The mod 2 cohomology of the linear groups over the ring of integers
Dominique Arlettaz, Mamoru Mimura, Koji Nakahata, Nobuaki Yagita
Abstract. This paper completely determines the ring structure, with *
*an explicit
description of the generators, of the mod 2 cohomology of the linear*
* groups
GL(Z) , SL(Z) and St(Z) and its module structure over the Steenrod a*
*lgebra.
1. Introduction
Recently, C. Weibel deduced from V. Voevodsky's proof [V] of the Milnor conject*
*ure the complete calculation
of the 2-torsion of the algebraic K-theory of the ring of integers Z (see [W]).*
* Of course, this has immediate
consequences on the mod 2 cohomology of the infinite general linear group GL(Z)*
* and more generally on
the understanding of the space BGL(Z)+ .
If p is a prime 3 or5 mod 8 , M. B"okstedt introduced in [B"ok] (see also [M] *
*and Section 4 of [DF]) a
space J(p) which is defined by the pull-back diagram
h0
J(p) ------! BO
?? ?
yf0p ?yc
b
Fp ------! BU ;
where Fp is the fiber of (p- 1) : BU ! BU (recall that Fp ' BGL(Fp)+ by Theorem*
* 7 of [Q2]), b
the Brauer lifting and c the complexification. The fibers of the horizontal map*
*s are homotopy equivalent
to the unitary group U . In fact, B"okstedt was more precisely interested in th*
*e covering space JK(Z; p) of
J(p) corresponding to the cyclic subgroup of order 2 of ss1J(p) ~=Z Z=2 . Afte*
*r completion at the prime
2 , he constructed a map
"': (BGL(Z)+)^2-! JK(Z; p)^2
which induces a split surjection on all homotopy groups. Recall that the local*
*ization exact sequence in
K-theory implies that
(BGL(Z[1_2])+)^2' (BGL(Z)+)^2x (S1)^2:
Therefore, "'provides a map
' : (BGL(Z[1_2])+)^2-! J(p)^2
________________________________________________________________
We would like to thank Christian Ausoni for his helpful comments on B"okstedt's*
* work [B"ok].
The third author thanks the Swiss National Science Foundation for financial sup*
*port.
1991 Mathematics Subject Classification: Primary 20 G 10; Secondary 19 D 55, 20*
* J 05, 55 R 40, 55 S 10.
1
which also induces a split surjection on all homotopy groups. Since the 2-torsi*
*on of K*(Z) is known by
[W], it is easy to check that "'and ' are actually homotopy equivalences. Conse*
*quently, we obtain for all
primes p 3 or5 mod 8 the pull-back diagram (see also Corollary 8 of [W])
h0
(BGL(Z)+)^2x (S1)^2------! BO^2
?? ?
y f0p ?yc
b
(Fp)^2 ------! BU^2
and the commutative diagram
j h
SU^2 ------! (BGL(Z)+)^2------! BO^2
?? ? ?
y i ?yfp ?yc
b
U^2 ------! (Fp)^2 ------! BU^2;
where both rows are fibrations: here fp and h denote the composition of the inc*
*lusion (BGL(Z)+)^2,!
(BGL(Z)+)^2x (S1)^2with f0pand h0 respectively ( fp is in fact the map induced *
*by the reduction mod p :
GL(Z) ! GL(Fp) ), and i is the 2-completion of the inclusion SU ,! U ' SU x S1.
This diagram enables us to calculate the mod 2 cohomology of BGL(Z)+ : we first*
* check easily (see Theorem
1) that there is a ring isomorphism
(*) H*(BGL(Z)+; Z=2) ~=H*(BO; Z=2) H*(SU; Z=2) :
Recall that H*(BO; Z=2) ~=Z=2[w1; w2; : :]:and H*(SU; Z=2) ~=(v3; v5; : :):wher*
*e deg(wj) = j and
deg(v2k-1) = 2k - 1 . The main objective of this paper is to describe the gener*
*ators of H*(BGL(Z)+; Z=2) :
the generators of the polynomial part are the Stiefel-Whitney classes, also den*
*oted by wj, coming from
H*(BO; Z=2) via the homomorphism induced by h , and we identify the exterior ge*
*nerators u2k-1 of
degree 2k - 1 in H*(BGL(Z)+; Z=2) in terms of the image of the homomorphism
f*p: H*(Fp; Z=2) ~=H*(BGL(Fp)+; Z=2) ! H*(BGL(Z)+; Z=2)
induced by the map fp for p 5 mod 8 (see Definitions 5 and 9). We can prove th*
*at the definition of
these classes does not depend on the choice of p by using the fact that they ar*
*e primitive cohomology
classes. The computation of the action of the Steenrod squares on the u2k-1's *
*enables us to deduce in
Corollary 12 that the above isomorphism (*) is actually an isomorphism of Hopf *
*algebras and of modules
over the Steenrod algebra (this last result can also be deduced by an indirect *
*argument from the above
diagram and Theorem 4.3 and Remark 4.5 of [M]). Moreover, we obtain an explicit*
* formula relating the
classes u2k-12 H*(BGL(Z)+; Z=2) , for k 2 , to the image of the homomorphism f*
**pfor all primes
3 or5 mod 8 (see Theorem 13). This provides a complete description of the mod *
*2 cohomology of the
group GL(Z) . In the remainder of the paper we compute the mod 2 cohomology of *
*the infinite special linear
group SL(Z) and of the infinite Steinberg group St(Z) (see Corollary 14, Theore*
*m 16 and Remark 17).
2
2. The mod 2 cohomology of the linear groups GL(Z) and SL(Z)
Theorem 1. There is a ring isomorphism
H*(BGL(Z)+; Z=2) ~=H*(BO; Z=2) H*(SU; Z=2) :
Proof. Let Q denote the subgroup of diagonal matrices in GL(Z) and : BQ ! BGL(*
*Z)+ the map
induced by the inclusion Q ,! GL(Z) . It is known by Theorem 22.7 of [Bor] that*
* the composition h :
BQ ! BO induces an injective homomorphism *h* : H*(BO; Z=2) ~=Z=2[w1; w2; : :]:*
*! H*(BQ; Z=2) ~=
Z=2[z1; z2; : :]:(with deg(zi) = 1 ) and that *h*(wj) = oej, the j-th elementar*
*y symmetric function in the
zi's. This implies that the infinite loop map h induces an injective homomorphi*
*sm h* : H*(BO; Z=2) !
H*(BGL(Z)+; Z=2) . Therefore, Theorem 15.2 of [Bor] shows that the Serre spectr*
*al sequence of the fibration
SU^2-j!(BGL(Z)+)^2h-!BO^2
collapses (see also Corollary 4.3 of [DF]) and we get additively the desired is*
*omorphism. Since (BGL(Z)+)^2
is an H-space, the maps and j produce an H-map
: BQ x SU^2-! (BGL(Z)+)^2
which induces an injective homomorphism
* : H*(BGL(Z)+; Z=2) -! H*(BQ; Z=2) H*(SU; Z=2) :
Moreover, the image of * is isomorphic to R H*(SU; Z=2) , where R is the suba*
*lgebra of H*(BQ; Z=2)
generated by the elementary symmetric functions oej. This provides the statemen*
*t of the theorem. _|_|
Our next goal is to show that the isomorphism given by the above theorem is act*
*ually an isomorphism of mod-
ules over the Steenrod algebra. Therefore, we first need to identify the genera*
*tors of H*(BGL(Z)+; Z=2) and
to understand the action of the Steenrod algebra on them. For j 1 let us write*
* wj2 H*(BGL(Z)+; Z=2)
for the image of the j-th universal Stiefel-Whitney class in H*(BO; Z=2) und*
*er the homomorphism
h* : H*(BO; Z=2) ! H*(BGL(Z)+; Z=2) . The action of the Steenrod algebra on th*
*e Stiefel-Whitney
classes is known by Wu's formula (see for instance [MT], Part I, p. 141). It re*
*mains to identify the exterior
generators of H*(BGL(Z)+; Z=2) . This will be done by using the homomorphism f**
*p: H*(Fp; Z=2) !
H*(BGL(Z)+; Z=2) induced by the map fp.
Let us first recall some properties of H*(Fp; Z=2) ~=H*(BGL(Fp)+; Z=2) . Accord*
*ing to Quillen's calcu-
lation and notation (see [Q2]), if p is a prime 5 mod 8 , then
H*(Fp; Z=2) ~=Z=2[c1; c2; : :]: (e1; e2; : :):;
where degcj= 2j and degek = 2k - 1 ; if p is a prime 3 mod 8 , then H*(Fp; Z=2*
*) is also generated
by the classes cj and ek (j 1 ; k 1) , but one has the relations
k-1X
e2k= c2k-1+ cjc2k-1-j
j=1
for k 1 , and H*(Fp; Z=2) is polynomial:
H*(Fp; Z=2) ~=Z=2[e1; e2; : :;:c2; c4; : :]:
(see also Section IV.8 of [FP]). In both cases, cj is the image under b* : H*(B*
*U; Z=2) ! H*(Fp; Z=2)
of the reduction mod 2 of the j-th universal Chern class in H2j(BU; Z) and a sp*
*ectral sequence argument
shows that * : H*(Fp; Z=2) ! H*(U; Z=2) ~=(v1; v2; : :):satisfies *(ek) = v2k-1*
* for k 1 . For a
3
prime p 3 or5 mod 8 , consider the homomorphism f*p: H*(Fp; Z=2) ! H*(BGL(Z)+;*
* Z=2) induced
by fp. For all j 1 , we showed in Lemma 1.4 of [A1] that
f*p(cj) = w2j
and established in [A2] for k 2 the non-vanishing of the exterior class f*p(ek*
*) if p 5 mod 8 , respectively
of the exterior class
k-1X
flk = f*p(ek) + w2k-1+ wjw2k-j-1
j=1
of degree 2k - 1 if p 3 mod 8 .
Let us mention the effect of the Steenrod squares on these cohomology classes.
k-1
Lemma 2. (a) In H*(SU; Z=2) , Sq2iv2k-1= i v2k+2i-1for k 2 , 1 i < k , and *
*Sq2i-1v2k-1= 0
for k 2 , 1 i k .
(b)In H*(BGL(Z)+; Z=2) , for any odd prime p , for k 1 and 1 i < k ,
k-1 * Xi k-j-1 2 * 2 *
Sq2if*p(ek) = i fp(ek+i) + i-j (wjfp(ek+i-j) + wk+i-jfp(ej)*
*) :
j=1
(c)In H*(BGL(Z)+; Z=2) , for k 1 and 1 i k ,
8
><0 , if p 1 mod 4 or if p 3 mod 4 and*
* k - i is odd,
Sq2i-1f*p(ek) = > i-1Xk-j-12 2
: i-j-1wjwk+i-j-1 , if p 3 mod 4 and k - i is even.
j=0
(d)In H*(BGL(Z)+; Z=2) , for any prime p 3 mod 8 and for k 1 ,
k-1 Xi k-j-1 2 2
Sq2iflk = i flk+i+ i-j (wjflk+i-j+ wk+i-jflj)
j=1
for 1 i < k and Sq2i-1flk = 0 for 1 i k .
Proof. Lemma 4 of [A2] gives the following information on the action of the Ste*
*enrod squares on the classes
ek 2 H*(Fp; Z=2) for k 1 : for any odd prime p and for 1 i < k ,
k-1 Xi k-j-1
Sq2iek = i ek+i+ i-j (cjek+i-j+ ck+i-jej) ;
j=1
and for 1 i k ,
8
><0 , if p 1 mod 4 or if p 3 mod 4 and k - i*
* is odd,
Sq2i-1ek = > i-1Xk-j-1
: i-j-1cjck+i-j-1, if p 3 mod 4 and k - i is even.
j=0
The formula (a) is well known but can be deduce from the previous equalities be*
*cause the composition
i** : H*(Fp; Z=2) ! H*(SU; Z=2) satisfies i**(ek) = v2k-1for k 2 and i**(cj) =*
* 0 for j 1 .
The statements (b) and (c) follow directly since Sq2if*p(ek) = f*p(Sq2iek) and *
*f*p(cj) = w2jfor j 1 .
In order to get (d), let us consider again the homomorphism * : H*(BGL(Z)+; Z=2*
*) ! H*(BQ; Z=2)
which is injective on Z=2[w1; w2; : :]:and trivial on exterior classes because *
*H*(BQ; Z=2) is polynomial. If
p 3 mod 8 , one has by the definition of flk
k-1X
Sq2iflk = Sq2if*p(ek) + Sq2iw2k-1+ Sq2i(wjw2k-j-1)
j=1
4
for 1 i < k . According to (b),
k-1 * Xi k-j-1 2 * 2 *
Sq2iflk = i fp(ek+i) + i-j (wjfp(ek+i-j) + wk+i-jfp(ej)) + (element of*
*Z=2[w1; w2; : :]:)
j=1
and consequently,
k-1 Xi k-j-1 2 2
Sq2iflk = i flk+i+ i-j (wjflk+i-j+ wk+i-jflj) + (element ofZ=2[w1;*
* w2; : :]:) :
j=1
Since the classes flk are exterior, they belong to the kernel of * and *(Sq2ifl*
*k) = 0 . However, the
injectivity of * on Stiefel-Whitney classes implies that the element of Z=2[w1;*
* w2; : :]:in the last formula
vanishes. The assertion (c) shows that Sq2i-1flk is an element of Z=2[w1; w2; :*
* :]:and one deduces similarly
that Sq2i-1flk = 0 . *
* _|_|
Our argument will be based on the understanding of the homomorphism
* : H*(BGL(Z)+; Z=2) ! H*(BGL(Z)+ x BGL(Z)+; Z=2) ~=H*(BGL(Z)+; Z=2) H*(BGL(Z*
*)+; Z=2)
induced by the H-space structure of BGL(Z)+ .
Lemma 3. (a) For any j 1 ,
Xj
*(wj) = ws wj-s:
s=0
(b) For any prime p 5 mod 8 and any integer k 2 ,
k-2X
*(f*p(ek)) = f*p(ek) 1 + 1 f*p(ek) + w2` f*p(ek-`) + f*p(ek-`) *
* w2`:
`=1
(c) For any prime p 3 mod 8 and any integer k 2 ,
k-2X
*(flk) = flk 1 + 1 flk+ w2` flk-`+ flk-` w2`:
`=1
Proof. Assertion (a) is known (see for instance [MT], Part I, p. 140). If deno*
*tes the H-space structure of
Fp, Proposition 2 of [Q2] implies that
iXk j Xk
*(f*p(ek)) = f*p(*(ek)) = f*p (c` ek-`+ ek-` c`) = w2` f*p(ek-`) + f**
*p(ek-`) w2`
`=0 `=0
for any odd prime p . If p 5 mod 8 , f*p(e1) vanishes since e1 is exterior and*
* one gets immediately (b).
If p 3 mod 8 , the definition of flk,
k-1X
flk = f*p(ek) + w2k-1+ wjw2k-j-1;
j=1
shows that
Xk
*(flk) = w2` f*p(ek-`) + f*p(ek-`) w2`+ (element ofZ=2[w1; w2; :*
* :]:) :
`=0
Since p 3 mod 8 , it turns out that f*p(e1) = w1 and consequently that
k-2X
*(flk) = flk 1 + 1 flk+ w2` flk-`+ flk-` w2`+ (element ofZ=2[w1; w2*
*; : :]:) :
`=1
5
However, the element of Z=2[w1; w2; : :]:in that formula must be trivial since *
**(flk) is exterior. This
implies the last assertion. *
* _|_|
Now, let p be a prime 5 mod 8 and k an integer 2 . Consider an integer m k ,*
* C the cyclic
group of order p - 1 and H*(BCm ; Z=2) ~=Z=2[x1; x2; : :;:xm ] (y1; y2; : :;:y*
*m ) with deg(xi) = 2 and
deg(yi) = 1 for 1 i m , endowed with the differential d defined by d(xi) = yi*
* and d(yi) = 0 . Then,
look at the homomorphism ae : H*(Fp; Z=2) ! H*(BCm ; Z=2) , introduced in [Q2],*
* p. 563-565, which is
injective in dimensions 2m (and in particular in dimensions 2k ) since its ke*
*rnel is the ideal generated
by the elements cj and ej for j > m , and which fulfills ae(cj) = sj and ae(ej)*
* = d(sj) for 1 j m ,
where sj denotes the j-th elementary symmetric function in x1; x2; : :;:xm . Fo*
*r k 1 , define the exterior
class m
X
k = xk-1jyj2 H2k-1(BCm ; Z=2) :
j=1
Lemma 4. For p 5 mod 8 and for any k 2 , the class ek 2 H2k-1(Fp; Z=2) satisf*
*ies
k-1X
ek = ae-1(k) + cjae-1(k-j):
j=1
X X X
Proof. Since sk = xi1xi2. .x.ik, one has d(sk) = xi1. .c.x*
*i`.x.i.kyi`. Then,
i1>> k-2X
>>>f*p(ek) + w2ju2k-2j-1; if p 5 mod 8,
< j=1
u2k-1= >
>>> k-1X k-2X
>>:f*p(ek) + w2k-1+ wjw2k-j-1+ w2ju2k-2j-1;if p 3 mod 8,
j=1 j=1
where f*p denotes the homomorphism H*(Fp; Z=2) ~= H*(BGL(Fp)+; Z=2) ! H*(BGL(*
*Z)+; Z=2)
induced by the reduction mod p : GL(Z) ! GL(Fp) .
Proof. If p 5 mod 8 , the statement is given by Proposition 6 (a). Observe in *
*particular that u2k-1can
be written as follows: u2k-1= Fk(f*p(e2); f*p(e3); : :;:f*p(ek)) , where Fk is*
* a polynomial with coefficients
in Z=2[w1; w2; : :]:. If p 3 mod 8 , consider again
k-1X
flk = f*p(ek) + w2k-1+ wjw2k-j-1
j=1
and define bu2k-1= Fk(fl2; fl3; : :;:flk) . It is obvious that bu2k-1is an ext*
*erior class and easy to check
as in the proof of Proposition 6 that j*(bu2k-1) = v2k-1. Moreover, observe th*
*at the homomorphism
* : H*(BGL(Z)+; Z=2) ! H*(BGL(Z)+ x BGL(Z)+; Z=2) acts on flk (for p 3 mod 8 )*
* and on f*p(ek)
(for p 5 mod 8 ) exactly in the same way according to Lemma 3 (b) and (c). Thu*
*s, the argument of the
proof of Corollary 7 implies that u2k-1(p) is also primitive if p 3 mod 8 . It*
* finally follows from Lemma
8 that
k-2X
u2k-1= bu2k-1= Fk(fl2; fl3; : :;:flk) = flk+ w2ju2k-2j-1: *
*_|_|
j=1
It is known that BGL(Z)+ ' BSL(Z)+ x BZ=2 and one deduces immediately the calcu*
*lation of the mod 2
cohomology of the space BSL(Z)+ (recall that H*(BSL(Fp)+; Z=2) is obtained from*
* H*(BGL(Fp)+; Z=2)
by dividing out e1 and c1):
Corollary 14. There is an isomorphism of modules over the Steenrod algebra
H*(BSL(Z)+; Z=2) ~=Z=2[w2; w3; : :]: (u3; u5; : :):;
where wk and u2k-1are also written for the image of wk and u2k-1under the homom*
*orphism induced
by the inclusion SL(Z) ,! GL(Z) . The formulas for u2k-1given by Theorem 13 do *
*still hold but observe
that the first Stiefel-Whitney class of SL(Z) is trivial.
Remark 15. The results of this section determine also the mod 2 cohomology of t*
*he groups GL(Z) and
SL(Z) because H*(BG+; Z=2) ~=H*(G; Z=2) for G = GL(Z) or SL(Z) .
9
3. The mod 2 cohomology of the Steinberg group St(Z)
The goal of this last section is to compute H*(St(Z); Z=2) by looking at the un*
*iversal central extension
Z=2 ~=K2(Z)- ! St(Z) -ss!!SL(Z)
and at the associated Serre spectral sequence
E*;*2~=H*(SL(Z); Z=2) H*(Z=2; Z=2) =) H*(St(Z); Z=2) :
Let us use the notation Q0 = Sq1 and Qr = Sq2rQr-1+ Qr-1Sq2r and observe that Q*
*r(w2) =
Sq2rSq2r-1. .S.q1w2 because Sq2rw2= 0 for r 2 and Sq1Sq2w2= 0 .
Theorem 16. (a) There is an isomorphism of modules over the Steenrod algebra
H*(St(Z); Z=2) ~=Z=2[w2; w3; : :]:=(w2; Qr(w2); r 0) (u3; u5; :*
* :):;
where wk and u2k-1denote the image of wk and u2k-1under the homomorphism ss* : *
*H*(SL(Z); Z=2) !
H*(St(Z); Z=2) .
(b) For k 2 ,
8
>>> k-2X
>>>f*p(ek) + w2ju2k-2j-1; if p 5 mod 8,
< j=4
u2k-1= >
>>> k-1X k-2X
>>:f*p(ek) + w2k-1+ wjw2k-j-1+ w2ju2k-2j-1;if p 3 mod 8,
j=4 j=4
where f*pis written here for the homomorphism H*(SL(Fp); Z=2) ! H*(St(Z); Z=2) *
*induced by the
reduction mod p : St(Z) ! St(Fp) ~=SL(Fp) .
Proof. Because H*(Z=2; Z=2) ~=Z=2[z] with degz = 1 , one can compute the differ*
*entials in the above
spectral sequence: d2(z) = w2, d3(z2) = Sq1d2(z) = Sq1w2 = w3, d5(z4) = Sq2d3(*
*z2) = Sq2w3 and
inductively, d2r+1(z2r) = d2r+1(Qr-1(z)) = Qr-1(d2(z)) = Qr-1(w2) = w2r+1+ (dec*
*omposable element of
Z=2[w2; w3; : :]:) by Wu's formula ([MT], Part I, p. 141). Therefore, the seque*
*nce (w2; Q0(w2); Q1(w2); : :):
is regular and we obtain Es;t1= 0 if t > 0 and E*;01~=H*(SL(Z); Z=2)=(w2; Qr(w2*
*); r 0) . This gives the
mod 2 cohomology of St(Z) as described by statement (a) and assertion (b) follo*
*ws directly from Corollary
14 (b) since w2 = w3= 0 . *
* _|_|
Remark 17. The above argument exhibits a surjective homomorphism from H*(BSL(*
*Z)+; Z=2) to
H*(BSt(Z)+; Z=2) . However, it is actually possible to find a nice map from BS*
*t(Z)+ to the space
BSpin inducing an injective homomorphism on mod 2 cohomology. More precisely, c*
*onsider the map " :
BSL(Z)+ ! BSL(R)+ induced by the inclusion Z ,! R and the map : BSL(R)+ ! BSL(*
*R)top' BSO
induced by the obvious map SL(R) ! SL(R)top, where the first group SL(R) is end*
*owed with the discrete
topology and SL(R)topwith the usual topology. Then, look at the following commu*
*tative diagram, where
the rows are fibrations in which the maps ff , ff0, ff00are the second Postniko*
*v sections of the corresponding
spaces ( BSpin is the fiber of ff00), the maps " and are the second Postnikov*
* sections of " and , and
10
the vertical maps on the left are the restrictions of " and to the fibers:
ss ff
BSt(Z)+ ------! BSL(Z)+ ------! K(K2Z; 2)
?? ? ?
y ?y" ?y"
ff0
BSt(R)+ ------! BSL(R)+ ------! K(K2R; 2)
?? ? ?
y ?y ?y
o ff00
BSpin ------! BSO ------! K(ss2BSO; 2)
The composition "is a homotopy equivalence because * "* : K2Z ! ss2BSO is an *
*isomorphism (see
Corollary 4.6 of [Br] or p. 25-26 of [Be]). Let us denote the composition " b*
*y O and its restriction
to BSt(Z)+ by O": BSt(Z)+ ! BSpin (note that O is the universal cover of the ma*
*p h defined in
the introduction and that the fiber of the 2-completion of "Ois SU^2because of *
*the last diagram of the
introduction). We get the commutative diagram
ss
BZ=2 ------! BSt(Z)+ ------! BSL(Z)+
?? ? ?
y ' ?y"O ?yO
o
BZ=2 ------! BSpin ------! BSO :
The ring structure of the mod 2 cohomology of BSpin is known by Proposition 6.5*
* of [Q1]:
H*(BSpin; Z=2) ~=Z=2[w"2; "w3; : :]:=(w"2; Qr(w"2); r 0) ;
where the w"k's are written here for the image of the universal Stiefel-Whitney*
* classes under the homo-
morphism o* : H*(BSO; Z=2) ! H*(BSpin; Z=2) . Since O* : H*(BSO; Z=2) ! H*(BSL(*
*Z)+; Z=2) is
injective, the map "Oinduces an injective homomorphism of modules over the Stee*
*nrod algebra
"O*: H*(BSpin; Z=2) ! H*(BSt(Z)+; Z=2) :
References
[A1] D. Arlettaz: Torsion classes in the cohomology of congruence subgroups, Ma*
*th. Proc. Cambridge Philos. Soc.
105 (1989), 241-248.
[A2] D. Arlettaz: A note on the mod 2 cohomology of SL(Z) , in: Algebraic Topol*
*ogy Poznan 1989, Proceedings,
Lecture Notes in Math. 1474 (1991), 365-370.
[Be] J. Berrick: An Approach to algebraic K-theory. (Pitman, 1982).
[B"ok]M. B"okstedt: The rational homotopy type of WhDiff(*) , in: Algebraic Top*
*ology, Aarhus 1982, Lecture Notes
in Math. 1051 (1984), 25-37.
[Bor]A. Borel: Topics in the homology theory of fibre bundles, Lecture Notes in*
* Math. 36 (1967).
[Br] W. Browder: Algebraic K-theory with coefficients Z=p , in: Geometric Appli*
*cations of Homotopy Theory I,
Evanston 1977, Lecture Notes in Math. 657 (1978), 40-84.
11
[DF] W. Dwyer and E. Friedlander: Conjectural calculations of general linear gr*
*oup homology, in Applications of
Algebraic K-theory to Algebraic Geometry and Number Theory, Boulder 1983, *
*Contemp. Math. 55 Part I
(1986), 135-147.
[FP] Z. Fiedorowicz and S. Priddy: Homology of classical groups over finite fie*
*lds and their associated infinite loop
spaces, Lecture Notes in Math. 674 (1978).
[M] S. Mitchell: On the plus construction for BGLZ[1_2] at the prime 2, Math. *
*Zeitschrift 209 (1992), 205-222.
[MT] M. Mimura and H. Toda: Topology of Lie groups I and II, Translations of Ma*
*th. Monographs 91 (AMS 1991).
[Q1] D. Quillen: The mod 2 cohomology rings of extra-special 2-groups and spino*
*r groups, Math. Ann. 194 (1971),
197-212.
[Q2] D. Quillen: On the cohomology and K-theory of the general linear groups ov*
*er a finite field, Ann. of Math.
96 (1972), 552-586.
[V] V. Voevodsky: The Milnor conjecture, preprint (1996), http://math.uiuc.edu*
*/K-theory/0170/.
[W] C. Weibel: The 2-torsion in the K-theory of the integers, preprint (1996),*
*http://math.uiuc.edu/K-theory/0141/.
Dominique Arlettaz Mamoru Mimura
Institut de mathematiques Department of Mathematics
Universite de Lausanne Faculty of Science
1015 Lausanne, Switzerland Okayama University
e-mail: dominique.arlettaz@ima.unil.ch Okayama, Japan 700
e-mail: mimura@math.okayama-*
*u.ac.jp
Koji Nakahata Nobuaki Yagita
Institut de mathematiques Faculty of education
Universite de Lausanne Ibaraki University
1015 Lausanne, Switzerland Mito
e-mail: koji.nakahata@ima.unil.ch Ibaraki, Japan
e-mail: yagita@mito.ipc.ibar*
*aki.ac.jp
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