly,
this identity implies that og*(ak) = 0 for all k 1. This completes the proof *
*of
Proposition 2-3.
We call power series g(w) as in Proposition 2-3 general symplectic orientati*
*ons.
Remark. Since there is no H-space structure on HP1 = BSp(1), the algebra
MSp*[P](HP1 ) does not have a structure of a coalgebra. Consequently, we cannot
talk about primitives in MSp*[P](HP1 ) unlike the case for MU*[P](CP1 ). Hence
there is no logarithm in this symplectic context. See the remark right after (2*
*-2).
In fact, the same method as in the first half of Proposition 2-3 applies to *
*classify
ring maps from MSp to MU. Let o : MSp[P] -! MU[P] be a ring map, and
let o* : MSp*[P](HP1 ) -! MU*[P](HP1 ) be the induced map. Let o*(wMSp ) =
g(wMU ) 2 MU*[P][ [ wMU ] ].
Proposition 2-4. There is a bijection between the set of ring spectra maps o :
MSp[P] -! MU[P] and the set of formal power series g(w) 2 MU*[P][ [ w ] ] of ho-
mogeneous degree 4 such that g(w) w mod (w2).
Proof. The proof is similar to the proof of Proposition 2-3 using the fact that*
* the
Kronecker pairing gives rise to an isomorphism
MU*[P](MSp[P]) ~=Hom MU[P] MU[P] (MSp[P]); MU[P] ;
* * *
and the ring maps o : MSp[P] -! MU[P] corresponds to algebra maps under the
above isomorphism. Here, MU[P] *(MSp[P]) ~=MU[P] *[ Q1; Q2; : :]:.
The statement in the second half of Proposition 2-3 can be generalized in the
following way.
MULTIPLICATIVE IDEMPOTENTS OF MSp
[2] 11
Lemma 2-5. Let o1; o2 : MSp[P]-! MSp[P] be two ring maps, and let o2*(w) =
P
w + k1 bkwk+1 be the power series associated to o2. Then o1 O o2 = o1 if a*
*nd
only if o1* annihilates all the higher coefficients of o2*(w), that is, if and *
*only if
o1*(bk) = 0 for all k 1.
Proof. LetPo1*(w) = g1(w) and o2*(w) = g2(w). Then (o1 O o2)*(w) = o1* g2(w) =
g1(w) + k1 o1*(bk)g1(w)k+1 . This is equal to o1*(w) = g1(w) if and only if
o1*(bk) = 0 for all k 1.
x3. Buhstaber Splitting and Ring Maps
of MU[2]factoring through MSp[2]
In x2, we saw that general complex orientations in MU*[P](CP1 ) control ring
maps MU[P]-! MU[P], and that general symplectic orientations in MSp*[P](HP1 )
control ring maps MSp[P] -! MSp[P]. In this section we study the relationship
between these two objects.
First we examine the relationship between the standard MSp-orientation w 2
MSp4(HP1 ) given in (2-4) and the standard MU-orientation x 2 MU2(CP1 )
given in (2-1). Let H1 be an infinite dimensional left H-vector space. The HP1*
* can
be thought of as the set of all the left H-lines L in H1 . The fibre of the tau*
*tological
left H line bundle i -! HP1 over L 2 HP1 is the set of all vectors in L. Si*
*nce
C H = C j.C, H1 can be thought of as an infinite dimensional C-vector space
and the set of all C-lines ` in H1 is CP1 . These C-lines form a tautological *
*complex
line bundle j -! CP1 . We have a canonical map : CP1 -! HP1 by mapping a
C-line ` 2 CP1 into the H-line L 2 HP1 containing `, that is, L = H.` = ` j.*
*`.
As left C-line bundles, j.j is isomorphic to the conjugate bundle __j. Thus it *
*follows
that *(i) = j __jas (left) complex bundles over CP1 .
Let aeSpU: MSp -! MU be the ring spectra map corresponding to the forgetful
functor. We consider the following maps:
aeSpU * 1 * * 1
(3-1) MSp*(HP1 ) --! MU (HP ) -! MU (CP ):
Let wMU = aeSpU*(wMSp ) 2 MU* (HP1 ). By a spectral sequence argument, we have
MU* (HP1 ) = MU* [ [ wMU ] ]. Since MU* (CP1 ) = MU* [ [ xMU ] ], the image of
the symplectic orientation wMSp 2 MSp4(HP1 ) under the above map is a power
series in xMU . This power series is identified in the next lemma.
Lemma 3-1. The image of the symplectic orientation wMSp under the map (3-1)
is given by
(3-2) * O aeSpU*(wMSp ) = x . __x2 MU4(CP1 );
MU
where __x= [-1]MU (x) = exp MU - log (x) . In particular, the induced map * :
MU* (HP1 ) -! MU* (CP1 ) is an injection and *(wMU ) = x . __x.
Proof. Elements x and __xare given by the following maps:
x : CP1 ,! MU(1) -! 2MU; __x: CP1 -(-1).--!CP1 ,! MU(1) -! 2MU;
12 HIROTAKA TAMANOI
Here (-1). denotes the homotopy inverse map. The product x . __x2 MU4(CP1 ) is
given by the composition of the upper arrows followed by the vertical map and t*
*he
map into 4MU in the following commutative diagram:
CP1 -(1;-1)O-----!CP1 x CP1----! MU(1) ^ MU(1)
? ? ?
(*) ?yj_j ?yjxj ?y
BU(2) ________ BU(2) ----! MU(2) - ---! 4MU:
In the above diagram, the vertical map labeled j __jis the classifying map for
the Whitney sum bundle j __j. Similarly, the vertical map labeled j x j is the
classifying map for the product bundle j x j. Since the composition i O : CP1 *
* -!
HP1 -! BU(2) is such that (i O )*(j2) = j __j, where i is the classifying map*
* for
i as a complex 2-dimensional vector bundle, the map i O is the classifying map
for the sum j __j. Hence we have the following commutative diagram:
CP1 ----! HP1 - ---! Mi = MSp(1) ----! 4MSp
?? ? ? ?
yj_j ?yi ?y ?yaeSpU
BU(2) ________BU(2) - ---! MU(2) ----! 4MU:
Sp
The element * aeU *(wMSp ) is represented by the composition of upper arrows
followed by the rightmost vertical arrow. By the commutativity of the above di-
agram, this is equal to the composition of the leftmost vertical arrow followed*
* by
the bottom arrows, which is equal to x . __xby the commutative diagram (*). This
proves (3-2). The injectivity of the map * is now obvious.
We consider Conner-Floyd Chern classes ci = cMUi of the quaternionic line bu*
*n-
dle i -! HP1 viewed as a 2-dimensional complex vector bundle. Recall that
*(i) = j __jfor the canonical map : CP1 -! HP1 . Thus by the natural-
ity of characteristic classes, the induced map * : MU* (HP1 ) -! MU* (CP1 ) is
such that * c1(i) = c1(j __j) = x + __xand * c2(i)P = c1(j)c1(__j) = x__x. S*
*ince
MU* (HP1 ) = MU* [ [ wMU ] ], we have c1(i) = i1 ffi(wMU )i for some unique*
*ly
determined ffi 2 MU2-4i. Note that the above summation starts with i = 1 by
dimensional reason. Since * is injective and *(wMU ) = x__xby Lemma 3-1, we have
c2(i) = wMU . By comparing the formula for * c1(i) , we have
X __
(3-3) x + __x= ffi(xx )i 2 MU2(CP1 );
i1
for some uniquely determined elements ffi 2 MU2-4i for i 1.
The standard MU-symplectic orientation wMU is such that wMU = x__x= -x2 +
(higher order terms) in MU* (CP1 ). Let (x) be a power series in x given by
s _________
(3-4) (x) = x - x_x 2 MU2[2][ [ x ] ]:
p _____________________________
Here by convention, square roots of the form 1 + (higher order terms in x) are
always taken as those with leading term 1. Note that (x)2 = -x__x. Let :
MU[2]-! MU[2]be the ring map whose associated power series is *(x) = (x).
This ring map was first considered by Buhstaber.
MULTIPLICATIVE IDEMPOTENTS OF MSp
[2] 13
Theorem 3-2 (Buhstaber Splitting) [B1, B2]. The Buhstaber map : MU[2]-!
MU[2]is a multiplicative idempotent and the spectrum split off as the image of *
* is
the isomorphic image of MSp[2]under the forgetful ring spectra map aeSpU.
More precisely, in the following natural cohomology transformations,
aeSpU* * * *
MSp*[2]( . ) ---! MU[2]( . ) -! MU[2]( . );
we have * O * = *, and * fixes an element in MU*[2]( . ) if and only if it is in
the image of aeSpU*. Furthermore, the map * annihilates all the higher coeffici*
*ents
of the power series *(x) and all the elements in MU*[2]in degrees congruent to 2
modulo 4.
The cohomologyptheory_MSp*[2]( . ) is complex oriented with the complex orie*
*n-
tation (x) = -x__x2 MSp2[2](CP1 ). Hence MSp*[2](CP1 ) ~=MSp*[2][ [(x)] ].
In short, the spectrum MSp[2]is a multiplicative summand of MU[2]. Let ss
and be the projection map onto this summand and the inclusion map from this
summand, respectively:
(3-5) : MU[2]-ss!MSp[2]-j!MU[2]:
Recall that mk = [ CPk ]=(k + 1) 2 MU*Qfor k 1. Modulo decomposables, the
power series (x) and the behavior of * : MU*[2]-!MU*[2]are described as follows.
p _______
Lemma 3-3. The power series *(x) = (x) = x -x =x 2 MSp*[2][ [ x ] ]
MU*Q[ [ x ] ] is such that
X
(x) x + m2k-1 x2k mod (decomposables );
(3-6) k1X
logMU -1 (T ) x + m2kT 2k+1 mod (decomposables ):
k1
Furthermore, * kills all the generators in degrees congruent to 2 mod 4, and
*(m2k) m2k modulo decomposables in MU*Qfor all k 1.
P
Proof. Since logMU (x) = x +P k1 mkxk+1 , its inverse power series exp MU (*
*x)
is such that exp MU (x) x - k1 mkxk+1 modulo decomposables.P So it fol-
lows that __x= [-1]MU (x) = exp MU(- logMU x) -x - 2 k1 m2k-1 x2k. Hence
p _______ q ________________________P P
(x) = x -x =x x 1 + 2 k1 m2k-1 x2k-1 x + k1 m2k-1 x2k modulo
P
decomposables. From this we have -1 (x) x - k1 m2k-1 x2k. Thus,
-1 -1 X -1 k+1
logMU (x) = (x) + mk (x)
k1
X X
x - m2k-1 x2k + mkxk+1 mod (decomposables )
k1 k1
X
x + m2kx2k+1 mod (decomposables ):
k1
14 HIROTAKA TAMANOI
By (2-3), we see that *(m2k) m2k modulo decomposables. Since MSp*[2]is
generated by elements having degrees congruent to 0 mod 4, * kills all the odd
degree elements. So we have *(m2k-1 ) = 0 for all k 1. This completes the proof
of Lemma 3-3.
Now we consider multiplicative cohomology transformations o* : MU*[2]( . ) -!
MU*[2]( . ) which factor through MSp*[2]( . ). These transformations are charac*
*ter-
ized as follows.
Proposition 3-4. Let o : MU[2]-! MU[2]be a ring spectra map corresponding to
a power series o*(x) = o (x) 2 MU*[2][ [ x ] ]. Then the ring map o factors th*
*rough
MSp[2]if and only if o (x) is of the form
r _________
(3-7) o (x) = x -g(xx_)_x2;
where g(w) 2 MSp*[2][ [ w ] ] is a power series whose leading term is w, and wi*
*th
coefficients in MSp*[2].
Proof. Suppose a ring map o from MU[2]to itself factors through MSp[2]as follow*
*s:
_o j
o : MU[2]-ss!MSp[2]-! MSp[2]-! MU[2];
for some ring map __ofrom MSp[2]to itself. By Proposition 2-3, to such __o, th*
*ere
corresponds a unique power series __o*(w) = g(w) 2 MSp*[2][ [ w ] ] in w, w*
*here
__o * 1 * 1 1
* : MSp[2](HP ) -! MSp[2](HP ). By pulling back this calculation to CP
by the map : CP1 -! HP1 , we have __o*(x__x)p_= g(x__x) in MSp*[2](CP1 ) *
* =
MSp*[2][ [ xMSp ] ], where xMSp = (x) = x -__x=x. Thus, o*(x) = j* O __oO ss*
**(x) =
p ______ p _________ p _________
j* O __o*(p -xx_)_=_j*_ -g(xx ) = -g(xx ). This last identity should be un*
*der-
stood as x -g(x__x)=x2 because it is a complex orientation. This proves (3-7).
Conversely, suppose that the power series associated to a ring map o : MU[2]*
*-!
MU[2]is of the form (3-7) for some power series g(w) 2 MSp*[2][ [w ] ]. Let __*
*og:
MSp[2]-! MSp[2]be the ring map corresponding to g(w) given by Proposition
2-3. Using the projection and the injection in (3-5), let og : MU[2]-! MU[2]be
a ring map given by og = j O __ogO ss factoring through MSp[2]. Then og*(x) =
p ______ p _________
j* O __og*O ss*(x) = j* O __og*(p_-xx_) = -g(xx ). Since og*(x) should have *
*leading
term x, the square root of -g(x__x) should be understood as the one with the
leading term x. But this implies that the effects of the ring maps o* and (og)**
* on
x are the same. Hence, by Proposition 2-1, we must have o = og = j O ___ogO ss *
*and
the map o factors through MSp[2].
We study more properties of those ring maps o : MU[2]-! MU[2]which fac-
tor through MSp[2]. Let logo(T ) be the power series obtained from logMU (T ) *
*by
applying o* to its coefficients. That is,
X
(3-8) logo(T ) = T + o*(mk)T k+1 2 MU*Q[ [ T ] ]:
k1
MULTIPLICATIVE IDEMPOTENTS OF MSp
[2] 15
Proposition 3-5. Let o : MU[2]-! MU[2]be the ring map factoring through
MSp[2]. Let g(w) 2 MSp*[2][ [ w ] ] be the power series associated to o as in (*
*3-7).
(i) Let o*(x) = o (x) 2 MU*[2][ [ x ] ] for some power series o (x). Then
(3-9) o*(__x) = -o (x) = o (__x):
Here, the power series o (__x) is the one obtained by replacing x by __xin o (x*
*).
(ii) The power series logo(T ) given in (3-8) is an odd power series in T . *
*That
is, o*(m2k-1 ) = 0 for all k 1. Furthermore, let '(Tp)_2_MSp*[2][_[ T ] ] be t*
*he power
series such that '(T ) T mod (T 2) and '(T ) = -g(-T 2) . Then '(T ) is an
odd power series such that o (x) = ' (x) and in MU*Qwe have
X X 2k+1
(3-10) T + o*(m2k)T 2k+1= '-1 (T ) + *(m2k) '-1 (T ) :
k1 k1
In other words, logo(T )P= log '-1 (T ) .
(iii) Let g(w) = w + k1 (-1)k2flkwk+1 2 MSp*[2][ [ w ] ]. Then we have *
*the
following congruence relation in MU*Qfor any k 1:
(3-11) o*(m2k) m2k - flk mod (decomposables ):
If, for some n, flk = 0 for k < n in the above expression of g(w), then in MU*Q,
(3-12) o*(m2k) = *(m2k) for k < n; and o*(m2n) = *(m2n) - fln:
(iv) The kernel of the induced map __o*: MSp*[2]-!MSp*[2]is contained in the
ideal generated by the higher coefficients of g(w). That is,
(3-13) Ker __o* (fl1; fl2; : :;:flk; : :): MSp*[2]:
P __
Proof. By (3-3), we have x + __x= i1 ffi(xx )i for some ffi 2 MU2-4i. Since*
* o*
annihilates all the elements in MU* in degrees congruent to 2 mod 4 because o*
factors through MSp*[2], applying o* to the above identity, we have o*(x)+o*(__*
*x) = 0.
Since o*(x) = o (x), we have o*(__x) = -o (x). On the other hand, replacing x b*
*y __x
in the power series o (x),
r _________ __ r _________ r _________
o (__x) = __x -g(xx_)__2= -x -x_ -g(xx_)__2= -x -g(xx_)_= -o (x):
x x x x2
p ________________________
Here, we are still using the convention that 1 + (higher order terms) = 1 +
(higher order terms). In the above calculation, since -__x=x starts with 1, we*
* can
move its square inside of the square root. This proves (3-9). For (ii), fr*
*om
(2-3), we have logo(T ) = logMU o -1(T ) . From the definition of __x, we *
*have
logMU_(__x) = - logMU(x). Let o (x) = T . Then from o (x) + o (__x) = 0, we h*
*ave
x = o -1(-T ). Thus logo(-T ) = logMU o -1(-T ) = logMU (__x) = - logMU(x) =
- logMU o -1(T ) = - logo(T ). This shows that the power series logo(T ) in (*
*3-8)
16 HIROTAKA TAMANOI
is an odd power series. Consequently, the coefficients of even powers of T va*
*n-
ishes, i.e., o*(m2k-1 ) = 0 for k 1. Next, we have logo(T ) = logMU o -1(T )*
* =
logMU -1 '-1 (T ) = log '-1 (T ) . This proves (3-10).
For (iii),Pgiven g(w) as above, the corresponding power series '(T ) is of t*
*he form
'(T ) T + k1 flkT 2k+1 modulo decomposables. (This is the reason for our
choicePof the multiplicative factor (-1)k2. in the coefficients of g(w).) So, '*
*-1 (T )
T - k1 flkT 2k+1, again modulo decomposables. Hence, modulo decomposables,
the R.H.S. of (3-10) is congruent to
X 2k+1 X X
'-1 (T ) + *(m2k) '-1 (T ) T - flkT 2k+1 + *(m2k)T 2k+1
k1 k1 k1
X
T + (m2k - flk)T 2k+1:
k1
Here, we used a congruence relation in Lemma 3-3. By (3-10), this means that
o*(m2k) m2k - flk for all k 1 modulo decomposables. This proves (3-11).
For some n, if flk = 0 for all k < n, then modulo the ideal (T 2n+3), we have
'-1 (T ) T - flnT 2n+1. Substituting this into the R.H.S. of (3-10), we obtain
Xn Xn
T + o*(m2k)T 2k+1 T - flnT 2n+1+ *(m2k)T 2k+1 mod (T 2n+3):
k=1 k=1
Comparing the coefficients of T 2k+1 for k n, we obtain (3-12).
For (iv), since '-1 (T ) T mod (fl1; : :;:flk; : :):in MSp*[2], reducing *
*(3-10)
modulo the same ideal extended to MU*Q, we have o*(m2k) *(m2k) for all k 1.
Since o* preserves the subring MSp*[2]of MU*[2]and * is the identity on this su*
*bring,
for any z 2 MSp*[2]we have o*(z) z modulo (fl1; : :;:flk; : :):. Thus, if o*(z*
*) = 0
for some z 2 MSp*[2], then z 0 modulo the ideal (fl1; : :;:flk; : :): MSp*[2].
This completes the proof of Proposition 3-5.
Remark. Later in Corollary 5-4, we will show that the inclusion relation in (3-*
*13)
is actually an equality when o is an idempotent.
x4. Constructing Multiplicative Idempotents in MSp[2]
In this section, we construct multiplicative idempotents acting on MSp[2]. *
*It
turns out that no polynomial generator in degree -4n such that 2n + 1 is a prime
power can be annihilated by multiplicative idempotents on MSp[2]with only 2
inverted. On the other hand, for any collection of polynomial generators of MSp*
**[2]
in non-prime-power degrees, there exists a multiplicative idempotent on MSp[2]
which annihilates exactly the given collection of polynomial generators, and no
more.
To describe our idempotents, recall that a cobordism class of a real 4n dime*
*n-
sional oriented closed manifold M4n can be taken as a polynomial generator of t*
*he
oriented cobordism ring SO*if and only if sn([M4n ]) = oen, where
aep if 2n + 1 is a power of the prime p,
(4-1) oen =
1 if 2n + 1 is not a prime power.
MULTIPLICATIVE IDEMPOTENTS OF MSp
[2] 17
Here, sn( . ) is a Pontryagin characteristic number which vanishes on decomposa*
*ble
elements [M2]. For example, sn([CP2n]) = 2n + 1 for all n 1, in other words,
sn(m2n) = 1 for all n 1.
aeSpU aeUSO
Since the composition of natural ring maps MSp - -! MU --! MSO is a
homotopy equivalence after inverting 2, we can use the same criteria to identify
polynomial generators of MSp*[2]over Z[2]= Z[ 1_2]. Since aeSpU*: MSp*[2]-!MU*[*
*2]is
an inclusion map, we regard MSp*[2]as a subring of MU*[2].
For each n 1, let n 2 MSp-4n[2]be a polynomial generator such that sn(n) =
oen. Since MSp*[2]is a subring of MU*Q= Q[ m1; m2; : :;:mn; : :]:, we can write
(4-2) n = oenm2n + h(n)(m1; m2; : :;:m2n-1 ) 2 MSp-4n[2]; n 1;
for some weighted homogeneous polynomial h(n) with Q-coefficients. The choice of
the generators n is not unique. Other choice of generators have different decom-
posable parts in terms of mk's. But we choose one set of such generators.
Recall that by Proposition 2-3, ring spectra maps o : MSp[2]-! MSp[2]and
power series g(w) correspond in 1 : 1 manner.
Proposition 4-1. The ring map o = o [n] : MSp[2]-! MSp[2]corresponding to
any power series g(w) of the form
(4-3) g(w) w + (-1)n2nwn+1 mod (nwn+2 )
has the following properties:
o*(k) = k for k < n;
(4-4) o*(n) = (1 - oen)n;
o*(k) k mod (n) \ (decomposables ) for k > n:
For k 6= n, o*(k) is a polynomial generator in degree -4k in MSp*[2], and the
kernel of o* : MSp*[2]-!MSp*[2]is such that Ker o* (n).
Furthermore, o = o [n] : MSp[2]-! MSp[2]is an idempotent if and only if
2n + 1 is not a prime power, which is the case if and only if Ker o* = (n).
Proof. When k < n, in the right hand side of (4-2) with n replaced by k, the ef*
*fect of
o* is the same as the effect of * by the first formula in (3-12) in Proposition*
* 3-5 (iii),
because of the form of the power series g(w) in (4-3). Thus o*(k) = *(k) = k
for k < n since * fixes elements in MSp*[2]. This proves the first formula in (*
*4-4).
The effect of o* on n is calculated using (3-12) and (4-2) as follows:
o*(n) = oen . o*(m2n) + h(n) o*(m1); : :;:o*(m2n-1 )
= oen . *(m2n) - n + * h(n)(m1; : :;:m2n-1 )
= * oenm2n + h(n)(m1; : :;:m2n-1 ) - oenn
= *(n) - oenn = (1 - oen)n:
Here, again we used the fact that *(n) = n because the Buhstaber idempotent
* fixes MSp*[2]. This shows the second formula in (4-4).
18 HIROTAKA TAMANOI
For the third formula, note that modulo the ideal (n), we have g(w) w. So
we have '-1 (T ) T modulo (n). Thus, from (3-10) we have o*(m2k) *(m2k)
mod (n) for all k 1. Hence o*(k) *(k) = k for all k 1, modulo the
ideal (n). By dimensional reason, when k > n, the difference o*(k) - k 2 (n)
must also lie in the ideal of decomposable elements. This proves the third form*
*ula
in (4-4).
Since the characteristic number sk( . ) vanishes on decomposable elements, (*
*4-4)
shows that sk o*(k) = sk(k) for all k different from n. So o*(k) for k 6= n can
be taken as polynomial generators of MSp*[2]in degree -4k.
By the second part of Proposition 2-3, the ring map o = o [n] associated to a
power series g(w) such that g(w) w+(-1)n2nwn+1 mod (n)\(decomposables )
is a multiplicative idempotent if and only if o* annihilates all the higher coe*
*fficients
of g(w), that is, if and only if o*(n) = 0. By the second formula in (4-4), th*
*is
happens if and only if oen = 1. By the definition of oen in (4-1), we see that *
*o [n]
is a multiplicative idempotent if and only if 2n + 1 is not a prime power. This
completes the proof of Proposition 4-1.
Note that for an integer n such that 2n + 1 is not a prime power, the above
multiplicative idempotent o [n] depends on the specific choice of the power ser*
*ies
g(w) of the form (4-3), although the kernel of the induced homomorphism o [n]* *
*on
the homotopy groups is completely determined by the choice of the indecomposable
element n.
The above multiplicative idempotents do not commute with each other in gen-
eral. So in general their compositions are not multiplicative idempotents, eit*
*her.
However, we can show that if we compose the above type of multiplicative idem-
potents in a specific order, the resulting map is also a multiplicative idempot*
*ent.
Theorem 4-2. Let n1 < n2 < . .<.nr be positive integers such that 2ni+ 1's are
not prime powers for 1 i r. Let ni 2 MSp-4ni[2]be a polynomial generator such
that sni(ni) = 1, and let o [ni] = o [ni] : MSp[2]-! MSp[2]be the multiplicative
idempotent associated to any power series g[ni](w) of the form
(4-5)
g[ni](w) w + (-1)ni2niwni+1 mod (ni) \ (decomposables ); 1 i r;
for 1 i r. Then the successive composition of these idempotents on the right
(4-6) o [n1; : :;:nr] = o [n1] O . .O.o [nr] : MSp[2]-! MSp[2]
is again a multiplicative idempotent such that
o [n1; : :;:nr]*(ni)= 0; for 1 i r;
(4-7)
o [n1; : :;:nr]*(k) k mod (n1; : :;:nr) \ (decomposables );
where the second formula holds for all k different from ni's. Thus, the kernel *
*of the
ring map o [n1; : :;:nr]* : MSp*[2]-!MSp*[2]is the ideal (n1; : :;:nr).
The associated power series g[n1;:::;nr](w) = o [n1; : :;:nr]*(w) has coeffi*
*cients in
the ideal (n1; : :;:nr) MSp*[2], and we have
Xr
(4-8) g[n1;:::;nr](w) = o [n1; : :;:nr]*(w) w + (-1)ni2niwni+1:
i=1
MULTIPLICATIVE IDEMPOTENTS OF MSp
[2] 19
modulo the ideal (n1; : :;:nr) \ (decomposables ).
Proof. We prove this theorem by induction on r. The case r = 1 is taken care of
by Proposition 4-1. Let r 1 and suppose that the above statements hold for a
sequence (n1; n2; : :;:nr) of the above type. Let nr+1 be such that 2nr+1 +1 is*
* not a
prime power and nr < nr+1. Let o [nr+1] : MSp[2]-! MSp[2]be a ring spectra map
associated to the power series of the form g(r+1)(w) = w+(-1)nr+12nr+1wnr+1+1 +
nr+1wnr+1+2 f(w) for some f(w) 2 MSp*[2][ [ w ] ]. By Proposition 4-1, o [nr+1]*
* is a
multiplicative idempotent. Note that the leading term of f(w) is of degree -4. *
*By
abbreviating (n1; n2; : :;:nr) by "n(r), we have
(*) g[n1;:::;nr;nr+1](w) = o [n1; : :;:nr; nr+1]*(w) = o ["n(r)]* O o [nr+1]*(*
*w)
n n +1 n +2
= o ["n(r)]* w + (-1) r+12nr+1w r+1 + nr+1w r+1 f(w)
nr+1+1
= g["n(r)](w) + (-1)nr+12o ["n(r)]*(nr+1) . g["n(r)](w)
nr+1+2 (r)
+ o ["n(r)]*(nr+1) . g["n(r)](w) . o ["n ]*f g["*
*n(r)](w) ;
where o ["n(r)]*f is obtained by applying oP["n(r)]* to the coefficients of f(w*
*). By
our inductive hypothesis, g["n(r)](w) w + ri=1(-1)ni2niwni+1 modulo the ideal
(n1; : :;:nr) \ (decomposables ) by (4-8), and o ["n(r)]*(nr+1) nr+1 modulo
(n1; : :;:nr)\(decomposables ) by (4-7). All the coefficients in the last term*
* in (*)
are in the ideal (n1; : :;:nr; nr+1) \ (decomposables ), because all the coeff*
*icients
of f(w) have strictly negative degrees and consequently nr+1 cannot appear on i*
*ts
own as a coefficient of a power of w. The second term from the last is congruen*
*t to
(-1)nr+12nr+1wnr+1+1 modulo the ideal (n1; : :;:nr; nr+1) \ (decomposables ).
Hence modulo the ideal (n1; : :::nr; nr+1) \ (decomposables ),
r+1X
g[n1;:::;nr+1](w) g["n(r)](w) + (-1)nr+12nr+1wnr+1+1 w + (-1)ni2niwni+1:
i=1
This proves the inductive step for (4-8).
Next, we prove the inductive step for (4-7). From the first and the second f*
*ormula
in (4-4), we have o [nr+1]*(ni) = ni for 1 i r and o [nr+1]*(nr+1) = 0. From
these formulae, we have
o [n1; : :;:nr+1]*(nr+1) = o ["n(r)]* o [nr+1]*(nr+1) = 0;
(r)
o [n1; : :;:nr+1]*(ni) = o ["n(r)]* o [nr+1]*(ni) = o ["n ]*(ni) = 0:
The last identity is due to the inductive hypothesis (4-7). This proves the ind*
*uctive
step for the first formula in (4-7).
At this point, note that the ring map o [n1; : :;:nr+1]* annihilates all the*
* higher
coefficients of the associated power series g[n1;:::;nr+1](w), since all the hi*
*gher coeffi-
cients are in the ideal (n1; : :;:nr+1) by (4-8). Thus, by Proposition 2-3, the*
* ring
spectra map o [n1; : :;:nr+1] is a multiplicative idempotent.
If k is any polynomial generator of degree -4k, then by Proposition 4-1 we
have o [nr+1]*(k) k mod (nr+1), which is also valid for k = nr+1. Thus,
(**) o [n1; : :;:nr+1]*(k) o ["n(r)]*(k) mod o ["n(r)]* (nr+1) :
20 HIROTAKA TAMANOI
By our inductive hypothesis, o ["n(r)]*(k) k mod (n1; : :;:nr) for all k, by
the second formula in (4-7). Letting k = nr+1, we see that o ["n(r)]*(nr+1) 2
(n1; : :;:nr; nr+1). So we have o ["n(r)]* (nr+1) (n1; : :;:nr+1). Com-
bining these formulae, (**) gives o [n1; : :;:nr; nr+1]*(k) k modulo the ideal
(n1; : :;:nr; nr+1) for all k. This completes the inductive step for the second
formula in (4-7).
We have now completed all the inductive steps and Theorem 4-2 is proved.
We now consider compositions of infinitely many multiplicative idempotents of
the form described in Proposition 4-1. For this, we have to deal with convergen*
*ce
in the cohomology group MSp*[2](MSp[2]) with respect to the filtration topology
defined as follows. Let (k): MSp(4k)[2]-!MSp[2]be the inclusion map of 4k-skele*
*ton
of MSp[2]into MSp[2]. This map induces a restriction map
(4k)
*(k): MSp*[2](MSp[2]) -! MSp*[2]MSp[2] :
Let F (k)= Ker *(k-1) MSp*[2](MSp[2]) for k 0. Then F (k)'s define a decreasing
filtration on MSp*[2](MSp[2]):
"
MSp*[2](MSp[2]) = F (0) F (1) . . .F (k) . . . n F (n)= F (1):
By the standard argument, it is easy to see thatTthis defines a complete Hausdo*
*rff
topology on this cohomology group, that is, kF (k)= {0}.
Lemma 4-3. Let o1; o2 : MSp[2] -! MSp[2]be two ring maps and let oi* :
MSp*[2](HP1 ) -! MSp*[2](HP1 ) be the induced maps for i = 1; 2. Suppose o1*(w)
o2*(w) modulo (wn+1 ) for some n 1. Then the corresponding ring spectra maps
o1 and o2 agree on the 4(n - 1)-skeleton. That is, we kave
(4n-4)
*(n-1)(o1) = *(n-1)(o2) 2 MSp*[2]MSp[2] ;
Proof. For i = 1; 2, let i : MSp[2] *(MSp[2]) -! MSp[2] * be the algebra map
corresponding to oi as in Lemma 2-2. From the proof of Lemma 2-3, the coefficie*
*nts
of the power series oi*(w) determine the images of the algebra generators Qk un*
*der
the maps i. By our assumption, we have 1(Qj) = 2(Qj) for 1 j n-1. Hence
1 and 2 agrees as MSp[2] *-module maps from MSp[2] * MSp(4n-4)[2]to MSp[2] *.
Due to the isomorphism
(4n-4) (4n-4)
MSp*[2]MSp[2] ~=Hom MSp[2] MSp[2] MSp[2] ; MSp[2] ;
* * *
the above agreement means that *(n-1)(o1) = *(n-1)(o2) : MSp(4n-4)[2]-!MSp[2].
That is, o1 and o2 agree on the 4(n - 1)-skeleton.
The above lemma can be used to control the convergence of a sequence of ring
maps in terms of the convergence of the associated power series.
MULTIPLICATIVE IDEMPOTENTS OF MSp
[2] 21
Proposition 4-4. Let oi : MSp[2]-! MSp[2]for i 1 be a sequence of ring
spectra maps such that the associated power series oi*(w) 2 MSp*[2][ [ w ] ] co*
*nverges
to g(w) as i -! 1. If o : MSp[2]-! MSp[2]is the ring map corresponding to g(w),
then the sequence of ring spectra maps oi converges to o with respect to the fi*
*ltration
topology in MSp*[2](MSp[2]).
Furthermore, if oi's are multiplicative idempotents, then their limit o is a*
*lso a
multiplicative idempotent.
Proof. Since the power series g(w) and oi*(w) agree more and more as i increase*
*s by
our assumption, the corresponding ring maps o and oi agree on higher and higher
skeletons by Lemma 4-3. Hence their difference o - oi belongs to higher and hig*
*her
filtration as i -! 1. By definition, this means that lim-!ioi = o with respect *
*to the
filtration topology in MSp*[2](MSp[2]).
Next, suppose that oi is a multiplicative idempotent for all i 1. Then by
Proposition 2-3, the induced ring map oi* : MSp*[2]-!MSp*[2]annihilates all the
higher coefficients of oi*(w) 2 MSp*[2][ [ w ] ] for all i. We let
X X (i)
o*(w) = w + (-1)k2flkwk+1 ; oi*(w) = w + (-1)k2flk wk+1 ; i 1:
k1 k1
For each k 1, there exists an integer Nk such that o*(w) oi*(w) modulo (wk+2 )
for all i Nk since the power series oi*(w) converge to o*(w). From Lemma 4-3,
the ring spectra maps o and oi agree on the 4k-skeleton of MSp[2]. Thus on the
kth coefficient flk of o*(w), we have o*(flk) = oi*(flk) for all i Nk. Since o*
*i is an
idempotent, we have oi*(fl(i)k) = 0 for all k 1. Since flk = fl(i)kfor i Nk, *
*we have
o*(flk) = oi*(flk) = oi*(fl(i)k) = 0. Since k is arbitrary, we see that o* ann*
*ihilates
all the higher coefficients of o*(w). Thus, by Proposition 2-3, o is a multipli*
*cative
idempotent.
Let L N be any subset of N consisting of integers ` such that 2` + 1 is not
a prime power. The cardinality of L may be finite or infinite. For each ` 2 L,
we choose an indecomposable element ` 2 MSp-4`[2]such that s`(`) = 1. Such
elements always exist by Milnor's criterion. For each ` 2 L, we consider a pow*
*er
series of the form (4-3) using ` in place of n, and let the corresponding ring
spectra map be o [`] = o [`] : MSp[2]-! MSp[2]for ` 2 L. We order integers in L
from the smallest one as follows:
L : `1 < `2 < . .<.`r < . .:.
We consider a sequence of finite compositions of the corresponding ring maps of
the following form:
(4-9) o (r)= o [`1] O o [`2] O . .O.o [`r] : MSp[2]-! MSp[2]
for r 1. Please note the order of the compositions. By Theorem 4-2, o (r)is
a multiplicative idempotent for all r 1. We consider the limit as r -! 1. To
describe our result, for each k 2 N \ L we choose an element k 2 MSp-4k[2]such
that sk(k) = 1.
22 HIROTAKA TAMANOI
Theorem 4-5. With the above notation, the multiplicative idempotents o (r)for
r 1 converge to a multiplicative idempotent o = oL : MSp[2]-! MSp[2]such that
ae o ( ) = 0; for` 2 L;
(4-10) * `
o*(k) k; mod (` | ` 2 L) \ (decomposables ):
Furthermore, the associated power series o*(w) 2 MSp*[2][ [ w ] ] is such that
X
(4-11) o*(w) w + (-1)`2`w`+1;
`2L
modulo the ideal (` | ` 2 L) \ (decomposables ) MSp*[2]. That is, all the hig*
*her
coefficients of the power series o*(w) are in the ideal (` | ` 2 L), and only i*
*nde-
composable coefficients which can appear are congruent to ` for some ` 2 L.
Proof. Since the multiplicative idempotent o [`r] fixes the 4(`r - 1)-skeleton *
*of
MSp[2]by Lemma 4-3, the sequence of multiplicative idempotents o (r)converges
to a multiplicative idempotent o with respect to the skeletal filtration topol*
*ogy
in MSp*[2](MSp[2]) by Proposition 4-4. The formulae (4-10) and (4-11) follow by
taking the limit of (4-7) and (4-8).
In Theorems 4-2 and 4-5, we dealt with compositions on the right of basic mu*
*lti-
plicative idempotents described in Proposition 4-1. Next we deal with compositi*
*ons
of these idempotents on the left and we describe differences and similarities b*
*etween
these two ways of composing idempotents.
Let 0 < n1 < n2 < . . .< nr be positive integers such that 2ni + 1 is not a
prime power for all 1 i r. As before, we choose indecomposable elements
ni 2 MSp-4ni[2]such that sni(ni) = 1, and multiplicative idempotents o [ni] as
in Proposition 4-1 for 1 i r. We consider the composition of these idempotents
on the left:
(4-12) o [nr; nr-1; . .;.n1] = o [nr] O o [nr-1] O . .O.o [n1] : MSp[2]-! MSp[2*
*]:
Unfortunately, unlike the successive compositions on the right as in (4-6), the
ring map (4-12) above is not an idempotent in general. To see the reason, it is
enough to consider the case r = 2. Let o [n1] = o1 and o [n2] = o2, and let
their associated power series be o1*(w) = g1(w) and o2*(w) = g2(w). Then the
power series associated to o [n2; n1] is given by (o2 O o1)*(w) = o2* o1*(w) =
o2* g1(w) = (o2*g1) g2(w) . Since all the higher coefficients of g1(w) are in*
* the
ideal (n1) and o2* moves elements within the ideal (n2), all the higher coeffi-
cients of o2*g1 (w) are in the ideal (n1; n2). So all the higher coefficients*
* of
(o2 O o1)*(w) are also in the ideal (n1; n2). Since o [n1]*(n1) = 0, the ring m*
*ap
o [n2; n1]* = o [n2]* O o [n1]* annihilates n1. But since o [n1]*(n2) n2 mod-
ulo the ideal (n1), and o [n2]* annihilates n2 and fixes n1 by degree reason,
o [n2; n1]* may not annihilate n2, and consequently o [n2; n1] may not be a
multiplicative idempotent. This is why we used compositions on the right as in
(4-6) rather than compositions on the left as in (4-12) to produce multiplicati*
*ve
idempotents. However, we can modify the choice of indecomposable elements ni's
so that the successive compositions as in (4-12) does give rise to an idempoten*
*t.
MULTIPLICATIVE IDEMPOTENTS OF MSp
[2] 23
Given indecomposable elements n1,_n2,_: : :, nr_as_above, we inductively_
construct indecomposable elements_ n1, n2 , : :,: nr such that sni( ni) = 1 f*
*or
1 i r as follows. First let n1 = n1. Then we let
8 __ __
>>>n2 = o [ n1](n2);
>><_ __ __
n3 = o [ n2] O o [ n1](n3);
(4-13) > .
>>> ..
>:__ __ __
nr = o [ nr-1] O . .O.o [ n1](nr);
*
*__
Here, from one line to the next line, we choose a multiplicative idempotent o [*
* ni] of
the form_described_in Proposition_4-1 using_the_newly constructed indecomposable
element ni, then we define ni+1 using o [ ni]. And we repeat this process.
We now show that the left compositions_of_basic_idempotents constructed for
these newly constructed elements n1; : :;: nr is an idempotent whose kernel is
precisely the ideal generated by indecomposable elements n1; : :;:nr given at t*
*he
beginning.
Theorem 4-6. Let n1, n2, : :,:nr be indecomposable elements in MSp*[2]such
that sni(ni) = 1 for positive integers 0 < n1_< n2 < . .<.nr such that 2ni + 1
is not_a prime power for 1 i r. We choose ni and multiplicative idempotents
o [ ni] as in (4-13) for 1 i r. Then
__
ni ni mod (n1; : :;:ni-1) for2 i r;
(4-14) __ __
( n1; : :;: nr) = (n1; : :;:nr) for 1 i r:
__
The successive left compositions of the basic multiplicative idempotents o [ n*
*i]
__ __ __ __
(4-15) o [ nr; : :;: n1] = o [ nr] O . .o.[ n1] : MSp[2]-! MSp[2]
is a multiplicative idempotent. Its kernel is the ideal generated by n1, : :,:n*
*r:
__ __ *
(4-16) Ker o [ nr; : :;: n1]* = (n1; : :;:nr) MSp[2]
__ __
The power series associated to the ring map o [ nr; : :;: n1] is such that
__ __ Xr n n +1
(4-17) o [ nr; : :;: n1]*(w) w + (-1) i2niw i
i=1
modulo the ideal (n1; n2; : :;:nr) \ (decomposables ).
Proof. We prove this theorem by induction on r. When r = 1, Theorem 4-6 is the
same as Proposition 4-1. Now we assume Theorem 4-6 for any r positive integers
0 < n1 < n2 < . .<.nr, and we prove Theorem 4-6 for any r + 1 positive integers
0 < n1 < n2 < . .<.nr < nr+1.
Suppose that we are given r + 1 indecomposable elements n1, : :,:nr, nr+1
such that sni(ni)_=_1 for 1 i r + 1, and that we have_constructed_indecom-
posable elements ni and multiplicative idempotents o [ ni] : MSp[2]-! MSp[2]f*
*or
1 i r + 1 as in (4-13).
24 HIROTAKA TAMANOI
__ __ __ __
The identity nr+1 = o [ nr; : :;: n1]*(nr+1) and (4-16) imply that nr+1
nr+1 modulo the ideal (n1; : :;:nr). This proves the inductive step for the fir*
*st
formula in (4-14).
Using this_newly obtained_congruence_relation_and_the second formula in (4-1*
*4),
we have ( n1; : :;: nr; nr+1) = (n1; : :;:nr; nr+1) = (n1; : :;:nr; nr+1). T*
*his
proves the inductive step for the second formula in (4-14). __
By the first formula in (4-4) in Proposition 4-1, we_have_o [ nr+1]*(ni) =
ni for 1 i r by degree reason. So we see that o [ nr+1]* preserves the
ideal (n1; :_:;:nr)._ Hence_using the inductive hypothesis (4-16) and the iden-
tity Ker o [__nr+1]*_=_( nr+1)_from Proposition_4-1, we_see that the_kernel of*
* the
ring map o [ nr+1; nr;_:_:;: n1]* = o [ nr+1]* O o [ nr; : :;: n1]* is prec*
*isely equal to
the ideal (n1; : :;:nr; nr+1). By what we have proved as the inductive step f*
*or
(4-14), this ideal is equal to (n1; : :;:nr; nr+1). This proves the inductive *
*step
for (4-16).
Using the inductive hypothesis (4-17), we have
__ __ __ __ __ __
o [ nr+1; nr; : :;: n1]*(w)= o [ nr+1]* O o [ nr; : :;: n1]*(w)
__ Xr n n +1
o [ nr+1]* w + (-1) i2niw i
i=1
__
modulo the ideal o [ nr+1]* (n1; : :;:nr) \ (decomposables_ ) = (n1; : :;:nr)*
* \
(decomposables ). Here we used the fact that o [ nr+1]* fixes the elements n1*
*, : :,:
nr by degree reason. We also have
__ n __ n +1 __
o [ nr+1]*(w) w + (-1) r+12 nr+1w r+1 mod ( nr+1) \ (decomposables )
w + (-1)nr+12nr+1wnr+1+1 ;
modulo (n1; : :;:nr; nr+1) \ (decomposables ), by the inductive step for (4-14*
*).
Combining these two congruences, we have
__ __ __ r+1X n n +1
o [ nr+1; nr; : :;: n1]*(w) w + (-1) i2niw i ;
i=1
modulo the ideal (n1; : :;:nr; nr+1)\(decomposables ). This proves the inducti*
*ve
step for (4-17).__ __ __
Since Ker o [ nr+1; nr; : :;:_n1]* =_(n1; : :;:nr;_nr+1) by what we have p*
*roved
as the inductive_step above,_o [ _nr+1;_ nr; : :;: n1]* annihilates_all the_hig*
*her coef-_
ficients of o [ nr+1; nr; : :;: n1]*(w). Hence the ring map o [ nr+1; nr; *
*: :;: n1] is
a multiplicative idempotent.
This completes all the inductive steps and the proof of Theorem 4-6 is now
complete.
Letting r -! 1, we obtain a statement corresponding to Theorem 4-5.
x5. General Multiplicative Idempotents in MSp[2]
The multiplicative idempotents on MSp[2]considered in Theorem 4-5 have as-
sociated power series g(w) of the form (4-11), in which an indecomposable eleme*
*nt
` in MSp*[2]can appear as the coefficient of w`+1 only when 2` + 1 is not a pri*
*me
power. In fact, we can show that any multiplicative idempotent acting on MSp[2]
has this property.
MULTIPLICATIVE IDEMPOTENTS OF MSp
[2] 25
Lemma 5-1. Let o : MSp[2]-! MSp[2]be a multiplicative idempotent, and let
X
(5-1) o*(w) = w + (-1)n2flnwn+1 2 MSp*[2][ [ w ] ]
n1
be the associated power series. If the coefficient fl` 2 MSp-4`[2]is indecompos*
*able for
some integer ` 2 N, then 2` + 1 is not a prime power and in fact s`(fl`) = 1.
If an indecomposable element ` 2 MSp-4`[2]is annihilated by the above multi-
plicative idempotent o , then the corresponding `th coefficient fl` is indecomp*
*osable
and thus 2` + 1 is not a prime power.
Proof. If a coefficient fl` is indecomposable for some ` 2 N, then we may write
fl` = c oe`m2`+ h(`)(m1; : :;:m2`-1)
for some c 6= 0 2 Z[ 1_2], and for some weighted homogeneous polynomial h(`)with
rational coefficients. Since o is a multiplicative idempotent, o* annihilates *
*all the
higher coefficients of o*(w) by Proposition 2-3. So we have o*(fl`) = 0. By Pro*
*po-
sition 3-5, modulo decomposable elements, we have
0 = o*(fl`) c oe`o*(m2`) c oe`(m2`-fl`) c oe`(m2`-c oe`m2`) = c oe`(1-c oe`)*
*m2`:
By considering the characteristic number s`( . ), we must have c oe`(1 - c oe`)*
* = 0.
Since c 6= 0 by our assumption, we must have coe` = 1. If ` is such that 2`+1 =*
* pj for
some odd prime p and j 1, then oe` = p and thus c = 1=p. But this is impossible
since c 2 Z[ 1_2]. Thus, we must have that 2` + 1 is not a prime power for any *
*prime.
In this case, oe` = 1 and consequently c = 1 2 Z[ 1_2]. Hence s`(fl`) = c oe` =*
* 1.
Next, if o*(`) = 0 for some indecomposable ` 2 MSp-4`[2], write
` = c oe`m2`+ h(`)(m1; : :;:m2`-1)
for some c 6= 0 2 Z[ 1_2]. We have 0 = o*(`) c oe`o*(m2`) c oe`(m2`- fl`) mod*
*ulo
decomposables by (3-11). By taking s`( . ), we have s`(fl`) = s`(m2`) = 1. Hence
fl` is indecomposable. By the first part of this Lemma, we see that 2` + 1 is n*
*ot a
prime power.
Let k be an integer such that the corresponding kth coefficient flk in o*(w)
is decomposable. We examine the behavior of o* on indecomposable elements in
degree -4k.
Lemma 5-2. LetPo : MSp[2]-! MSp[2]be a multiplicative idempotent and let
o*(w) = w + n1 (-1)n2flnwn+1 be its associated power series. Let
(5-2) Lo = {` 2 N | s`(fl`) 6= 0}:
Then for each k 2 N \ Lo, there exists an indecomposable element k 2 MSp-4k[2]
such that sk(k) = oek and o*(k) = k.
Proof. For each k 2 N \ Lo, we choose an arbitrary indecomposable element 0k2
MSp-4k[2]such that sk(0k) = oek. We may write
0k= oekm2k + h(k)(m1; : :;:m2k-1 ):
26 HIROTAKA TAMANOI
If we let k = o*(0k), then o*(k) = (o* O o*)(0k) = o*(0k) = k. We calculate
sk(k). From (3-11), in MUQ * we have o*(m2k) m2k-flk modulo decomposables.
Since k 2 N \ Lo, the element flk itself is a decomposable element by the defin*
*ition
of Lo. Hence o*(m2k) m2k modulo decomposables for k 2 N \ Lo. Thus, we have
k = o*(0k) oekm2k modulo decomposables. This implies that sk(k) = oek. This
completes the proof.
Combining the previous two lemmas, we obtain the following description for
general multiplicative idempotents.
Theorem 5-3.P Let o : MSp[2]-! MSp[2]be a multiplicative idempotent and let
o*(w) = w + n1 (-1)n2flnwn+1 2 MSp*[2][ [ w ] ] be the associated power ser*
*ies.
We let Lo = {` 2 N | s`(fl`) 6= 0}. Then the following statements hold.
(i) For any integer ` 2 Lo, 2` + 1 is not a prime power and s`(fl`) = 1.
(ii)For any integer k 2 N \ Lo, there exists an indecomposable element k 2
MSp-4k[2]such that sk(k) = oek, and we have
MSp*[2]= Z[ 1_2][ fl`; k | ` 2 Lo; k 2 N \ Lo ]:
Furthermore, the multiplicative idempotent o* : MSp*[2]-!MSp*[2]is given *
*by
aeo (fl ) = 0; ` 2 L
(5-3) * ` o
o*(k) = k; k 2 N \ Lo:
Thus, Ker o* is the ideal generated by the indecomposable elements fl` for
` 2 Lo.
(iii)For any k 2 N \ Lo, flk is a decomposable element in the ideal (fl` | ` 2*
* Lo).
Proof. (i) is proved in Lemma 5-1. For (ii), the equation sk(k) = oek follows f*
*rom
Lemma 5-2. Also, the elements fl`'s and k's form a complete set of generators
of MSp*[2]~=MSO*[2]in view of Milnor's criterion. Since o is a multiplicative
idempotent, from Proposition 2-3 we have o*(fln) = 0 for all n 1, in particula*
*r for
n 2 Lo. This proves the first formula in (5-3). The second formula in (5-3) fol*
*lows
from Lemma 5-2. For (iii), if k 2 N \ Lo, then flk is in the ideal of decomposa*
*ble
elements by the definition of the set Lo. Since o*(flk) = 0 by Proposition 2-3,*
* flk is
in the ideal (fl` | ` 2 Lo) by (ii). This completes the proof of Theorem 5-3.
Corollary 5-4. Let o : MSp[2]-! MSp[2]be a multiplicative idempotent. Then the
kernel of the induced map o* : MSp[2] * -! MSp[2] * is precisely the ideal gene*
*rated
by the higher coefficients of the associated power series
X
o*(w) = w + (-1)n2flnwn+1 2 MSp*[2][ [ w ] ]:
n1
Furthermore, as a set of generators of this ideal, we may take indecomposable c*
*oef-
ficients {fl` | ` 2 Lo} of o*(w).
Proof. This follows from (ii) of Theorem 5-3.
MULTIPLICATIVE IDEMPOTENTS OF MSp
[2] 27
Corollary 5-5. Let o : MSp[2]-! MSp[2]be any multiplicative idempotent and
let o : MSp[2] *(MSp[2]) -! MSp[2] *be the corresponding algebra map. Let I =
(Q1; : :;:Qn; : :): MSp[2] *(MSp[2]) be the augmentation ideal. Then the sequen*
*ce
I -o! MSp[2] * o*-!MSp[2] *is exact. In other words, Ker o* is generated by ele*
*ments
o(Qk) for k 1.
Proof. This follows from the fact that the higher coefficients of the associate*
*d power
series o*(w) are exactly the images of Qj's of the associated algebra map o, as
shown in the proof of Proposition 2-3.
So far in this section, we have assumed that we are given a multiplicative i*
*dem-
potent and we deduced its properties from the associated power series. We consi*
*der
the converse, and ask the following question.
Question. Let o :PMSp[2]-! MSp[2]be the ring map associated to a given power
series g(w) = w + n1 (-1)n2flnwn+1 2 MSp*[2][ [ w ] ]. What conditions on f*
*ln's
are necessary and sufficient so that o is a multiplicative idempotent?
Given a power series g(w), we define a set Lg as Lo given in (5-2). From The*
*orem
5-3, we know that in order that o is a multiplicative idempotent, all the higher
coefficients of g(w) have to belong to the ideal (fl` | ` 2 Lg), and s`(fl`) = *
*1 for
` 2 Lg. It turns out that these conditions are not enough. See Theorem 5-9 below
for precise conditions. Our necessary and sufficient condition described there*
* is
constructive and it is expressed inductively. Our basic observation is the foll*
*owing.
P
Lemma 5-6. Given a power series g(w) = w + n1 (-1)k2flnwn+1 , let
XN
(5-4) g(N)(w) = w + (-1)nflnwn+1
n=1
be the degree N +1 truncated polynomial of g(w) for N 1. Let o; o (N): MSp[2]-!
MSp[2]be the ring spectra maps associated to g(w) and g(N)(w), respectively. Th*
*en
o is a multiplicative idempotent if and only if o (N)is a multiplicative idempo*
*tent
for all N 1.
Proof. Since the power series g(w) and g(N)(w) agree mod (wN+2 ), the correspon*
*d-
ing ring maps o and o (N)agree on 4N-skeleton of MSp[2]by Lemma 4-3. Thus,
(*) o*(fln) = o*(N)(fln) for 1 n N:
If o is a multiplicative idempotent, then o* annihilates all the higher coeffic*
*ients
of g(w) and we have o*(fln) = 0 for all n 1. This implies that for any N 1,
o*(N)(fln) = 0 for 1 n N due to (*). This means that o*(N) annihilates all
the higher coefficients of the associated power series g(N)(w). Hence o (N) is*
* a
multiplicative idempotent for any N 1.
Conversely, if o (N)is a multiplicative idempotent for all N 1, then (*) im*
*plies
that o* annihilates all the higher coefficients of g(w). Thus o is a multiplic*
*ative
idempotent.
In view of this lemma, we can construct power series corresponding to multi-
plicative idempotents inductively in terms of polynomials adding one term at a
28 HIROTAKA TAMANOI
time. But not arbitrary monomials can be added to obtain multiplicative idempo-
tents. To understand the restriction on monomials which can be added, we prepare
a lemma.
Lemma 5-7. Let o1 : MSp[2]-!PMSp[2]be a ring map with the associated power
series g1(w) = o1*(w) = w + n1 (-1)n2ffinwn+1 . Suppose that ffiN is a dec*
*om-
posable element for some N 1 and let g2(w) = w + (-1)N 2flN wN+1 + . . .be a
power series for some flN 2 MSp-4N[2]. Let o be a ring spectra map correspondi*
*ng
to the composed power series g(w) = g2 g1(w) . Then the effects of o* and o1* on
MSp*[2]are related by the following formula:
X X 2k+1
(5-5) T + o*(m2k)T 2k+1= '-12(T ) + o1*(m2k) '-12(T ) ;
k1 k1
p __________
where '2(T ) = -g2(-T 2) T mod (T 2), and T is a formal variable. In part*
*ic-
ular, o and o1 agree on the 4(N - 1)-skeleton of MSp[2], and on degree 4N,
(5-6) o*(flN ) = o1*(flN ) - sN (flN )flN :
Suppose o1 is a multiplicative idempotent. Then, for o to be a multiplicat*
*ive
idempotent, it is necessary that o1*(flN ) = sN (flN )flN . In other words, if*
* flN is
decomposable, then for o to be an idempotent it is necessary that flN 2 Ker o1*
**. If
flN is indecomposable, then for o to be an idempotent it is necessary that 2N *
*+ 1 is
not a prime power, sN (flN ) = 1, and flN 2 Im o1*.
p _________ p __________
Proof. Let '(T ) = -g(-T 2) and '1(T ) = -g1(-T 2) . Then we have '(T ) =
'2 '1(T ) . From (3-10), we have logo1(T ) = log '-11(T ) . This implies that
logo(T ) = log '-1 (T ) = log '-11'-12(T ) = logo1 '-12(T ) . This proves (*
*5-5).
Since '-12(T ) = T -flN T 2N+1 +. .,.equating the coefficients of T 2k+1's, *
*we have
o*(m2k) = o1*(m2k) for k N - 1; and o*(m2N ) = o1*(m2N ) - flN :
Thus, o* and o1* agree on elements in degree -4k for 1 k N - 1. If
flN = c m2N + h(N)(m1; : :;:m2N-1 ) for some c 2 Z[ 1_2] and for some weighted
homogeneous polynomial h(N) with rational coefficients, then a similar calculat*
*ion
as in the proof of Proposition 4-1 shows that o*(flN ) = o1*(flN ) - c flN . S*
*ince
c = sN (flN ), we obtain (5-6).
If o is a multiplicative idempotent, then o* annihilates all higher coeffici*
*ents of
N-1X k k+1 N N+1 N+2
g(w) = g2 g1(w) w+ (-1) 2ffikw +(-1) 2(ffiN +flN )w mod (w ):
k=1
Since o1 is a multiplicative idempotent and ffiN is decomposable, ffiN must b*
*elong to
the ideal generated by ffi1, : :,:ffiN-1 by Corollary 5-4. Since o is an idem*
*potent,
o* annihilates ffi1, : : :,ffiN-1 . Thus, ffiN is also annihilated by o*. Si*
*nce o* also
annihilates ffiN + flN because o* annihilates all the higher coefficients of *
*g(w), it
follows that o*(flN ) = 0. Combining with (5-6), we have o1*(flN ) = sN (flN )*
*flN .
Since o1 is assumed to be an idempotent, sN (flN )2 = sN (flN ). Thus, sN (flN*
* ) =
0; 1. Hence if flN is decomposable, we must have o1*(flN ) = 0. And if flN *
*is
MULTIPLICATIVE IDEMPOTENTS OF MSp
[2] 29
indecomposable, we must have sN (flN ) = 1, which can happen only when 2N + 1
is not a prime power by Theorem 5-3 (i). In this case, o1*(flN ) = flN implies*
* that
flN 2 Im o1*.
We apply this lemma to our present context of constructing multiplicative id*
*em-
potents inductively.
P N-1
Lemma 5-8. Let g(w) = w + n=1 (-1)n2flnwn+1 2 MSp*[2][ [ w ] ]. For some
flN 2 MSp-4N[2], we let "g(w) = g(w) + (-1)N 2flN wN+1. Let o; "o: MSp[2]-! M*
*Sp[2]
be the ring maps corresponding to g(w) and "g(w), respectively. Then
"ois an idempotent () o is an idempotent and o*(flN ) = sN (flN )flN 2 MSp**
*[2]
(
o is an idempotent and either (a) flN 2 Ker o* or
()
(b) 2N + 1 is not a prime power, sN (flN ) = 1, and flN 2 Im o*
Proof. Let h(w) be a power series such that "g(w) = h g(w) . h(w) is of the form
h(w) = w + (-1)N 2flN wN+1 + (higher order terms):
Suppose "ois a multiplicative idempotent. Since g(w) is a truncation of "g(w), *
*o is a
multiplicative idempotent by Lemma 5-6. By applying Lemma 5-7 with g1(w) and
g2(w) replaced by g(w) and h(w), we see that o*(flN ) = sN (flN )flN . In MSp*[*
*2], this
condition is equivalent to the one stated above due to Theorem 5-3 (i).
Conversely, suppose o is a multiplicative idempotent and the element flN is
such that o*(flN ) = sN (flN )flN . Then by (5-6), the ring map "ohas the prop*
*erty
"o*(flN ) = o*(flN ) - sN (flN )flN = 0. Since the power series g(w) and "g(w*
*) agree
modulo (wN+1 ), the corresponding ring maps o and "o agree on the 4(N - 1)-
skeleton by Lemma 4-3 and we have "o*(fln) = o*(fln) for 1 n N - 1. Since o is
assumed to be a multiplicative idempotent, o*(fln) = 0 for 1 n N - 1. Hence
"o*annihilates all the higher coefficients of the associated power series "g(w)*
*. Thus
"ois a multiplicative idempotent.
Combining Lemma 5-6 and Lemma 5-8, we get the following characterization for
those power series g(w) which correspond to multiplicative idempotents.
Theorem 5-9. Let o : MSp[2]-! MSp[2]be a ring spectra map with its associated
P
power series g(w) = w + k1 (-1)k2flkwk+1 . Let o (n): MSp[2]-! MSp[2]be the
P n
ring map associated to the truncated polynomial g(n)(w) = w + k=1(-1)k2flkwk+1
for all n 1. We let o (0)be the identity map. Then
(5-7) o is an idempotent () o*(n-1)(fln) = sn(fln)fln for alln 1
This condition is equivalent to the following conditions for all n 1:
(1) fln 2 Ker o*(n-1)and fln is decomposable, if 2n + 1 is a prime power,
(2) fln 2 Ker o*(n-1), if 2n + 1 is not a prime power and fln is decomposab*
*le,
(3) fln 2 Im o*(n-1)and sn(fln) = 1, if 2n + 1 is not a prime power and fln*
* is
indecomposable.
30 HIROTAKA TAMANOI
Proof. Suppose o is a multiplicative idempotent. Then by Lemma 5-6, o (n)are all
multiplicative idempotents for all n 0. Applying Lemma 5-8 to the pair o (n)and
o (n-1), we see that o*(n-1)(fln) = sn(fln)fln for all n 1.
Conversely, assume that o*(n-1)(fln) = sn(fln)fln for all n 1. We show that*
* o (n)
is a multiplicative idempotent for all n 0 by induction on n. When n = 0, then
o (0)is an identity map, and in particular it is an idempotent. Now assume that
o (n-1)is a multiplicative idempotent for n 1. By Lemma 5-8 applied to o (n-1)
and o (n)together with our hypothesis o*(n-1)(fln) = sn(fln)fln, we see that o *
*(n)is
also a multiplicative idempotent. This completes the inductive step and we have
shown that o (n)is a multiplicative idempotent for all n 1. By Lemma 5-6, this
implies that o is also a multiplicative idempotent.
By Theorem 5-3 (i), sn(fln) 6= 0 if and only if 2n + 1 is not a prime power,*
* and
in this case we actually must have sn(fln) = 1. From this we obtain the above m*
*ore
detailed conditions.
x6. Multiplicative Indecomposable Splittings of MSp[2]
In this final section, we split off minimal ring spectra from MSp[2]. Altho*
*ugh
such subspectra are not unique, they are all homotopically equivalent to each o*
*ther.
First we clarify what we mean by minimal ring spectra.
Definition 6-1. A ring spectrum E is said to be multiplicatively indecomposable
if any multiplicative idempotent acting on E is an identity map.
When a ring spectrum E is multiplicatively indecomposable, we cannot split o*
*ff
a subring spectrum. In this sense, E is "minimal".
We apply Theorem 4-5 for the largest possible L N. We let
^L= {` 2 N | 2` + 1 is not a prime power};
(6-1) ^
P = {k 2 N | 2k + 1 is a prime power}:
For each ` 2 ^L, we choose an indecomposable element ` 2 MSp-4`[2]such that
s`(`) = 1. The choice of these indecomposable elements is not unique.
Theorem 6-2. For any choice of indecomposable elements ` 2 MSp-4`[2]for ` 2 ^L,
there exists a multiplicative idempotent o : MSp[2]-! MSp[2]and indecomposable
elements k 2 MSp-4k[2]for k 2 ^Psuch that
ae0; if 2n + 1 is not a prime power
(6-2) o*(n) =
n if 2n + 1 is a prime power:
Let LSpo = o (MSp[2]) be the multiplicative summand split off from MSp[2]as the
image of o . Then its homotopy groups form a polynomial algebra given by LSpo **
* =
Z[ 1_2][ n | n 2 ^P], and the spectrum LSpo is a multiplicatively indecomposabl*
*e ring
spectrum.
Proof. The first half of Theorem 6-2 is a special case of Theorem 4-5. To find
elements k for k 2 ^Pstated above we first choose arbitrary element 0k2 MSp-4k[*
*2]
such that sk(0k) = oek. We then let k = o*(0k). It then follows that sk(k) = oek
and o*(k) = k for k 2 ^P, as in the proof of Lemma 5-2.
MULTIPLICATIVE IDEMPOTENTS OF MSp
[2] 31
To see that the ring spectrum LSp = LSpo is multiplicatively indecomposable,
let O : LSp -! LSp be an arbitrary multiplicative idempotent. We can extend
it to a multiplicative idempotent of MSp[2]by letting Ob : MSp[2]-ss!LSp -O!
LSp -j!MSp[2], where ss and j are the projectionPand the inclusion maps. Let the
associated power series be bO*(w) = w + n1 (-1)n2flnwn+1 2 MSp*[2][ [ w ] ].*
* By
Proposition 3-5 (iii), in MUQ * we have bO*(m2n) m2n - fln modulo the ideal of
decomposable elements for any n 1. If k 2 ^P, then flk is decomposable by Lemma
5-1 and we have bO*(m2k) m2k modulo decomposables for k 2 ^P. Thus, O*(k)
has the same sk( . ) number as k, and so O*(k) is an indecomposable element in
LSp* for all k 2 ^P. It follows that O is a multiplicative idempotent which ind*
*uces an
isomorphism on homotopy groups. Hence O2 = O and O is invertible. It follows th*
*at
O is an identity map. This proves that LSp is multiplicatively indecomposable.
Remark. (i) The homotopy group of the BP spectrum at a prime p is given by
BP* = Z(p)[v1; v2; : :;:vn; : :]:, where |vn| = 2(pn - 1). If we use Hazewinkel*
* gener-
ators rather than Araki generators (see [R]), then in fact vn 2 MU2(pn-1) is in*
*tegral
and vn = pmpn-1 + (decomposables ) for all n 1. Hence if k 2 ^Pis of the form
2k + 1 = pj for some odd prime p and for some j 1, we may let k = o*(vj) using
the jth Hazewinkel generator of BP* for the odd prime p.
(ii) When localized at an odd prime p, the spectrum MSp(p) splits into a wedge
sum of many suspension copies of the BP spectrum. The above theorem says
that when only 2 is inverted in MSp, copies of BP spectra for all odd primes are
bound together in LSp and they cannot be separated, since LSp is multiplicative*
*ly
indecomposable.
Although LSpo is uniquely determined by the multiplicative idempotent o , the
ring map o itself is not canonically determined by the set of generators `'s ch*
*o-
sen for each ` 2 ^L. The situation is similar to Lemma 4-1: the multiplicati*
*ve
idempotent depends on our choice of power series of the form (4-3) for a given *
*n.
However, any two such multiplicatively indecomposable ring spectra can be
shown to be equivalent, as follows.
Proposition 6-3. Let o1; o2 : MSp[2]-! MSp[2]be multiplicative idempotents
such that Lo1 = Lo2 = ^L. Let LSpo1 and LSpo2 be the ring spectra split off as
the images of the idempotents o1, o2, respectively. Then these two ring spectra*
* are
equivalent.
Proof. For i = 1; 2, let ssi and ji be the projection and the inclusion maps for
LSpoi, that is, MSp[2]-ssi!LSpoi ji-!MSp[2]. Then the composition of ring spect*
*ra
maps LSpo1 -j1!MSp[2]-ss2!LSpo2 induces an isomorphism on homotopy groups,
because the indecomposable generators in LSp*o1are mapped into indecomposable
generators of LSp*o2. Hence it is a homotopy equivalence.
From now on, we just refer to the spectrum LSp without making the multiplica-
tive idempotent o explicit. As remarked above, when localized at p, MSp and MU
split into wedge sums of many suspension copies of the BP spectrum. A similar
decomposition exists when only 2 is inverted.
32 HIROTAKA TAMANOI
Corollary 6-4. We have the following decompositions of spectra:
_ _
(6-3) MSp[2]~= I2|I|LSp; MU[2]~= J 2|J|LSp;
where I ranges over all finite (possibly empty) sequences of integers from 2 ^L:
I = (i1 i2 . . .ir | ik 2 2 ^Lfor1 k r; r 0);
and J ranges over all finite (possibly empty) sequences of integers which are e*
*ither
odd or from 2 ^L:
J = j1 j2 . . .jr | jk 2 2 ^L[ (2N - 1) for 1 k r ;
P P
and |I| = k ik and |J| = k jk.
The proof is standard and straightforward, so it is omitted.
Remark. In [BM], a ring spectrum oddMU is constructed as the image of the multi-
plicative idempotent "2 on MU[2]which annihilates m2k-1 for all k 1 and which
is an identity on m2k for all k 1. Since the composition of ring spectra maps
aeSpU ssodd
MSp[2]--! MU[2]-! MU is an equivalence, we also have a decomposition of
the form (6-3) for oddMU.
References
[A1] Adams, J. F., Lectures on generalized cohomology, Lecture Notes in Math., *
*vol. 99, Springer-
Verlag, Berlin, 1969, pp. 1-138.
[A2] Adams, J. F., Stable Homotopy and Generalized Homology, University of Chic*
*ago Press,
Chicago, Ill, 1974.
[BM] Baker, A. and Morava, J., MSP localized away from 2 and odd formal group l*
*aws, preprint
(1995).
[B1] Buhstaber, V. M., Two-valued formal groups I, Math. USSR Izv. 9 (1975), 98*
*7-1006; II 10
(1976), 271-308.
[B2] Buhstaber, V. M., Topological applications of the theory of two valued for*
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