HOMOLOGICAL SYMBOLS AND THE QUILLEN
CONJECTURE
MARIAN F. ANTON
Abstract. We formulate a "correct" version of the Quillen con-
jecture on the cohomology of linear groups by defining an unstable
form of Milnor K-theory and show that this version can be solved
by a finite process.
1. Introduction
Let R be a commutative ring with unit and GL1(R) = U(R) the
group of invertible elements in R. Recall that the Milnor K-theory
MK(R) of R is the quadratic algebra generated by the group of units
U(R) with the relations u v = 0 for u + v = 1. Mimicking this con-
struction we define a new algebra MH(R) called the algebra of homo-
logical symbols of R as a quadratic algebra generated by the homology
HU(R) of the group of units with relations to be determined.
The role of this new algebra is to approximate the homology algebra
H(R) of all the general linear groups GLn(R) of finite rank n over R i.e.
the direct sum of all the modules HGLn(R) taken over all nonnegative
integers n with the product induced by the matrix block multiplication.
The diagonal embeddings of the n-fold products of U(R) with itself
in GLn(R) induce a canonical algebra homomorphism from the tensor
algebra on HU(R) to H(R). If the relations in MH(R) vanish in H(R)
under this homomorphism, then we get a canonical homomorphism of
algebras from MH(R) to H(R).
Definition 1.1. The algebra of homological symbols MH(R) is the
quadratic algebra generated by HU(R) with relations those elements
of the 2-fold tensor product of HU(R) with itself which vanish in
HGL2(R) under the canonical homomorphism.
One challenging problem is to calculate the quadratic algebra MH(R)
for any ring R and any homology theory H. To hint to the depth of
this problem let H(R, 1) be the colimit of HGLn(R) under the maps
induced by the tail inclusions GLn(R) into GLn+1(R) and similarly
____________
Date: October 31, 2007.
1
2 MARIAN F. ANTON
MH(R, 1) be the corresponding colimit of the modules MH(R, n) of
the "rank n elements" in MH(R). Then the induced canonical homo-
morphism from MH(R, 1) to H(R, 1) is an isomorphism for suitable
H and R as a consequence of the Bloch-Kato conjecture [12].
In terms of this algebra some of the previous results and conjectures
in the unstable range can also be reformulated. For instance, if we
replace in Definition 1.1 the homology HGL2(R) by the homology of
an etale model for the classifying space BGL2(R) for certain rings R
and homology theories H then we obtain a new algebra say the algebra
of etale homological symbols HM (R) (see also Definition 3.5 ) which
comes with a natural comparison map MH(R) ! HM (R). Then our
results in [2, 3] can be reformulated by saying that for suitable choices
for R and H we conjecture that this comparison map is an isomorphism
and we prove that if our conjecture holds then Quillen's conjecture fails
in high ranks. Recall that Quillen's conjecture [11] states that the mod
` cohomology of GLn(R) is a free module over a certain ring of Chern
classes for any regular prime ` and suitable rings R containing 1=` and
the `-roots of unity. Our conjecture applied to the case where R is the
ring from Quillen's conjecture and H is the mod ` homology can be
verified for ` = 2, 3 [1, 3, 10].
Based on this evidence, the Conjecture 4.2 should be the "correct"
conjecture and the aim of this paper is to offer a group theoretical
approach to this conjecture modulo some computer implementation to
be done in a follow up paper. This would settle Quillen's conjecture
one prime at a time extending the results to ` = 5, 7, .... More precisely,
our main result is:
Theorem 1.2. Let R = Z[1=`, i] where ` is an odd regular prime
number, i a primitive `-root of unity and H the mod ` homology theory.
Then there is an effective finite process to calculate the relations of
MH(R) in homological degree two.
An "effective finite process" means that there are finitely many ex-
plicit words we need to check in a finitely presented explicit group. A
more precise formulation is given in Theorem 6.1. We will exemplify
this process for ` = 3 reproving our previous result but the method used
in this paper is general and new. We hope that the algebra MH(R)
which looks like some unstable version of Milnor K-theory would be
an interesting object to study for other problems as well. For instance,
H can be taken to be a generalized homology theory at the classifying
space level and there are no restrictions on R.
The paper is organized as follows. After reviewing some group ho-
mology in x2, we formulate our conjecture and observe that for a given
HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 3
prime ` there are only finitely many relations to calculate xx3,4. Then
we replace GL2(R) by a group SE2 and prove in x5 that SE2 is finitely
presented, a key result formulated explicitly as Theorem 5.2. In x6 we
give an explicit finite method based on Hopf's formula to verify these
relations in homology degree two, exemplifying by the case ` = 3. In
the final section x7 we make some remarks on the relations in the higher
homological degrees.
2. Group homology preliminaries
Let G be a group and B*G the (normalized) bar complex with BiG
the free abelian group generated by the set of symbols [x1|...|xi] where
x1,..., xi are non-identity elements of G. The boundary operator is
given by the formula
Xs-1
@[x1|...|xs] = [x2|...|xs]+ (-1)j[x1|...|xjxj+1|...|xs]+(-1)s[x1|...|xs-1]
j=1
where [x1|...|xjxj+1|...|xs] equals zero if xjxj+1 is the identity of G.
By definition, the group homology of G with Z- respectively Z=`-
coefficients is the homology of the complex B*G respectively B*G Z=`
and is denoted by H*(G, Z) respectively H*(G, Z=`) where ` is a prime
number. There are various ways to produce cycles representing ele-
ments in these homology groups.
First of all, the complex B*G is endowed with a shuffle product
X
[x1|...|xi] ^ [xi+1|...|xi+s] = (-1)oe[xoe(1)|...|xoe(i+s)]
where the sum is over all the permutations oe of i + s letters that shuffle
1,..., i with i + 1,..., i + s (i.e. oe-1 (1) < ... < oe-1 (i) and oe-1 (i + 1)*
* <
... < oe-1 (i + s)) and (-1)oeis the signature of oe. With respect to the
shuffle product, B*G is an (anti-)commutative, associative, and unital
graded algebra. We remark that
Lemma 2.1. The Leibniz formula
@([x1|...|xi] ^ [xi+1|...|xi+s])=(@[x1|...|xi]) ^ [xi+1|...|xi+s]
+(-1)i[x1|...|xi] ^ (@[xi+1|...|xi+s])
holds if only if xjxk = xkxj for all j i < k.
The proof is immediate. In particular, we have
Corollary 2.2. The element of BiG defined by
= [x1] ^ [x2] ^ ... ^ [xi]
4 MARIAN F. ANTON
for x1,x2,..., xi elements of G commuting with one another is a cycle
that modulo the boundaries is i-linear and skew-symmetric in x1,..., xi.
One way to find cycles mod ` is to define for each non-negative integer
k and each `-torsion element v of G an element in B2kG by the formula
X
[v](k)= [vi1|v|vi2|v|...|vik|v]
where the sum is taken over all integers i1,i2,..., ik from 1 to ` - 1 with
the convention that the sum is the identity element [ ] if k = 0. An
easy calculation shows that [v](1)^ [v](1)= 2[v](2)where v` = 1. By
induction we establish that
Lemma 2.3. For any `-torsion element v of G and any non-negative
integers k and s the following equation
` '
r + s (k+s)
[v](k)^ [v](s)= [v]
s
holds in B*G.
Again an easy calculation shows that @[v](1)= `[v] where v` = 1 and
by using Lemma 2.3 and Lemma 2.1 we can establish by induction the
following:
Proposition 2.4. For each `-torsion element v of G and positive in-
teger k the following equation
@[v](k)= `[v](k-1)^ [v]
holds in B*G. In particular, [v](k)is a cycle mod `.
Remark 2.5. For a given group G, the short exact sequence of com-
plexes
0 ! B*G Z=` 1-`-!B*G Z=`2 ! B*G Z=` ! 0
where the homomorphism 1 ` is the multiplication by ` on the second
factor, induces a long exact sequence of homology groups
fi
! Hi(G, Z=`) ! Hi(G, Z=`2) ! Hi(G, Z=`) -! Hi-1(G, Z=`) ! ...
where fi is a Bockstein homomorphism. By a diagram chasing, Propo-
sition 2.4 implies that modulo the boundaries
fi[v](k)= [v](k-1)^ [v]
for any `-torsion element v of G and positive integer k.
HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 5
By Lemma 2.1, if A is an abelian group, then B*A is a differential
graded algebra inducing the same structure on B*A Z=` and hence,
inducing a graded algebra structure on H*(A, Z=`). Moreover let `A
denote the `-torsion subgroup of A and (`A) the algebra of divided
powers generated in degree two by `A over the field with ` elements
F`. Then Lemma 2.3 and Proposition 2.4 shows that the map `A !
H2(A, Z=`) induced by v 7! [v](1)can be extended to a graded algebra
homomorphism
(2.1) (`A) ! H*(A, Z=`).
Similarly, let (A Z=`) denote the exterior algebra generated in degree
one by A Z=` over F`. Then Corollary 2.2 shows that the map
A Z=` ! H1(A, Z=`) induced by a 1 7! [a] extends to a graded
algebra homomorphism
(2.2) (A Z=`) ! H*(A, Z=`).
Theorem 2.6 ([4] p. 126). If ` is an odd prime and A is an abelian
group, then the maps (2.1) and (2.2) induce a natural isomorphism of
graded algebras
(`A) (A Z=`) H*(A, Z=`).
For later use, let G and K be two groups and recall the K"unneth
isomorphism [5], p. 218,
(2.3) H*(G, Z=`) H*(K, Z=`) H*(G x K, Z=`)
induced by the natural map
[x1|...|xi] [xi+1|...|xi+s] 7! [x1 x 1|...|xix 1] ^ [1 x xi+1|...|1 x xi+s]
where xj is an element of G for j i and an element of K for j > i.
If both G and K are abelian, the left hand side of (2.3) is a graded
algebra under the rule
(a b)(c d) = (-1)|b||c|(a ^ c) (b ^ d)
where a, c are (homogenous) elements of H*(G, Z=`) with |c| the degree
of c and similarly for b, d in H*(K, Z=`). In this setting, (2.3)is a graded
algebra isomorphism.
3. An algebra of homological symbols
For the rest of this paper we make the following:
Notation 3.1. Let R = Z[i, 1=`] be a ring of S-integers where ` is a
regular odd prime number and i is a primitive `-root of unity.
6 MARIAN F. ANTON
The motivation for introducing the concept of "homological symbol"
comes from the following construction. Denote by GLn the discrete
group of invertible n x n-matrices over R and by BGLn its (natural)
classifying space. Recall that the group homology of GLn is the same
as the singular homology of BGLn. This classifying space fits into a
diagram
fn et
BGLxn1-'n!BGLn -! BGLn
where 'n is the classifying space map induced by the canonical inclusion
GLxn1 GLn and fn is the natural etale approximation map at the
prime ` as constructed by [7]. The above diagram induces a diagram
in homology
fn* et
(3.1) Hi(GLxn1, Z=`) 'n*-!Hi(GLn, Z=`) --! Hi(BGLn , Z=`)
whose terms will be denoted from the left to the right by Tin, Hin, and
Hetin. As i and n run over all non-negative integers, these terms form
three double graded algebras denoted respectively by T**, H**, and
Het**, where the product is induced by the matrix-block multiplication
GLj x GLk ! GLj+k.
It is immediate that T** can be identified via the K"unneth isomor-
phism (2.3)applied to G = K = GL1 with a tensor algebra generated
in rank one by H*(GL1, Z=`). Moreover, by the Dirichlet Unit The-
orem, GL1 is the product of a cyclic group containing i and a free
abelian group of rank r = (` - 1)=2 generated by the fundamental
units say e1, e2, ..., er. Hence, by the Theorem 2.6 we have that
Proposition 3.2. Each element of H*(GL1, Z=`) is represented by a
formal sum of cycles of the form
[i](k)^
with Z=`-coefficients, where {u1, ..., ui} runs over all the subsets of
{i, e1, ..., er} and k over all non-negative integers.
Notation 3.3. Via the composition of the maps (3.1)we have a graded
algebra homomorphism denoted by g from T** to Het**.
To describe the kernel of the map g let ff : GLx21! GLx21send each
pair of units u x v to u-1v x uv and ff* denote by abuse the composed
map
H*(GL1, Z=`) 2 H*(GLx21, Z=`) ff*-!H*(GLx21, Z=`) T**
where the first isomorphism is a K"unneth isomorphism. If z is an ele-
ment of H*(GL1, Z=`) represented by [i](j)^ as in Proposition
HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 7
3.2, a short calculation shows that
(3.2) ff*([i](k)^ =z) [i-1 x i](k)^ [i x i](j)
^
^.
where homology classes are eventually denoted by their representative
cycles.
Theorem 3.4. [3, 8] The graded algebra homomorphism g : T** ! Het**
is surjective and its kernel is generated as a two-sided ideal by rank two
elements of the form (3.2)where i, k run over all non-negative integers
such that i-k is positive and even, {u1, ..., ui} runs over all the subsets
of {i, e1, ..., er}, and z runs over a set of generators for H*(GL1, Z=`).
Based on this theorem, the purpose of this paper is to study the
following algebra:
Definition 3.5. Let HM**be the algebra of etale homological sym-
bols at ` defined as the bi-graded algebra generated in rank one by
H*(GL1, Z=`) with relations in rank two given by (3.2) subject to the
conditions in the Theorem 3.4.
Remark 3.6. Theorem 3.4 can be reformulated by saying that g : T** !
Het**induces a natural isomorphism of bi-graded algebras HM** Het**.
The point of the definition is that HM**is a quadratic algebra (with
respect to the rank) that mimics a similar construction for the Milnor
K-theory but starting with H*(GL1, Z=`) instead of GL1. To emphasis
this point of view, (3.2) correspond to some "homological Steinberg
relations at `" via the K"unneth isomorphism
H*(GLx21, Z=`) H*(GL1, Z=`) H*(GL1, Z=`)
induced by the map sending
Xk
[u x v](k)7! [u](h) [v](k-h)and [a x b] 7! [a] 1 + 1 [b]
h=0
where a, b, u, v are elements of GL1 with u` = v` = 1.
4. A vanishing conjecture
From the diagram (3.1), the homomorphism g : T** ! Het**factorizes
through a graded algebra homomorphism ' : T** ! H** induced by
the canonical inclusions. A natural question is whether ' induces a
natural graded algebra homomorphism HM**! H**, i.e. whether '
maps the relations in HM**to zero in H**. Equivalently, we ask whether
8 MARIAN F. ANTON
the algebra of etale homological symbols is the same as the algebra of
general homological symbols as defined in Definition 1.1. To answer
this question for a given `, only finitely many relations need to be
verified in H**.
To see this let SLn be the subgroup of GLn consisting of matrices
with determinant 1 and construct a commutative diagram
GLx21 -ox1--!SL2 x GL1
? ?
ff?y ~?y
GLx21 --'-! GL2
` '
u-1 0
where o : GL1 ! SL2 sends a unit u to the diagonal matrix 0 u
and ~ sends a matrix A and a unit u to their matrix by scalar product.
By passing to homology we see that o*(y) = 0 implies
'*ff*(y z) = ~*(o*(y) z) = 0
for all z in H*(GL1, Z=`) and conversely, by a spectral sequence argu-
ment [3] if the above equation holds for all z it follows that o*(y) = 0
in H*SL2.
Proposition 4.1. [3] The relations in HM**vanish in H** under the
map ' induced by the canonical inclusions, if and only if for each subset
{u1, ..., ui} of {i, e1, ..., er} and each integer k with i - k positive and
even there are two chains y and z in B*SL2 such that
[o (i)](k)^ = @y + `z.
To analyze the last equation, let SE2 be the group generated by the
disjoint union of GL1 and R subject to the relations [6]
(4.1) dxed0edye = -dx + ye
(4.2) u-1dxe = dxu2eu
(4.3) duedu-1edue = -u-1
for u, -1 2 GL1 and x, y, 0 2 R together with the relations in GL1,
where the elements of R are written between d and e to distinguish`them'
x 1
from those of GL1. It is immediate that by sending dxe to -1 0
and u to o (u) for x 2 R and u 2 GL1 we get a well-defined group
homomorphism SE2 ! SL2 which is an isomorphism if R is Euclidean
[6]. Moreover the map o : GL1 ! SL2 factorizes through the canon-
ical inclusion GL1 ! SE2. In this slightly more general context, we
formulate a vanishing conjecture:
HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 9
Conjecture 4.2. There are chains y and z in B*SE2 such that the
equations in Proposition 4.1 (same restrictions) hold with each o (u)
replaced by the image of u 2 GL1 in SE2 under the canonical inclusion.
In other words, [i](k)^ 's vanish in H*(SE2, Z=`) under the
stated conditions. If this conjecture is true for a given `, it is obvious
that only finitely many verifications are needed.
Remark 4.3. SE2 is generated by d0e, d1_`e, and i. A proof will be given
in the next section.
Remark 4.4. One way to find evidence for the Conjecture 4.2 in the
homology degree two is via Hopf's formula [4]:
K \ [F, F ]
(4.4) H2(SE2, Z) ___________,
[F, K]
where according to Remark 4.3, F is the free group generated by d0e,
d1_`e, and i and K is the kernel of the canonical map F ! SE2. For A, B
subsets of F , the notation [A, B] means the subgroup of F generated
by the commutators [a, b] = aba-1b-1 with a 2 A and b 2 B. The
isomorphism (4.4)is induced by the map B2SE2 ! F sending [g|h] to
OE(g)OE(h)OE(gh)-1 where g, h are in SE2 and OE is a right inverse of the
canonical map F ! SE2. In particular, if gh = hg, then maps
to the commutator [OE(g), OE(h)] modulo [F, K].
5. A finite presentation for SE2
To begin the analysis of the conjecture as suggested by Remark 4.4
we simplify the notations by setting a = d0e and b = d1_`e and we will
refer to (4.1) as type I relations, to (4.2) as type II relations and to
(4.3)as type III relations. Also, we introduce further notations:
Notation 5.1. For each i mod ` define in SE2
-1 = a2, bi = iribiri, (b0 = b)
and for each i 2 {1, ..., r} define ui = a-1w-1iwhere
`-2Y
wi = (-1)r-1(bb-1i)`a (bjia)`-jb(`-1)i(bb-1i)`.
j=0
Finally, we define in SE2
2+1_(r2+r) 2
` = (-1)rir 2 (u1u2...ur) .
With these notations and terminology, the goal of this section is to
prove that SE2 has a finite presentation:
10 MARIAN F. ANTON
Theorem 5.2. Under the assumption that -1, i, 1 - ii for i integer
from 1 to r generate the group of units GL1, the group SE2 is generated
by i, a, and b subject to the following 1_8(9`2 + 4` + 59) relations:
(5.1) -b = b(-1), -i = i(-1)
(5.2) (a(ba)`)3 = 1, ((ba)`a-1)3 = -1
(5.3) (-1)2 = 1, i` = 1, uiuj = ujui, uii = iui
(5.4) w2i= -1, iai = a, bsu-1i(abs+i)2 = -uibsabs+2i
`-1Y
2 -1
(5.5) bsabt = btabs, bsa = `(bsa)` ` , b (ab ) = a
=1
where i, j run over integers from 1 to r, and s, t over integers mod `.
Remark 5.3. The assumption that GL1 is generated by the cyclotomic
units as in the Theorem 5.2 is equivalent with the assumption that the
class number of Q(i + i-1 ) is h+ = 1, by [14], p. 145. By the same
source, h+ = 1 for all ` 67 for instance.
In particular, this theorem implies Remark 4.3. The proof of the
theorem is given by a sequence of lemmas. First we prove that the
relations occurring in Theorem 5.2 hold in SE2. In what follows we
tacitly assume the relations in GL1.
Lemma 5.4. In SE2, the element -1 is central while the elements bia
for i mod ` commute with one another. In particular, (5.1) and the
first relation in (5.5) hold.
Proof. By type I relations -1 = a2 as in Notation 5.1. Furthermore, by
type II relations the unit -1 is central in SE2 and this fact combined
with the previous statement is equivalent to (5.1). Also, the type II
relations together with the fact that i2r = i`-1 = i-1 imply that
ii ri 1 ri
(5.6) d__ e = i d__ei = bi
` `
for each i mod `. The fact that the elements bia commute with one
another follows now from the type I relations. This property is the first
relation in (5.5).
Lemma 5.5. The unit 1 - ii in GL1 is represented in SE2 by the word
ui as defined in Notation 5.1 for all i from 1 to r. In particular, the
relations (5.3) hold.
Proof. In the ring R we have the following equation:
`-1
__1___ 1 X ij
= __ (` - j)i
1 - ii ` j=0
HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 11
which by type I relations, (5.6), and the fact that -1 is central implies
`-2
1 Y `-j r
(5.7) d______e = (bija) b(`-1)i(-1) .
1 - ii j=0
Similarly, by type I relations we have
`ii `-1
(5.8) diie = d___e = (bia) bi
`
which plugged in the equation
d1 - iied0ediie = -d1e
and after multiplying the equation on the right by a gives:
d1 - iiea(bia)` = -(ba)`.
Solving this equation by using Lemma 5.4 and -1 = a2 gives
(5.9) d1 - iie = (bb-1i)`a.
Plugging (5.7)and (5.9)in the type III relation
(1 - ii)-1 = -d1 - iied(1 - ii)-1ed1 - iie
we deduce that
`-2Y
(1 - ii)-1 = (-1)r-1(bb-1i)`a (bjia)`-jb(`-1)i(bb-1i)`a.
j=0
Using now Notation 5.1, we conclude the proof of the lemma.
Corollary 5.6. The unit element ` in GL1 is represented in SE2 by
the word given in Notation 5.1.
Proof. It follows from Lemma 5.5 and the equation
`-1Y Yr
` = (1 - ii) = (-ir+i)(1 - ii)2
i=1 i=1
which holds in the group GL1.
Remark 5.7. If a ring element x is written as
N
1 X i
(5.10) x = _____ i
`2k+1 =1
where k, i1,..., iN are nonnegative integers, then by type II and type I
relations we have
(5.11) dxe = `kbi1abi2a...biN-1abiN`k(-1)N-1 .
12 MARIAN F. ANTON
In particular, combining this equation with Lemma 5.5, we conclude
that a, b, i generate SE2 under the assumption of Theorem 5.2.
Proposition 5.8. In SE2 all the relations (5.1) - (5.5) hold.
Proof. The relations (5.2)are obtained by combining the type I relation
d-1ed0ed1e = -d0e with (5.8) for i = 0 and the type III relations
applied to the units 1 and -1.
The relations iai = a and w2i= -1 follow from the type II relations
applied to i, a, and ui, the later in view of Lemma 5.5, and a2 = -1.
This proves the first two relations in (5.4). Applying type I and II
relations to the identity
1_ j i 2 2 j+i 1 j j+2i
i (1 - i ) + __i = __(i + i ),
` ` `
we obtain in view of Lemma 5.5, the last relation in (5.4).
The second relation in (5.5)comes from type I and II relations:
ii ii`2 `2-1
bi = d__ e = d____e = `(bia) bi`
` `3
combined with Corollary 5.6. The last relation in (5.5) follows in a
similar manner from the cyclotomic equation in R. This completes the
proof in view of Lemma 5.5 and 5.4.
Now we show that all the relations in SE2 are a consequence of those
in Theorem 5.2. To this end we take the equation (5.11)as a definition
for its left hand side. In what follows we assume that GL1 is generated
by the cyclotomic units as in Remark 5.3. In agreement with Lemma
5.5 we define 1 - ii to be the word ui and thus GL1 is generated by
-1, i, and ui for i from 1 to r. Also we assume that the relations in
Theorem 5.2 hold for the rest of the section. In particular, by (5.3)we
can assume that all the relations in GL1 hold in SE2.
Lemma 5.9. For all i we have uiaui = a and in particular, `a` = a.
Proof. By Notation 5.1, u-1i= wia, so that
u-1iau-1i= wia2wia = -w2ia = a
since a2 = w2i= -1 is a central element of order two by (5.1), (5.3),
and (5.4). The second relation `a` = a follows now from the definition
of ` as in Notation 5.1, the first relation, and (5.4).
Corollary 5.10. For every unit u in GL1 we have uau = a.
Lemma 5.11. The definition (5.11) is invariant under the following
three elementary operations applied to (5.10): (1) permuting i1,...,iN ,
HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 13
(2) multiplying the numerator and denominator by `2, and (3) adding
1 + i + ... + i`-1 to the numerator.
Proof. When permuting i1,...,iN we use (5.5) to show that the right
hand side of (5.11) is invariant. Under the operation (2), we show that
the same formula is invariant by induction on N. The case N = 1:
2 -1 k+1
`kbi1`k = `k+1(bi1a)` a `
can be reduced to `-1a-1 = a-1` and this last relation follows from
Lemma 5.9. The induction step follows by (5.5). Under the operation
(3), the formula (5.11) becomes
`kbi1a...biNaba...b`-1`k(-1)N+`-1 .
By (5.5), ba...b`-1 can be replaced by a2 and then we can use a2 = -1
to prove the invariance.
Lemma 5.12. Any two representations of a ring element x as a sum
of the form (5.10) are related by a sequence of elementary operations
as defined in Lemma 5.11.
This lemma is a consequence of the irreducibility of the cyclotomic
polynomial 1 + t + ... + t`-1 over the integers. The proof is left as an
exercise in elementary algebra.
Proposition 5.13. The type I relations in SE2 are a consequence of
Theorem 5.2 relations.
Proof. Let two ring elements be represented as
N M
1 X i 1 X j
x = _____ i , y = _____ i ~.
`2k+1 =1 `2t+1~=1
Then their sum x + y can be represented as
N M
1 X i X 2i j
x + y = _____( i + ` i ~)
`2k+1 =1 ~=1
where we assume that i = k - t is a nonnegative integer. By Lemma
5.12 any other similar representation of x + y is related to the above
one by a sequence of elementary transformations. By Lemma 5.11 we
can use the above representation to verify the type I relation:
2i `2i-1 k N+`2M-1
dx + ye = `kbi1a...biNa(bj1a)` ...(bjM a) a ` (-1) .
By (5.5) and Lemma 5.11 applied to the `2i-power factors, the above
formula equals
`kbi1a...biNa`-ibj1a...abjM `k-i(-1)N+M-1 .
14 MARIAN F. ANTON
By Lemma 5.9, we have a`-i = `ka`t and thus, the formula above is
-dxed0edye proving the proposition.
Lemma 5.14. If the type II relations are satisfied by a unit u in GL1
and two ring elements x and y in R, then the type II relation will be
satisfied by u and the ring element x + y.
Proof. By Proposition 5.13 and Corollary 5.10 we have the following
calculation:
dx + ye = -dxeadye = -udxu2euaudyu2eu
= udxu2 + yu2eu = ud(x + y)u2eu.
Observe that the xu2 + xu2 and (x + y)u2 are related by elementary
transformations and Lemma 5.11 applies.
Lemma 5.15. If the type II relations are satisfied by two units u and v
in GL1 and for all ring elements x in R, then the type II relations are
satisfied by the product uv and the inverse u-1 and all ring elements.
Proof. Just plug-in y = xu2 in the equation vdxv2ev = dxe, multiply
on the left and right by u and use uv = vu which holds in GL1. So that
uv satisfies type II relations for any y and a similar argument works
for u-1.
Lemma 5.16. The units i and ui for i from 1 to r satisfy the type II
relations for any ring element x.
Proof. In view of Lemma 5.14 it is enough to check the case when
x = ij`-2k-1. For i this case follows directly by definitions and the
relations in GL1:
ij k k k k ij-2
id_____ ei = ` ibji` = ` bj-2` = d_____ e.
`2k+1 `2k+1
For ui we have to prove that:
iju2i k ij - 2ij+i + ij+2i k ij
uid_____ eui = ui` d__________________e` ui = d_____ e.
`2k+1 ` `2k+1
By using the definition (5.11)and Proposition 5.13 we reduce the iden-
tity above in a similar manner as in the proof of Proposition 5.8 to
-uibjabj+2i(bj+iabj+i)-1a-1ui = bj.
But this is a rearrangement of the last relation in (5.4).
Proposition 5.17. In SE2 the type II relations follow from the rela-
tions in Theorem 5.2.
HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 15
Proof. It follows from Lemma 5.15, Lemma 5.16 and the the way GL1
is generated.
Lemma 5.18. The relation (5.9) is a consequence of the Theorem 5.2
relations.
Proof. We represent 1 - ii by the following sum:
1 i-1 i+1 `-1
1 - ii = __(2` + `i + ... + `i + `i + ... + `i ).
`
Applying the defining formula (5.11), we get
2-1
d1 - iie = (ba)2`(b1a)`...(bi-1a)`(bi+1a)`...(b`-1a)`a-1(-1)` .
By using (5.5), the last formula becomes
(ba)`(bab1a...b`-1a)`(bia)-`a-1 = (ba)`a2`(bia)-`a-1
which in view of a2 = -1 and (5.5)again, implies (5.9).
Lemma 5.19. The units -1, 1, and ui for i from 1 to r satisfy the
type III relations.
Proof. In view of Proposition 5.13, the first two equations to verify are:
` 3 ` -1 3 3 -1 -1 3
-1 = d1e3 = d_ e = ((ba) a ) and 1 = d-1e = (-ad1e a )
`
and these follow from (5.2). Again by Proposition 5.13 the relation
(5.7)holds and in view of Notation 5.1 can be rewritten as
1 -1 -1 -` -1 -`
d______e = -a (bbi ) wi(bbi )
1 - ii
Combining this equation with Lemma 5.18, the type III equation
-u-1i= duiedu-1ieduie
can be rewritten as u-1i= wia which is true by definition.
Proposition 5.20. In SE2 the type III relations follow from Theorem
5.2 relations.
Proof. The main observation is the fact that by using Proposition 5.17,
if a unit u satisfies the type III relation then so does uv2 for any unit
v. Indeed,
duv2edu-1v-2 eduv2e = v-1 duedu-1eduev-1 = -u-1v-2 .
Since 1, -1 and ui generate GL1 up to squares, the conclusion now
follows from Lemma 5.19.
This finishes the proof of Theorem 5.2.
16 MARIAN F. ANTON
6. Evidence for the conjecture 4.2
Let 1 ! K ! F ! SE2 ! 1 be the finite presentation as given
for example by Theorem 5.2. In order to verify that a two dimensional
cycle oe for SE2 as in Proposition 4.1 is a boundary mod ` we first write
this cycle as a word w in F via the explicit map given in Remark 4.4.
Now observe that F mod [F, F ] is torsion free, so that if w = x` has a
solution x in F mod [F, F ] knowing that w is an element in [F, F ] \ K
and thus w = 1 mod [F, F ], then x 2 [F, F ]. We need only to verify if
w is an element in [F, K]K` by checking if w is trivial in F=[F, K]K`
where K` is the normal subgroup with relators the `-powers of the
relators in K. This means that there is an element x in K such that
w = x` mod [F, K]. By the previous observation, x must be in [F, F ]
so that w is trivial in ([F, F ] \ K=[F, K]) Z=` and conclude by the
universal coefficients that this is equivalent to the Conjecture 4.2. To
summarize, we have with Notation 5.1,
Theorem 6.1. Conjecture 4.2 holds in homological degree two if and
only if the following 1_2(r2 + r) commutators [i, us] and [ui, uj] vanish in
F=[F, K]K` for all 1 s r and 1 i < j r.
Remark 6.2. The fundamental units eiin the Conjecture 4.2 can always
be replaced by the the cyclotomic units ui as in Remark 5.3 because
even if h+ is not 1 it is relatively prime to ` due to the fact that ` was
assumed to be regular.
Remark 6.3. To exemplify the process in Theorem 6.1, we implement
the algorithm using GAP [9] as follows. Start with the free group "f"
generated by "z","a","b","b1",...,"b(l-1)","-1","w1",...,"wr","u1",...,"ur",
"l" and a list "k" consisting of all the relators defined by Notation 5.1
and Theorem 5.2. Then form the list "c" of all the commutators "[x,y]"
where "x" is "z", "a", or "b" and "y" runs through the list "k". Fi-
nally, we concatenate the list "c" and the list "pk" of all the `-powers
in "k" and obtain the list "ck". To check whether a commutator as in
Theorem 6.1 is trivial in the factor group "h:=f/ck" we use a rewriting
system for "h".
Definition 6.4. To speed up the verifications for ` = 3 we define a
group G given by the generators s, r, t, u, v and the relators:
s3r-2 , (sr)2r-2 , t3r-2 , r2t3, (t-1sr)3r-2 , (tr-1 s)3r-2 , s3u-2, (su)2u-2,
(su)2u-2, v3u-2, u2v3, (v-1 su)3u-2, (vu-1s)3u-2, t-1stsr2u-1s-3v,
t-1stsrs2ru-1s-2vs-3, rt-1s3u-1s-1v-1 s-1vu-1,
rsr-2 s-1t-1tr-1 s-2v(u-1s-3v)2u-1.
HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 17
Lemma 6.5. Let ` = 3 and SE2 be presented as in Theorem 5.2.
The map sending s, r, t, u, v respectively to -i2, a, -a(ba)3, -w1,
w1(ab-1b1a)-3 defines a group homomorphism from G to SE2.
Proof. It is enough to use the presentation of SE2 for ` = 3, add the new
generators s = -i2, r = a, t = -a(ba)3, u = -w1, v = w1(ab-1b1a)-3
and check each relation defining G in the new presentation of SE2. The
routine is the following:
f:=FreeGroup("z","a","b","b1","b2","-1","w1","u1","3",
"s","t","v");;
k:=[f.6^-1*f.2^2,
f.4^-1*f.1*f.3*f.1,
f.5^-1*f.1^2*f.3*f.1^2,
f.7^-1*(f.3*f.4^-1)^3*f.2*(f.3*f.2)^3*(f.4*f.2)^2*f.5
*(f.3*f.4^-1)^3,
f.8*f.7*f.2,
f.9^-1*f.6*f.1^2*f.8^2,
f.6*f.3*f.6^-1*f.3^-1,
f.6*f.1*f.6^-1*f.1^-1,
(f.2*(f.3*f.2)^3)^3,
((f.3*f.2)^3*f.2^-1)^3*f.6^-1,
f.6^2,
f.1^3,
f.8*f.1*f.8^-1*f.1^-1,
f.7^2*f.6^-1,
f.1*f.2*f.1*f.2^-1,
f.3*f.8^-1*(f.2*f.4)^2*f.5^-1*f.2^-1*f.3^-1*f.8^-1*f.6^-1,
f.4*f.8^-1*(f.2*f.5)^2*f.3^-1*f.2^-1*f.4^-1*f.8^-1*f.6^-1,
f.5*f.8^-1*(f.2*f.3)^2*f.4^-1*f.2^-1*f.5^-1*f.8^-1*f.6^-1,
f.3*f.2*f.4*f.3^-1*f.2^-1*f.4^-1,
f.3*f.2*f.5*f.3^-1*f.2^-1*f.5^-1,
f.4*f.2*f.5*f.4^-1*f.2^-1*f.5^-1,
f.2^-1*f.3^-1*f.9*(f.3*f.2)^9*f.9^-1,
f.2^-1*f.4^-1*f.9*(f.4*f.2)^9*f.9^-1,
f.2^-1*f.5^-1*f.9*(f.5*f.2)^9*f.9^-1,
f.3*f.2*f.4*f.2*f.5*f.2^-1,
#additional generators
f.6*f.1^2*f.10^-1,
f.6*f.2*(f.3*f.2)^3*f.11^-1,
f.7*(f.2*f.3^-1*f.4*f.2)^-3*f.12^-1];
h:=f/k;
RequirePackage("kbmag");
18 MARIAN F. ANTON
R:=KBMAGRewritingSystem(h);;
MakeConfluent(R);
F:=FreeStructureOfRewritingSystem(R);
#checking relations in G
r1:=ReducedWord(R,F.10^3*F.2^-2);
r2:=ReducedWord(R,(F.10*F.2)^2*F.2^-2);
r3:=ReducedWord(R,F.11^3*F.2^-2);
r4:=ReducedWord(R,F.2^2*F.11^3);
r5:=ReducedWord(R,(F.11^-1*F.10*F.2)^3*F.2^-2);
r6:=ReducedWord(R,(F.11*F.2^-1*F.10)^3*F.2^-2);
r7:=ReducedWord(R,F.10^3*F.7^-2);
r8:=ReducedWord(R,(F.10*F.7)^2*F.7^-2);
r9:=ReducedWord(R,F.12^3*F.7^-2);
r10:=ReducedWord(R,F.7^2*F.12^3);
r11:=ReducedWord(R,(F.12^-1*F.10*F.7)^3*F.7^-2*F.6);
r12:=ReducedWord(R,(F.12*F.7^-1*F.10)^3*F.7^-2*F.6);
r13:=ReducedWord(R,F.11^-1*F.10*F.11*F.10*F.2^2*F.7^-1
*F.10^-3*F.12*F.6);
r14:=ReducedWord(R,F.11^-1*F.10*F.11*F.10*F.2*F.10^2
*F.2*F.7^-1*F.10^-2*F.12*F.10^-3*F.6);
r15:=ReducedWord(R,F.2*F.11^-1*F.10^3*F.7^-1*F.10^-1
*F.12^-1*F.10^-1*F.12*F.7^-1);
r16:=ReducedWord(R,F.2*F.10*F.2^-2*F.10^-1*F.11^-1
*F.10^-1*F.11*F.2^-1*F.10^-2*F.12*(F.7^-1
*F.10^-3*F.12)^2*F.7^-1*F.6);
Theorem 6.6. Conjecture 4.2 is true for ` = 3.
Proof. According to the above lemma it is enough to apply Remark
6.3 to G as defined in Definition 6.4 rather than SE2. The word w =
urs-1r-1 u-1s of G maps to [u-11, i] in SE2 and it will be verified to be
the identity.
f:=FreeGroup("s","r","t","u","v");;
k:=[f.1^3*f.2^-2,
(f.1*f.2)^2*f.2^-2,
f.3^3*f.2^-2,
f.2^2*f.3^3,
(f.3^-1*f.1*f.2)^3*f.2^-2,
(f.3*f.2^-1*f.1)^3*f.2^-2,
f.1^3*f.4^-2,
(f.1*f.4)^2*f.4^-2,
HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 19
f.5^3*f.4^-2,
f.4^2*f.5^3,
(f.5^-1*f.1*f.4)^3*f.4^-2,
(f.5*f.4^-1*f.1)^3*f.4^-2,
f.3^-1*f.1*f.3*f.1*f.2^2*f.4^-1*f.1^-3*f.5,
f.3^-1*f.1*f.3*f.1*f.2*f.1^2*f.2*f.4^-1*f.1^-2*f.5*f.1^-3,
f.2*f.3^-1*f.1^3*f.4^-1*f.1^-1*f.5^-1*f.1^-1*f.5*f.4^-1,
f.2*f.1*f.2^-2*f.1^-1*f.3^-1*f.1^-1*f.3*f.2^-1*f.1^-2
*f.5*(f.4^-1*f.1^-3*f.5)^2*f.4^-1];
i:=x->x^-1;;
ik:=List(k,i);
c1:=ListN(f.1*k*f.1^-1,ik,\*);
c2:=ListN(f.2*k*f.2^-1,ik,\*);
c3:=ListN(f.3*k*f.3^-1,ik,\*);
c4:=ListN(f.4*k*f.4^-1,ik,\*);
c5:=ListN(f.5*k*f.5^-1,ik,\*);
c:=Concatenation(c1,c2,c3,c4,c5);
p:=x->x^3;;
pk:=List(k,p);
ck:=Concatenation(c,pk);
h:=f/ck;
RequirePackage("kbmag");
R:=KBMAGRewritingSystem(h);;
MakeConfluent(R);
F:=FreeStructureOfRewritingSystem(R);
w:=ReducedWord(R,F.4*F.2*F.1^-1*F.2^-1*F.4^-1*F.1);
gap> w;
Since for ` = 3 there is only one cycle oe = * to verify, the
Conjecture 4.2 is true for ` = 3.
7. The general case of Theorem 1.2
To state and prove the Theorem 1.2 in general we need only to find
a finite process to verify the Conjecture 4.2 in all homological degrees.
This was done in the previous section for all cycles of homological di-
mension two. To extend this process to higher homological dimensions
we can follow [13]. For the rest of this section denote G = SE2. For
each c 1, let us start with c finite presentations
1 ! Ki ! Fi ! G ! 1
20 MARIAN F. ANTON
for i = 1, ..., c that could be as in Theorem 5.2. Let F = F1 ? ... ? Fc
be the free product of all Fi0s and identify each Fi and Ki with its
canonical image in F for i = 1, ..., c. Define inductively
[a1, ..., ac] = [[a1, ..., ac-1], ac]
for ai group elements and [S1, ..., Sc] the group generated by [a1, ..., ac]
with ai 2 Si for Si subgroups of a group S. In particular, define
flc(S) = [S, ..., S] (c times, c 1).
Let K be the kernel of the canonical projection F ! G induced by the
canonical projections Fi ! G for i = 1, ..., c. By the main result in [13]
we can identify the homology group H2c(G, Z) with a subgroup
([K1, ..., Kc] \ N)flc+1(K) F
H2c(G, Z) __________________________ ______________________.
[K1, ..., Kc, F ]flc+1(K) [K1, ..., Kc, F ]flc+1(K)
for some suitable normal subgroup N F . With little effort, we can
show that the last factor group is finitely presented. In this way 2c-
cycles in H2c(G, Z=`) can be represented by words in finitely presented
groups. To check (2c + 1)-cycles in H2c+1(G, Z=`) we can shift the
dimensions via the short exact sequence
IG ! ZG ! Z
where ZG ! Z is the augmentation map and IG the augmentation
ideal. Since the calculations are involved it will be postponed to a
future paper.
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Department of Mathematics, University of Kentucky, Lexington,
KY 40506-0027, U.S.A. and I.M.A.R., P.O. Box 1-764, Bucharest, RO
014700
E-mail address: anton@ms.uky.edu
*