UNSTABLE K-COHOMOLOGY ALGEBRA IS
FILTERED LAMBDA-RING
DONALD YAU
Abstract. Boardman, Johnson, and Wilson gave a precise for-
mulation for an unstable algebra over a generalized cohomology
theory. Modifying their definition slightly in the case of complex
K-theory by taking into account its periodicity, we prove that an
unstable algebra for complex K-theory is precisely a filtered ~-ring,
and vice versa.
1. Introduction
Lambda operations in complex K-theory were first introduced by
Grothendieck. These operations should be thought of as exterior power
operations; in fact, for an element ff in K(X) which comes from an
actual vector bundle on X with X a finite complex, ~i(ff) is the element
represented by the ith exterior power of that vector bundle. It was
realized early on that these operations generate all K-theory operations
(see [6], for example). A ~-ring is, roughly speaking, a commutative
ring with operations ~i which behave exactly like ~-operations on K-
theory of spaces. It is, therefore, natural to think that ~-rings have
all the algebraic structures to capture the unstable K-theory algebra
of spaces. There is, however, a more precise notion of an unstable
algebra, which we now recall.
In their seminal article [3] Boardman, Johnson, and Wilson gave a
precise definition for an unstable E*-cohomology algebra, where E* is
a generalized cohomology theory, such as K-theory, satisfying some
reasonable freeness conditions. Given E* denote by E_k the kth space
in the -spectrum representing E*. Of course, operations on Ek(-)
____________
Date: August 3, 2002.
1991 Mathematics Subject Classification. 55N20,55N15,55S05,55S25.
Key words and phrases. Unstable algebra, lambda-ring.
1
2 DONALD YAU
are just the elements of E*E_k. There are functors
Uk(-) = FAlg(E*E_k, -) :FAlg ! Set
from the category FAlg of complete Hausdorff filtered E*-algebras and
continuous E*-algebra homomorphisms to sets. Thanks to the extra
structures on the spaces E_k, the functor U whose components are Uk
to graded sets becomes a comonad on the category FAlg. Then these
authors define an unstable E-cohomology algebra to be a U-coalgebra.
This definition applies to K-theory in particular.
The purpose of this note is to show that these two notions of un-
stable algebras for K-theory (almost) coincide. This is, perhaps, not
surprising and is even intuitively obvious. But the author feels that it is
still worthwhile to record this result and to identify the two competing
notions of unstable algebras for K-theory.
We need to modify the above definition of an unstable E-cohomology
algebra slightly in the case of K-theory by taking into account its 2-
periodicity. Now the base point component of the 0th space in the
-spectrum representing K-theory is the classifying space BU of the
infinite unitary group. Since K0(pt) = Z it makes sense to consider
the functor
U(-) = U0(-) = FRing(K(BU), -) :FRing ! Set
from the category FRing of complete Hausdorff filtered rings and con-
tinuous ring homomorphisms. This functor can again be lifted to a
comonad on the category of filtered rings. In what follows, an unstable
K-cohomology algebra is by definition a U-coalgebra for this comonad
U.
We will define precisely what a filtered ~-ring is below (see Defini-
tion 2.4). This is basically a filtered ring with a ~-ring structure for
which in the expression ~i(r), both the ~-variable and the r-variable
are continuous.
We are now ready to state the main result of this note.
UNSTABLE K-COHOMOLOGY ALGEBRA IS FILTERED LAMBDA-RING 3
Theorem 1. For a complete Hausdorff filtered ring R, an unstable K-
cohomology algebra structure on R is exactly a filtered ~-ring structure
on R, and vice versa.
One advantage of having a result like this is that in order to study
unstable algebras in the sense of Boardman-Johnson-Wilson, one has to
be able to unravel the enormous amount of information encoded in a U-
coalgebra and should compare it to more familiar structures whenever
possible. Theorem 1 does this for K-theory, identifying U-coalgebras
with the well-studied ~-rings.
1.1. Organization of the paper. The rest of this paper is organized
as follows.
In x2 we recall the definitions of filtered rings and ~-rings. Then we
describe the filtered ring K(BU). The main aims of the section are
to define a filtered ~-ring and to observe that the completion of the
K-theory of a space is such.
x3 begins by recalling the notions of comonads and their coalgebras
which are necessary in order to define unstable K-cohomology algebra.
We define the modified comonad U for K-theory, taking into account
its periodicity, and define an unstable K-cohomology algebra as a coal-
gebra over this comonad. Particular attention is paid as to how this
comonad arises from the extra structure of the space BU. We also ob-
serve that the completion of the K-theory of a space X is an unstable
K-cohomology algebra.
In x4, the final section, we then show that an unstable K-cohomology
algebra is a filtered ~-ring and vice versa, proving Theorem 1.
2. Filtered ~-ring
All rings considered in this paper are assumed to be commutative,
associative, and have a unit.
4 DONALD YAU
2.1. Filtered ring. More information about filtered objects can be
found in Boardman [2].
A filtered ring is a ring R equipped with a directed system of ideals
IaR. Directedness means that for every IaR and IbR, there exists an
IcR which is contained in the intersection IaR \ IbR. The filtration
induces a filtration topology on R, which is Hausdorff (resp. complete)
if and only if the natural map
c :R ! lim-R=IaR
is injective (resp. surjective). If R is not already complete Hausdorff,
one can always complete it by taking the inverse limit R^ = lim R=IaR,
which is complete and Hausdorff with the induced filtrations. The ring
R^ is called the completion of R.
We mention two elementary but very useful facts about the comple-
tion.
o The map c has the following universal property: If f :R ! S is
a continuous ring homomorphism to a complete Hausdorff fil-
tered ring S, then f factors through c uniquely by a continuous
ring homomorphism f^ :R^ ! S.
o The image of R in its completion is dense.
From now on, all filtered rings will be required to be complete and
Hausdorff, unless otherwise specified.
A filtered ring homomorphism is a ring homomorphism which is con-
tinuous with respect to the filtration topology. The category of com-
plete Hausdorff filtered rings and continuous ring homomorphisms is
denoted by FRing.
When we discuss the comonad U we will need to use the not-at-all
obvious fact that completed tensor product is the coproduct in the cate-
gory of complete Hausdorff filtered rings. More precisely, suppose that
R = (R, {IaR}) and S = (S, {IbS}) are complete Hausdorff filtered
UNSTABLE K-COHOMOLOGY ALGEBRA IS FILTERED LAMBDA-RING 5
rings. Then their completed tensor product is defined as
R S
(2.1) R b S = lim-__________________________________ab.
a, bker(R S ! (R=I R) (S=I S))
Then R bS is a complete Hausdorff filtered ring and is the coproduct
of R and S in the category FRing (see [2, Lemma 6.9] for a proof).
2.2. ~-ring. More information about ~-rings can be found in [1, 4].
A ~-ring is a ring R equipped with functions
~i: R ! R (i 0)
which satisfy the following conditions. For any integers i, j 0 and
elements r and s in R:
o ~0(r) = 1
o ~1(r) = r
o ~i(1) = 0 for i > 1
P i
o ~i(r + s) = k=0 ~k(r)~i-k(s)
o ~i(rs) = Pi(~1(r), . .,.~i(r); ~1(s), . .,.~i(s))
o ~i(~j(r)) = Pi,j(~1(r), . .,.~ij(r)).
The polynomials Pi and Pi,jare defined as follows. Consider the vari-
ables ,1, . .,.,i and j1, . .,.ji. Denote by s1, . .,.si and oe1, . .,.oei, re-
spectively, the elementary symmetric functions of the ,'s and the j's.
The polynomial Pi is defined by the requirement that the expression
Pi(s1, . .,.si; oe1, . .,.oei) be the coefficient of ti in the finite product
Yi
(1 + ,m jnt).
m,n=1
Similarly, if s1, . .,.sij are the elementary symmetric functions of
,1, . .,.,ij, then the polynomial Pi,jis defined by the requirement that
the expression Pi,j(s1, . .,.sij) be the coefficient of ti in the finite prod-
uct
Y
(1 + ,l1. .,.ljt).
l1<... 0), corresponding to the
operations ~i in K-theory (see, e.g., [6, Thm. 4.15]). The variable ~i
lies in filtration exactly 2i, and the filtered ring structure on K(BU) is
generated this way.
2.4. Filtered ~-ring.
UNSTABLE K-COHOMOLOGY ALGEBRA IS FILTERED LAMBDA-RING 7
Definition 2.4. A filtered ~-ring is a complete Hausdorff filtered ring
R = (R, {IaR}) with a ~-ring structure such that the following two
conditions hold.
o The ~i (i > 0) is an equicontinuous family of functions. That
is, for every filtration ideal IaR, there exists an IbR such that
whenever r 2 IbR, one has ~i(r) 2 IaR for every i > 0.
o For every element r 2 R and every filtration ideal IaR, there
exists an integer N > 0 (depending on r and a) such that
P k Q k
whenever l=1 ilel N, one has l=1 ~il(r)el2 IaR.
Thus, a filtered ~-ring is essentially a filtered ring with a ~-ring
structure in which the expression ~i(r) is continuous in both the ~-
and the r-variables.
A filtered ~-ring map is a continuous ring homomorphism which
commutes with the ~-operations.
In order that these algebraic gadgets do model the K-theory of spaces
(completed if necessary), we have to show that K(X)^ is a filtered ~-
ring for any CW space X.
Proposition 2.5. For any CW space X, the completion K(X)^ has
a canonical filtered ~-ring structure for which the completion map
c :K(X) ! K(X)^ is a ~-ring map.
Proof. The completion K(X)^ is clearly a complete Hausdorff filtered
ring. Its universal property implies that the composite map
i c ^
K(X) ~-! K(X) -! K(X)
factors through c uniquely via a map which we call ~i. Once it is
shown that these ~i make K(X)^ a filtered ~-ring, it is automatically
true that c is a map of ~-rings.
Now since the image of K(X) in its completion is dense and since
K(X) with its ~-operations is a ~-ring, it follows immediately that
K(X)^ with its ~i is also a ~-ring. We must still prove that these
8 DONALD YAU
~i have the required continuity properties in Definition 2.4 to make
K(X)^ a filtered ~-ring. Again, since K(X) is dense in K(X)^, it
suffices to show that the ~-operations on K(X) have these continuity
properties.
To see that the ~i on K(X) are equicontinuous, pick any filtration
ideal IaX = ker(K(X) ! K(Xa)) corresponding to a finite subcom-
plex Xa and pick any element ff 2 IaX. Then ff is represented by a map
ff :X ! BU whose restriction to Xa is nullhomotopic. If r 2 K(BU)
is any operation at all, represented as a map r :BU ! BU, then the
composite r O ff :X ! BU is still nullhomotopic when restricted to Xa.
That is, the filtration ideal IaX is, in fact, closed under any K-theory
operations, including the ~-operations. This proves that {~i}i>0 is an
equicontinuous family of functions on K(X).
To demonstrate the other continuity property, let ff be an element of
K(X) and let IaX be a filtration ideal. The element ff is represented by
a map ff :X ! BU, which induces a continuous ring homomorphism
ff*: K(BU) ! K(X)
sending an element r 2 K(BU) to the element r(ff) in K(X) repre-
sented by the composite
X ff-!BU -r! BU.
That the required continuity property holds now follows from the con-
tinuity of ff* and the filtered ring structure on K(BU) (see x2.3).
Corollary 2.6. If f :X ! Y is a map of CW spaces, then the induced
map f*^ :K(Y )^ ! K(X)^ is a filtered ~-ring map.
3. Unstable K-cohomology algebra
The main purpose of this section is to define the comonad U (see
eq. (3.2)). Its coalgebras are by definition the unstable K-cohomology
algebras. We will also see that the K-theory of a space (completed if
necessary) is a U-coalgebra.
UNSTABLE K-COHOMOLOGY ALGEBRA IS FILTERED LAMBDA-RING 9
We begin by recalling the concepts of comonads and their coalgebras.
3.1. Comonad and coalgebra. The reader can consult MacLane's
book [5] for more information on this topic.
A comonad on a category C is a functor F :C ! C equipped with
natural transformations j :F ! Id and :F ! F 2, called counit and
comultiplication, satisfying the counital and coassociativity conditions:
F j O = IdR = jF O and F O = F O .
The natural transformations and j are often omitted from the no-
tation and we speak of F as a comonad.
If F is a comonad on a category C, then an F -coalgebra structure on
an object X of C is a morphism , :X ! F X in C, called the structure
map, satisfying the counital and coassociativity conditions:
(3.1) F , O , = O , and j O , = IdX .
We sometimes abuse notation and say that X is an F -coalgebra, leaving
the structure map , implicit.
A map of F -coalgebras g :(X, ,X ) ! (Y, ,Y ) consists of a morphism
g :X ! Y in C such that ,Y O g = F g O ,X .
3.2. The comonad U for K-theory. The discussion in this section
follows closely x8 of Boardman, Johnson, and Wilson [3], except that
we take into account the periodicity of K-theory and consider only the
degree 0 part. We will first define the comonad U and then discuss its
ring structure (when applied to a filtered ring), filtration, comultipli-
cation and counit. After that we will define unstable K-cohomology
algebra and observe that the argument in Boardman-Johnson-Wilson
shows that the K-theory of any space (completed if necessary) is such.
We will use the Yoneda lemma many times without explicitly men-
tioning it.
10 DONALD YAU
3.2.1. Definition of U. Let, then, R = (R, {IaR}) be an arbitrary com-
plete Hausdorff filtered ring. Define the functor U (to Set only at the
moment) to be
(3.2) UR = FRing(K(BU), R),
the set of continuous ring homomorphisms from K(BU) to R.
3.2.2. Ring structure on UR. Let X be an arbitrary CW space.
There are maps
~, OE :BU x BU ' BU
which induce the natural addition (by ~) and multiplication (by OE)
structure on the K-theory K(X) of a space X. These maps satisfy
certain associativity, commutativity, etc. conditions which make K(X)
a commutative ring with unit. Thanks to the Kunneth homeomorphism
K(BU x BU) ~= K(BU) bK(BU) [2, Thm. 4.19], they induce in K-
theory the following maps:
~*, OE*: Z[[~1, ~2, . .].] ' Z[[~1, ~2, . .].] b Z[[~1, ~2, . .].].
Therefore, for each integer k 1, we can write
X
~*(~k) = r0ff r00ff
ff
for some elements r0ffand r00ffin K(BU). By precomposition this be-
comes the Cartan formula for a sum
X
~k(x + y) = r0ff(x)r00ff(y) (x, y 2 K(X))
ff
(3.3) X
= ~i(x)~j(y)
i+j=k
where we have used the usual convention ~0(x) = 1. Since eq. (3.3)
holds for an arbitrary space X and any elements x and y in K(X), we
must have that
X
(3.4) ~*(~k) = ~i ~j.
i+j=k
UNSTABLE K-COHOMOLOGY ALGEBRA IS FILTERED LAMBDA-RING 11
A similar reasoning, using the property
~k(xy) = Pk(~1x, . .,.~kx; ~1y, . .,.~ky),
leads to the formula
(3.5) OE*(~k) = Pk(~1 1, . .,.~k 1; 1 ~1, . .,.1 ~k).
Now suppose that f and g are elements of UR = FRing(K(BU), R).
Their sum and product, f + g and fg, both have the form
f bg multiplication
K(BU) ! K(BU) bK(BU) --! R bR --- - - - - !R,
where for f + g (resp. fg) the left-hand map is ~* (resp. OE*). Then eq.
(3.4)and (3.5)imply that on the element ~k 2 K(BU) these maps can
be expressed as the Cartan formulas
X
(f + g)(~k) = f(~i)g(~j)
(3.6) i+j=k
(fg)(~k) = Pk(f(~1), . .,.f(~k); g(~1), . .,.g(~k)).
It follows that the additive and multiplicative identities, 0UR and 1UR ,
of UR are given by the maps
0UR :~k 7! 0 (k > 0)
8
(3.7) < 1 ifk = 1
1UR :~k 7!
: 0 ifk > 1.
The second equality follows from the fact that Pk(x1, . .,.xk; 1, 0, . .,.0) =
xk.
3.2.3. Filtration on UR. The ring UR = FRing(K(BU), R) is filtered
by the ideals
IaUR = ker(UR ! U(R=IaR))
(3.8)
= {f 2 UR : f(~k) 2 IaR for allk > 0}.
12 DONALD YAU
It is easy to see that the surjective map UR ! U(R=IaR) has kernel
IaUR. Thus, since R is complete Hausdorff, it follows that
UR = FRing(K(BU), R)
= lim FRing(K(BU), R=IaR)
a
= lim(UR)=IaUR
a
That is, UR is also a complete Hausdorff filtered ring. Note that the
indexing set for the filtration of UR is the same as that for R.
So far we have seen that U is a functor on the category FRing of
complete Hausdorff filtered rings.
3.2.4. Comultiplication and counit. In order to make U a comonad on
FRing, we still need the natural transformations :U ! U2 and
j :U ! Id. Let us begin with the former.
There is a filtered ring map
æ :K(BU) -! U(K(BU))
g 7! (f 7! f O g)
Using the formula ~i~j(x) = Pi,j(~1x, . .,.~ijx), we can express the
map æ in terms of the elements ~i 2 K(BU) as follows:
æ(~j)(~i) = ~iO ~j
(3.9)
= Pi,j(~1, . .,.~ij).
Here ~iO ~j is the element in K(BU) represented by the composition
of the K-theory operations ~i and ~j. Now if f is an element in UR,
then its image under the comultiplication map R :UR ! U2R is
the composite map
(3.10) R f = (Uf) O æ :K(BU) ! U(K(BU)) ! UR.
As for the counit j :U ! Id, it is defined by
(3.11) jR f = f(~1);
that is, jR is simply the evaluation map at ~1.
UNSTABLE K-COHOMOLOGY ALGEBRA IS FILTERED LAMBDA-RING 13
3.2.5. Unstable K-cohomology algebra. The proof in Boardman-Johnson-
Wilson [3, Thm. 8.8(a)] that their U is a comonad on the category of
complete Hausdorff filtered E*-algebras carries over almost without
change to show the following.
Proposition 3.12. The functor U defined in eq. (3.2)together with
and j above is a comonad on the category FRing of complete Hausdorff
filtered rings.
Following Boardman, Johnson, and Wilson we now make the follow-
ing definition.
Definition 3.13. An unstable K-cohomology algebra is a U-coalgebra
for the comonad U in Proposition 3.12. A map of unstable K-
cohomology algebras is a map of U-coalgebras.
Now given any CW space X, composition of maps yields a continuous
map
O :K(BU) x K(X) ! K(X)
which gives, after completion, the map
K(BU) x K(X)^ ! K(X)^.
Taking its adjoint we obtain a map
(3.14) æX :K(X)^ ! FRing(K(BU), K(X)^) = U(K(X)^).
The argument of [3, Thm. 8.11(a)] in Boardman-Johnson-Wilson,
which shows that their analogous map æX : E*(X)^ ! U(E*(X)^)
makes E*(X)^ a U-coalgebra, now gives
Proposition 3.15. The map æX in eq. (3.14)makes K(X)^ an unsta-
ble K-cohomology algebra.
This proposition is also a consequence of Proposition 2.5 and Theo-
rem 1 in the Introduction.
14 DONALD YAU
Corollary 3.16. If f :X ! Y is a map of CW spaces, then the in-
duced map f*^ :K(Y )^ ! K(X)^ is a map of unstable K-cohomology
algebras.
4. Identifying unstable K-cohomology algebra with
filtered ~-ring
In this final section we will prove Theorem 1 in the Introduction.
So let R = (R, {IaR}) be an arbitrary complete Hausdorff filtered
ring. Suppose that R has the structure of an unstable K-cohomology
algebra, , :R ! UR. We must show that this gives a filtered ~-ring
structure on R. Recall that
UR = FRing(K(BU), R) = FRing(Z[[~1, ~2, . .].], R).
We define the operations
~i: R ! R (i 0)
by setting ~0 1 and for i > 0,
(4.1) ~i(r) def=(,r)(~i) (r 2 R).
We claim that these operations make R into a filtered ~-ring; that is,
a ~-ring structure on R together with the two continuity properties in
Definition 2.4. The argument is divided into six steps, the first one for
the continuity properties and the rest for the ~-ring structure.
Step 1. We first check the continuity properties in Definition 2.4. To
see that the family {~i}i>0 is equicontinuous, let IaR be a filtration
ideal. We must show that there exists an IbR such that ~i(IbR) IaR
for every i > 0. Since , is continuous, given IaUR, there exists IbR such
that ,(IbR) IaUR. That is, if r 2 IbR, then ~i(r) = (,r)(~i) 2 IaR
for every i > 0. This shows that {~i}i>0 is an equicontinuous family of
functions on R.
To check the second continuity property in Definition 2.4, let r be an
element in R and let IaR be a filtration ideal. The element ,r 2 UR
UNSTABLE K-COHOMOLOGY ALGEBRA IS FILTERED LAMBDA-RING 15
is a continuous ring homomorphism from K(BU) to R. Thus, given
IaR, there exists an integer N > 0 such that whenever ff 2 K(BU) has
P k
filtration strictly great than N, then (,r)(ff) 2 IaR. Now if l=1 ilel
Q k e P k
N, then the filtration of the element l=1~ill2 K(BU) is l=1 2ilel>
N, and so we have
Yk _Yk !
~il(r)el = (,r) ~elil2 IaR.
l=1 l=1
This proves the desired continuity property.
Step 2. We check that ~1 is the identity map on R. Recall that the
counit jR :UR ! R is the evaluation map at ~1. Since there is an
equality IdR = jR ,, it follows that for any element r in R, we have
that
~1(r) = (,r)(~1)
= (jR ,)(r)
= r.
So ~1 is the identity map on R.
Step 3. We check that ~i(1) = 0 for any i > 1. Denoting the
multiplicative identity of UR by 1UR (see eq. (3.7)), we have that for
i > 1,
~i(1) = (,1)(~i)
= 1UR (~i)
= 0
as desired.
Step 4. Now we show the Cartan formula for a sum of any two
elements x and y in R. Using the additivity of , and eq. (3.6), we
16 DONALD YAU
calculate
~k(x + y) = ,(x + y)(~k)
= (,x + ,y)(~k)
X
= {(,x)(~i)}{(,y)(~j)}
i+j=k
X
= ~i(x)~j(y)
i+j=k
This proves the Cartan formula for a sum.
Step 5. The Cartan formula for ~k(xy) is proved similarly, using the
multiplicativity of , and eq. (3.6).
Step 6. Finally, we show that ~i~j(x) = Pi,j(~1x, . .,.~ijx). Let x
be an element in R. Then, using the coassociativity of , (see eq. (3.1))
and eq. (3.9)and (3.10), we calculate
~i~j(x) = ~i(,x(~j))
= {,(,x(~j))}(~i)
= { R (,x)(~j)}(~i)
= {U(,x) O æ}(~j)(~i)
= (,x)Pi,j(~1, . .,.~ij)
= Pi,j(~1x, . .,.~ijx)
as desired.
We have shown that the operations ~i in eq. (4.1)make R into a ~-
ring which also satisfies the two continuity properties in Definition 2.4.
Therefore, the unstable K-cohomology algebra structure on R gives a
filtered ~-ring structure on R. This proves half of Theorem 1.
The above argument can easily be reversed to show that a filtered
~-ring structure on R yields, via eq. (4.1), an unstable K-cohomology
algebra structure on R. Indeed, the two continuity properties in the
UNSTABLE K-COHOMOLOGY ALGEBRA IS FILTERED LAMBDA-RING 17
definition of a filtered ~-ring make sure that both the proposed struc-
ture map , :R ! UR and ,(r) :K(BU) ! R for any r 2 R are
continuous. The Cartan formulas for ~n(x + y) and ~n(xy) imply the
additivity and multiplicativity, respectively, of the structure map ,,
and the property about ~i~j(x) leads to the coassociativity of ,. That
, is counital follows from the condition that ~1 is the identity on R.
The proof of Theorem 1 is complete.
References
[1]M. F. Atiyah and D. O. Tall, "Group representations, ~-rings and the J-
homomorphism", Topology 8 (1969) 253-297.
[2]J. M. Boardman, "Stable operations in generalized cohomology", Handbook of
Algebraic Topology (ed I. M. James, North-Holland, Amsterdam, 1995), pp.
585-686.
[3]J. M. Boardman, D. C. Johnson, and W. S. Wilson, Ü nstable operations in
generalized cohomology", Handbook of Algebraic Topology (ed I. M. James,
North-Holland, Amsterdam, 1995), pp. 687 - 828.
[4]D. Knutson, ~-rings and the representation theory of the symmetric group,
Lecture Notes in Mathematics 308 (Springer, Berlin-New York, 1973).
[5]S. MacLane, Categories for the working mathematician, Graduate Texts in
Mathematics (Springer, Berlin, 1971).
[6]H. Toda, Ä survey of homotopy theory", Adv. Math. 10 (1973) 417 - 455.
E-mail address: dyau@math.uiuc.edu
Department of Mathematics, University of Illinois at Urbana-
Champaign, 1409 W. Green Street, Urbana, IL 61801, USA