On a Conjecture of Quillen at the Prime 3
Marian F. Anton
November 18, 1997
1 Introduction
The goal of thisparticle_is to study the mod 3 cohomology of the discrete
group GLn(Z[1_3; 31]). The motivation for this study is a conjecture made by
Quillen. Namely, let l be a prime number and let A be the ring of Sintegers
in a number field containing a primitive lth root of unity, and assume that
l is a unit in A. Then we have the following
Quillen's Conjecture: [18, p. 591] H*(BGLn(A); Fl) is a free module
over the ringH*(BGLn(Ctop); Fl).
Here, the module structure is induced by an arbitrary imbedding of the
ring A in the field C of complex numbers, and GLn(Ctop) denotes the topo
logical group GLn(C).
It is known that H*(BGLn(Z[1_2]); F2) satisfies this conjecture for n = 2
[Mitchell, 17] and n = 3 [Henn, 11] but not for n 32 [Dwyer, 6]1. We show
as a result of our study that
p __
1.1. Theorem. H*(BGLn(Z[1_3; 31]); F3) satisfies Quillen's conjecture for n =
2, but not for n 27.
p __
To be more precise, choose a primitive third root of unity i = 31 in
C, and let A denote the subring Z[1_3; i] of C, fixed throughout the entire
article. Let Dn(A) be the group of diagonal matrices in GLn(A) and n :
BDn(A) ! BGLn(A) the classifying space map induced by the inclusion
of Dn(A) in GLn(A). Then, according to HennLannesSchwartz [13, p.
______________________________
1n 32 was "improved" to n 14 by Henn and Lannes
1
2
51], Quillen's conjecture implies that H*(BGLn(A); Fl) is "detected on the
diagonal matrices", i.e. the induced map
*n: H*(BGLn(A); Fl) ! H*(BDn(A); Fl)
is a monomorphism. In this context, the second part of Theorem 1.1 follows
immediately from
1.2. Theorem. H*(BGLn(A); F3) is not detected on the diagonal matrices
for n 27.
The proof of Theorem 1.2 follows almost the same pattern as in [Dwyer,
6]; in particular it depends on an explicit calculation in the case n = 2:
1.3. Theorem. H*(BGL2(A); F3) P (c2; c4) (e1; e3) (e01; e03).
Here, the indices of the generators give their cohomological degrees and
P , denotes the polynomial, exterior algebra over the field F3 with three
elements.
In the course of proving 1.2 we show how to deduce the first part of
Theorem 1.1 as well (see 3.6, proof of step 2).
It is unknown what is the minimum value of n for which *nfails to be
injective. Also, we have to point out that for n 27, the problem of finding
explicit elements in the kernel of *nstill remains open.
This article is organized as follows. In x2 we construct a homotopy fibre
square 2.9 in order to study theetale approximation to BGLn(A) at the
prime 3. In x3, we use Theorem 1.3 to show that if *nis injective, then the
mod 3 cohomology of BGLn(A) is bounded from above by the cohomology
of itsetale approximation. Then we combine this result with the homotopy
fibre square 2.9 in order to prove Theorem 1.2. In x4 we prove two lemmas
needed in x3. It remains to prove Theorem 1.3. In x5 we prove this theorem
in high dimensions based on a theorem of Brown [5], while the proof in low
dimensions is finished in x6, using a complex constructed by Alperin [1] and
a theorem of Henn [12].
The author is very grateful to William Dwyer for his guidance and also
to Avner Ash for pointing Alperin's complex out to him.
Note: For the rest of this article let k denote F3. Also, let H*() denote
H*(; F3) whenever it is not otherwise stated, and the same for H*().
3
2 A homotopy fibre square
We approach the mod 3 cohomology of BGLn(A) by "approximating" this
space at the prime 3. In order to do this we start with a diagram 2.2 of
rings involving the ring A itself, and after recalling the concepts of "etale
topological type" and "etale approximation", we construct from diagram
2.2 a diagram 2.9 of simplicial sets, involving theetale approximation to
BGLn(A). The main goal of this section is to show that diagram 2.9 is a
homotopy fibre square.
2.1. A commutative diagram. It is elementary to show that the ring A
is a euclidean domain and that in this ring, the prime 7 splits as a product
p1p2 of two conjugate primes. This decomposition leads to a commutative
diagram of rings and ring homomorphisms
f2
A ! ^Ap2
? ?
f1?y ?y (2.2)
^Ap1! C
Here A^pis the padic completion of A with respect to the prime p = p1 or
p2 and we label the primes p1, p2 in such a way that fi composed with the
canonical map from A^pito its residual field F7 induces a group homomor
phism Ax ! Fx7sending i 7! 2 mod 7 for i = 1 and i 7! 4 mod 7 for i = 2.
The maps ^Api! C are abstract embeddings of ^Apiin C making the diagram
commutative (e.g. by standard field theory we can extend the inclusion of A
in C to an embedding of A^pin C ).
Also, from the Dirichlet Unit Theorem for instance, we can see that the
multiplicative group Ax of invertible elements in A is a direct product of
cyclic groups
Ax = <1> x x <1  i> (2.3)
where the generators are indicated in brackets "< >". Similarly, we have ([20],
p. 74)
^Axp= Fx7x pro pgroup (2.4)
4
2.5. Etale topological type of A. Recall that a simplicial object in a
category C is a contravariant functor from the category of standard sim
plices __n= {0; 1; 2; :::; n}, n 0, to C. The opposite category of commutative
rings with identity can be enlarged to the category of schemes (as defined in
[10], p. 74) and then to the category of simplicial schemes. For instance, a
commutative ring B can be regarded as an affine scheme X = Spec(B) and a
scheme X as a simplicial scheme sending __nto X and all maps __n! __mto the
identity map of X (the constant functor). In this way, diagram 2.2 with the
arrows reversed can be regarded in the category of locally noetherian sim
plicial schemes, which is the domain of the "etale topological type" functor
( )et(as defined in [9], p. 36).
The target of this (covariant) functor is the category of prosimplicial set*
*s,
which can be regarded as a generalization of the category of CWcomplexes.
( )et: {locally noetherian simplicial schemes} ! {prosimplicial sets}
Recall that a proobject in a category C is a functor from a (small) left
filtering category to C ([4], III, 8.1).
Applying ( )etto 2.2 we get the following commutative diagram
pt ! (C)et ! (A^p2)et
? ?
? ?
y y (2.6)
(A^p1)et! (A)et
where "pt" is a point and pt ! (C)etis a homotopy equivalence ([9], 4.5, and
[2], p. 115, 125). From this diagram we have
2.7. Proposition. The map
(f1)et_ (f2)et: (A^p1)et_ (A^p2)et! (A)et
given by 2.6 induces an isomorphism on mod 3 cohomology.
Proof.For any noetherian ring B we have a natural isomorphism
H*((B)et) H*et(Spec(B); k)
([9], p. 49) where H*etis theetale cohomology (defined as in [16], p. 84), and
k is the constant sheaf. Because Hqet(Spec(B); k) = 0 if B = A or A^p and
q 2 [15], we can use the following short exact sequence ([16], 4.11)
1 ! Bx =(Bx )3 ! H1et(Spec(B); k) ! 3torsion ofP ic(B) ! 0
5
to conclude the proof.
Indeed, P ic(B) = 0 if B = A or A^p (being principal ideal domains),
and Bx =(Bx )3 is generated by i and 1  i if B = A, and by 3 mod 7 if
B = A^p, according to 2.3 and 2.4. Using these generators, we find out that
the induced map
H1((A)et) ! H1((A^p1)et_ (A^p2)et) H1((A^p1)et) H1((A^p2)et)
can be identified by naturality with an automorphism of the vector space
2 0
k k given by the invertible matrix (see also 2.1).
1 1 __
__
2.8. Etale approximations to BGLn(A) and BSLn(A). We will detect
some of the mod 3 cohomology of BGLn(A) and BSLn(A) by "approximat
ing" these spaces at the prime 3. But first let's recall some definitions.
(i) If S ! V T is a diagram of simplicial sets or simplicial schemes,
then Hom(S; T )V is the simplicial set sending __nto the set of natural trans
formations S [n] ! T over V . (S [n] is the simplicial set or simplicial
scheme sending __mto the disjoint union of copies of Sm , the image of __mvia
S, indexed by maps __m! __nin the category )
Let G be a group scheme over a scheme W and let X be a simplicial
scheme over W .
(ii) BG is the (classifying) simplicial scheme over W sending __nto the
nfold fibre product of G with itself over W ([9], p. 8). The "distinguished"
point of Hom(X; BG)W is induced by the section W ! G ("identity") of
the group scheme G over W , and its connected component will be denoted
by Hom0(X; BG)W .
(iii) ([7], p. 250) If X, G, and W are locally noetherian, then
Hom3((X)et; (BG)et)(W)et holimlim!Hom(Sff; Tfi)Vfi
fi ff
where {Tfi! Vfi} is a proobject of Kan fibrations naturally associated to
the fibrewise 3completion of (BG)etover (W )et, and (X)etis thought of as
a prosimplicial set sending each object ff of a left filtering category to Sff.
(iv) ([7], p. 250) If X, G, and W are locally noetherian, then there exists
a natural map
Hom(X; BG)W ! Hom3((X)et; (BG)et)(W)et
6
The connected component that contains the image of the distinguished point
defined in (ii) will be denoted by
Hom03((X)et; (BG)et)(W)et
Let R = Z[1_3] and let B be any noetherian Ralgebra. The simplicial set
Hom03((B)et; (BG)et)(R)etwill be denoted by Xn(B) if G = GLn;R, and by
Yn(B) if G = SLn;R. (GLn;R is the group scheme over Spec(R) defined by
the ring R[xij; 1 i; j n][t]OE(t . det(xij)  1), and SLn;R is defined by the
same ring, but with t = 1). Essentially from definitions, we have a natural
homotopy equivalence
BG(B) ! Hom0(Spec(B); BG)Spec(R)
where G can be taken to be GLn;R or SLn;R ([7], p. 277). Composing this
map with the natural map provided by (iv) we get a natural transformation
On : BGLn(B) ! Xn(B) or n : BSLn(B) ! Yn(B) :
Definition. The pairs (On; Xn(B)) and ( n; Yn(B)) are called "etale approx
imations" at the prime 3 to BGLn(B) and BSLn(B), respectively.
Because theetale approximations are functorial, diagram 2.2 gives rise to
a commutative diagram of simplicial sets
Xn(A) ! Xn(A^p2)
? ?
? ?
y y (2.9)
Xn(A^p1) ! Xn(C)
Now, we prove that this diagram is a homotopy fibre square. In order to
do this, we need the following
2.10. Proposition. ([6], 2.7) Let B be a noetherian Ralgebra containing i
and let U ! (B)etbe a map of prosimplicial sets which induces an isomor
phism on mod 3 cohomology. Then the induced map
Xn(B) ! Hom03(U; (BGLn;R)et)(R)et
is a homotopy equivalence.
7
Using this proposition, the idea is to show that diagram 2.9 can be ob
tained up to homotopy by applying Hom03(; (BGLn;R)et)(R)etto the diagram
pt ! S1
? ?
? ?
y y (2.11)
S1 ! S1 _ S1
Because the 1dimensional sphere S1 is a trivial prosimplicial set, the lim!in
ff
definition (iii) 2.8 can be ignored. Hence 2.11 gives rise to a homotopy fibre
square and the proof is done.
Indeed, observe that A^p being a complete discrete valuation ring, the
canonical map from ^Apto its residual field F7 induces a homotopy equivalence
([9], 4.5, and [2], p. 115, 125)
(F7)et! (A^p)et (2.12)
Also, S1 can be mapped into (F7)et isomorphically on mod 3 cohomol
ogy (e.g. by sending the generator of ss1(S1) to the Frobenius element of
ss1((F7)et), [8], 3.2).
In this way, diagram 2.11 can be mapped into diagram 2.6 by maps com
patible with both diagrams and such that:
(a) pt ! (C)etis a homotopy equivalence;
(b) S1 ! (A^p)etis an isomorphism on mod 3 cohomology;
(c) S1 _ S1 ! (A)etis an isomorphism on mod 3 cohomology by (b) and
by Proposition 2.7.
Applying Hom03(; (BGLn;R)et)(R)etto these maps and using Proposition
2.10 we get homotopy equivalences via which diagram 2.9 can be replaced
by a homotopy fibre square as it was stated.
2.13. Proposition. There are natural homotopy equivalences of kcomplete
simplicial sets
k1 BGLn(F7) ! Xn(A^p)
k1 BGLn(Ctop) ! Xn(C)
Also, if On : BGLn(A) ! Xn(A) induces an isomorphism on mod 3
cohomology then it induces a homotopy equivalence
k1 BGLn(A) ! Xn(A):
8
Here, k1 X means the kcompletion of the simplicial set X in the sense
of [4].
Proof.The map (a) from the previous proof and the homotopy equivalence
2.12 induce homotopy equivalences
Xn(A^p) ! Xn(F7) and Xn(C) ! Hom03(pt; (BGLn;R)et)(R)et Xn(pt)
by 2.10. Also, there are natural homotopy equivalences
__
k1 BGLn(F7) ! Xn(F7) and k1 BGLn(Ctop) k1 BGLn(F 7) ! Xn(pt)
__
by [6], 2.10, 2.11, and [9], 12.4, where F7 denotes the algebraic closure of F7.
Because BGLn(Ctop) is simply connected, k1 BGLn(Ctop) is kcomplete ([4],
VII, 5.1). Using [9], 12.2, we have a homotopy fibre square
__
k1 BGLn(F 7) ! k1 BGLn(Ctop)
? ?
? ?
y y (2.14)
k1 BGLn(Ctop) ! k1 (BGLn(Ctop))x2
According to [4], II, 5.3, k1 applied to this square yields another homo
topy fibre square k1 (2.14). Therefore, we may choose a map (2:14) !
k1 (2:14)_between squares with_the map on upper left corner defined to be
k1 BGLn(F 7) ! k1 k1 BGLn(F 7)_and_the map on the other corners to be
the identity. Hence, k1 BGLn(F 7) is kcomplete. In this way, we conclude
the first part of 2.13.
In order to prove the second part, we apply k1 to the homotopy fibre
square 2.9 to conclude as before that Xn(A) is kcomplete and use [4], I,
__
x5. __
Remark 2.15. In the same way, we can show that the similar commutative
diagram
Yn(A) ! Yn(A^p2)
? ?
? ?
y y
Yn(A^p1)! Yn(C)
is a homotopy fibre square.
9
3 Proof of Theorem 1.2
It is known that H*Xn(A) is a lower bound for H*BGLn(A), [8], and we
show that if *nis injective, then it is an upper bound as well. But first, let's
compute H*Xn(A).
3.1. An EilenbergMoore spectral sequence (EMSS). By Proposition
2.13, Xn(C) is simply connected. Hence, the EMSS applied to the homotopy
fibre square 2.9 :
Es;r2= T orr;sH*Xn(C)(H*Xn(A^p1); H*Xn(A^p2)) ) H*Xn(A) (3.2)
converges [23].
At this point, we can use Quillen's computations ([19], p. 575) to observe
that H*BGLn(F7) P (c2; :::; c2n) (e1; :::; e2n1) is a free module over
H*BGLn(Ctop) P (c2; :::; c2n), the module structure being induced from
an embedding Fx7 ! Cx . (, P and the subscripts of the generators are
explained in the Introduction). Hence, using Proposition 2.13, we conclude
that the spectral sequence 3.2 collapses and
H*Xn(A) H*GLn(F7) H*BGLn(Ctop)H*GLn(F7)
P (c2; :::; c2n) (e1; :::; e2n1) (e01; :::; e02n1) (3.3)
where the module structures are induced by the embeddings ^Api! C, i = 1,
2, in diagram 2.2. Also, by duality, we obtain a monomorphism
ff : H*Xn(A) ! H*GLn(F7) H*GLn(F7) (3.4)
which will be used in 3.6.
Remark 3.5. Applying EMSS to the homotopy fibre square 2.15 in the case
n = 2, we conclude in the same way that
H*Y2(A) P (c4) (e3) (e03)
In what follows, n is the symmetric group with n elements and if M is
a nmodule then Mn is the submodule of ninvariants and Mn is the
quotient module of ncoinvariants.
Studying the induced map
O*n * *n *
H*Xn(A)  ! H BGLn(A)  ! (H BDn(A))n
we observe an interesting phenomenom
10
3.6. Lemma. *nand *nO*nhave the same image.
This lemma follows at once as soon as we prove
Step 1. If *nand *nO*nhave the same image for n = 2, then they have
the same image for any n.
Step 2. O*2is an isomorphism (and so, *2and *2O*2have the same image).
Proof of Step 1. Regard the composed map
On*n* ff 2
n : (H*BDn(A))n  ! H*Xn(A)  ! (H*GLn(F7)) (3.7)
as a restriction of the map
a a a
t : H*( BDn(A))1 ! H*( Xn(A)) ! H*( BGLn(F7))2 (3.8)
n0 n0 n0
where`1 is the union of all n, ff is the monomorphism defined in 3.4, and
means disjoint union. The advantage of this point of view is that t is an
algebra homomorphism, the multiplication being induced from the matrix
block multiplication.
If *2and *2O*2have the same image, then, by duality, 2* and O2*2* have
the same kernel, or, equivalently, 2*(ker(2)) = 0 (ff is injective). But we
have
3.9. Lemma. (4.1) ker(t) is generated as an ideal by ker(2).
It follows that n*(ker(n)) = 0 for any n, and again by duality we get
Step 1.
Proof of Step 2. According to ([8], 6.3) O*2is injective. The fact
that it is an isomorphism follows from the fact that there exists an abstract
isomorphism H*X2(A) H*BGL2(A) of graded rings which are finite in
each dimension. But by 3.3, this is what Theorem 1.3 says. Also, O*2being a
natural isomorphism, from 3.3 we deduce the first part of Theorem 1.1.
From the homotopy fibre square 2.9, we get the homotopy fibre square
Map(Bm ; Xn(A)) ! Map(Bm ; Xn(A^p2))
? ?
? ?
y y
Map(Bm ; Xn(A^p1)) ! Map(Bm ; Xn(C))
11
where Map is the space of unpointed maps and m is the cyclic group of
order 3m with m any natural number.
3.10. Proposition. ([6], 3.5) The induced map
om;n : [Bm ; Xn(A)] ! [Bm ; Xn(A^p1)] x [Bm ; Xn(A^p2)]
is injective, where "[,]" stands for the homotopy classes of unpointed maps.
3.11. Proposition. ([14]) Let be a group of virtually finite cohomological
dimension, and let be a finite 3group. Then the natural map
Rep(; ) ! [B; k1 B]
is a bijection.
Here, Rep(; ) is the set of conjugacy classes of group homomorphisms
! .
Now, suppose that *nis injective. Then by Lemma 3.6 it follows that O*n
is surjective. But it is known that O*nis injective ([8], 6.3), so that O*nis an
isomorphism. Hence, by Proposition 2.13 we can naturally identify
[Bm ; Xn(A^p)] [Bm ; k1 BGLn(F7)] and
[Bm ; Xn(A)] [Bm ; k1 BGLn(A)]
Moreover, both GLn(F7) and GLn(A) satisfy the finiteness condition of
Proposition 3.11 ([21], p. 124). Therefore, om;n can be identified with the
map
o0m;n: Rep(m ; GLn(A)) ! Rep(m ; GLn(F7)) x Rep(m ; GLn(F7))
which is induced by the canonical maps A ! A^pi! F7, i = 1; 2. In con
clusion, if *nis injective then o0m;nis injective by Proposition 3.10. But we
have
mp__
3.12. Lemma. (4.2) If o0m;nis injective and 3m1 n, then Z[1_3; 3 1] is a
principal ideal domain.
p __
Because Z[1_3; 811] is not a principal ideal domain ([25], p. 353), we con
clude that *nis not injective for n 27.
12
4 Proofs
4.1. Proof of Lemma 3.9. The domain of the map t in 3.8, is the symmet
ric algebra over k generated by the vector space H*BD1(A) with H*BDn(A) =
(H*BD1(A))n (K"unneth Formula). So that, we can describe t as an alge
bra homomorphism by its restriction 1 in 3.7. But, by naturality, 1 is
computable as the composed map below
f1*f2* x 2
H*D1(A) d*! (H*D1(A))2  ! H*(F7 )
where d : D1(A) ! D2(A) is the diagonal map.
Recall that fi sends i 7! 32 mod 7 for i = 1 and i 7! 34 mod 7 for i = 2.
Hence, we can identify
[H*(Fx7)]x2 [P (a) (b)] x [P (a) (b)] and
H*BD1(A) P (x) (y) (y)
in such a way that f*1x f*2is given by (a; a) 7! (x; x), and (b; b) 7! (y; y + *
*y).
Here, a, a, and x have degree 2, and b, b, y, and y have degree 1.
Choosing the dual basis
aj aj1b aj aj1b xj xj1y xj1y xj1yy
uj vj uj vj j jj jj j
we can easily describe the duals f1*, f2*, and d* as follows
f f d
_____1*___2*___________________________*_________________________
 P
j  uj uj a b
 a+b=j
 P
jj vj vj (a jb + ja b)
 a+b=j
 P
jj 0 vj (a b + a b)
 a+b=j
 P P
j 0 0 (ja jb ja jb) + (a b + a b)
 a+b=j+1 a+b=j

and hence we can compute 1 and t.
A slightly better description of t can be obtained by introducing the
notations
j j jj  jj jj
uj uj vj vj
13
(observe that all u's have even degrees and all v's have odd degrees ). Namely,
we can identify
a
H*( BDn(A))1 P (uo; uo) (vo; vo)
n0
a
and H*( BGLn(F7))2 P (uo; uo) (vo; vo) ([19], p. 570)
n0
in such a way that t is given as an algebra homomorphism by
!
P ur P vr P vr P ur
t = ua ub uavb vaub vbva
a+b=r a+b=r a+b=r a+b=r+1
The notation uo, for instance, means the infinite sequence u0, u1, u2,... .
We conclude the proof of Lemma 3.9 by showing that the elements
X
qn = (urus1 + vsvr) ; n 2 ;
r+s=n
which obviously belong to ker(2), generate ker(t) as an ideal.
Indeed, we can express each u0un1 (n 2) in terms of qn and monomials
containing u with < n  1. It follows that any polynomial F in uo, vo,
vo, uo, multiplied by a suitable power of u0 can be expressed as a polynomial
in uo, vo, vo, and qn, n 2. Because t(u0) = u0u0 is not a zero divisor, it's
enough to show that any polynomial F in uo, vo, vo belongs to ker(t) only if
F = 0.
Suppose that F 6= 0 but F still belongs to ker(t). In this case, we can
uniquely expand the polynomial as follows
X
F = ufi00:::ufinnvIvJ
where I and J are finite sets of natural numbers and vI, for instance, means
vi1:::virif I = {i1; :::; ir} and i1 < ::: < ir, or vI = 1 if I is empty.
Let's introduce the following lexicographical order: uff00:::uffnnvI0vJ0 is
the leading term of the polynomial F if for any other term ufi00:::ufinnvIvJ
there exists j0 in J0 \ J biger than all j in J \ J0, or J = J0 and there exists
i0 in I0 \ I biger than all i in I \ I0, or J = J0 and I = I0 and there exists
0 0 such that ff0 > fi0 and ff = fi for all > 0. If "Fis the image of F
14
after applying t and substituting ur = 1 for all r 0, then the leading term
of "Fwith respect to the same lexicographical order is
u"ff00uff11:::uffnnvI0vJ0
`
where the disjoint union I0 J0 has "ff0 ff0 elements. Therefore, "Fcannot
be zero, contradiction.
mp__
4.2. Proof of Lemma 3.12. Any ideal a of O = Z[1_3; 3 1], being a free
Amodule of rank 3m1 , can be naturally regarded as an element of
Rep(3m; GL3m1A), and similarly, F7 A a as an element of
Rep(3m; GL3m1F7). Therefore, if o0m;nis injective for n 3m1 then, when
mp_
ever F7 A a and F7 A b are isomorphic over F7( 3 1 ) for two specified
homomorphisms A ! F7, then a and b are isomorphic as Omodules. The
proof can be concluded by the density argument that any ideal a is isomor
phic over O to an ideal a0in O of index prime to 7 (Tchebotarev Theorem).
5 Proof of Theorem 1.3
5.1. Reduction to SL2(A). In Remark 3.5, using anetale approximation
( 2; Y2(A)) to BSL2(A), we have detected a subalgebra H*Y2(A) P (c4)
(e3) (e03) imbedded in H*BSL2(A) via *2. The injectivity of *2follows
from the injectivity of O*2([8], 6.3) and the exact sequence
1 ! SL2(A) ! GL2(A) det!Ax ! 1
If we manage to show that *2is onto, then
H*BSL2(A) P (c4) (e3) (e03) (5.2)
and combining this result with the exact sequence above and the K"unneth
Formula, we get
H*BGL2(A) H*BSL2(A) H*(Ax ) P (c2; c4) (e1; e3) (e01; e03)
5.3. A MayerVietoris sequence. By a method explained in ([22], p.110)
we can decompose SL2(A) as an amalgam SL2Z[i] * SL2Z[i], where is
a b
the congruence subgroup of all in SL2Z[i] with c 0 mod ss. Here
c d
15
ss = 1i is a prime above 3 in Z[i]. With this amalgam we have an associated
MayerVietoris exact sequence
::: ! Hn ! HnSL2Z[i] HnSL2Z[i] ! HnSL2Z[1_3; i] ! Hn1::: (5.4)
which will be used to prove 5.2.
5.5. Surjectivity of *2in high dimensions. At this point, it is convenient
to apply a theorem of Brown [5] to compute the high dimensional cohomology
of = , SL2Z[i], or SL2(A).
For this, consider A*() the category having all nontrivial elementary
abelian 3subgroups E of as "objects" and all group homomorphisms in
duced by a conjugation by an element in as "morphisms". Assume that
has a finite virtual cohomological dimension (v:c:d:) and that the maximum
rank of E over Z=3 in A*() is 1. Then the 3primary component H^*(3)of
the Farell cohomology H^* of with any coefficients is determined by the
category A*() as follows [5]
Y
H^*(3) ^H*(N (E))(3) (5.6)
(E)
where we choose one representative E in each isomorphism class of A*()
and denote by N (E) the normalizer of E in .
It is easy to compute v:c:d:, [3], and it is elementary to list a set of
representatives E, their centralizers C (E) in , and their automorphism
groups Aut(E) in A*() . In our case we have
16
_________v:c:d:___E_____C_(E)___Aut(E)___

  i 0  Z=6  0
  0 i2  
   
  i 0  
 2  2  Z=6  0
  ss i  
   
  i 1  Z=6  0
  0 i2  
   
________________________________________________
  i 0  
SL2Z[i]  2  2  Z=6  Z=2
  0 i  
   
  0 1  Z=6  0
  1 1  
   
________________________________________________
  i 0  
SL2(A)  3  2 Z=6 x Z  Z=2
  0 i  
where E is always Z=3 and we have pointed out only its generator.
Recall that there is a natural map Hi() ! H^i() which becomes an
isomorphism for i > v:c:d:, [5]. Also, H*(; k) = H*(; k)(3)and from the
exact sequence
1 ! C (E) ! N (E) ! Aut(E) ! 1
we have H*(N (E); k) H*(C (E); k)Aut(E)because Aut(E) in our cases is
always 0 or Z=2 ([20], p. 126).
In conclusion, from 5.6 with kcoefficients, we get
H*>2 [H*>2(Z=3)]x3 [P (a2) (b1)]x3 [degree > 2] (5.7)
H*>2SL2Z[i] [H*>2(Z=3)]Z=2x H*>2(Z=3)
[P (a22) (a1b1)] x [P (a02) (b01)] [degree > 2] (5.8)
H*>3SL2(A) [H*>3(Z=3 x Z)]Z=2
P (a22) (a2b1) (a2b01) [degree > 3] (5.9)
If we compare 5.9 with 5.2 we get the surjectivity of *2for * > 3.
17
We announce here two lemmas which are needed in 5.12:
5.10. Lemma. H1(; Z) Z=3 Z=3 (0 or Z=2):
5.11. Lemma. H2(; k) k k k
For the convenience of the reader, their proofs are delayed to 6.7 and 6.4,
respectively.
5.12. Surjectivity of *2in low dimensions. To finish the proof of the
surjectivity of *2in all dimensions, it is enough to show that H1SL2(A)
H2SL2(A) = 0 and H3SL2(A) k2. For this, let's return to the Mayer
Vietoris sequence 5.4.
Case * = 1. The amalgam SL2Z[i] * SL2Z[i] is made of the inclusion i
a b a ssb
of in SL2Z[i] and the map j given by ! ([22], p. 110).
ssc d c d
1 1
Meanwhile, H1(SL2Z[i]; Z) Z=3 being generated by ([24], p. 42).
0 1
It is easy to check that
1 1
1 1 1 1 i2 0 1 1 i2 0
j =
0 1 0 1 0 i 0 1 0 i
" t#1
1 0 1 1
and j =
ss 1 0 1
and hence i1 j1 : H1(; Z) ! H1(SL2Z[i]; Z)2 is onto. From 5.4 with
Zcoefficients follows that H1(SL2(A); Z) = 0 and so H1(SL2(A); k) = 0.
Case * = 2. Because H2(SL2Z[i]; Z) Z=4 is known ([1], p. 376), from
5.4 with Zcoefficients and Lemma 5.10 follows that
Z=4 Z=4 ! H2(SL2(A); Z) ! (0 or Z=2)
is exact. Hence, by the universal coefficient theorem, we get
H2(SL2(A); k) = 0.
Case * = 3. From ([1], p. 376) immediately follows that H2SL2Z[i] k
and, in agreement with 5.8, H3SL2Z[i] H4SL2Z[i] k2. Also, H2 k3
by Lemma 5.11, H2SL2(A) = 0 by the previous case, H3 H4 k3 by
18
5.7, H4SL2(A) k, and H5SL2(A) = 0, both by 5.9. So that the sequence
5.4 with kcoefficients becomes
0 ! k3 ! k4 ! k ! k3 ! k4 ! H3SL2(A) ! k3 ! k2 ! 0
The exactness of this sequence implies at once that H3SL2(A) k2.
6 Low dimensional cohomology of
In order to prove Lemma 5.11, we define and study an action of on a
contractible CWcomplex X constructed by Alperin [1], and use a theorem
of Henn [12] to compute the second mod 3 cohomology of . Also, in 6.7 we
give a proof of Lemma 5.10 based on a presentation of SL2Z[i] in terms of
generators and relations as in [24]. In what follows, R stands for Z[i].
6.1. Alperin's complex X. This complex is the geometric realization X
of a poset consisting of all sets {L0; L1; :::; Ln} of independent lines in R2,
with n 1, ordered by inclusion. By "independent lines" we understand
ai bi
subspaces Li = (ai; bi)R of R2 with det invertible in R for any
aj bj
i < j. In particular, SL2(R) acts cellwise on X by left multiplication of the
ai
generators . It is shown in [1] that X is a contractible 2dimensional
bi
CWcomplex and the orbit space SL2(R)OX is the geometric realization of
the poset consisting of
ae oe ae oe
0 1 0 1 1
M = ; ; N = ; ; ;
1 0 1 0 1
ae oe ae oe
0 1 1 1 0 1 1 1
P1 = ; ; ; ; and P2 = ; ; ; 2 :
1 0 1 i 1 0 1 i
where each line is indicated by one of its generators. Using this complex, we
can prove Lemma 5.11. Indeed, acts on X via the inclusion SL2(R).
But X being contractible, we have H*() H*(X) [5], where H* means the
equivariant mod 3 cohomology. This is the motivation to study the action
of on X, leading to our result.
6.2. The orbit space OX. We construct this orbit space starting with
1 0 0 1 0 1
SL2(R)OX. First, observe that ffl = , ff = , fi = ,
0 1 1 0 1 1
19
i2 0
and fl = 2 form an independent system of representatives for the
i i
right cosets of in SL2(R). The system is also complete because the image
of via the natural surjective map SL2(R) ! SL2(F3) has index four. By
elementary calculations we obtain
orbits for vertices: [M] = [ffM]; [fiM] = [flM]; [N] = [fiN] =
1 1 i2 1
[ffN]; [flN]; [P1] = [fiP1] = [flP1] = [ffP1]; [P2] = [fiP2] =
0 1 0 i
i 1 i2 0
[ffP2] = 2 [flP2].
0 i2 i  i i
orbits for 1simplices: [M N]; [M ffN]; [fiM N];
i 0
[fiM flN]; [M P1] = 2 [M ffP1]; [fiM P1] = [flM flP1];
0 i
i2 0 i2 0
[M P2] = [M ffP2]; [fiM P2] = 2 [fiM flP2];
0 i i  i i
1 1
[N P1] = [ffN ffP1] = [fiN fiP1]; [flN P1]; [N P2] =
0 1
1 1
[ffN ffP2] = [fiN fiP2]; [flN flP2].
0 1
orbits for 2simplices: [M N P1]; [M ffN ffP1];
[fiM N P1]; [fiM flN P1]; [M N P2]; [M ffN ffP2];
[fiM N P2]; [fiM flN flP2].
All isotropy groups of cells are Z=2 except M = fiM = flN = P1 =
P2 = flNflP2= flNP1 = Z=6.
The conclusion is that the orbit space OX has the homotopy type of
the sphere S2. Moreover, if Xsingis the subspace of X which consists of
all the points fixed by an element of order 3 in , then OXsinghas the
homotopy type of a discrete space with three points. In this way, from the
exact sequence
H1(OXsing) ! H2(O(X; Xsing)) ! H2(OX) ! H2(OXsing)
we deduce that
H2(O(X; Xsing)) k (6.3)
6.4. Proof of Lemma 5.11. From 6.2 we have that acts cellwise on
a contractible CWcomplex X with finite stabilizers of cells. Let fl3 be the
20
maximum rank r such that (Z=3)r can be imbedded in , and let limibe the
iderived functor of the inverse limit functor. Then, the equivariant mod 3
cohomology of Xsingcan be computed using the following spectral sequence
([12], p. 274):
Ei;j2= limiA*()Hj(BC (E)) ) Hi+j(Xsing)
with Ei;j2= 0 if i fl3 and j is arbitrary.
Because the category A*() is already well known from 5.5, we conclude
that fl3 = 1 and
H*(Xsing) [P (a2) (b1)]x3 (6.5)
Now, we can compute H2() H2(X) using the exact sequence
H1(X) ! H1(Xsing) ! H2(X; Xsing) !
! H2(X) ! H2(Xsing) ! H3(X; Xsing) (6.6)
where H1(X) H1() k2 by Lemma 5.10, H1(Xsing) H2(Xsing) k3
by 6.5, H2(X; Xsing) H2(O(X; Xsing)) k by 6.3, and H3(X; Xsing)
H3(O(X; Xsing)) = 0 because dim X = 2. In other words, 6.6 becomes
k2 ! k3 ! k ! H2(X) ! k3 ! 0
and hence H2(X) k3.
6.7. Proof of Lemma 5.10. We give a grouptheoretic proof based on a
presentation of SL2Z[i] in terms of generators and relations as follows ([24],
p. 41):
1 1 1 i 1 0 i2 0
.
Let 0 be the subgroup of generated by T , U, J, L, CT 3C, and
CU1 T C. Then, using the relations CL = L2C, UL = LT , T L = LT 1U1 ,
U1 L = LT 1, and T 1L = LUT , we can "move" all L's from the right to
the left, such that any element of can be written in the form
0 n0 m n 0
fl0CU T C:::CUm T CU T Cfl0
21
where fl0 and fl00are in 0. Suppose that there exists an element in which
is not in 0. Then we can find one such element which written in the above
form has the minimum number of C's. But (CU1 T C)m = CUm T mCJm1
and (CT 3C)n = CT 3nCJn . So that, by a suitable change of fl00in 0, we can
replace CUm T nC by CT m+nC and this one by CT fflC, where ffl = 0, 1, or 1.
Also, C2 = J, CT C = T 1CT 1, and CT 1C = T CT J. Therefore, in any
case, the minimality condition on C's fails. This nonsense shows that in fact
0 = . In other words, can be generated by T , U, J, L, JCT 3C, and
JCU1 T C. In ab we have
1 0 1 0 i2 0 1 0 i 0
= 2 0;
ffss2 1 ffss1 0 i ffss 1 i
1 fiss 1 fi i 0 1 fi i2 0
= 2 0;
0 1 0 1 0 i 0 1 0 i
1 0 1 3 1 0
JCT 3C = 0; (JCU T C) = 0;
3 1 3ss 1
1 3 2
U T; T 3= 0; J 0; and
0 1
1 0 1 1 1 i 1 0 1 1 1 0 3
L = 2 T 0:
iss 1 0 1 0 1 ss 1 0 1 ss 1
Hence, ab is generated by at most two elements of order 3 and at most one
of order 2. According with 5.12, Case * = 1, ab maps onto Z=3 Z=3. In
conclusion,
H1(; Z) Z=3 Z=3 (0 or Z=2):
References
[1]Alperin, R.: Homology of SL2(Z[!]), Comment. Math. Helvetici 55
(1980) 364377
[2]Artin, M.; Mazur, B.: Etale Homotopy, Lecture Notes in Math. 100,
Springer, Berlin, 1969
22
[3]Borel, A.; Serre, JL.: Cohomologie d'immeubles et de groupes S
arithmetiques, Topology 15 (1976), 211232
[4]Bousfield, A.K.; Kan, D.M.: Homotopy Limits, Completions and Local
izations, Lecture Notes in Math. 304, Springer, Berlin, 1972
[5]Brown, K.S.: High dimensional cohomology of discrete groups, Proc.
Natl. Acad. Sci. USA 73 (1976), 17951797
[6]Dwyer, W.G.: Exotic cohomology for GLn(Z[1_2]), preprint (Notre Dame)
[7]Dwyer, W.G.; Freidlander, E.M.: Algebraic and etale Ktheory, Trans.
Amer. Math. Soc. 272 (1985), 247280
[8]Dwyer, W.G.; Freidlander, E.M.: Topological models for arithmetic,
Topology 33 (1994), 124
[9]Friedlander, E.M.: Etale Homotopy of Simplicial Schemes, Princeton
Univ. Press, Princeton, 1982
[10]Hartshorne, R.: Algebraic Geometry, Springer, New York, 1993
[11]Henn, HW.: The cohomology of SL(3; Z[1_2]), preprint (Heidelberg)
[12]Henn, HW.: Centralizers of elementary abelian psubgroups, the Borel
construction of the singular locus and applications to the cohomology of
discrete groups, Topology 36 (1997), 271286
[13]Henn, HW.; Lannes, J.; Schwartz, L.: Localizations of unstable A
modules and equivariant mod p cohomology, Math. Ann. 301 (1995),
2368
[14]Lee, CN.: On the unstable homotopy type of the classifying space of a
virtually torsionfree group, preprint (Northwestern)
[15]Mazur, B.: Notes on etale cohomology of number fields, Ann. scient. Ec.
Norm. Sup., 4e serie, 6 (1973), 521552
[16]Milne, J.S.: Etale Cohomology, Princeton Univ. Press, Princeton, 1980
[17]Mitchell, S.A.: On the plus construction for BGLZ[1_2] at the prime 2,
Math. Z. 209 (1992), 205222
23
[18]Quillen, D.: The spectrum of an equivariant cohomology ring: II, Annals
of Math. 94 (1971), 573602
[19]Quillen, D.: On the cohomolgy and Ktheory of the general linear groups
over a finite field, Annals of Math. 96 (1972), 552586
[20]Serre, JP.: Corps locaux, Hermann, Paris, 1968
[21]Serre, JP.: Cohomologie des groups discrets, Princeton Univ. Press,
Princeton, 1971, pp.77169
[22]Serre, JP.: Arbres, amalgames, SL2, Asterisque 46 (1977)
[23]Smith, L.: Homological algebra and the EilenbergMoore spectral se
quence, Trans. A.M.S. 129 (1967), 5893
[24]Swan, R.G.: Generators and relations for certain special linear groups,
Advances in Mathematics 6 (1977), 177
[25]Washington, L.: Introduction to Cyclotomic Fields, Springer, Berlin,
1982
University of Notre Dame
Notre Dame, IN, 46556
Institute of Mathematics
Bucharest, RO