INFINITE SUMS OF ADAMS OPERATIONS AND
COBORDISM
IMMA G'ALVEZ AND SARAH WHITEHOUSE
Abstract. In recent work, various algebras of stable degree zero opera-
tions in p-local K-theory were described explicitly [5]. The elements are
certain infinite sums of Adams operations. Here we show how to make
sense of the same expressions for MU(p)and BP , thus identifying the
Ä dams subalgebraö f the algebras of operations. We prove that the
Adams subalgebra is the centre of the ring of degree zero operations.
1.Introduction
This paper builds on recent work on stable operations in p-local complex
K-theory [5]. That work described all operations in terms of Adams opera-
tions. In a sense, this was done by specifying which infinite sums of Adams
operations are allowed. Here we explore these expressions for MU(p)and
BP .
We begin by briefly discussing the definition and basic properties of Adams
operations for these theories. These facts are well known, but we hope it
is helpful to gather them together here. The main results of [5] are then
recalled and the analogues for MU(p)and BP are proved, thus identifying
the Ä dams subalgebraö f the algebras of operations for these theories.
The final sections are devoted to showing that this subalgebra is precisely
the centre of the ring of stable degree zero operations. This can be compared
to old results of Novikov (for MU) and Araki (for BP ), [8, 2]. These state
that the Adams operations form the centre of the group of stable degree zero
multiplicative operations.
2. Adams Operations
We begin by summarizing some standard results on stable cohomology
operations.
Proposition 1. [3, Section 14] Let E = MU, BP or KU. A stable multi-
plicative cohomology operation ` : E ! E is uniquely determined by its value
`(xE ) on the orientation class xE 2 E2(CP 1).
We remark that the same is true for the Adams summand of p-local K-
theory, which we denote by G.
____________
Date: 11thJune 2004.
2000 Mathematics Subject Classification. Primary: 55S25; Secondary: 55N22, 19*
*L41.
Key words and phrases. K-theory - operations - cobordism.
1
2 IMMA G'ALVEZ AND SARAH WHITEHOUSE
In order to have stable Adams operations, we will work p-locally, for a
prime p.
Definition 2. Let E = MU(p), BP , KU(p)or G. A stable Adams operation
ffEis defined for each ff 2 Zx(p)as the unique multiplicative operation given
on the orientation class xE by
[ff]E (xE ) * 1
ffE(xE ) = _________2 E (CP ) = E*[[xE ]] ,
ff
where [ff]E denotes the formal sum.
Lemma 3. ffEacts as multiplication by ffn on E2n.
Proof.Let the log seriesPof the formal group law corresponding to E be
given by logE(xE ) = 1i=0mixi+1E, where mi 2 E2i Q. The operation ffE
transforms this into logE(( ffE)-1(xE )), see [9, Appendix B]. This is
ff -1 1
logE ( E ) (xE ) = logE expE __logEffxE
ff
1
logE ffxE X i i+1
= _________= ff mixE .
ff i=0
Thus on coefficients, mi 7! ffimi. Since E* is torsion free and since ffEis
multiplicative, this is sufficient to determine ffEon the coefficient groups.
For E = MU, BP , KU or G, the representation of the ring of operations
given by their actions on coefficient groups is faithful:
E*(E) ,! End (E*).
For MU this result is due to Novikov [8, Lemma 5.2]; for KU, see [6, The-
orem 2]. The BP and G versions follow. In particular, we see that Adams
operations multiply as expected: ffE fiE= fffiE. We also note that, since ffE
is multiplicative, it is a grouplike element with respect to the coproduct of
E*(E), ( ffE) = ffE ffE.
If e denotes the connective cover of E, the map e ! E leads to a map
E*(E) ! e*(e). Thus we may define an Adams operation ffefor e, with all
the expected properties, as the image of ffEunder this map.
3. Operations in p-local K-theory
Let KU(p)be the p-local periodic complex K-theory spectrum and let ku(p)
be the connective version. As above, for these theories we have stable Adams
operations, ffKUand ffkurespectively, for each ff in Zx(p).
Let p be an odd prime and let q be primitive modulo p2. This ensures that
q is primitive modulo pr for all r 1, which is to say that the powers of q
are dense in the p-adic units Zxp. Clearly, the polynomial ring generated by
the Adams operation qKU is a subring of the ring of all stable degree zero
INFINITE SUMS OF ADAMS OPERATIONS AND COBORDISM 3
KU(p)operations. This does not give all operations; it is countable and the
operation algebra is known to be uncountable [1].
Definition 4. Let p be an odd prime and let q be primitive modulo p2. Let
E = MU(p)or KU(p)and let e = MU(p)or ku(p). We define two families of
polynomials.
Q n-1
1) Let 'en= Q i=0( qe- qi) 2 Z(p)[ qe].
2) Let En= n-1i=0( qE- qi) 2 Z(p)[ qE], where qi is the i + 1-st term of
the sequence
1, q, q-1, q2, q-2, q3, q-3, q4, . . ..
For p odd, the following result of [5] shows how to view the ring of all
degree zero stable KU(p)operations as a completion of the polynomial ring
Z(p)[ qKU]. It also gives a description for the connective case, corresponding
to a different completion of a polynomial ring. These results may be viewed
as specifying which infinite sums of Adams operations are admissible.
Theorem 5. [5, Theorems 2.2 and 6.2]
1) Elements of ku0(p)(ku(p)) can be expressed uniquely as infinite sums
X
an'kun,
n 0
where an 2 Z(p).
2) Elements of KU0(p)(KU(p)) can be expressed uniquely as infinite sums
X
an KUn,
n 0
where an 2 Z(p).
Explicit product and coproduct formulas and many related results can be
found in [5]. Analogues of these results hold for p = 2, but they are more
complicated. We give only the connective version.
Definition 6. Define elements ikun2 ku0(2)(ku(2)), for n 0, by
m-1Y
iku2m+1= ( -1ku- 1) ( 3ku- 32i+1),
i=0
m-1Y Xm 2m
`i(3)`i(3 ) ku
iku2m= ( 3ku- 32i) + ____________2ii2m-2i+1,
i=0 i=1 2`i(3 )
Q r-1
where `r(X) = i=0(X - 32i).
Theorem 7. [5, Theorem 8.2] The elements of ku0(2)(ku(2)) can be expressed
uniquely as infinite sums X
anikun,
n 0
where an 2 Z(2).
4 IMMA G'ALVEZ AND SARAH WHITEHOUSE
4.Infinite Sums of Adams Operations for MU(p)
In this section, we show that the infinite sums of Adams operations of the
previous section are also defined for MU(p).
P 1
Proposition 8. Let p be an odd prime. The infinite sums n=0an'MUn,
where an 2 Z(p), are well-defined operations in MU(p)-cohomology.
Proof.Clearly, finite sums of the 'MUn are well-defined operations. To see
that the same is true for the infinite sums, it suffices to show that 'MUn ! 0
as n ! 1 in the usual filtration topology of MU*(p)(MU(p)).
This may be checked by considering the action on coefficient groups. By
Lemma 3, qMU acts as multiplication by qi on ß2i(MU(p)). It follows that
'MUn acts as zero on ß2i(MU(p)) for i < n.
Proposition 9. For any odd prime p, the map
ku0(p)(ku(p)) ! MU0(p)(MU(p))
given by
X1 X1
an'kun7! an'MUn
n=0 n=0
is an injective algebra map.
P 1
Proof.Consider n=0 an'MUn in MU0(p)(MU(p)) and suppose that am 6= 0,
withPm chosen minimal. This operation acts on ß2m (MU(p)) 6=Q0 as the finite
sum mn=0an'MUn = am 'MUm and thus as multiplication by am m-1i=0(qm -
qi) 6= 0. Thus the operation is non-trivial. So the map is injective.
It is easy to see that we have an algebra map: the product of two infinite
sums is determined in both the source and the target by the products of
Adams operations. (See [5, Prop. 2.7] for an explicit formula.)
The source ku0(p)(ku(p)) is a completed bialgebra. We note that the injec-
tive map above also respects the coproduct, since the coproduct of a general
infinite sum is determined by the fact that the Adams operations are group-
like; see [5, Prop. 2.9] for an explicit formula. So the image of ku0(p)(ku(p))*
* is
a subbialgebra of MU*(p)(MU(p)).
There is also an injective algebra map KU0(p)(KU(p)) ! MU0(p)(MU(p)).
This is because the injection of Proposition 9 can be composed with the
inclusion of bialgebras
KU0(p)(KU(p)) ,! ku0(p)(ku(p)),
resulting from the covering map ku(p)! KU(p). An explicit formula for this
map is given in [5, Prop. 7.1], expressing each KUn in terms of the 'kum.
Again the injection is well-behaved with respect to the coproduct. This
time we have a conjugation map Ø and this is also respected by the inclusion,
-1
as it is determined for both E = KU(p)and E = MU(p)by Ø qE= qE .
INFINITE SUMS OF ADAMS OPERATIONS AND COBORDISM 5
So far the discussion in this section has all been for odd primes. However,
it is easy to see that the analogues for p = 2 also hold. One defines iMUn 2
Z(2)[ 3MU, -1MU] in the obvious way, just as in Definition 6.
P 1
Proposition 10. (1) The infinite sums n=0 aniMUn, where an 2 Z(2),
are well-defined operations in MU(2)-cohomology.
(2) The map ku0(2)(ku(2)) ! MU0(2)(MU(2)) given by
1X X1
anikun7! aniMUn
n=0 n=0
is an injective algebra map.
Proof.The key properties of ikunare that it acts as zero on coefficient groups
ß2i(ku(2)), for all i < n, and its action is non-zero on ß2n(ku(2)). (See [5,
Section 8] for a proof and related formulas.) Using this, one may check that
the infinite sums are well-defined MU(2)operations just as in Proposition 8
and that we have the claimed injection of algebras just as in Proposition 9.
5. Infinite Sums of Adams Operations for BP
We record here the BP analogues of the results of the preceding section,
omitting proofs since these are easy modifications of the MU(p)versions.
For p an odd prime, the spectra KU(p)and ku(p)each split into p-1 copies
of spectra which we denote by G and g, respectively1. As before we choose
q primitive modulo p2 and now we let ^q= qp-1. Thus the powers of ^qare
dense in 1 + pZp. Now we recall from [5] the description of degree zero stable
operations for g.
Definition 11. Define ^'gn2 Z(p)[ qg] g0(g), for n 0, by
n-1Y
^'gn= ( qg- ^qi).
i=0
Also define ^'BPn2 Z(p)[ qBP] BP 0(BP ) in the same way.
Theorem 12. [7, Theorem 2.2], [5, Theorem 4.4] Elements of g0(g) can be
expressed uniquely as infinite sums
X
an'^gn,
n 0
where an 2 Z(p).
Proposition 13. Let p be an odd prime.
P 1
(1) The infinite sums n=0an'^BPn, where an 2 Z(p), are well-defined
operations in BP -cohomology.
____________
1The notations E(1) and e(1) and L and l are also used.
6 IMMA G'ALVEZ AND SARAH WHITEHOUSE
(2) The map ' : g0(g) ! BP 0(BP ) given by
X1 X1
an'^gn7! an'^BPn
n=0 n=0
is an injective algebra map.
The same remarks as in the preceding section about the coproduct and
the comparison with the periodic operations may be made here.
Now let p = 2. Again we define iBPn 2 Z(2)[ 3BP, -1BP] just as in Defini-
tion 6.
Proposition 14. Let p = 2.
P 1
(1) The infinite sums n=0aniBPn, where an 2 Z(2), are well-defined
operations in BP -cohomology.
(2) The map ' : ku0(2)(ku(2)) ! BP 0(BP ) given by
X1 X1
anikun7! aniBPn
n=0 n=0
is an injective algebra map.
6. Diagonal Operations
We will need to discuss operations which act diagonally on coefficient
groups and so we introduce the following notation.
Definition 15. Let E = MU, MU(p)or BP . We write DE for the subring
of E0(E) consisting of operations whose action on each coefficient group E2i
is multiplication by an element ~i of the ground ring, (that is an element of
Z for E = MU, an element of Z(p)for E = MU(p)or E = BP ).
We next recall some results from [8] on MU operations. For each fi-
nite non-decreasing sequence of positive integers, ffP= (ff1, ff2, ff3, . .)., *
*there
is an operation S~ff2 MU*(MU), of degree 2|ff| = 2 iffi. Here S~ff=
OE(~oeff), where OE : MU*(BU) ! MU*(MU) is the Thom isomorphism, oeff2
MU*(BU) is the Conner-Floyd characteristic class associated to ff and ~oeffis
defined by ~oeff(,) = oeff(-,).
Proposition 16. Let E = MU, MU(p)or BP . Then Z (E0(E)) = DE .
Proof.Clearly, DE is contained in the centre Z (E0(E)). To prove the reverse
inclusion, we first let E = MU. Since E* is torsion free, an operation ` is
zero if and only if `* 1Q is zero on E* Q = Q[CP 1, CP 2, . .].. So we can
work with monomials in the classes [CP n]. We first consider the operation
~S(m)2 E2m (E). We let S(m) = [CP m]S~(m)2 E0(E). This operation has the
property that it acts as zero on any decomposable class of degree 2m and
as multiplication by (m + 1) on [CP m]. Let ` 2 Z(E0(E)). The relation
S(m)` = `S(m) then tells us that `*[CP m] = ~m [CP m] for some ~m 2 Z.
More generally, for ff = (ff1, ff2, . .,.ffr), we have a Landweber-Novikov
operation ~Sff2 E2|ff|(E). By [8, Lemma 5.5], (S~ff)*([CP n]) = ~ff[CP n-|ff|],
INFINITE SUMS OF ADAMS OPERATIONS AND COBORDISM 7
where ~ff2 Z and ~ff6= 0. Let CP ff= CP ff1x CP ff2x . .x.CP ffrand let
Sff= [CP ff]S~ff2 E0(E). Then the relation (Sff`)*[CP |ff|] = (`Sff)*[CP |ff|]
tells us that ` acts on [CP ff] as multiplication by ~|ff|2 Z. Thus ` 2 DE .
The argument is just as above for MU(p)and that for BP is similar, but
i-1
here we need only work with monomials in the [CP p ], since BP* Q =
i-1
Q[CP p | i 1].
We remark that, if one considers the algebra of all operations, rather than
only degree zero ones, then the centre consists of just the constant operations.
7. Congruences
In this section p is an odd prime and we compare two families of congru-
ences, one related to the connective Adams summand g and the other to
BP . The results will be used in the next section to give a new description
of the centre of BP 0(BP ).
Clearly, we may identify the ring ofQdiagonal operations DBP with a sub-
ring of the infinite direct product k 0Z(p). In what follows we will often
use this identification without further comment.
We begin by noting the analogue for the connective Adams summand g
of [4, Theorem 11].Q This characterizes g0(g) as a subring of the infinite
direct product k 0Z(p)by a family of congruences which involve Gaussian
polynomials.
n
Definition 17. The Gaussian polynomial i 2 Z[t] is defined, for non-
negative integers n and i, by
~ ~ i-1
n Y 1 - tn-k
= ________.
i k=01 - ti-k
n
Also let i adenote the value of this polynomial at t = a.
Theorem 18. If OE 2 g0(g) acts on g2(p-1)i= Z(p)as multiplication by
~i2 Z(p), then
Xn n-i~ ~
n ffi (n)
(-1)n-i^q( 2 ) ~i 0 mod p p,
i=0 i ^q
for all n 0, where ffip(n) = n + p(n!). Moreover every sequence satisfying
these congruences arises from a unique stable operation.
Proof.A Z(p)-basis for G0(g) is given in [5, Prop. 4.2]. Using [4, Prop. 8] this
may be expressed in terms of Gaussian polynomials. The result follows from
the fact that g0(g) is the Z(p)-linear dual of G0(g).
Definition 19. Let
n-in
(-1)n-i^q( 2 )i^q
Cn,i= _________________ 2 Q,
pffip(n)
and let Cn = (Cn,i)i 0, a sequence of rational numbers.
8 IMMA G'ALVEZ AND SARAH WHITEHOUSE
We note that Cn,n = p-ffip(n)and Cn,i= 0 when i > n. Also note that
ffip(pr) = 1 +Pp + p2 + . .+.pr. The nth congruence of Theorem 18 may be
expressed as ni=0Cn,i~i2 Z(p)or, adopting vector notation, Cn . ~ 2 Z(p).
The rest of this section will be devoted to comparing the congruences of
Theorem 18 with a system of congruences characterizing DBP .
Next we recall that, using duality, an operation ` : BP ! BP is deter-
mined by the left BP*-module map ~`= <`, > : BP*(BP ) ! BP*, where
< , > is the Kronecker pairing. The action of the operation on coefficient
groups `* : BP* ! BP* is obtained by composition with the right unit map:
`* = ~`jR . (See, for example, [3, 11.22].)
We will adopt the BP notation of [9]. Thus BP* = Z(p)[v1, v2, . .]., where
the vi are Araki's generators, |vi| = 2pi- 2, and BP*(BP ) = BP*[t1, t2, . .].,
where |ti| = 2pi-2. We also use the elements li2 BP* Q, related recursively
to the vi by the formula of Araki. As usual, because all objects are torsion
free, one may calculate rationally. We will do this implicitly, so that, for
example, we write ~`(li) to mean (~` 1Q)(li).
We suppose that we have ` 2 DBP , acting on BP2(p-1)ias multiplication
by ~i2 Z(p). (So, in the notation of Sect. 6, ~ = (~0, ~p-1, ~2(p-1), . .)..) By
consideration of the relation `* = ~`jR , one may calculate recursively ~`(x)
for x 2 BP*(BP ) in terms of the sequence ~ = (~i)i 0. Since, in ~`(x), the
coefficient of each monomial in the generators vi must lie in Z(p), one finds
congruences imposed on the terms of this sequence.
We now give more details of this system of congruences. Let ff, fi, fl, ffi,*
* ffl
be finite sequences of non-negative integers. If ff = (ff1,Pff2, . .,.ffm ) the*
*n vff
denotes the monomial vff11vff22.v.f.fmmand let ||ff|| = mi=1ffi(1 + p + . .+.
ff|
pi-1) = _|v__2(p-1).
P
Let Efffi,fl2 Z(p)be determined by jR (vff) = fi,flEfffi,flvfitfl. Then, s*
*ince
~`jR (vff) = `*(vff), we find that
(
X X ~||ff||if ff = ffl,
(20) Efffi,flDflffi(~) =
fi+ffi=fflfl 0 otherwise,
P fl fl
where ~`(tfl) = ffiDffi(~)vffi. Now (20) determines Dffi(~) recursively as a
finite rational linear combination of the ~i. On the other hand, we have
Dflffi(~) 2 Z(p), for all fl and ffi, and these integrality conditions form our
family of congruences.
For example, D(1)(1)(~) = a(~1-~0_p), where a 2 Zx(p). So ~1-~0_p2 Z(p). Noti*
*ce
that this is the congruence C1 . ~ 2 Z(p)of Theorem 18.
INFINITE SUMS OF ADAMS OPERATIONS AND COBORDISM 9
In order to compare the congruences of Theorem 18 and the BP ones, we
need some further notation.
Q
Definition 21.QLet ~ denote an element in k 0Z(p)and define the following
subsets of k 0Z(p).
Sg1 = ~ | Ci. ~ 2 Z(p)for all 0 i,
Sgn = ~ | Ci. ~ 2 Z(p)for 0 i n,
fl
SBP1 = ~ | Dffi(~) 2 Z(p)for all fl,,ffi
0 BP 0
SBPn = ~ | 9 ~ 2 S1 with ~i= ~i for0 i n .
Thus g0(g) ~= Sg1 by Theorem 18 and DBP ~= SBP1 by the preceding
discussion. Also, clearly
"1
Sg0 Sg1 . . .Sgn Sgn+1 . . .Sg1 = Sgn,
n=0
"1
SBP0 SBP1 . . .SBPn SBPn+1 . . .SBP1 = SBPn.
n=0
Lemma 22. For all n 0, Sgn SBPn.
Proof.Let ~ 2 Sgn. Given the form of the congruences of Theorem 18, it
is easy to see that there is some ~0 2 Sg1 ~= g0(g) such that ~i = ~0ifor
0 i n. Then, using the injection ' of Proposition 13, we can find an
operation ` 2 BP 0(BP ) inducing ~0. Thus ~02 SBP1. But then ~ 2 SBPn.
In due course, we will establish the reverse inclusion, using induction on
n. Note that the congruence Cn . ~ 2 Z(p)has the form
p-ffip(n)(~n + y) 2 Z(p),
where y is a Z(p)linear combination of ~n-1, . .,.~0. We will see that it is
enough to find a BP congruence like this for each n. Although the following
lemma is rather elementary, for convenience we record the precise formulation
we need, leaving the proof to the reader.
Lemma 23. For each r 0, let cr = (cr,i)i 0 be a sequence of rational
numbers such that
cr,i2 p-ffip(r)Z(p), for 0 i r - 1,
cr,r2 p-ffip(r)Zx(p),
cr,i= 0, for i > r.
Let bcrbe another sequence satisfying the same conditions as cr.
10 IMMA G'ALVEZ AND SARAH WHITEHOUSE
Let n 0 and let
( )
Y fifi
Sn = ~ 2 Z(p)ficr . ~ 2 Z(p)for 0 r n,
k 0
( )
Y fifi
Tn = ~ 2 Z(p)ficr . ~ 2 Z(p)for 0 r n - 1, bcn. ~ 2 Z(p).
k 0
If Sn Tn then Sn = Tn.
Of course, the Cr of Definition 19 satisfy the hypotheses of the cr in the
lemma.
Definition 24. Let ` 2 DBP , with `* = ~ 2 SBP1. Define V~ : BP*(BP ) !
Z(p)as the composite V~ = ß`~, where ß : BP* ! Z(p)is the algebra map
determined by ß(v1) = 1 and ß(vi) = 0 for all i > 1.
Thus, for x 2 BP*(BP ), V~(x) is a finite rational linear combination of
the ~i and the fact that it lies in Z(p)is one of our BP congruences. We will
see that it is enough to consider only those BP congruences arising in this
way.
P r
LemmaP25. Let x, y 2 BP*(BP ). Suppose V~(x) = i=0ai~i and V~(y) =
s P r P s
j=0bj~j. Then V~(xy) = i=0 j=0aibj~i+j.
Proof.This is easy to check, using that jR is a ring homomorphism, that ~`
is a left BP*-module homomorphism and that ~`jR = `* = ~.
n
We recall the standardnnotation ßn =Qp - pp . We write, for n 1,
~ßn= in_p= 1 - pp -1 2 Zx(p)and ~ffn= ni=1~ßi2 Zx(p).
Lemma 26. V~(1) = ~0 and for i 0,
i+1X k
(27) V~(ti+1) = p-(i+1)~ff-1i+1~ffip(pi)- p-kf~f-1kV~(tpi+1-k).
k=1
Proof.It is easily checked using induction and the Araki formula that, for k
k-1) -1
1, the coefficient of vffip(p1in lk is p-kf~fk 2 Q. We recall that jR (li+1) =
P i+1 pk
k=0lkti+1-k and thus
i+1X k
~ffip(pi)li+1= `*(li+1) = ~`jR (li+1) = lk~`(tpi+1-k).
k=0
i)
Now (27) follows by equating coefficients of vffip(p1.
Now we produce our special BP congruence.
Proposition 28.PFor each n 0, there is an element dn 2 BP*(BP ) such
that V~(dn) = nj=0dn,j~j, where dn,j2 p-ffip(n)Z(p)for 0 j n - 1 and
dn,n2 p-ffip(n)Zx(p).
INFINITE SUMS OF ADAMS OPERATIONS AND COBORDISM 11
Proof.Firstly, we note that it is enough to prove this for n = piforPeach i 0.
For suppose that we have found dpi, for i 0, then, writing N = Mk=0akpk,
where 0 ak pQ- 1, it is straightforward to check using Lemma 25 that
we may put dN = Mk=0dakpk.
Now we will construct dpiinductively. We adopt the induction hypothesis
that such an element exists and moreover can be chosen of the form ti+1+
pri+1, for some ri+1 2 BP*(BP ). For i = 0, we may take d1 = t1, since
V~(t1) = p-1f~f-11(~1 - ~0). Now we assume that we have constructed dpj for
0 j < i and we explain how to construct dpi.
For the remainder of this proof, for a and b finite rational linear combina-
tions of the ~k, we write a ~ b to mean that a and b differ by some element
i) i
of the form a0~0 + . .+.api~pi, where ak 2 p-ffip(pZ(p)for 0 k p - 1
i)+1
and api2 p-ffip(p Z(p). Note that, if a ~ b and if a satisfies the conditions
required for V~(dpi), then so does b.
Using Lemma 25, it is not hard to see that we can correct for the first
term on the right-hand side of the equation (27). In fact, we can find ~r02
i) pi+1
BP*(BP ), a Z(p)linear combination of dffip(p1, . .,.d1 , such that
Xi k
(29) V~(ti+1- p~r0) ~ - p-kf~f-1kV~(tpi+1-k).
k=1
Let 1 k i. By our induction hypothesis, dpi-k= ti-k+1+ pri-k+1, for
some ri-k+1 2 BP*(BP ). It follows that
k pk pk k+1
(30) dppi-k= (ti-k+1+ pri-k+1) = ti-k+1+ p ~ri-k+1,
for some ~ri-k+12 BP*(BP ).
Now we set
Xi
dpi= ti+1- p~r0- p ~ff-1k~ri-k+12 BP*(BP ).
k=1
Clearly dpi has the form required by the induction hypothesis. Using (29)
and (30), a calculation shows that
Xi k
V~(dpi) ~ - p-kf~f-1kV~(dppi-k).
k=1
Now, using the induction hypothesis and Lemma 25, one checks that
V~(dpi) has the required form.
Now fix a choice of dn 2 BP*(BP ) as in Proposition 28 and let CBPn =
(dn,i)i 0. Then the BP congruence V~(dn) 2 Z(p)may be written CBPn. ~ 2
Z(p).
Proposition 31. For all n 0, SBPn = Sgn.
12 IMMA G'ALVEZ AND SARAH WHITEHOUSE
Proof.This willQbe proved by induction on n. It is true for n = 0, since
SBP0 = Sg0= k 0Z(p). Now we assume that SBPn-1= Sgn-1. Let
( )
Y fifi
TnBP = ~ 2 Z(p)fiCr . ~ 2 Z(p)for 0 r n - 1, CBPn. ~ 2 Z(p).
k 0
By Lemma 22, Sgn SBPn. Now if ~ 2 SBPn, then ~ 2 SBPn-1= Sgn-1, by
the induction hypothesis, and so Cr . ~ 2 Z(p)for 0 r n - 1. Also,
if ~ 2 SBPn, there is ~0 2 SBP1 such that ~i = ~0ifor 0 i n. By
Proposition 28, CBPn . ~0 2 Z(p). But CBPn . ~0 = CBPn . ~. So SBPn TnBP.
Thus we have
Sgn SBPn TnBP.
Applying Lemma 23 with cr = Cr and bcr= CBPr gives Sgn= TnBP. So
SBPn = Sgn.
8. The Centre of the Ring of Degree Zero Operations
Definition 32. (1) We define the Adams subalgebra of MU(p)opera-
tions, AMU(p), by
( 1 )
X fifi
AMU(p) = an'MUn fian 2 Z(p) MU0(p)(MU(p)), for p odd,
( n=0
X1 fi )
AMU(2) = aniMUn fifian 2 Z(2) MU0(2)(MU(2)).
n=0
(2) We define the Adams subalgebra of BP operations, ABP , by
( 1 )
X fifi
ABP = an'^BPnfian 2 Z(p) BP 0(BP ), for p odd,
( n=0
X1 fi )
ABP = aniBPnfifian 2 Z(2) BP 0(BP ), for p = 2.
n=0
Each Adams subalgebra is the image of an injective algebra map con-
structed in Section 4 or 5 and it follows that there are isomorphisms:
AMU(p) ~=ku0(p)(ku(p)), ABP ~= g0(g), for p odd,
AMU(2) ~=ku0(2)(ku(2)), ABP ~= ku0(2)(ku(2)), for p = 2.
In Section 6, we identified the centre of the ring of all degree zero stable
operations with the ring of diagonal operations. In this section we show that
this is precisely the Adams subalgebra. Clearly, the Adams subalgebra is
contained in the diagonal operations; we need to see that this inclusion is in
fact an equality. We begin with BP at odd primes, since this case follows
directly from the results of the previous section.
INFINITE SUMS OF ADAMS OPERATIONS AND COBORDISM 13
Theorem 33. Let p be an odd prime. The centre of the ring of degree zero
BP operations is the Adams subalgebra:
0
Z BP (BP ) = ABP .
Proof.By Proposition 16, the centre is equal to the ring of diagonal oper-
ations, so it is enough to show that DBP = ABP . By Proposition 13 and
Theorem 18, ABP ~= g0(g) ~=Sg1. Also DBP ~= SBP1. Here the isomorphisms
are given by sending operations to their actions on coefficients. By Proposi-
tion 31, SBPn = Sgnfor all n and so SBP1 = Sg1. So we have a commutative
diagram
and thus DBP = ABP .
Now we need to explain how to prove the corresponding result for BP
when p = 2 and for MU(p)for all primes. The ideas involved are the same as
those for BP at odd primes. Therefore we explain how the arguments need
to be modified, without giving all the details.
We begin by noting that, as in [6] and [4], we can characterize ku0(ku)
and ku0(p)(ku(p)) by systems of congruences. Thus, letting an operation in
ku0(ku) act on ß2i(ku) as multiplication by ~i2 Z, we have
( )
Y fifiiXvi,j
ku0(ku) ~=Sku1:= ~ 2 Z fi ____~j 2 Z for all i 0.
k 0 j=0 d(i)
Q
Here d(i) = p primepflp(i), where flp(i) = ffip(bi=(p - 1)c). Descriptions of
the integers vi,jcan be found in [6] and [4]. (More precisely, these references
give algorithms for the vi,j, but not a global formula.) We also have
( i )
ku(p) Y fifiXvi,j
ku0(p)(ku(p)) ~=S1 := ~ 2 Z(p)fi ____~j 2 Z(p)for all i 0.
k 0 j=0 d(i)
Using duality and [4, Props. 3, 20], it is possible to write an equivalent syst*
*em
of p-local ku congruences explicitly, in terms of Gaussian polynomials, just
as we did in the previous section for the Adams summand g. However, for
our purposes here, it is enough to note that we have one congruence for each
non-negative integer n and this takes the form p-flp(n)(~n+y) 2 Z(p), where y
is a Z(p)linear combination of ~0, ~1, . .,.~n-1. Of course, we may also define
ku(p)
Skunand Sn in the obvious way, by imposing the relevant congruences for
0 i n.
Theorem 34. Let p = 2. The centre of the ring of degree zero BP operations
is the Adams subalgebra:
0
Z BP (BP ) = ABP .
14 IMMA G'ALVEZ AND SARAH WHITEHOUSE
Proof.As before, the centre is the ring of diagonal operations, so we need to
ku(2)
show that ABP = DBP . Using Proposition 14, ABP ~= ku0(2)(ku(2)) ~=S1 .
ku(2) BP
Again DBP ~= SBP1. For each n 0, the inclusion Sn Sn follows
from Proposition 14. To establish the reverse inclusions, it is necessary
to produce a suitable BP congruence for each non-negative integer n, of
ku(2)
the form described above for the congruences defining S1 . Noting that,
for p = 2, flp(n) = ffip(n), we see that such a congruence is provided by
Proposition 28 (which does hold for p = 2). Just as in the previous section,
ku(2) BP ku(2)
we conclude that SBPn = Sn for all n 0, and therefore S1 = S1 .
The result follows.
Now we turn to the results for MU(p). We have MU* = Z[x1, x2, . .].,
where |xi| = 2i, and MU*(MU) = MU*[b1, b2, . .]., where |bi| = 2i. Now let
` 2 DMU act on ß2i(MU) as multiplication by ~i 2 Z. Just as for BP , we
can identify DMU with a set of sequencesPcharacterized by a family of con-
gruences. Thus we set ~`(bfl) = ffi flffi(~)xffi, and the flffi(~) are deter*
*mined
recursively as finite rational linear combinations of the ~i by ~`jR = `* = ~.
Then we define
( )
Y fifi
SMU1 = ~ 2 Z fi flffi(~) 2 Z for all,fl, ffi
k 0
and we have DMU ~=SMU1. Similarly, we let
( )
MU(p) Y fififl
S1 = ~ 2 Z(p)fi ffi(~) 2 Z(p)for all fl,,ffi
k 0
MU(p)
and we see that DMU(p) ~=S1 .
Theorem 35. For any prime p, the centre of the ring of degree zero MU(p)
operations is the Adams subalgebra:
0
Z MU(p)(MU(p)) = AMU(p).
Proof.By Proposition 16, the centre is the ring of diagonal operations and
we need to show that AMU(p) = DMU(p). Using Propositions 9 and 10, for all
primes p, we have
ku(p)
AMU(p) ~=ku0(p)(ku(p)) ~=S1 .
MU(p)
Also, DMU(p) ~=S1 . Again the isomorphisms are given by sending opera-
MU(p) ku(p)
tions to their actions on coefficients. So we need to show that S1 = S1 .
We will explain how to exploit the results for BP to show this.
Suppose that ` 2 DMU(p) acts on ß2n(MU(p)) as multiplication by ~n.
Using the same arguments as for BP , it will be enough to find a particular
family of MU(p)congruences satisfied by the ~i. Specifically, we need, for
each n 0, an MU(p)congruence of the form p-flp(n)(~n + y) 2 Z(p), where
INFINITE SUMS OF ADAMS OPERATIONS AND COBORDISM 15
ku(p)
y is a Z(p)linear combination of ~0, ~1, . .,.~n-1. This is because S1 is
given by such a system of congruences, as explained above.
Now MU(p) *(MU(p)) is a polynomial extension of BP*(BP ) and we write i :
BP*(BP ) ,! MU(p) *(MU(p)) for the inclusion. The elements dk of BP*(BP ),
constructed in Proposition 28, give us elements i(dk) in MU(p) *(MU(p)). Let
V~ : MU(p) *(MU(p)) ! Z(p)be given by composing ~`: MU(p) *(MU(p)) !
MU(p) * with the algebra map MU(p) * ! Z(p)which sends xp-1 to 1 and
xj to zero for all j 6= p - 1. Then, for n = k(p - 1) we can deduce the
required congruence easily from the corresponding BP one, using V~(i(dk)) =
V~(dk) 2 Z(p), where ~ = (~0, ~p-1, ~2(p-1), . .)..
For p = 2, this gives all the required congruences. Finally, let p be an odd
prime and let n = k(p - 1) + j, with 0 j p - 2. Since flp(n) = ffip(k), we
need a congruence of the form p-ffip(k)(~n + y)2 Z(p), where y is some Z(p)
linear combination of ~0, . .,.~n-1. By definition of V~, the sum over r of the
coefficients of xrp-1in ~`(i(dk))iis V~(i(dk)). Now we have ~`(2b1) = (~1-~0)x1
P j j j j
and it follows that ~`(2jbj1) = l=0(-1)j-l l~l x1. Using this, it may be
checked that a congruence of the required type is provided by the sum over
r of the coefficients of the terms xj1xrp-1in ~`2jbj1i(dk).
We end with a result about integral MU.
Theorem 36. Z (MU0(MU)) ~=ku0(ku).
Proof.By Proposition 16, Z (MU0(MU)) = DMU . Also DMU ~= SMU1 and
ku0(ku) ~=Sku1. Now we have
" MU " ku
SMU1 = S1 (p) and Sku1= S1 (p).
p p
MU (p) ku(p)
The proof of Theorem 35 shows that, for every prime p, S1 = S1 and
so SMU1 = Sku1.
References
[1]J. F. Adams and F. W. Clarke, Stable operations on complex K-theory, Illino*
*is J.
Math. 21 (1977), 826-829.
[2]S. Araki, Multiplicative operations in BP cohomology, Osaka J. Math. 12 (19*
*75),
343-356.
[3]J. M. Boardman, Stable operations in generalized cohomology. Handbook of al*
*gebraic
topology. Edited by I. M. James. (North-Holland, Amsterdam 1995), 585-686.
[4]F. Clarke, M. D. Crossley and S. Whitehouse, Bases for cooperations in K-Th*
*eory,
K-theory 23 (2001), 237-250.
[5]F. Clarke, M. Crossley and S. Whitehouse, Algebras of operations in K-theor*
*y, to
appear in Topology, preprint available from Mathematics arXiv at
http://front.math.ucdavis.edu/math.KT/0401414.
[6]K. Johnson, The action of the stable operations of complex K-theory on coef*
*ficient
groups, Illinois J. Math. 28 (1984), 57-63.
[7]W. Lellmann, Operations and co-operations in odd-primary connective K-theor*
*y, J.
London Math. Soc. 29 (1984), 562-576.
16 IMMA G'ALVEZ AND SARAH WHITEHOUSE
[8]S. P. Novikov, The methods of algebraic topology from the viewpoint of cobo*
*rdism
theory, Math. USSR-Izv. 1 (1967), 827-913.
[9]D. C. Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals*
* of Math-
ematics Studies No. 128. (Princeton University Press, Princeton NJ 1992).
Computing, Communications Technology and Mathematics, London Met-
ropolitan University, Holloway Road, London N7 8DB, UK.
E-mail address: i.galvezicarrillo@londonmet.ac.uk
Pure Mathematics, University of Sheffield, Sheffield S3 7RH, UK.
E-mail address: s.whitehouse@sheffield.ac.uk