EXOTIC COHOMOLOGY FOR GL n(Z[1=2])
W. G. Dwyer
x1. Introduction
Let denote the ring Z[1=2], Gn the group GL n() of invertible n x n matrices
over , and Dn the group of diagonal matrices in Gn. The inclusion Dn ! Gn
induces a classifying space map n : BDn ! BGn and a cohomology homomorphism
*n: H *(B Gn; F2) -! H *(B Dn; F2) :
Say that the mod 2 cohomology of Gn is detected on diagonal matrices if *nis
injective. In [15, 14.7] Quillen made a conjecture which specializes in the ca*
*se of
the ring to the following statement (see [9, p. 51]):
1.1 Conjecture. For any n 1 the mod 2 cohomology of Gn is detected on
diagonal matrices.
There is some evidence for this conjecture. Mitchell [14] and Henn [10] have
proved it for n 3. Voevodsky has announced a proof of the mod 2 Quillen-
Lichtenbaum Conjecture for Z, and from [5] and [14] it follows that *nis inject*
*ive
on the image of H *(B GL (); F2) -! H *(B Gn; F2). In particular, 1.1 is true i*
*n the
stable range.
The aim of this paper, though, is to give a disproof of Conjecture 1.1.
1.2 Theorem. The mod 2 cohomology of G32 is not detected on diagonal matrices.
Given the remarks above, it is a consequence of 1.2 that there exists an ele*
*ment
in the cohomology of G32 which is not in the image of H *(B GL (); F2). In fac*
*t,
1.2 is proved by a technique distantly related to the one which Quillen uses in*
* [15,
p. 592] to show that for various other number rings S and primes p the restrict*
*ion
map H *(B GL (S); Fp) ! H *(B GL n(S); Fp) is not surjective.
Further developments. Very recently, Henn and Lannes have improved upon 1.2 by
showing that the mod 2 cohomology of G14 is not detected on diagonal matrices.
Moreover, Henn proves in [11, 0.6] that the Poincare series pn(t) of the kernel*
* of *n
has a pole at t = 1 of order n - n0 + 1, where n0 is the smallest natural number
such that *n0is not injective. Combining these results shows that for any n 14
the cohomology of Gn is far from being detected on diagonal matrices.
______________
This research was partially supported by National Science Foundation Grant D*
*MS95-05024
Typeset by AM S-T*
*EX
1
2 W. G. DWYER
A generalization. Theorem 1.2 is a consequence of a result which is slightly m*
*ore
general. Recall that if P and G are groups and ff; fi : P ! G are homomorphisms,
then ff is said to be conjugate to fi if there is an element g 2 G such that gf*
*fg-1 = fi.
Let aeR : Gn ! GL n(R) and aeF3 : Gn ! GL n(F3) be the obvious homomorphisms.
Two homomorphisms ff; fi : P ! Gn are said to become conjugate over R (resp.
become conjugate over F3) if aeR ff and aeR fi (resp. aeF3ff and aeF3fi) are co*
*njugate.
1.3 Theorem. Suppose that the mod 2 cohomology of Gn is detected on diagonal
matrices. Let P be a finite 2-group with homomorphisms ff; fi : P ! Gn. Then ff
is conjugate to fi if and only if ff becomes conjugate to fi over R and over F3.
To obtain 1.2 from 1.3, let n denote the group of 2n-th roots of unity. The
smallest n with the property that the ideal class group of (n) is nontrivial is*
* 6,
and the degree of (6) over (equivalently, the rank of (6) as a -module) is
OE(26) = 25 = 32. Let P = 6, considered as a subgroup of (6)x , and let I be a
nonprincipal ideal in (6). It is then not hard to use the multiplicative action*
*s of P
on I and on (n) itself to construct two nonconjugate homomorphisms P ! G32
which become conjugate over R and over F3.
The proof of 1.3 is homotopy theoretic. If X and Y are spaces, let [X; Y ]de*
*note
the set of (unpointed) homotopy classes of maps X ! Y ; if P and G are groups,
let {P; G} denote the set of conjugacy classes of homomorphisms P ! G. The
classifying space functor gives a bijection {P; G} ~= [B P; BG]. For each n 1 *
*we
construct a space Xn together with a map
On : BGn ! Xn
such that the following three statements hold.
1.4 Proposition. If the mod 2 cohomology of Gn is detected on diagonal matrice*
*s,
then O*n: H *(Xn; F2) ! H *(B Gn; F2) is an isomorphism.
1.5 Proposition. If O*nis an isomorphism, then for any finite 2-group P the map
On . (-) : [B P; BGn]-! [B P; Xn] is a bijection.
1.6 Proposition. Let P be a finite 2-group with homomorphisms ff; fi : P ! Gn.
Then On . (B ff) is homotopic to On . (B fi) if and only if ff becomes conjugat*
*e to fi
over R and over F3.
Together these imply 1.3.
Section 2 contains a description of the machinery which is used to construct*
* the
space Xn, x3 has proofs of the above three propositions, and x4 contains the de*
*riva-
tion of 1.2 from 1.3. Throughout the paper, unspecified homology and cohomology
is to be taken with mod 2 coefficients. The symbol ^BG denotes the 2-completion*
* [1]
of the classifying space of G. The usual topological groups GL n(R) and GL n(C)
are denoted GL topn(R) and GL topn(C).
The idea for this paper originated in conversations with S. Mitchell, and the
approach depends heavily on his calculations from [14]. The construction of Xn
goes back to work with E. Friedlander [4]. Some of the arguments below can be
generalized, but we have decided to concentrate on proving 1.2.
EXOTIC COHOMOLOGY FOR GL n(Z[1=2]) 3
x2. Etale homotopy theory
The space Xn promised in x1 is constructed usingetale homotopy theory [8],
which gives a covariant mechanism for assigning a space (more accurately a pro-
space) Aet to any reasonable scheme or simplicial scheme A. We build Xn as a
certain space of maps between twoetale homotopy types (2.6), in imitation of the
way in which GL n() can be described as a certain set of maps between schemes.
Etale homotopy types. Here are some examples ofetale homotopy types. In
the examples, the symbol ` stands for a prime number.
2.1 Fields and complete local rings. If k is a field, then Spec (k)et is a pro-*
*space
of type K(ss; 1), where ss is the Galois group over k of the separable algebraic
closure of k. If S is a complete local ring with residue class field k, then t*
*he
map Spec (k)et ! Spec (S)et induced by S ! k is an equivalence. For instance,
Spec (C)et is contractible, Spec (R)et is equivalent to B Z=2, and both Spec (F*
*3)et
and Spec(Z3)et are equivalent to the profinite completion of a circle.
If S is a commutative ring, let GL n;S denote the rank n general linear group
scheme over S. Applying the usual bar construction to GL n;S gives a classifyi*
*ng
object B GL n;S [8, 1.2], which is a simplicial scheme. See [8, x8] for the fo*
*llowing
three examples.
2.2 General linear groups over algebraically closed fields.. If k is an algebra*
*ically
closed field of characteristic zero, the pro-space (B GL n;k)et is equivalent t*
*o the
profinite completion of the space BGL topn(C). If k is the algebraic closure o*
*f a finite
field, then the `-completion of (B GL n;k)et at any prime ` not equal to char(k*
*) is
equivalent to the Bousfield-Kan `-completion tower {(Z=`)sB GL topn(C)}s.
2.3 General linear groups over other fields. If k is a field of characteristic *
*zero with
algebraic closure k, the natural sequence
(B GL n;k)et-! (B GL n;k)et-! Spec(k)et
is a fibration sequence of pro-spaces. In effect, if ss is the Galois group of *
*k over k,
(B GL n;k)et is the Borel construction of the natural action of ss on (B GL n;k*
*)et. If k
is a finite field with algebraic closure k and ` 6= char(k), there is a similar*
* fibration
sequence
{(Z=`)s(B GL n;k)et}s -! {(Z=`)os(B GL n;k)et}s -! Spec(k)et ;
where {(Z=`)os(-)}s denotes fibrewise `-completion over Spec(k)et.
2.4 General linear groups over number rings. Suppose that R is a number ring, a*
*nd
that S is the ring R[1=`]. Let k denote either the algebraic closure of the quo*
*tient
field of S, or the algebraic closure of one of the residue class fields of S (n*
*ote that
none of these residue fields have characteristic `). Then the natural sequence
{(Z=`)s(B GL n;k)et}s -! {(Z=`)os(B GL n;S)et}s -! Spec(S)et :
is a fibration sequence. There are identical fibration sequences if S is replac*
*ed by
a complete local ring of residue characteristic different from `.
4 W. G. DWYER
2.5 Number rings. Theetale homotopy type of a number ring is not easy to pin
down; see [5, 2.1] for a description of its untwisted "integral" homology. We *
*will
settle for a partial description of Spec ()et, where, as usual, = Z[1=2]. Cho*
*ose
an embedding Z3 -! C. There are induced commutative diagrams of rings and of
pro-spaces
C ---- R Spec (C)et ----! Spec (R)et
x? x ?? ??
? ?? y y :
Z3 ---- Spec (Z3)et ----! Spec()et
Since Spec(C)et is contractible, the diagram induces a map
Spec(Z3)et_ Spec(R)et-! Spec()et :
Pick an equivalence B Z=2 ! Spec (R)et, and a map S1 ! Spec (Z3) which sends
the generator of ss1S1 to the Frobenius automorphism of F3 over F3. What results
is a map
S1 _ BZ=2 ! Spec()et :
By [5], this map induces an isomorphism on mod 2 cohomology (cf. [6, x2]).
2.6 Etale approximations to B GL n(S). Let S be an algebra over . The ho-
momorphism ! S induces a map Spec (S)et ! Spec ()et. We set ` = 2 and
let B GLetn(S) denote the basepoint component of the space which in [4, 2.3] is
called Hom `(Spec (S)et; (B GL n; )et) ; in other words, B GLetn(S) is the ba*
*sepoint
component of the function space of maps over Spec()etfrom Spec(S)etto the fibre-
wise 2-completion of (B GL n; )et. (The function space is pointed because the m*
*ap
B GL n; ! Spec() has a natural section provided by the basepoint of the fibrew*
*ise
bar construction.) Since B GL n(S) can be identified as the basepoint component*
* of
the space of maps Spec (S) ! B GL n; over Spec () [4, 4.2], functoriality give*
*s a
map
On;S : BGL n(S) ! BGL etn(S)
(see [4, 2.5, pf. of 4.2]). We will describe this map below in some particular *
*cases.
2.7 Remark. Suppose that B is a space (i.e. a trivial pro-space) together with*
* a
map B ! Spec()et. Let "holim " denote the homotopy inverse limit functor from
pro-spaces to spaces [8, x6], and E be the fibration over B obtained by the fol*
*lowing
homotopy pullback diagram
E ----! holim{(Z=2)os(B GL n; )et}s
?? ?
y ?y :
B ----! holim Spec ()et
The homotopy fibre of E ! B is ^BGL topn(C). Essentially by definition, the spa*
*ce
Hom `(B; (B GL n; )et) is equivalent to the space of sections of the map E ! *
*B.
EXOTIC COHOMOLOGY FOR GL n(Z[1=2]) 5
It follows from a spectral sequence argument [4, 2.11] that if S is a -algeb*
*ra
and f : B ! Spec (S)et is a map of pro-spaces which induces an isomorphism on
mod 2 cohomology, then f induces an equivalence
BGLetn(S) -'!Hom `(B; (B GL n; )et) :
2.8 The complex numbers. Since Spec(C)et is contractible, it follows from 2.7 t*
*hat
B GLetn(C) is equivalent to ^BGL topn(C). The same calculation works for any se*
*para-
bly closed field of characteristic not 2. The map On;C is essentially the compo*
*site
of the usual map B GL n(C) ! BGL topn(C) with the 2-completion map.
2.9 The real numbers. Since Spec (R)et is equivalent to B Z=2, it follows from *
*2.7
that B GLetn(R) is equivalent to the basepoint component of the space of sectio*
*ns
of a fibration over B Z=2 with ^BGL topn(C) as the fibre. By naturality, this f*
*ibration
is the one associated to the action of Z=2 on ^BGL topn(C) by complex conjugati*
*on.
It follows from [7] that the natural map from ^BGL topn(R) to this space of sec*
*tions
is an equivalence. The map On;R is the essentially the composite of the usual m*
*ap
B GL n(R) ! BGL topn(R) with the 2-completion map.
2.10 The field F3 . Consider the commutative diagram
B GL n(F3 ) ----! colim nB GL n(F3 ) = BGL (F3 )
? ?
On;F3?y ?ycolimn(On;F3)
BGL etn(F3 )----! colimn B GLetn(F3 )
in which the colimit maps are induced by the usual matrix block inclusions. The
right hand vertical arrow induces an isomorphism on mod 2 cohomology (see [4,
4.5, 8.6]). The horizontal maps induce surjections on mod 2 cohomology; for the
upper one see [16] and for the lower one 2.7. By diagram chasing, On;F3 induces
a surjection on mod 2 cohomology. Since B GL n(F3 ) and B GLetn(F3 ) have mod 2
cohomology rings which are abstractly isomorphic ([16], 2.7) and finite in each
dimension, On;F3 must be an isomorphism on mod 2 cohomology. It follows that
B GLetn(F3 ) is equivalent to ^BGL n(F3).
2.11 The field F3. There is a map S1 ! Spec (F3)et which sends a generator of
ss1(S1) to Frobenius automorphism of F3 over F3. This map is an isomorphism on
cohomology with finite coefficients, in particular, on mod 2 cohomology. It fol*
*lows
from 2.7 that B GLetn(F3) is equivalent to the space of sections of a fibration*
* over
S1 with fibre ^BGL topn(C) ' B GLetn(F3 ) (2.8). By naturality, this is the fi*
*bration
associated to the action of on B GLetn(F3 ) and so its space of sections is (*
*by
definition) the homotopy fixed point set (B GLetn(F3 ))h. Consider the commutat*
*ive
diagram of spaces with an action of (the action in the left hand column is tr*
*ivial):
BGL n (F3) ---u-! B GL n(F3 )
? ?
On;F3?y On;F3?y :
B GLetn(F3) ---v-! B GLetn(F3 )
6 W. G. DWYER
The space B GLetn(F3 ) is 2-complete (2.7). The map On;F3 induces an equivalence
^BGL n(F3 ) ! B GLetn(F3 ) (2.10). Quillen [16] shows that the map B GL n(F3) !
(^BGL n(F3 ))h induced by the map u gives an isomorphism on mod 2 cohomology.
It follows that the map
On;F3 et ' et h
BGL n (F3) ---! BGL n (F3) -! (B GL n (F3 ))
also induces an isomorphism on mod 2 cohomology, and that B GLetn(F3) is equiv-
alent to ^BGL n(F3).
Remark. The conjecture that On;S induces an isomorphism on mod 2 cohomology
is a very strong unstable analogue of the mod 2 Quillen-Lichtenbaum Conjecture
for the ring S. This conjecture is true for a finite field (cf. 2.11) or the *
*algebraic
closure of a finite field (cf. 2.10). It is unknown whether or not it is true*
* for the
fields R and C. The results in this paper show that it is false for the ring .
x3. The space Xn and its properties
We define Xn to be the space B GLetn() from 2.6, and On : BGn ! Xn the map
On; : BGL n() ! BGL etn(). In this section we prove 1.4, 1.5 and 1.6.
(Co)homology`of`Xn.` Let B Do, B Go and Xo denote respectively the spaces
BDn, BGn and Xn. The index n in these coproducts runs over all non-
negative integers, where for n = 0 the spaces involved are contractible. There *
*are
maps
BDo -! B Go -O!Xo :
Under matrix block sum all three of these spaces are homotopy associative H-spa*
*ces;
B Go and Xo are homotopy commutative. The maps and O respect the multipli-
cations.
There is a natural identification
B D1 ' B(Z x Z=2) ' S1 x BZ=2 :
Let e 2 H 1S1 and fik 2 H kB Z=2 be generators. We denote the classes e fik-1
and 1 fik in H kB D1 by ak and bk respectively. Let aGk = *(ak), bGk = *(bk),
aXk = (O)*(ak), bXk = (O)*(bk). Since H *B Do is the free F2-algebra on the
elements ak (k 1) and bk (k 0), the image of * is the subalgebra of H *B Go
generated by the classes aGk and bGk, while the image of (O)* is generated by t*
*he
classes aXk and bXk.
3.1 Proposition. The algebra H *Xo is the free commutative F2-algebra on the
classes aXk (k 1) and bXk (k 0) subject to the following relations:
(1) (aXk)2 = 0 for k odd, and
(2) aXkbX0+ aXk-1bX1+ . .+.aX1bXk-1= 0 for k even.
EXOTIC COHOMOLOGY FOR GL n(Z[1=2]) 7
3.2 Lemma. There is a homotopy fibre square
Xn ----! B GLetn(R)
?? ?
y ?y :
BGLetn(Z3) ----! B GLetn(C)
Proof. This is a consequence of 2.5 and 2.7.
3.3 Remark. The square from 3.2 can up to homotopy be rewritten in the following
way:
Xn ----! ^BGLtopn(R)
?? ?
y u?y :
B^GL n(F3) ---v-! ^BGLtopn(C)
The rewriting is justified by 2.9, 2.8, 2.1 and 2.7. By naturality the map u is*
* the
usual one induced by the map R ! C. The map v is a little more problematical,
but it is clear from 2.5 that the restriction of v to the diagonal matrices in *
*GL n(F3)
is induced by the ordinary diagonal inclusion {1}n ! B GL n(C). The fact that
the mod 2 cohomology of B GL n(F3) is detected on diagonal matrices [14, x3] [1*
*6]
means that it is easy to compute the map on mod 2 homology or cohomology
induced by v.
Proof of 3.1. This is essentially given by Mitchell in [14, 4.6]; his space JK(*
*Z) can
be identified with colimn Xn, where the maps in the colimit are the block inclu*
*sions
given by product with a point in X1. The role of diagram [14, 4.1] is played by*
* the
homotopy fibre square from 3.3 above.
3.4 Proposition. The classes aGk, k 1 and bGk, k 0 in H *B Go satisfy the
analogs of relations (1) and (2) from 3.1.
Proof. See [14, pf. of 7.1]. The place to look for these relations is in H *B G*
*L 2(),
and Mitchell computes this homology explicitly [14, x6].
Proof of 1.4. By 3.1 and 3.4, the homology map (Onn)* is surjective and its ker*
*nel
is equal to the kernel of (n)*. By duality, the cohomology map (Onn)* is inject*
*ive
and its image is equal to the image of *n. This implies that *nis injective if*
* and
only if O*nis an isomorphism.
Maps into Xn. Let P be a finite 2-group. We are interested in studying the set
[B P; Xn] of unbased homotopy classes of maps from B P to Xn. The homotopy
fibre square from 3.3 gives a map
(3.5) [B P; Xn] ! [B P; ^BGLn (F3)] x [B P; ^BGLtopn(R)] :
8 W. G. DWYER
3.6 Lemma. The map 3.5 is injective.
Proof. It follows from 3.3 that there is a homotopy fibre square of mapping spa*
*ces
Map (B P; Xn) ----! Map (B P; ^BGLtopn(R))
?? ?
y ?y :
Map (B P; ^BGLn (F3))----! Map (B P; ^BGLtopn(C))
By an elementary argument, it is enough to show that each component of the space
Map (B P; ^BGLtopn(C)) is 1-connected. By [7] the space`Map (B P; ^BGLtopn(C*
*)) is
equivalent to the 2-completion of the disjoint union aeBC(ae(P )), where ae r*
*uns
through a set of representatives for the conjugacy classes of homomorphisms P !
GL n (C), and C(ae(P ))is the centralizer of ae(P ). By elementary representat*
*ion
theory each one of these centralizers is isomorphic to a product of complex gen*
*eral
linear groups, and so is connected.
The following theorem is derived in [13, x1] from Carlsson's work in [2] and*
* [3].
We state it only for the prime 2, although it holds for any prime.
3.7 Theorem. Let be a group of virtually finite cohomological dimension, and
let P be a finite 2-group. Then the natural map
{P; } ~=[B P; B] -! [B P; ^B]
is a bijection.
Proof of 1.5. The space Xn is 2-complete (3.3), so the fact that O*nis an isomo*
*r-
phism implies that Xn is equivalent to ^BGn. Since Gn is a group of virtually f*
*inite
cohomological dimension [17, p. 124] the result follows from 3.7.
Proof of 1.6. Consider the commutative diagram
B aeF3
[B P; BGL n (R)] -BaeR---[B P; BGn]----! [B P; BGL n (F3)]
?? ? ?
y ?y ?y :
[B P; BGLetn(R)] ---- [B P; Xn] ----! [B P; BGLetn(F3)]
By 2.9, 2.11, 3.7 and [7], the right and left vertical arrows are bijections. T*
*he result
follows from 3.6.
x4. Exploiting the class group
In this section we derive 1.1 from 1.3 by showing how class groups of cyclot*
*omic
extensions of can be used to construct the necessary homomorphisms from finite
2-groups into GL 32().
Recall that n denotes the multiplicative group of 2n-th roots of unity. We m*
*ake
a distinction between [n], which is the group ring of n over , and (n), which
is the integral closure of in the field obtained from Q by adjoining n. There
is a surjection [n] ! (n) (cf. [18, 2.6]) which is not an isomorphism unless
EXOTIC COHOMOLOGY FOR GL n(Z[1=2]) 9
n = 0. Because of this surjection, though, two modules over (n) are isomorphic
as (n)-modules if and only if they are isomorphic as [n]-modules.
4.1 Remark. If M is a -module and ff : n ! Aut (M) is a homomorphism, let
Mffdenote the [n]-module obtained by letting n act on M via ff. Given two
homomorphisms ff; fi : n ! Aut (M), it is clear that ff is conjugate to fi if a*
*nd
only if the [n]-modules Mffand Mfiare isomorphic.
4.2 Lemma. Suppose that I is a nonzero ideal in (n). Then there are isomor-
phisms of (n)-modules
R I ~=R (n)
:
F3 I ~=F3 (n)
Proof. Since the quotient (n)=I is a torsion group, the first isomorphism resul*
*ts
from taking the inclusion I ! (n) and tensoring with R. For the second, note
that by the Chebotarev density theorem there are an infinite number of prime id*
*eals
in (n) isomorphic (as modules) to I. Up to isomorphism, then, I can be taken to
be a prime ideal of residue characteristic different from 3. Tensoring the incl*
*usion
I ! (n) with F3 again gives the required isomorphism.
Recall that (n) is free as a module over of rank OE(2n) = 2n-1 [18, 2.5]. T*
*he
same is true of any nonzero ideal in (n).
4.3 Lemma. If (n) has a nonzero ideal which is not principal, then there exist
two nonconjugate homomorphisms ff; fi : n ! GOE(2n)which become conjugate both
over R and over F3.
Proof. Let I be such a nonzero ideal. Choosingnbases for I and for (n) allows
both to be identified (as -modules) with OE(2 ). The multiplicative actions of n
on I and on (n) thus give two homomorphisms ff; fi : n ! GL OE(2n)(). These
homomorphisms are not conjugate, because I and (n) are not isomorphic as
[n]-modules (equivalently, as (n)-modules). The homomorphisms are conju-
gate over R or over F3, because (4.2) the two modules become isomorphic when
tensored with R or with F3.
Proof of 1.2. According to 4.3, it is enough to show that (6) is not a principal
ideal domain. By [18, p. 353] the ideal class group of Z(6) is cyclic of order*
* 17.
Since the ideal class group of (6) is the quotient of the ideal class group of *
*Z(6)
by the subgroup generated by the prime ideals of Z(6) which lie above 2, it is
enough to show that there is only one prime P in Z(6) above 2, and that P is
principal. This is a standard calculation; see [18, 1.4] or [12, p. 73].
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Department of Mathematics, University of Notre Dame, Notre Dame, Indiana
46556 USA
E-mail address: dwyer.1@nd.edu