THE COHOMOLOGY OF THE MORAVA STABILIZER
GROUP S2 AT THE PRIME 3
VASSILY GORBOUNOV, STEPHEN F. SIEGEL, AND PETER SYMONDS
Abstract. We compute the cohomology of the Morava stabilizer
group S2 at the prime 3 by resolving it by a free product Z=3*Z=3
and analyzing the "relation module."
1. Introduction and Statement of the Main Result
The applications of the main theorem of this paper in homotopy
theory are due to the Morava Change of Rings Theorem [7]. Let p be
a prime, and denote by Sn the group of units in the maximal order
of a cyclic division algebra over Qp of index n and Hasse invariant 1_n.
The Morava Theorem says essentially that the cohomology of Sn with
coefficients in a certain representation describes the Bousfield localiza-
tion functor LK(n). This is the localization of stable homotopy theory
with respect to the spectrum of the n-th Morava K-theory, K(n) [2].
The functors LK(n) play an important role in homotopy theory [10 ]. At
present the case n = 1 is completely understood for all primes p. The
next case, n = 2, has been partially investigated for primes p 5 (see
for example [15 ], [16 ]). The functor LK(2) for small primes is harder to
study because the group S2 is of infinite cohomological dimension.
In this paper we deal with the prime 3 only. As the first step of
the analysis of LK(2) one needs to compute the continuous cohomology
of a certain canonical subgroup S02of S2 with trivial coefficients. We
compute these cohomology groups in the course of the proof of the
Theorem 1.1 below. This "almost" computes, according to the Morava
Change of Ring Theorem, the homotopy groups of the localization of
the Toda Smith complex V (1); for more details see [9]. The rest of the
calculation of ss*LK(2)S0 consists of two Bockstein spectral sequences.
We will not pursue this here.
Theorem 1.1. H*c(S2; F3 ) is freely generated as a graded-commutative
F3 -algebra by elements Z (in degree 1), C,E (in degree 3), and X (in
3)2
degree 4). Its Poincare series is (1+t)(1+t_1-t4.
____________
The research of the second author was supported by an NSF postdoctoral
fellowship.
1
2 VASSILY GORBOUNOV, STEPHEN F. SIEGEL, AND PETER SYMONDS
We obtain this by first calculating the cohomology of the canonical
subgroup, Sl, of S2.
Theorem 1.2. H*c(Sl; F3 ) has generators e1, e2 (in degree 1), x1, x2,
a (in degree 2), c1, c2 (in degree 3). The product of any two generators
with different subscripts is 0 and in addition there are relations
a2 = ae1 = ae2 = ac1 = ac2 = 0; c1e1 = ax1; c2e2 = ax2:
2+t3
Its Poincare series is 1+t+t____1-t.
A computation of H*c(Sl; F3 ) was sketched in [9], but the multi-
plicative structure given there does not agree with the one above. The
question of the cohomology of Sl was first reopened by Henn in con-
nection with a deep theorem of his on the cohomology of profinite
groups [4], and he also obtained the result stated here. Our calculation
proceeds by more classical methods.
The structure of H*c(Sl; F3 ) can also be described as follows: it was
shown in [3] that if j is the quotient map
Sl -j!Sl=Sl 0~=Z=3 Z=3;
(where Sl 0denotes the commutator subgroup [Sl; Sl] of Sl), then
there exists a homomorphism Z=3 * Z=3 -i!Sl such that ji is onto.
The image R = j*H*(Z=3Z=3) H*(Sl) is mapped isomorphically
onto H*(Z=3 * Z=3) by i* (R is the subring generated by 1, e1, e2, x1,
x2). The kernel R0 = Ker i* (generated by a, c1, c2, as an R-module)
is additively like R but with the degrees increased by 2. The structure
of R0 as an R-module is as given above and R02 = 0 : this determines
the ring H*c(Sl) ~=R R0.
2. Background Information
We briefly recall some facts about the group of units of a maximal
order in a division algebra. A full account can be found in [11 ], for
example. Consider a cyclic algebra D over Qp of index n and Hasse
invariant _1n. It can be constructed as follows. Let W be the totally
unramified extension of Qp of degree n (so W ~= Qp(i) where i is a
(pn - 1)-st root of unity). The Galois group of the extension W=Qp is a
cyclic group of order n: it is generated by the Frobenius homomorphism
oe. We form the crossed product algebra of W and Gal(W=Qp ). This
amounts to introducing a variable S which commutes with W according
to the formula wS = Swoeand satisfies Sn = a 2 Qpx . To define D we
set Sn = p. D is a division algebra over Qp of rank n2.
Let O be the maximal order in D: it is generated by S and the
integers of W. Its maximal ideal is OS and O=OS ~= Fpn. We are
COHOMOLOGY OF THE MORAVA STABILIZER GROUP 3
interested in three groups contained in O. The first is the group of
units of O, which we denote by Sn. The second is the subgroup of
strict units in Sn . It consists of the elements a 2 Sn such that a 1
mod S. We denote it by S0n. The third is the kernel of the reduced norm
restricted to S0n. We denote this subgroup by Sl. It is a pro-p group
because it is p-filtered and compact ([6], II, 2.1.3).
Let Hr denote the subgroup of Sl consisting of elements congruent
to 1 modulo Sr. By definition H1 = Sl. According to [12 ], there are
injective maps aer : Hr=Hr+1 ! O=OS ~=Fpn given by aer(1 + aSr) a
modS, (a 2 O). They are also surjective unless n|r, in which case the
image consists of those elements of trace 0 over the prime field.
From now on we shall only consider p = 3 and n = 2. Then
|H1=H2| = 9, |H2=H3| = 3, |H3=H4| = 9, and according to [12 ],
[H1; H1] = H2, [H1; H2] = H3, [H2; H2] = H4. Now W contains an
8-th root of unity i, and so z = iS-1_22 D is a cube root of unity.
Thus X = z and Y = zS are two elements of Sl of order 3. Their
images x = j(X ), y = j(Y) generate Sl=Sl 0~= Z=3 Z=3, (where
Sl0= [Sl; Sl] = H2). We now define i : Z=3 * Z=3 ! Sl to be the
map which takes the generators X and Y to X and Y. This map is in
fact injective [3], but we do not need to know that here.
There is a group D of automorphisms of Sl which has order 8 and
is generated by (i) conjugation by S, which interchanges X and Y, and
(ii) conjugation by Si, which interchanges X and X 2 and fixes Y.
Note that the action of D lifts to an action on Z=3 * Z=3.
The natural cohomology theory for a profinite group is the cohomol-
ogy on continuous cochains [13 ], denoted by H*c, and that is what we
use here. It agrees with the usual cohomology on a finite group.
Any maximal finite subgroup of Sl is cyclic of order 3, so the Krull
dimension of H*c(Sl; F3 ) is one [8], [7].
3. Resolutions
The fact that ji is onto implies that Im i is dense in the pro-3 topol-
ogy. So we have an epimorphism of pro-3 groups
Z=3"* Z=3 -!Sl (b denotes pro-3 completion ):
Let K denote the kernel of this map. The kernel of Z=3 * Z=3 -!
Z=3 Z=3 is free on the four generators [Xi; Yj], (1 i; j 2) [14 ].
The completion of the corresponding short exact sequence remains ex-
act, since Z=3 Z=3 is finite. This leads to a diagram with exact rows
and columns:
4 VASSILY GORBOUNOV, STEPHEN F. SIEGEL, AND PETER SYMONDS
K --- ! F^4 -- - ! Sl 0
fl ? ?
fl ? ?
fl y y
K --- ! Z=3"* Z=3 -- - ! Sl (3.1)
? ?
? ?
y y
Z=3 Z=3 _______Z=3 Z=3:
Now K is a closed subgroup of a free pro-3 group (i.e. the pro-3 com-
pletion of a free group), so is itself a free pro-3 group ([13 ], Cor. 2 to I,
Prop. 24).
According to the theory in ([6], V 2.5.7), Sl0 is equi-3-valued (with
the usual 3-adic valuation), so H*c(Sl0) ~=*H1 (Sl0). Sl0=[Sl0; Sl0] =
H2=H4 has order 27 and is easily seen to have exponent 3, so it has
rank 3, and thus so does H1c(Sl0).
Let N denote Sl0=[Sl0; Sl0] as F3 E-module (where E = Sl=Sl0~=
Z=3 Z=3). Then H2c(Sl0) ~=N by duality.
We want to be able to describe modules such as N explicitly. For
this purpose note that if we set X = x - 1 and Y = y - 1, then
F3 E ~=F3 [X; Y|X3 = Y3 = 0]. The augmentation ideal I is generated
by X and Y , and, because E is a p-group, I is the radical.
Since [H1; H2] = H3 and |H2=H3| = 3, we must have N=IN ~= F3 . If
we consider the image of ann N in I=I2 ~= E, we see that it cannot be
1-dimensional, since then it could not be invariant under the group of
automorphisms D. Hence ann N = I2 and N ~= F3 E=I2.
Now consider the spectral sequence
Hpc(Sl0; Hq(K)) ) Hp+q(^F4):
There are only two rows, since K is a free pro-3 group, and we deduce
that
Hrc(Sl0; H1 (K)) = 0; r 2;
H1c(Sl0; H1 (K)) ~=H3 (Sl0) ~=F3 ;
and there are short exact sequences
0 -! H1c(Sl0) -! H1 (^F4) -! E0;11-!0; (3.2)
0
0 -! E0;11-!H1c(K)Sl -!N -! 0: (3.3)
0
Sequence (3.2) shows that E0;11~=F3 . Let M denote H1c(K)Sl as an
F3 E-module, so we have
0 -! F3 -! M -! N -! 0:
COHOMOLOGY OF THE MORAVA STABILIZER GROUP 5
4. The Structure of M
The short exact sequence 0 ! F3 ! M ! N ! 0 shows that either
M ~= N F3 , or M is generated by one element. In the latter case,
M ~= F3 E= ann M, and we shall assume that this holds for the rest of
this section. The image of ann M in I2=I3 is a subspace S of codimen-
sion 1, which must be invariant under the action of the group of auto-
morphisms, D. Since I2=I3 has basis {X2; XY; Y 2}, it is easy to check
that the only possibilities for S are S1 = ; S2 = ,
and S3 = . Let Mi = F3 E=Si. We claim that, in fact,
M ~= M1, but this will only become apparent later. Notice, however,
that if we extend the field to F9 , then all three modules differ only by
an automorphism of the group algebra, for if
OE2 : X 7! X + Y; Y 7! X - Y;
then OE2(S1) = S2, and so MOE22~=M1. Similarly, if
OE3 : X 7! X + iY; Y 7! X - iY; (where i2 = 1)
then OE3(S1) = S3, and so MOE33~=M1.
5. The Cohomology of M1
Let p be an odd prime, k a field of characterisitc p, and E =
an elementary abelian p-group of order p2. (We will only need the case
where p = 3 and k = F3 , but it is just as easy to prove the result of
this section more generally.) The group algebra kE is the truncated
polynomial algebra k[X; Y | Xp = Y p = 0], where X = x - 1 and
Y = y - 1. A minimal projective resolution P !fflk of the trivial kE-
module k may be constructed as follows. First let Pn be the free left
kE-module on the n + 1 symbols er;swhere r + s = n and r; s 0. For
notational convenience we set er;s= 0 if r < 0 or s < 0. Then define
@(er;s) = X1+(p-2)(r+1)er-1;s+ (-1)rY 1+(p-2)(s+1)er;s-1;
where (n) is defined to be 0 if n is even and 1 if n is odd, and set
ffl(e0;0) = 1. We have
H*(E; k) = Hom kE(P; k) = k[x1; x2] k k[e1; e2]
where x1 = e*0;2(by which we mean x1(e0;2) = 1 and x1(er;s) = 0 for
(r; s) 6= (0; 2)), x2 = e*2;0, e1 = e*0;1, and e2 = e*1;0. The generators
e1 and e2 in H1(E; k) ~= Hom (E; k) correspond to maps E -! k with
kernels and respectively, and x1 and x2 are their Bocksteins.
Now let M1 = kE=(Xp-1; Y p-1). (If p = 3 and k = F3 , this is
consistent with the definition of M1 given above.)
6 VASSILY GORBOUNOV, STEPHEN F. SIEGEL, AND PETER SYMONDS
Proposition 5.1. As a right H*(E; k)-module, H*(E; M1) is gener-
ated by elements ff (degree 0), ffi1; ffi2 (degree 1), and fi (degree 2), subje*
*ct
to the relations
ffe1 = ffe2 = ffi2e1 = ffi1e2 = 0;
ffi1e1 = ffx1; ffi2e2 = ffx2;
fie1 = -ffi2x1; fie2 = ffi1x2:
In particular, H*(E; M1) is a free k[x1; x2]-module on ff; ffi1; ffi2; fi.
Proof. We compute an explicit basis for Hn (E; M1) = Zn (E; M1)=Bn(E; M1).
First note that M1 is a commutative ring, and Hom kE(Pn; M1) is a free
left M1-module on the generators fni= e*i;n-i(0 i n). Let a denote
the image in M1 of an element a in kE. Then Xp-1 = Y p-1= 0, so for
f 2 Hom kE(Pn; M1) and r; s 0 we have
f@(er;s) = (r)X f(er-1;s) + (-1)r(s)Y f(er;s-1): (5.1)
P n
Suppose first that n = 2m is even. Let f = i=0ffifni(ffi 2 M1)
and for notational convenience set ff-1 = ffn+1 = 0. Then equation
(5.1) implies that f 2 Zn (E; M1) if and only if
(j)ffj-1X + (-1)j(n + j + 1)ffjY = 0 whenever 0 j n + 1.
But this occurs if and only if ffjX = 0 = ffjY for all even j, or equiva-
lently ffj 2 (Xp-2 Y p-2) for all even j. Hence
Xn
Zn (E; M1) = { ffjfnj| ffj 2 M1 if j is odd, ffj 2 (Xp-2 Y p-2) if j is}even:
j=0
A similar calculation yields
Xn
Bn(E; M1) = { ffjfnj| ffj 2 (X ; Y) if j is odd, ffj = 0 if j is}even:
j=0
Hence the images of the n + 1 elements ffmk= Xp-2 Y p-2e*2k;2m-2k(0
k m) and fimk = e*2k+1;2m-2k-1(0 k < m) of Zn (E; M1) form a
basis in Hn (E; M1).
Now suppose that n = 2m + 1 is odd. Let S = {(i; j) 2 M1 M1 |
iX = jY } and T = {(iY ; iX ) | i 2 M1}. Working as in the even
case, we get
Xm
Zn (E; M1) = { (ffifn2i+ fiifn2i+1) | (ffi; fii) 2 S};
i=0
Xm
Bn(E; M1) = { (ffifn2i+ fiifn2i+1) | (ffi; fii) 2 T }:
i=0
COHOMOLOGY OF THE MORAVA STABILIZER GROUP 7
Now S=T is 2-dimensional and is spanned by the images of (Xp-2 ; 0)
and (0; Y p-2). Hence the images of the n+1 elements flmk = Xp-2 e*2k;2m+1-2k; *
*ffimk =
Y p-2e*2k+1;2m-2k(0 k m) of Zn (E; M1) form a basis in Hn (E; M1).
We now turn to the module structure. Recall that composition with
ffl is a chain map Hom kE(P; P ) ! Hom kE(P; k) and this map induces an
~=
isomorphism in cohomology H* Hom kE (P; P ) ! H*(E; k). The action
H*(E; M1) k H*(E; k) ! H*(E; M1)
is induced by the map on the cochain level
Hom kE(P; M1) k Hom kE (P; P ) ! Hom kE(P; M1)
given by composition. So we must first lift xi, eito maps "xi2 Z2 Hom kE (P; P *
*),
"ei2 Z1 Hom kE (P; P ). This is accomplished by setting x"1(er;s) =
er;s-2, "x2(er;s) = er-2;s, "e1(er;s) = (-1)r+s+1Y (p-2)(s+1)er;s-1, "e2(er;s) =
(-1)r+1X(p-2)(r+1)er-1;s.
Now let ff = [ff00], ffi1 = -[fl00], ffi2 = -[ffi00], and fi = [fi10]. We ha*
*ve
ffxi1xj2= [ffi+jj], ffi1xi1xj2= -[fli+jj], ffi2xi1xj2= -[ffii+jj], and fixi1xj2=
[fii+j+1j] (i; j 0). These are easily verified; the first, for example, just
follows from the fact that
ff00"xi1"xj2(er;s) = ff00(er-2j;s-2i) = Xp-2 Y p-2((r; s) = (2j; 2i)) = ffi+jj(*
*er;s):
(Here we are using the computer science notation where, for a propo-
sition P, (P) = 1 if P and (P) = 0 otherwise.) This proves that
ff, ffi1, ffi2, and fi generate H*(G; M1) as a k[x1; x2]-module and that
H*(G; M1) is a free k[x1; x2]-module on those four generators.
We now turn to the relations. It is routine to check that the genera-
tors satisfy these relations; for example, the last of these follows from
the fact that
fi10"e1(er;s) = (-1)r+s+1Y (p-2)(s+1)fi10(er;s-1) = Y p-2((r; s) = (1; 2)) = ff*
*i00"x1(er;s):
Now let A* denote the graded H*(E; k)-ring defined abstractly by these
4 generators and 8 relations. Since H*(E; M1) satisfies the relations,
there is a surjective H*(E; k)-homomorphism A* i H*(E; M1). We
wish to show that this homomorphism is an isomorphism, and to do this
it suffices to show that dim k(An) dim kHn (E; M1) for all n. This will
follow if we can show that A* is generated as a k[x1; x2]-module by the
four generators, because we know that H*(G; M1) is a free k[x1; x2]-
module on those generators. Hence for each 2 {ff; ffi1; ffi2; fi} and
i 2 {1; 2} we must show that ei is in the k[x1; x2]-submodule of A*
generated by ff; ffi1; ffi2; fi. But this is exactly what the 8 relations_tell
us. |__|
The following lemma is easy to check, but will be useful.
8 VASSILY GORBOUNOV, STEPHEN F. SIEGEL, AND PETER SYMONDS
Lemma 5.2. We can replace fi by any other element of H2(E; M1)
linearly independent of ffx1 and ffx2 and get the same relations.
6. The Final Calculation for Sl
We assume for now that M = M1: this will be justified later. Con-
sider the spectral sequence
Hp(E; Hqc(Sl0; H1 (K))) ) Hp+q(Sl; H1 (K))
Again it has only two rows.
H*-2(E) -d2!H*(E; M):
H*(E) is a torsion-free F3 [x1 ; x2] module, hence so is Ker d2. But
this spectral sequence shows that H*(Sl; H1 (K)) is finitely generated
over H*(E), and hence over H*c(Sl), which has Krull dimension 1.
It maps on to Ker d2, forcing a common bound on dim Ker d2 in each
degree. Thus Ker d2 = 0 and we have:
0 ! H*-2(E) ! H*(E; M) ! H*c(Sl; H1 (K)) ! 0:
(6.1)
Now d2(1) is not a linear combination of ffx1 and ffx2 otherwise it
would be annihilated by e1 and e2. By Lemma 5.2 we may assume that
d2(1) = fi.
We have proved:
Proposition 6.2. H*c(Sl; H1 (K)) is an H*(E)-module on generators
ff (degree 0), ffi1, ffi2 (degree 1) with relations
ffe1 = ffe2 = ffi1e2 = ffi2e1 = ffi1x2 = ffi2x1 = 0;
ffi1e1 = ffx1; ffi2e2 = ffx2:
Now consider the spectral sequence
H*c(Sl; H*(K)) ) H*(Z=3"* Z=3 ):
One has H*c(Z=3"* Z=3 ) ~= H*(Z=3 * Z=3). (Consider the the short
exact sequence F4 ! Z=3 * Z=3 ! Z=3 Z=3 and its pro-3 com-
pletion, and use the Comparison Theorem.) The map i* : H*c(Sl) !
H*(Z=3 * Z=3) is an isomorpism in degree 1, by construction. As the
right hand side is generated by elements of degree 1 and their Bock-
steins, i* is onto in all degrees and the spectral sequence becomes the
short exact sequence
0 -! H*-2c(Sl; H1 (K)) -2! H*(Sl) -! H*(Z=3 * Z=3) -! 0;
(6.3)
COHOMOLOGY OF THE MORAVA STABILIZER GROUP 9
of right H*(E)-modules. Set a = d2ff, ci = d2ffii. Identify ei and xi
with their images under j*. All that remains is to check that Im j* =
R ~=H*(Z=3 * Z=3), i.e. that e1e2 = e1x2 = e2x1 = x1x2 = 0:
The spectral sequence
H*(E; H*c(Sl0)) ) H*(Sl)
has four rows. We know that d2(E*;12) E*;02~=H*(E) is contained in
Ker(ji)*, which is generated as an H*(E)-module by e1e2, e1x2, e2x1,
and x1x2. Since H1c(Sl0) is dual to N as an E-module it is isomorphic
to I3, and we can calculate dim E0;12= 1 and dim E1;12= 3. (Use
dimension shifting: F3 E=I3 has invariants of dimension 3.) But E1;02
yields all of H1c(Sl) and thus dim d2(E0;12) = 1. This accounts for
e1e2. Also dim E2;0= 2 and dim H2c(Sl) = 3, so dim Ker (d2: E1;12-!
E3;02) 1. The image of this map must have dimension 2, which
accounts for e1x2 and e2x1. Finally x1x2 is the Bockstein of e1x2.
All that remains is to justify our assertion that M ~= M1. We do this
by carrying out the above calculation for each of the other possibilities
and obtaining a contradiction.
If M ~= F3 N, then the short exact sequence (6.1) shows that
dim Hnc(Sl; M) = dim Hn (E; N) + 2. This is impossible because N has
complexity 2, yet H*c(Sl) has Krull dimension 1.
If M ~= M2, then the structure of H*(E; M2) as an H*(E)-module
is like that of H*(E; M1), but twisted by OE2. Let us denote the new
module structures by *. Then
u * v = u(OE*2v); u 2 H*(E; M); v 2 H*(E):
Thus a * x1x2 = a(x1 + x2)(x1 - x2) = ax21- ax226= 0. But the
argument that x1x2 = 0 is still valid since it only depends on the
additive structure of H*c(Sl; M). The case M ~= M3 is similar.
7. The Cohomology of S2
Remark 7.1. [9] If p > 3 and n = 2 then Sl is torsion-free so H*(Sl)
has finite cohomological dimension by [8]. It contains an open sub-
group H2, which is a Poincare duality group of dimension 3. So by
([13 ], V4.7) Sl is also a Poincare duality group of dimension 3. Since
dim H1c(Sl) = 2, there is only one possible multiplicative structure,
namely that with generators e1, e2 (degree 1), f1, f2 (degree 2), and
relations e1e2 = 0, e1f1 = e2f2, f21= f22= f1f2 = 0.
Remark 7.2. The map
nrd x log +
S02i (1 + 3Z3 ) -!~ 3Z3
=
10 VASSILY GORBOUNOV, STEPHEN F. SIEGEL, AND PETER SYMONDS
is split by x 7! exp (1_2x), the image being central in S02. Thus S02~=
Sl x Z+3and H*c(S02) ~=H*(Sl) (z), degz = 1.
The group S2 is the semi-direct product of S02and Z=8 (the splitting
is obtained by lifting the elements of Fx9 to roots of unity in W). There-
fore H*c(S2; F3 ) is isomorphic to the subring of H*c(Sl; F3 ) invariant
under conjugation by the eighth root of unity i 2 W.
We must track the action of i through our calculation. Notice that
conjugation by i has the effect X 7! Y 7! X 27! Y2 7! X . Thus ei1=
-e2; ei2= e1, and the same for their Bocksteins, xi1= -x2; xi2= x1.
Regarding all groups as i-modules now, (3.3) shows that H0(E; M) ~=
E0;11~= F3 . From short exact sequence (3.2) we see that i acts on
H0(E; M) as multiplication by det H1c(F^4)(i)= detN(i). But i per-
mutes the explicit generators of ^F4transitively, hence detH1c(F^4) = -1.
To calculate detN(i), note that under the map ae2 of Section 2, con-
jugation by i acts as the identity, whilst under ae3 it corresponds to
multiplication by -i on F9 , which has determinant 1 over F3 . This
proves that det N(i) = -1, and consequently from (6.1), ffi = -ff.
Sequence (6.3) then shows that ai = -a.
The relations involving the elements ci now force ci1= -c2; ci2=
-c1. Clearly zi = z, and this completes the calculation of the action
of i.
It is now fairly straightforward to calculate the invariants of this
action, especially if one notes that, if X = x21+ x22, then X is invariant
and H*c(Sl; F3 ) is free over F3 [X]. We obtain Theorem 1.1 by setting
Z = z; C = c1 - c2; E = e1x1 + e2x2; X = x21+ x22.
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Department of Mathematics, Northwestern University, Evanston
IL 60208-2730
E-mail address: vgorb@math.nwu.edu, siegel@math.nwu.edu, symonds@math.nwu.edu