Integral cohomology and Chern classes
of the special linear group over the ring of integers
By DOMINIQUE ARLETTAZ
Institut de mathematiques, Universite de Lausanne, 1015 Lausanne, Swi*
*tzerland
e-mail: dominique.arlettaz@ima.unil.ch
CHRISTIAN AUSONI
Departement Mathematik, HG, ETH-Zentrum, 8092 Z"urich, Switzerland
e-mail: ausoni@math.ethz.ch
MAMORU MIMURA
Department of Mathematics, Faculty of Science, Okayama University, Okayam*
*a, Japan 700
e-mail: mimura@math.okayama-u.ac.jp
and NOBUAKI YAGITA
Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan
e-mail: yagita@mito.ipc.ibaraki.ac.jp
Abstract
This paper is devoted to the complete calculation of the additive structure of *
*the 2-torsion of the integral
cohomology of the infinite special linear group SL(Z) over the ring of integers*
* Z . This enables us to
determine the best upper bound for the order of the Chern classes of all integr*
*al and rational representations
of discrete groups.
____________________
1. Introduction
The Hopf algebra structure of the mod 2 cohomology of the infinite special and *
*general linear groups SL(Z)
and GL(Z) over the ring of integers Z has been completely determined in [AMNY] *
*as a module over the
Steenrod algebra. For instance, H*(SL(Z); Z=2) ~=H*(BSL(Z)+; Z=2) is the tensor*
* product of a polynomial
algebra with an exterior algebra:
H*(BSL(Z)+; Z=2)~=H*(BSO; Z=2) H*(SU; Z=2)
~=Z=2[w2; w3; : :;:wj; : :]: Z=2(u3; u5; : :;:u2k-1;*
* : :):;
where deg(wj) = j and deg(u2k-1) = 2k - 1 .
The first goal of this paper is to investigate the mod 2 cohomological Bockstei*
*n spectral sequence
E*1(BSL(Z)+) ~=H*(BSL(Z)+; Z=2) =) (H*(BSL(Z)+; Z)=torsion) Z=2
________________________________________________________________________________
2000 Mathematics Subject Classification: Primary 20 G 10; Secondary 19 D 55, 55*
* R 40, 55 T 99, 57 R 20.
1
of the space BSL(Z)+ (see Sections 1 - 5 of [Brd]). By using the mod 2 Bockstei*
*n spectral sequences of
the spaces BSO and BSL(Fp)+ (for a prime p 5 mod 8 ) and the maps h : BSL(Z)+ *
*! BSO and
fp : BSL(Z)+ ! BSL(Fp)+ induced by the inclusion Z ,! R and by the reduction mo*
*d p respectively, we
compute the terms E*r(BSL(Z)+) and the differentials dr for all r 1 (see Theor*
*em 4.3 and Corollary
4.4). Of course, this detects the 2-torsion of the integral cohomology H*(BSL(Z*
*)+; Z) ~=H*(SL(Z); Z)
of the special linear group SL(Z) . Theorem 4.7 actually provides an explicit a*
*dditive presentation of the
2-torsion of H*(BSL(Z)+; Z) by generators and relations (see also Remark 4.8 fo*
*r some partial information
on the multiplicative structure). It turns out that H*(BSL(Z)+; Z) contains no *
*cyclic direct summand of
order 4 and that the set of all non-trivial elements of Z=2(u4i+1; i 1) is in *
*one-to-one correspondence with
an additive basis of H*(BSL(Z)+; Z)=torsion(see Corollary 4.6). Moreover, we ar*
*e able to understand the
effect of the induced homomorphisms h* : H*(BSO; Z) ! H*(BSL(Z)+; Z) and f*p: H*
**(BSL(Fp)+; Z) !
H*(BSL(Z)+; Z) on the 2-torsion elements: Theorem 4.9 asserts in particular tha*
*t h* is injective on the
elements of order 2 and that f*pis injective on all cyclic direct summands of o*
*rder 2r with r 3 .
Notice that it is easy to extend these results to the integral cohomology H*(GL*
*(Z); Z) ~=H*(BGL(Z)+; Z)
of the general linear group GL(Z) because of the homotopy equivalence BGL(Z)+ '*
* BSL(Z)+ x BZ=2
(see for example [Ar1], Lemma 1.2).
As a consequence, we obtain the exact order of all Chern classes cn(SL(Z)) 2 H2*
*n(SL(Z); Z) of the inclusion
SL(Z) ,! GL(C) (see Proposition 5.2 and Theorem 5.3) and deduce the best upper *
*bound for the order of
the Chern classes of all integral and rational representations of discrete grou*
*ps (see Corollary 5.6).
The paper is organized as follows. Sections 2 and 3 present the mod 2 Bockstei*
*n spectral sequence for
the spaces BSO and BSL(Fp)+ respectively. The mod 2 Bockstein spectral sequence*
* and the 2-torsion
of the integral cohomology of BSL(Z)+ are computed in Section 4. Finally, Secti*
*on 5 is devoted to the
investigation of the order of the Chern classes of integral and rational repres*
*entations of discrete groups.
2. The mod 2 Bockstein spectral sequence for BSO
The determination of H*(BSL(Z)+; Z=2) is based on cohomological calculations in*
*volving the pull-back
diagram
(BSL(Z)+)^2---h---! BSO^2
?? ?
yfp ?yc
(BSL(Fp)+)^2---b---! BSU^2;
where (-)^2denotes the completion at the prime 2 , p any prime 3 or5 mod 8 , h*
* the map induced
by the inclusion Z ,! R , fp the map induced by the reduction mod p : Z!! Fp, c*
* the complexification
and b the Brauer lift, and where the homotopy fibers of both horizontal maps ar*
*e SU^2(for the details of
that construction, see [AMNY], where the argument is presented for GL(Z) instea*
*d of SL(Z) , or Chapter
3 of [Au]; notice also that an unstable version of this computation is given in*
* [He]). In order to go through
the mod 2 Bockstein spectral sequence for BSL(Z)+ , we shall first consider the*
* mod 2 Bockstein spectral
sequence for the spaces BSO and BSL(Fp)+ , and the homomorphisms induced in coh*
*omology by the maps
h and fp.
2
Let us start by looking at the mod 2 Bockstein spectral sequence
E*1(BSO) ~=H*(BSO; Z=2) ~=Z=2[w2; w3; : :;:wj; : :]:=) (H*(BSO; Z)=torsi*
*on) Z=2 :
Its first differential is d1= Sq1 and we know by Wu's formula (see for instance*
* [MT], Part I, p.138, Theorem
5.12) that for i 1 , Sq1(w2i) = w2i+1, Sq1(w2i+1) = 0 , and that Sq1(w2j) = 0 *
*for j 1 . Thus, we may
deduce that
E*2(BSO) ~=Z=2[w22; w24; : :;:w22i; : :]:
is concentrated in degrees 0 mod 4 . Since dr is of degree 1 , it is then obvi*
*ous that dr= 0 for all r 2 .
Consequently, we have proved the following result.
Proposition 2.1.
(a) The mod 2 Bockstein spectral sequence for BSO has the property that E*1(BSO*
*) ~=Z=2[wj; j 2]
and E*r(BSO) ~=E*1(BSO) ~=Z=2[w22i; i 1] for all r 2 .
(b) All non-trivial 2-torsion elements of H*(BSO; Z) have order exactly equal t*
*o 2 .
(c) H*(BSO; Z)=torsion~=Z[p4i; i 1] , where p4i is of degree 4i and represe*
*nts an element of
H4i(BSO; Z) whose reduction mod 2 is w22i2 H4i(BSO; Z=2) .
Remark 2.2. The additive and multiplicative structures of the 2-torsion of H*(B*
*SO; Z) has been obtained
a long time ago in [Brn], Theorem 1.5, and [F], Theorem 1 (see also [Bo], Theor*
*em 24.7 and Proposition 25.6,
[CV], Theorem 1, and [ThE], Theorem A). For completeness, let us recall here it*
*s additive structure, which
can also be determined by the argument we shall use in the next sections (see L*
*emma 3.6 and Theorems 3.7
and 4.7): if denotes the graded Z-algebra Z[q2i+1; i 1] Z[p4i; i 1] with de*
*g(q2i+1) = 2i + 1 and
deg(p4i) = 4i , then the 2-torsion subgroup of H*(BSO; Z) is additively isomorp*
*hic to the graded -module
generated by
{GA | A running over all non-empty finiteNsubsets1of= N - {0}}
with relations generated by X
{2GA ; q2i+1GA-{i}} :
i2A
P t
Here, the element G{i1;:::;it}2 H*(BSO; Z) is of degree 2( j=1ij) + 1 and redu*
*ces mod 2 to the class
P t *
j=1w2ij+1w2i1. .w.2i(j-1)w2i(j+1).w.2.it2 H (BSO; Z=2) .
3. The mod 2 Bockstein spectral sequence for BSL(Fp)+
In this section, let us consider the space BSL(Fp)+ for any prime number p 5 *
*mod 8 . Its mod 2
cohomology is H*(BSL(Fp)+; Z=2) ~=Z=2[c2; c3; : :;:ck; : :]: Z=2(e2; e3; : :;:e*
*k; : :):with deg(ck) = 2k
and deg(ek) = 2k - 1 (see Theorem 1 of [Q]). The first differential of its mod *
*2 Bockstein spectral sequence
E*1(BSL(Fp)+) ~=Z=2[ck; k 2] Z=2(ek; k 2) =) (H*(BSL(Fp)+; Z)=torsion*
*) Z=2
is trivial since d1(ck) = Sq1(ck) = 0 and d1(ek) = Sq1(ek) = 0 for all k 2 acc*
*ording to Lemmas 3 and 4
of [Ar4]. Thus,
E*2(BSL(Fp)+) ~=E*1(BSL(Fp)+) :
In order to understand the higher differentials dr, let us recall the definitio*
*n of dr (see Section 1 of [Brd]).
If x 2 Enr(BSL(Fp)+) , then there is an element ex2 Hn(BSL(Fp)+; Z=2r) such tha*
*t the homomorphism
r : Hn(BSL(Fp)+; Z=2r) ! Hn(BSL(Fp)+; Z=2) induced by the natural surjection Z=*
*2r!! Z=2 sends
3
exonto x . Let fir : Hn(BSL(Fp)+; Z=2r) ! Hn+1(BSL(Fp)+; Z) denote the Bockstei*
*n homomorphism
associated with the short exact sequence
r
0 -! Z .2-!Z -! Z=2r- ! 0
and red2: Hn+1(BSL(Fp)+; Z) ! Hn+1(BSL(Fp)+; Z=2) the reduction mod 2. Then, t*
*he differential
dr: Enr(BSL(Fp)+) ! En+1r(BSL(Fp)+) is defined by
dr(x) = red2(fir(ex)) :
Let us apply this to the case of the space BSL(Fp)+ .
Definition 3.1. For any integer r 2 , let Nr = {k 2 N | v2(k) = r - 2} , where*
* v2(-) is the 2-adic
valuation (in other words, Nr= {k = 2r-1i + 2r-2| i 0} ).
Remark 3.2. For any prime p 5 mod 8 and any integer r 2 , Nr= {k 2 N | v2(pk-*
* 1) = r} . In order
to check this, it is sufficient to show that v2(pk - 1) = v2(k) + 2 for any pos*
*itive integer k . Let us write
Xk k
p = 4m + 1 with m odd. Then pk- 1 = 4tmt. For t 2 , one has
t=1 t
i k j ik(k - 1) . .(.k -jt + 1)
v2 t 4tmt = 2t + v2 _________________t! 2t + v2(k) - v2(t!) v2(k) + t + 1*
* v2(k) + 3 ;
since v2(t!) t - 1 . This implies that v2(pk- 1) = v2(4km) = v2(k) + 2 .
Lemma 3.3. Let p be any prime 5 mod 8 and r be any integer 2 . If k 2 Nr, the*
*n the class ek
belongs to E2k-1s(BSL(Fp)+) for all s r and dr(ek) = ck 2 E2kr(BSL(Fp)+) .
Proof. If k belongs to Nr, then r = v2(pk - 1) by Remark 3.2. Thus, according t*
*o Section 3 of [Q], ek
is the image of an element eek2 H2k-1(BSL(Fp)+; Z=2r) under the homomorphism r *
*and consequently,
ek 2 E2k-1r(BSL(Fp)+) . Moreover, it follows from Lemma 5 of [Q] that dr(ek) = *
*red2(fir(eek)) = ck. _|_|
We get the complete calculation of the mod 2 Bockstein spectral sequence for th*
*e space BSL(Fp)+ .
Theorem 3.4. For any prime p 5 mod 8 , the mod 2 Bockstein spectral sequence f*
*or BSL(Fp)+ satisfies:
(a) E*2(BSL(Fp)+) ~=E*1(BSL(Fp)+) ~=Z=2[ck; k 2] Z=2(ek; k 2) and d2(ek) = c*
*k whenever
k 2 N2= {odd positive integers} ,
(b) for any r 3 , E*r(BSL(Fp)+) ~=Z=2[ck; k 2 Ns fors r] Z=2(ek; k 2 Ns fors*
* r) and
dr(ek) = ck whenever k 2 Nr,
(c) E*1(BSL(Fp)+) = 0 .
By looking at the differential graded Z=2-algebras
Fs= Z=2[ck; 2 k 2 Ns] Z=2(ek; 2 k 2 Ns) ; ffis(ek) = ck; fors *
* 2 ;
one can write the Er-terms of the mod 2 Bockstein spectral sequence of BSL(Fp)+*
* as follows.
*
* N
Corollary 3.5. For any prime p 5 mod 8 and for any integer r 2 , E*r(BSL(Fp)+*
*) ~= sr Fs with
the differential dr= ffir on Fr and dr= 0 on Fs when s > r .
The knowledge of the mod 2 Bockstein spectral sequence determines the additive *
*structure of the 2-torsion
of the integral cohomology H*(BSL(Fp)+; Z) since the elements of the image of d*
*r detect the elements of
order 2r in H*(BSL(Fp)+; Z) . Let us start with the following observation.
4
Lemma 3.6. Consider a set N and the differential graded Z=2-algebra
DN = Z=2[xn; n 2 N] Z=2(yn; n 2 N) ;
where the differential is a derivation ffi given by ffi(yn) = xn . Then the ima*
*ge of ffi is the Z=2[xn; n 2 N]-
module generated by
X Y
{HA = xa yb| A running over all non-empty finiteNsubsets}of
a2A b2A-{a}
with the relations generated by
X
{ xaHA-{a}| A running over all non-empty finiteNsubsets}of:
a2A
Proof. Let us write P for the polynomial tensor factor Z=2[xn; n 2 N] of DN . *
*Since ffi2 = 0 , one has
ffi(P) = 0 and the factQthat ffi is a derivation shows that DN!! Im(ffi) is a *
*morphism of P -modules. The
elements of the form a2Aya, where A runs over all non-empty finite subsets of*
* N , generate DNQ as a
P -module. Therefore, the imagePof ffi is generated,Qas a P -module, by the ele*
*ments HA = ffi( a2Aya) . We
get obviously the relations a2AxaHA-{a}= ffi2( a2Aya) = 0 and there are no o*
*ther relations because
H*(DN ; ffi) ~=Z=2 . *
* _|_|
Let us deduce the following explicit description of the additive stru*
*cture of the 2-torsion of
H*(BSL(Fp)+; Z) . Observe in particular that there is no direct summand of orde*
*r 2 in H*(BSL(Fp)+; Z) .
Theorem 3.7. Let p be any prime 5 mod 8 and consider the graded Z-algebra
= Z[ak; k 2] Z(bk; k even 2) ;
where deg(ak) = 2k and deg(bk) = 2k - 1 .
As a graded abelian group, the 2-torsion subgroup of H*(BSL(Fp)+; Z) is additiv*
*ely isomorphic to the
graded -module generated by
{HA;r| r 2 ; A running over all non-empty finiteNsubsetsrof}
with relations generated by
8 2rH ;
>>> A;r
< akHA;r for all k 2 Ns with 2 s < r,
bkHA;r for all k 2 Ns with 2 s r,
>>>X
: akHA-{k};rfor all r 2, A Nr.
k2A
P t
The element H{k1;:::;kt};r2 H*(BSL(Fp)+; Z) is of degree 2( j=1kj) - t + 1 and*
* reduces mod 2 to the
P t
class j=1ckjek1. .e.k(j-1)ek(j+1).e.k.t2 H*(BSL(Fp)+; Z=2) .
Proof. Let us denote by Pr the polynomial tensor factor Z=2[ck; 2 k 2 Nr] of F*
*r for r 2 . According
to Corollary 3.5, the image of dr is
O
Im(dr) = Im(ffir) ( Fs)
s>r
N
and Lemma 3.6 implies that Im(dr) is the (Pr ( s>rFs))-module generated by the*
* HA;r's, where A
runs over all non-empty finite subsets of Nr, with the relations given by Lemma*
* 3.6. By definition of ,
the generators ak and bk of the Z-algebra are in one-to-one correspondence wit*
*h the classes ck and ek
respectively, which generate the Z=2-algebra
O
P2 ( Fs) ~=Z=2[ck; k 2] Z=2(ek; k even 2) :
s>2
5
The assertion then follows by gluing together the information on Im(dr) for r *
*2 . _|_|
Remark 3.8. The additive structure of H*(BSL(Fp)+; Z) has been already calculat*
*ed in [Hu], but in a
completely different way.
4. The mod 2 Bockstein spectral sequence for BSL(Z)+
Finally, let us investigate the mod 2 Bockstein spectral sequence
E*1(BSL(Z)+) ~=Z=2[wj; j 2] Z=2(u2k-1; k 2) =) (H*(BSL(Z)+; Z)=torsion*
*) Z=2
for the mod 2 cohomology of the space BSL(Z)+ . Since the induced homomorphism *
*h* : H*(BSO; Z=2) !
H*(BSL(Z)+; Z=2) sends the Stiefel-Whitney classes wj 2 Hj(BSO; Z=2) onto the c*
*orresponding classes
wj2 Hj(BSL(Z)+; Z=2) , we have again Sq1(w2i) = w2i+1, Sq1(w2i+1) = 0 for i 1 *
*and we know from
Lemma 12 of [AMNY] that Sq1(u2k-1) = 0 for k 2 . Therefore, we obtain the E2-t*
*erm as follows:
E*2(BSL(Z)+) ~=E*2(BSO) Z=2(u2k-1; k 2) ~=Z=2[w22i; i 1] Z=2(u2k-1; *
*k 2) :
Because of the naturality of the mod 2 Bockstein spectral sequence with respect*
* to h*, we may deduce from
Section 1 that all higher differentials dr are trivial on Z=2[w22i; i 1] , r *
*2 .
Lemma 4.1. For all positive integers r and i , one has dr(u4i+1) = 0 in E4i+2r(*
*BSL(Z)+) .
P
Proof. This is true if r = 1 . For r 2 , dr(u4i+1) is of the form dr(u4i+1) = *
* sw(s) u(s) , where w(s)
is a product of classes w22i( i 1 ) and u(s) a product of classes u2k-1( k 2 *
*). According to Proposition
7 of [AMNY], the classes u2k-1 ( k 2 ) are primitive cohomology classes, in ot*
*her words, *(u2k-1) =
u2k-1 1 + 1 u2k-1, where * is the coproduct H*(BSL(Z)+; Z=2) ! H*(BSL(Z)+ x BS*
*L(Z)+; Z=2)
provided by the H-space structure of BSL(Z)+ . In particular, it follows from t*
*he fact that E*r(BSL(Z)+)
is a differential Hopf algebra (see Proposition 4.7Pof [Brd]) that *(dr(u4i+1))*
* = dr(*(u4i+1)) and conse-
quently that dr(u4i+1) is primitive. However, for sw(s) u(s) to be primitive*
*, it is necessary to have
u(s) primitive, which is only possible if u(s) = 1 or u(s) = u2k-1for some k . *
*In both cases, the element
w(s) u(s) cannot lie in degree 4i + 2 since deg(w(s)) 0 mod 4 . Consequently,*
* the sum must be empty
and we get dr(u4i+1) = 0 . *
* _|_|
Now, let us consider any prime p 5 mod 8 and the homomorphism
f*p: H*(BSL(Fp)+; Z=2) ~=Z=2[ck; k 2] Z=2(ek; k 2) -! H*(BSL(Z)+; Z*
*=2)
induced by the reduction mod p . We shall replace the generators u4i-1(for i *
*1 ) of the exterior
subalgebra Z=2(u2k-1; k 2) of H*(BSL(Z)+; Z=2) by
2i-2X
"4i-1= f*p(e2i) = u4i-1+ w2ju4i-2j-12 H4i-1(BSL(Z)+; Z=2)
j=2
(see Theorem 13 of [AMNY]). Thus, the first two terms of the mod 2 Bockstein sp*
*ectral sequence of BSL(Z)+
can be expressed as
E*1(BSL(Z)+) ~=H*(BSL(Z)+; Z=2) ~=Z=2[wj; j 2] Z=2("4i-1; i 1) Z=2(u4i*
*+1; i 1) ;
E*2(BSL(Z)+) ~=Z=2[w22i; i 1] Z=2("4i-1; i 1) Z=2(u4i+1; i 1)*
* :
Let us consider again the sets of integers Nr= {k = 2r-1i + 2r-2| i 0} introdu*
*ced in Definition 3.1.
6
Lemma 4.2. For any r 3 , if k 2 Nr, then "2k-1 belongs to E*s(BSL(Z)+) for all*
* s r and
dr("2k-1) = w2k.
Proof. By naturality of the mod 2 Bockstein spectral sequence with respect to f*
**p, this follows from the
equality
dr(ek) = ck fork 2 Nr
given by Lemma 3.3, from the definition "2k-1= f*p(ek) (where k is even since k*
* 2 Nr with r 3 ) and
the formula f*p(ck) = w2k(see Lemma 1.4 of [Ar3]). *
* _|_|
This argument implies also the vanishing of d2, because w2k= 0 in E*2(BSL(Z)+) *
*when k belongs to
N2= {odd positive integers} . Let us summarize the information we obtain on E*r*
*(BSL(Z)+) .
Theorem 4.3. The mod 2 Bockstein spectral sequence for BSL(Z)+ satisfies:
(a) E*1(BSL(Z)+) ~=H*(BSL(Z)+; Z=2) ~=Z=2[wj; j 2] Z=2("4i-1; i 1) Z=2(u4i+*
*1; i 1) and
d1(w2i) = w2i+1, d1(w2i+1) = 0 , and d1 is trivial on all classes "4i-1and *
*u4i+1(i 1) .
(b) E*2(BSL(Z)+) ~=E*3(BSL(Z)+) ~=Z=2[w2k; k even 2] Z=2("2k-1; k even 2) Z=2*
*(u4i+1; i 1) .
(c) For any r 3 ,
E*r(BSL(Z)+) ~=Z=2[w2k; k 2 Ns fors r] Z=2("2k-1; k 2 Ns fors r) Z=2(*
*u4i+1; i 1)
and dr("2k-1) = w2kwhenever k 2 Nr.
(d) E*1(BSL(Z)+) ~=Z=2(u4i+1; i 1) .
The calculation of the differentials in that mod 2 Bockstein spectral sequence *
*enables us to split its E1-term
as a tensor product of differential graded Z=2-algebras:
O
E*1(BSL(Z)+) ~=D1 ( Ds) D1 ;
s3
where
D1= Z=2[w2i+1; i 1] Z=2(w2i; i 1) ; ffi1(w2i) = w2i+1;
Ds= Z=2[w2k; k 2 Ns] Z=2("2k-1; k 2 Ns) ; ffis("2k-1) = w2k; for*
*s 3 ;
D1 = Z=2(u4i+1; i 1) ; ffi1 = 0 :
The spectral sequence can then be described in the following simple way.
Corollary 4.4. The mod 2 Bockstein spectral sequence for BSL(Z)+ satisfies:
N
(a) E*1(BSL(Z)+) ~=D1 ( s3 Ds) D1 and the first differential is d1 = ffi1 on*
* D1 and d1 = 0 on
Ds when 3 s 1 . N
(b) E*2(BSL(Z)+) ~=E*3(BSL(Z)+) and for r 3 , E*r(BSL(Z)+) ~=( sr Ds)D1 with *
*the differential
dr= ffir on Dr and dr= 0 on Ds when r < s 1 .
(d) E*1(BSL(Z)+) ~=D1 .
Remark 4.5. Let us mention that the mod 2 Bockstein spectral sequence for the g*
*roup SL3(Z[1_2]) has
been recently computed by H.-W. Henn in Section 4.3 of [He].
The following interesting observations are immediate consequences of Theorem 4.*
*3 (b) and (d) and Corollary
4.4 (b) and (d).
7
Corollary 4.6.
(a) There is no cyclic direct summand of order 4 in H*(BSL(Z)+; Z) .
(b) The set of all non-trivial elements of Z=2(u4i+1; i 1) is in one-to-one co*
*rrespondence with an additive
basis of H*(BSL(Z)+; Z)=torsion.
By applying again Lemma 3.6, we get an explicit description of the additive str*
*ucture of the 2-torsion of the
integral cohomology H*(BSL(Z)+; Z) . In order to formulate the main result of t*
*his section, let us use again
the notation introduced in Remark 2.2 and Definition 3.1: N1= N-{0} and Nr= {k *
*= 2r-1i+2r-2| i 0}
for r 3 .
Theorem 4.7. Consider the graded Z-algebra
= Z[!k;1; k 2 N1] Z[!k;r; k 2 Nr withr 3] Z(z2k-1; k 2) ;
where deg(!k;1) = 2k + 1 , deg(!k;r) = 2k when r 3 and deg(z2k-1) = 2k - 1 .
As a graded abelian group, the 2-torsion subgroup of H*(BSL(Z)+; Z) is additive*
*ly isomorphic to the
graded -module generated by
{JA;r| r = 1 orr 3 ; A running over all non-empty finiteNsubsetsr*
*of}
with relations generated by
8 2rJ ;
>>> A;r
< !k;sJA;r for all s < r and all k 2 Ns, when r 3,
z2k-1JA;r for all k 2 Ns with 3 s r, when r 3,
>>>X
: !k;rJA-{k};rfor r = 1 and r 3, A Nr.
k2A
P t
The element J{k1;:::;kt};12 H*(BSL(Z)+; Z) is of degree 2( j=1kj) + 1 and*
* reduces mod 2 to
P t
the class j=1w2kj+1w2k1. .w.2k(j-1)w2k(j+1).w.2.kt2 H*(BSL(Z)+; Z=2) . For r*
* 3 , the element
P t
J{k1;:::;kt};r2 H*(BSL(Z)+; Z) is of degree 2( j=1kj) - t + 1 and reduces *
*mod 2 to the class
P t 2 * +
j=1wkj"2k1-1. .".2k(j-1)-1"2k(j+1)-1.".2.kt-12 H (BSL(Z) ; Z=2) .
Proof. For r = 1 or r 3 , let us call Pr the polynomial tensor factor of Dr. B*
*ecause of the splitting of
E*r(BSL(Z)+) given by Corollary 4.4, the image of dr is
O
Im(dr) = Im(ffir) ( Ds) D1
s>r
N
and Lemma 3.6 implies that Im(dr) is the (Pr ( s>rDs) D1 )-module generated b*
*y the JA;r's, where
A runs over all non-empty finite subsets of Nr, with the relations provided by *
*Lemma 3.6. Now, let us
denote by the graded Z-algebra = Z[!k;1; k 2 N1] Z[!k;r; k 2 Nr withr 3] Z*
*(z2k-1; k 2)
whose generators are in one-to-one correspondence with those of
O
P1 ( Ds) D1 ~=
s3 2
Z=2[w2k+1; k 1] Z=2[wk; k even 2] Z=2("2k-1; k even 2) Z=2(u2k-*
*1; k odd 3)
as follows: !k;1corresponds to w2k+1, !k;rto w2kwhen r 3 and k 2 Nr, z2k-1to *
*"2k-1when k
is even and to u2k-1 when k is odd. The assertion of the theorem then follows b*
*y gluing together the
information on the elements of order 2r in H*(BSL(Z)+; Z) given by the determin*
*ation of Im(dr) for
r 1 . *
* _|_|
Remark 4.8. The isomorphism established in Theorem 4.7 is an additive isomorphi*
*sm: for instance, for
3 s < r , k 2 Ns and A a non-empty finite subset of Nr, the product of the ele*
*ments of H*(BSL(Z)+; Z)
8
corresponding to J{k};sand JA;runder that isomorphism is non-trivial, even if t*
*he reduction mod 2 of
J{k};sis w2k, which is the generator of Ds corresponding to the generator !k;so*
*f (that product is
actuallyPanQelement of order 2s in H*(BSL(Z)+;PZ)Qwhich reduces mod 2 to the re*
*duction mod 2 of
( k2A!k;r i2A-{k}z2i-1) J{k};s, where ( k2A!k;r i2A-{k}z2i-1) 2 ). More gene*
*rally, the above
mod 2 Bockstein spectral sequence calculation provides the following multiplica*
*tive relations mod 2 between
the additive generators of the 2-torsion subgroup of H*(BSL(Z)+; Z) given by Th*
*eorem 4.7.
If 1 s < r , then it is obvious that
iX Y j
JA;rJB;s !k;r z2i-1 JB;s mod 2 :
k2A i2A-{k}
If r 3 and if red2denotes again the reduction mod 2, one has
iY j
red2(JA;rJB;r)=red2(JA;r) ffir "2j-1
i j2B
Y j
= ffir red2(JA;r) "2j-1
i j2B
X Y Y j
= ffir w2k "2i-1 "2j-1 :
k2A i2A-{k} j2B
The fact that the classes "2i-1are exterior implies that
i X Y j X i Y *
* j
red2(JA;rJB;r) = ffir w2k "2i-1 = w2kffir *
*"2i-1
k2A such thati2A-{k}[B k2A such that i2A-{k}[B
A-{k}\B=; A-{k}\B=;
and finally that X
JA;rJB;r !k;rJA-{k}[B;r mod 2 :
k2A such that
A-{k}\B=;
If r = 1 , the formula X
JA;1JB;1 !k;1JA-{k}[B;1 mod 2
k2A such that
A-{k}\B=;
does still hold but the classes "2i-1should be replaced by w2iin the argument.
Finally, the above computation helps us to understand, at the prime *
*2 , the homomorphisms
h* : H*(BSO; Z) ! H*(BSL(Z)+; Z) and f*p: H*(BSL(Fp); Z) ! H*(BSL(Z)+; Z) induc*
*ed by the
inclusion Z ,! R and by the reduction mod p , when p 5 mod 8 .
Theorem 4.9.
(a) The homomorphism h* : H*(BSO; Z) ! H*(BSL(Z)+; Z) is injective on the tor*
*sion classes of
H*(BSO; Z) .
(b) For every generator of infinite order p2k in H2k(BSO; Z) with k even, h*(p2*
*k) is an element of order
2r if k belongs to Nr, r 3 (up to odd torsion).
(c) For p 5 mod 8 , the image of any generator of any cyclic direct summand*
* of order 4 in
H*(BSL(Fp)+; Z) under the homomorphism f*p: H*(BSL(Fp)+; Z) ! H*(BSL(Z)+; Z*
*) has order
2 in H*(BSL(Z)+; Z) .
(d) For any r 3 , the homomorphism f*p(with p 5 mod 8 ) is injective on all c*
*yclic direct summands
of order 2r in H*(BSL(Fp)+; Z) .
9
Proof. Assertion (a) is obvious since the Stiefel-Whitney classes wj 2 Hj(BSO;*
* Z) correspond to the
elements wj2 Hj(BSL(Z)+; Z) via h*. The fact that the reduction mod 2 of p2k is*
* w2k2 H2k(BSO; Z=2) ,
by Proposition 2.1 (c), and that dr("2k-1) = w2kif k 2 Nr, according to Theorem*
* 4.3 (c), implies (b).
According to Theorem 3.7, the generators of the cyclic direct summands of order*
* 2r in H*(BSL(Fp)+; Z)
belong to the -module generated by the HA;r's, where A = {k1; : :;:kt} is a fin*
*ite subset of Nr and one
has:
iXt j Xt
f*p(red2(HA;r)) = f*p ckjek1. .e.k(j-1)ek(j+1).e.k.t= w2kj"2k1-1. .".2k(j*
*-1)-1"2k(j+1)-1.".2.kt-1
j=1 j=1
since f*p(ckj) = w2kjby Lemma 1.4 of [Ar3]. If r = 2 , N2 = {odd positive integ*
*ers} and Assertion (c)
follows from the equality
d1(wkj-1wkj"2k1-1. .".2k(j-1)-1"2k(j+1)-1.".2.kt-1) = w2kj"2k1-1. .".2k(j-1*
*)-1"2k(j+1)-1.".2.kt-1
when kj is odd. If r 3 , then Assertion (d) is a consequence of
iYt j Xt
dr "2kj-1 = w2kj"2k1-1. .".2k(j-1)-1"2k(j+1)-1.".2.kt-1
j=1 j=1
in E*r(BSL(Z)+) . *
* _|_|
5. Chern classes of integral representations of groups
For n 1 , let us call cn(SL(Z)) 2 H2n(SL(Z); Z) the n-th Chern class of the in*
*clusion oe : SL(Z) ,!
GL(C) , i.e., cn(SL(Z)) = oe*(_cn) , where _cndenotes the n-th universal Chern *
*class of degree 2n in
H*(BGL(C); Z) ~=Z[_c1; _c2; : :;:_cn; : :]:and oe* : H*(BGL(C); Z) ! H*(BSL(Z);*
* Z) ~=H*(SL(Z); Z) the
homomorphism induced by oe . The order of cn(SL(Z)) has been only determined up*
* to a factor 2.
Definition 5.1. For any positive even integer n , let En be the denominator of *
*Bn_n, where Bn is the n-th
Bernoulli number; for instance, B2= 1_6, B4= -_1_30, B6= 1_42, and E2= 12 , E4=*
* 120 , E6= 252 .
Proposition 5.2. The Chern classes cn(SL(Z)) are all torsion classes. If n is o*
*dd, then cn(SL(Z)) is of
order 2 in H2n(SL(Z); Z) . If n is even, then the order of cn(SL(Z)) in H2n(SL(*
*Z); Z) is equal to 2En
when n 2 mod 4 , and to En or 2En when n 0 mod 4 .
Proof. See [EM1], Section 5, [EM2], Main Theorem, and [Ar1], Einleitung and Kor*
*ollar 2.5. _|_|
The determination of the exact order of cn(SL(Z)) in the case where n 0 mod 4 *
*now follows from the
mod 2 Bockstein spectral sequence calculations presented in Theorem 4.3 and Cor*
*ollary 4.4.
Theorem 5.3. For any positive even integer n , the Chern class cn(SL(Z)) is a t*
*orsion class of order 2En
in H2n(SL(Z); Z) .
Proof. The odd-primary part of the order of cn(SL(Z)) is given by Proposition 5*
*.2. For all integers r 3 , we
know from Lemma 4.2 that dr("2n-1) = w2nwhen n 2 Nr. Therefore, the fact that r*
*ed2(cn(SL(Z))) = w2n
for all n 1 (see [MT], Part I, p.137, Theorem 5.11) implies that for n even, c*
*n(SL(Z)) is of order 2r
in Hn(SL(Z); Z) when n 2 Nr (r 3) . On the other hand, according to von Staudt*
*'s theorem (see [BS],
p.384, Theorem 4), 2t divides En if and only if 2t-1divides n . Thus, if n 2 Nr*
*= {n 2 N | v2(n) = r-2} ,
10
then the 2-primary part of En is 2r-1. Consequently, the 2-primary parts of the*
* order of cn(SL(Z)) and
of the integer 2En coincide for any even integer n 2 . *
* _|_|
Remark 5.4. The same result holds for the Chern classes cn(GL(Z)) of the genera*
*l linear group GL(Z)
since there is a homotopy equivalence BGL(Z)+ ' BSL(Z)+ x BZ=2 (see for instanc*
*e [Ar1], Lemma 1.2).
The knowledge of the order of the Chern classes of SL(Z) produces the following*
* result on the Chern classes
of the linear groups over the field of rationals Q .
Corollary 5.5. The Chern classes cn(SL(Q)) and cn(GL(Q)) are all torsion classe*
*s. If n is odd, they
are of order 2 . If n is even, the order of cn(SL(Q)) and of cn(GL(Q)) is equal*
* to 2En .
Proof. Since the order of cn(SL(Q)) and of cn(GL(Q)) is a positive multiple of *
*the order of cn(SL(Z)) , a
lower bound for it is given by Proposition 5.2 and Theorem 5.3. The assertion t*
*hen follows from Theorem
11 of [Ar2]. *
* _|_|
For any complex representation ae : G ! GL(C) of any discrete group G , the Che*
*rn classes of ae are
cn(ae) = ae*(_cn) 2 H2n(G; Z) , where ae* is the induced homomorphism H2n(BGL(C*
*); Z) ! H2n(BG; Z) ~=
H2n(G; Z) . Of course, the above calculations produce the following consequence*
* for any integral represen-
tation ae : G ! GL(Z) ,! GL(C) or any rational representation ae : G ! GL(Q) ,!*
* GL(C) of any discrete
group G .
Corollary 5.6. The best upper bound for the order of the n-th Chern class cn(ae*
*) of any integral or rational
representation ae of any discrete group G is equal to 2 when n is odd and to 2E*
*n when n is even.
Proof. Since ae is an integral representation, respectively a rational represen*
*tation, cn(ae) is the image of
cn(GL(R)) under the induced homomorphism H2n(GL(R); Z) ! H2n(G; Z) , where R = *
*Z , respectively
R = Q . Consequently, the order of cn(ae) divides the order of cn(GL(R)) which*
* has been obtained in
Proposition 5.2, Theorem 5.3, Remark 5.4 and Corollary 5.5. It turns out that t*
*he order of cn(SL(Z)) is
the best possible upper bound since one can choose G = SL(Z) and ae the inclusi*
*on into GL(C) . _|_|
Remark 5.7. The assertion of Corollary 5.6 is of particular interest because th*
*e best upper bound for the
order of the n-th Chern class of any integral representation of any finite grou*
*p is smaller, i.e., only equal to
En (see [EM1], Theorem 4.12, and [ThC], p. 89).
Acknowledgements. The third and fourth authors would like to thank the Swiss Na*
*tional Science Foundation
for financial support when they visited the University of Lausanne and the JAMI*
* of the Johns Hopkins
University for its hospitality during the last stage of writing the manuscript.
11
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