Publicacions
Ma emá iques,
Vol
34
(1990),
151-174
.
COHOMOLOGY
OF
LIE
GROUPS
MADE
DISCRETE
A
bs ac
PERE
PASCUAL
GAINZA
We
gi e
a
su ey
o
he
wo k
o
Milno , F iedlande ,
Mislin,
Suslin,
and
o he
au ho s
on
he F iedlande -Milno
conjec u e
on
he
homology
o
Lie
g oups
made
disc e e
and
i s
ela ion
o
he
algeb aic
K- heo y
o
ields
.
Le
G
be
a
Lie
g oup
and
le
G6
deno e
he
same
g oup
wi h
he
disc e e
opology
.
The
na u al
homomo phism
G
6
-+
G
induces
a
con inuous
map
be ween
classi ying
spaces
E
.F iedlande
and
J
.
Milno
ha e
conjec u ed
ha
l
induces
isomo phisms
o
homology
and
cohomology
wi h
ini e
coe icien s
.
The
homology
o
BG
6
is
he
Eilenbe g-McLane
homology
o
he
g oup
G
6
,
ha d
o
compu e,
and
one
o
he
in e es s o
he conjec u e
is
ha
i
pe mi s
he
compu a ion
o
hese
g oups
wi h
ini e
coe icien s
h ough
he
compu a ion
(much
be e
unde s ood)
o
he
homology
and cohomology
o
BG
.
The
Eilenbe g-McLane
homology
g oups
o
a
opological
g oup
G
a e
o
in e es
in
a
a ie y o
con ex s
such
as
he
heo y
o
olia ions
[3],
[11],
he
scisso s
cong uence
[6]
and
algeb aic
K- heo y
[26]
.
Fo
example,
he
Hae lige
classi ying
space
o
he heo y
o
olia ions
is
closely
ela ed
o
he
g oup
o
homeomo phisms
o
a
opological
mani old,
o
which
one can
p o e
analogous
esul s
o
he
F iedlande -Milno
conjec u e
wi h
en i e
coe Ficien s,
c
.
[20],
[32]
.
These
no es
a e
an
exposi ion
o
he
con ex
o
he
conjec u e,
some
known
esul s
mainly due
o
Milno ,
F iedlande , Mislin
and
Suslin,
and
i s
applica ion
o
he
s udy
o
he
g oups
K
;(C),

i
>
0
.
We
hank
F
.
Guillén,
V
.
Na a o
Azna and
A
.
Roig
o
many
help ul
con-
e sa ions
abou
his
heme
.
*Pa ially
suppo ed
by
CICYT
n
.
0348-86
:
BG
6
----->
BG
.
152

P
.PASCUAL
GAINZA
wi h
ibe
G,
and
a
igh
G-ac ion
sa is ying
1
.
C1assi ying
spaces
(1
.1)

Le
G
be
a
opological
g oup
.
Remembe
ha
a
p incipal
G-bundle
consis s o
a
con inuous
map
p
:
E
-'
B
E
x
G
such
ha
he e
is
an open
co e ing
{Uj
o
B
and
homeomo phisms
cp
a
:
U
a
x
G
-
)
p-1(U
.)
psP
«
=
p
¡u
.
,
P
.
(b,
9)
-
coa
(b,
e)g
.
The
no ion
o
equi alence
o
p incipal
G-bundles
is
he
ob ious
one
.
De lni ion
.
A
classi ying
space
o
G
is
a
opological
space
BG
wi h
a
p incipal
G-bundle
EG

)
BG
such
ha
EG
is
con ac ible
and
is
uni e sal
in
he
ollowing
sense
:
i
p
E
-+
B
is
any
p incipal
G-bundle
hen
he e
is
a
con inuous
map
B-~BG
such
ha
p
is
.
he
ibe
p oduc
E
------------
EG
B

;
BG
.
The
exis ence
o
classi ying
spaces
may
be
p o ed
by
B own's
ep esen a ion
heo em
( o
CW-complexes
[38]
(11
.33))
o
by
gi ing
a
speci ic
cons uc ion
.
In
he
ollowing
pa ag aphs
we
p esen
Segal's
cons uc ion,
[30],
o
which
we
ha e
o
assume
ha
G
is
an
ANR,
as
e i ied
by
Lie
g oups
.
This
cons uc ion
co esponds o
he
nonhomogeneous
no malized
ba
cons uc ion
o
disc e o
g oups
.
The
analogous
o
nonno malized
ba
cons uc ions
would
be
he
Milno
classi ying
space
(c
.
.
[15])
and
ha
o
Dold-Lasho
(c
.
.
[4]),
see
[33]
.
(1
.2)

Le
C
be
a
opological
ca ego y
(Le
.
Ob
C
and
Mo
C
a e
opological
spaces
and
he
s uc u al
maps
a e
con inuous),
Segal
de ines he
simplicial
opological
space
NC,
called
he
ne e
o
C
,
whose
n-simplexes
a e he
elemen s
COHOMOLOGY
OF
LIE
GROUPS
MADE
DISCRETE

153
( 1,
. .
.,
.)
o
(Mo
C)
"'
o
which
i is
de ined
he
composi ions
;+1
o
;
,
wi h
bounda y
and
degene acy
maps
as
usual
.
De lni ion
.

BC
:=11
NC
¡l .
(Obse e
ha
we
ake
he
hick
geome ic
ealisa ion
iden i ying
only
bound-
a y
maps
[31])
.
To
a
opological
g oup
G
we
can
associa e
he
ca ego ies
:
-
G
:
i
has
only
one
objec ,
e
E
G,
and
Mo G
=
G,
-
G
:ObC=GandMo C=GxG,
so
we
ob ain he
classi ying
spaces
BG
and
BG
.

Obse e
ha _

i
91,
92
a e
objec s
o
G
he e
is
one and
_only
one
mo phism
om
g1
o
92
ha
is
an
isomo phism,
so
i
ollows
ha
G
is
equi alen
o
he
i ial
ca ego y
wi h
one
objec
and one
mo phism
and om
he
gene al
heo y o
classi ying
spaces
we
can
deduce
ha
B
G
is
con ac ible
.
The
unc o
G
--+
G
sending
he
mo phism
(9
1
,9
2
)
o
_
o
he
mo phism
9,
1
92
o
G,
gi es
ise
o
a
con inuous
map
B
G
,
BG
.
Obse e
also
ha
BG
is
a
G- ee
space
and
ha
B
GIG
=
BG
.
We
w i e
EG
=
B
G
.
P oposi ion
.
I
(G,
e)
is
an
ANR,
hen
EG
--+
BG
is
a
p incipal
G-bundle
.
(1
.3)

We
will
gi e
now
an
app oxima ion
o
he
uni e sali y
o
his
p inci-
pal
G-bundle
.
Remembe
ha
o a
p incipal
G-bundle
p
:
E
-3
B
he e
a e
associa ed
ansi ion
unc ions
gap
:u
.nup
%
G,
sa is ying
he
usual
cocicle
condi ion
de ined
in
he
ollowing
way
:
The
maps
~p
=
w«
1
°wp
:(u
.nu#)xC
(Ua
nup)xC
a e
compa ible
wi h
he
p ojec ion
p,
so
hey
de ine
maps
h~p
:(U,,nup)xC
-
:
G
and
as
he
W
a
a e
G-equi a ian ,
we
ha e
W
.(b,
h
.,i(b,
g))
=
Sp#(b, g)
=
(Po(b,
e)g
=
`o
.(b,
h«p(b,
e))g
=,p
.
(b,
hap(b,
e)g)
15
4

P
.PASCUAL
GAINZA
Le
.
so
we
can
de ine
Obse e
ha
h
.p(b,
g)
=
h
.p(b,
e)g

,
9
.p(b)
=
h«p(b,e)
E
G
I is
well
known
ha
he
equi alence
classes
o
p incipal
G-bundles
a e de e -
mined
by
he
ansi ion
unc ions
(c
.
by example
[381
(11
.16))
.
The
in o ma-
ion
gi en
by
hese unc ions
may
be
in e p e ed
in
he
con ex
o
classi ying
spaces
in
he
ollowing
way
:
Le
p
:
E
-4
X
be a
p incipal
G-bundle
and
le
U
=
(U,,)
be an open
co e
o
X
wi h
associa ed
ansi ion
unc ions
g,,p
.
To
he
couple
(X,
L
)
we
associa e
he
ollowing
opological
ca ego y
XU
:
he
objec s
a e he
pai s
(x,
U,,),
wi h
x
E
U
a
,
and
he e
is
a
unique
mo phism
(x,
U,,)
-~
(y,
Up)
i
x
=
y,
i
.e
. ,
ObXU=LjU
.
. ,
Mo
X
U=
Ll
Ua
n
Up
(a,p)
(NXu)n
=

Ll

Uao
n
. .
.
n
U
Q

(C o
. . .
a
.)
Bg
:
BXU
;
BG
he
sum
being
o e
all
(n
+
1)-uples
wi h
Uaon
. .
.nU,,,
Lemma
.
The
ansi ion
unc ions
de ine
a con inuous unc o
9
.
XU

G
hence
he e
is
a con inuous
map
Simila ly,
i
V=
{V,
=
p
-1
(U,,)}
,
he
i ializa ions
(pa)
de ine
a
unc o
E
V
:
G
,
and
we
ob ain
a
con inuous
map
BEV
EG
.
Finally,
he
p ojec ion
p
gi es
ise
o
a
unc o
E
---+
XU
so
ha
he
ollowing
diag am
commu es
The
inclusion
U
->
X
de ines a
mo phism
be ween
simplicial
spaces
NX
U
X,
and
we
ob ain
he
commu a i e
diag am
is
a
homo opy
equi alence
.
is
an
isomo phism
.
The
esul s
abo e
gi e
:
COHOMOLOGY
OF
LIE
GROUPS
MADE
DISCRETE

155
BE
-
;
EG
BE
)
E
BX
U
EU
--,
X
.
P oposi ion
.
(c
.[30,
4
.1])

I
U
is
a
nume able
co e ing,
hen
he
na u al
map
BXU
i
X
Fo
a
pa acompac
space
each
co e ing
is
nume able
so
we
deduce
Co olla y
.
EG
->
BG
is
a
classi ying
space
o
G
o
pa acompac
spaces
.
I
X
is
no
pa acompac
we
s ill
ha e
a
ela ion
be ween
X
and
Xu
P oposi ion
.
(c
.
[5,
p
.
85])
The
induced
mo phism
H*(X,
Z)
)
H*(BX
U
,
Z)
(1
.4)

I
we
deno e
by
KG(X)
he
se
o
equi alence
classes o
p incipal
G-bundles wi h
base
X
,
emembe
ha
a
cha ac e is ic
class
is
a
na u al
ans o ma ion
o
unc o s
c
:
KG(-)

>
H*(-,
Z)
.

15
6

P
.PASCUAL
GAINZA
P oposi ion
.
The
map
ha
sends
a
cha ac e is ic
class
c
o
he
elemen
c(EG)
o
H*(BG,
Z)
is
a
bijec i e
co espondence
.
In
ac ,
i
c
is
a
cha ac e is ic
class
and
E
-
i
X
is
a
p incipal
G-bundle
we
ha e
(no a ions
as (1
.3))
induces
an
injec ion
such
ha
he
sequence
e*
(c(E))
=
g*(c(EG)),
bu
¿s
u
is
an
isomo phism,
so
c(E)
and
his
iden i y
de e mine
c(EG)
com-
ple ely
.
Recip ocally,
i
co
E
H*(BG,
Z)
and
E
-->
X
is
a
p incipal
G-bundle,
we
can
de ine
he
cha ac e is ic
class
c(E)
by
,su*
(c(E»
=
g*(co)
.
The
p oo
ha
he
class
so
de ined
is
independen
o
he
i ializa ions
and
he
co e ing
Umay
be
seen
in
[51,
pp
.86-88
.
(1
.5)

Rema k
ha
o
some
g oups
G
i is
possible
o
compu e
he
coho-
mology
o
BG
by
Bo el's
heo em
on
he spec al
sequence
o
he
ib a ion
EG
--+
BG
(c
.
[38],
15
.62)
.
2
.
The
Weil
homomo phism
:
The
case
o
a ional
coe icien s
In
his
and
he
ollowing
§
we
deno e
by
G
a
Lie
g oup
.
In
his
pa ag aph
we
show
ha
he
isomo phsim
conjec u e
has
no
sende
i
we
conside
a ional
coe icien s
.
(2
.1)
Le
M
be
a
di e en iable
mani old,
p
:
E
-->
M
a
di e en iable
p incipal
G-bundle
and
x
E
E
.
The
map
G->E
x
:
g
=
TeG
-
.
T
.,
E,
TxE
dp*+
T
p
1
x
1M
->
0
is
exac
.
De lni ion
.

A
connec ion
on
E
is
a
g- alued
1- o m
8
E
A
1
(E,g)
such
ha
:
i)
B
zo
,
=ld,
ii)
R*B
=
Ad(g
-i
)
o
0
,
whe e
R
g
:
E-iE
deno es
he
ac ion
o g
on
E
.
COHOMOLOGY
OF
LIE
GROUPS
MADE
DISCRETE

157
By
example,
in
he
i ial
bundle
M
x
G
-->
M
one
has
he
Mau e -Ca an
connec ion
de ined
by
B
(x,g)
=
(Lg'i
0
7 2)
*
,
whe e
7 2
:
M
x
G
->
M
is
he
p ojec ion,
and
L
g
-1
:
G
-->
G
is
he
asla ion
by
he
le
de ined
by
he
in e se
o
g
.
Using
his
connec ion
and an
a gumen
o
pa i ions
o
uni y
o e
i iallizing
open
se s
one
can
p o e
easily
ha
any
(di e en iable)
p incipal
G-bundle
o e
a
pa acompac
mani old
has
a
connec ion
.
The
cu a u e
2
o a
connec ion
B
is
he
g- alued
2- o m
de ined
by
52( 1,
2)
=
d8(h 1,
h 2),
whe e
h
is
he ho izon al
componen
o
.
52
is
in a ian
by
he
G-ac ion
.
(2 .2)

G
ac s
on
he
symme ic
algeb a
o
g*
,
S'(g*)
by
(9P)( 1,-
. .
k)=P(Ad(g-1) l,-
.
.,Ad(g-1) k)
.
Le
I
k
(G)
be
he
G-in a ian
subse
o
S
k
(g*)
.
The
p oduc
on
S'(g*)
in-
duces
an
algeb a
s uc u e
on
I*(G)
.
Le
9 be
a
connec ion
on
E
wi h
cu a u e
o m
52
,
hen
S2k
E
A
2k
(E,g®k),
and
as
2
is
in a ian
and
ho izon al,
P(S2k)
is
an
in a ian
ho izon al
2k- o m
so
he e
is
a
2k- o m
on
M
which
maps
o
P(S2k)
,
we
will
deno e
i
by
he
same symbol
.
We
ha e
he
classical
esul
(c
.
[18],
cap
XII)
:
Theo em
(Weil
homomo phism)
.

P(52
k
)
E
A
2k
(M)
is
a
closed
o m
.
Le WE(P)
be
he
co esponding
de
Rham
cohomology
class
.
Then
:
i)
WE(P)
is
independen
o
¡he
connec ion
,
i
only
depends on
he
isomo -
phism
class
o
E
.
ii)
wE
:
I*(G)
--->
Hd
R
(M)
is
an
algeb a
homomo phism
.
iii)
i
:
N
->
M
is
a
di
e en iable
map
be ween mani olds
hen
w *E
=
*-E
.
(2
.3)

Al hough
BG
is
no
in
gene al
a
di e en iable
mani old
i is
possible
o de ine
a
Weil
homomo phism
I*
(G)

>
H*(BG)
.
Fo
ha
one
obse es
ha
NG
is
a
simplicial di e en iable
mani old
and
ha
one
can
ex end
he
no ions
abo e
o
his
mo e
gene al
con ex
:
a
p incipal
G-bundle
o e
asimplicial
mani old
M
is
a
simplicial
G-mani old
E
and a
mo phism
E
-->
M
such
ha
i
is
a
p incipal
G-bundle
in
each
deg ee,
E
n
->
M,
.
A
connec ion
on
E
is
a
connec ion
on
O
n
x
&
,
o
all
n
,
compa ible
15
8

P
.PASCUAL
GAINZA
wi h
he
mo phsm
o
0
and
E
(sea
[5],
6
.2,
o
mo e
de ails)
and
one
de ines
he
cu a u a
o m
9
as in (2
.2)
.
I
P
E
I*(G),
P(Q
k )
is
a
closed
2k- o m
on
M
and
de ines
a
cohomology
class
e i ying
i)-iii)
o
he
abo e
heo em
.
On
he
ibe
bundle
NG
->
NG
we
can
de ine
he
ollowing
connec ion
:
Le
B
o
be
he
Mau e -Ca an
connec ion
o
he
ibe
bundle
G
-+
p
,
and
q
;
:
A,,
x
NG
n
-->
G
he
i- h
p ojec ion
o
G
n+1
in o
G
;
we
de ine
he
canonical
connec ion
by
Now
we
ha e
:
WE(P)
E
H
2k
(II
M
II,
R),
B
=
o
ll
o
+
- -
-
+
nOn
.
Theo em
.

(c
.[5,
6
.13])

The e
is
a
canonical
homomo phism
w
:
I*(G)
%
H*(BG,R)
such
hai
i
P
E
I*(G),
w(P)
is
he
2k- o m
on
NG
ep esen ed
by
P(SZ
k
),
whe e
52 is
he
cu a u a
o
¡he
canonical
connec ion
.

w
sa is ies
:
i)
w(P)(E)
=
WE(P),
whe e
wE
is
he
mo phism
de ined
in
(2
.2)
and
w(P)(E)
is
he cha ac e is ic
class
co esponding
o
w(P)
by (1
.4)
.
ii)
w
:
I*(G)
---,
H*(BG,R)
is
an
algeb a
homomo phism
.
iii)
i
H
--->
G
is
a
mo phism
o Lie
g oups,
¡he
diag am
is
cominu a i e
.
I*(G)
>
I*(H)
H*(BG)
H*(BH)
Rema k
.

See
[2]
o
ano he
p esen a ion
o
he
Weil
homomo phism
in
his
gene al
con ex
.
(2
.4)

Remembe
ha
a
connec ion
Bon
a
p incipal
G-bundle
E
--->
M
is
said
o
be
la
i
i s
cu a u a
o m
anishes
.
A
p incipal
G-bundle
admi ing
a
la
connec ion
will
be
callad
la
.
Thé
la
bundles
a e
cha ac e ized
in
e mS
o
ansi ion
unc ions
by
he
ollowing
esul
P oposi ion
.

(c
.[5,
3
.22])

A
p incipal
G-bundle
E
--)
M
is
lai
i
and
only
i
he e
is
a
i iallizing
open
co e U
=
{U
a
}
o
M
such
ha ¡he ansi ion
unc ions
gap
:
U
a
nUp
->
G
a e
cons an
.
Co olla y
.
A
p incipal
G-bundle
is
la
i
and
only
i
admi s a
G
6
- educ ion
.
In
e ms
o
he
Weil
homomo phism
and
using
he
commu a i e
diag am
(&),
we
deduce
COHOMOLOGY
OF
LIE
GROUPS
MADE
DISCRETE

159
Co olla y
.
The
composi ion
I*(G)
_
w
_+
H*(BG,R)
-->
H*(BG6,R)
is
iden ically
ze o
.
(2
.5)

I
G
is
a
compac
Lie
g oup
we
ha e
he
ollowing
esul
o
Ca an
Theo em
.
(c
.
[5,
8
.1])
Le
G
be
a
compac
Líe
g oup,
hen
he
Weil ho-
momo phism
w
:
I*(G)

H*(BG,
R)
is
an
isomo phism
.
Hence
om
(2
.4)
i
ollows
Co olla y
.
Le
G
be
a
compac
Lie g oup,
hen
he
mo phism
H*(BG,
Q)

H*(BG
6
,
Q)
is
ze o
.
(2
.6)

I
G
is
a complex
Lie
g oup,
he e
is
a
Che n-Weil
homomo phism
I¿(G)
--+
H*(BG,
C)
simila
o
he
Weil
homomo phism
.
I
G
is
semisimple wi h
ini ely
many
connec ed
componen s
hen
his
homomo phism
is
bijec i e
as
one
can
p o e
by
using
(2
.5)
applied
o
a
maximal
compac
subg oup
K
o
G
(c
.
.
[22],
lemma
12),
so
we
ha e
:
P oposi ion
.
I
G
is
a
complex
semisimple
Líe g oup
wi h
ini ely
many
connec ed
componen s,
hen
¡he
mo phism
as
ze o
.
H*(BG,
Q)
-->
H*(BG
6
,
Q)
In
he
appendix
o
[22],
he
eade
can
see
some
o he cases
whe e
7 7 *
wi h
a ionnal
coe icien s
is
ze o
and
p o iding
e idence
o
he
impo ance
o
he
ini e
coe icien s
in
he
F iedlande -NIilno
conjec u e
.
16
6

P
.PASCUAL
GAINZA
ii)
I
k
is
an
inini e
ield,
¡he
mo phisms
Hi
(GL
n
(k),
Z)
-,
Hi
(GL(k),
Z)
a e
isomo phisms
o
0
<
i
<
n
.
Mo eo e ,
he
mo phisms
H
n
(GLn
(k),
Z)
--=,
H
n
(GLn
+
l
(k),
Z)

...
a e
isomo phisms
and
¡he
homology
p oduc
k*
®
.
. .
®
k*
=
H
i
(GLk(k),
Z)
®n
%
Hn
(GLn(F)
,
Z
)
induces
an
isomo phisms
KM(k)

Hn(GL(k))
l
Hn-i
(GL
n
(k)),
whe e
KM
deno es
Milno
K- heo y
.
In
he
ollowing
pa ag aph
we
will
only
use
ii)
o
k
=
R,
C
whose
p oo
is
much
mo e
elemen a y
as
Suslin
ema ks
.
(5
.2)

Suslin
ealizes
Bg
in
he
ollowing
way
:
ix
a
le
in a ian
iemannian
me ic
on
G
and
le
G
e
be
he
ball o
adius
e
cen e ed
a e
E
G
.
Le
BG
E
be
he
geome ic
ealisa ion o
he
simplicial
se
whose
p-simplexes
a e he
p-uples
[gl,
.
.
. ,
gp]
such
ha
GE
n
gl
G
E
n
.. .
n
g
,
. . .
gp
G
E
:~
0
wi h
he
usual
ace
and
degene acy
ope a o s
.
Then
BG
E
-
-
;
BGá

)
BG
H
n
(GL(k),
Z)
is
a
homo opy
ib a ion
(c
.
[36,
4
.1])
.
Wi h
his
p esen a ion
o
Bg
Suslin
p o es
:
Theo em
.

([36,
4
.3])
.
Le k
='R
o
C
.
Fo
e
suicienily
small,
he
inclusion
BGL
n
(k)
--%
BGLn(k)6
--~
BGL(k)á
induces
¡he
ze o
mo phism
in
H
*
(-,
Z/m)
.
Idea o
p oo
.

By
he
igidi y
heo em
H
i
(GL
(Oi,
mi)
,
Z/m)
=
0
whe e
m
z
is
he
maximal
ideal o
O
x
.

Xn~i

)
(GLn)x'
-
2

,
GLn
de ine
ma ices
a
j
E
GL
n
(Ch
;)
.
Le
u
n
,¡
be
he
chain
such
ha
COHOMOLOGY
OF
LIE
GROUPS
MADE
DISCRETE

16
7
Conside
he
simplicial
scheme
BGL
n
/k
,
and
le
Xn
;
be
he
hénselianiza ion
o
i s
i- h
componen
(BGL,)i
=
(GL
n
)
Xi
in
he
uni
sec ion
and
Cn
;
be
he
co esponding
coo dina e
ing
.
Fo
ixed
n
,
he
schemes
Xn
;
make
a
simplicial
scheme
and
he
maps
[al,
.
. .
.
a¡]
E
C
;
(GLn
(an,i)
,
Z
/
m
)

,
whe e
C
*
(-,
Z/m)
is
he
s anda d
complex
.
By
induc ion
on
i
and
(*)
o
Ch
.i,
we
may
p o e
he
exis en e
o
chains
Cn,i
E
Ci-~1
(GL
(On
;)
,
Z/m)
The
ing
o
con inuous
unc ions
0n'

~°o
GL
n
(k)`
is
henselian
hence
he e
is
a
canonical
map
oh,
;1
O
;°
;
.
Le
ces°¡
be
he
images
co esponding
o c
n,
; .
The
g oup
GL
(O
;°!")
may
be
iden i ied
o
he
g oup
o
ge ms
o
con inuous
maps
(GL
n(k)) x
'
->
GL(k),
hence,
o ixed
N
>
0
so
ha
he
c
c°
;
a e
de ined
in
(GL(k)E)
.x',

i <_
N,
and
o
su icien ly
small
e
,
we
will
ha e
mo phisms
Ci
(BGL
n (k)
E
,
Z/m)
-~
Ci+i
(GL(k),
Z/m)
ha
de ine
a
homo opy
o
ze o
by
he cons uc ion
o
he
cn,i
.
Co olla y
.
The
map
BGL(k)
6
-
BGL(k)
induces
isomo phism
in
homology
wi h
ini e
coe
cien s
.
Idea o
p oo
.

As
we
ha e
he
ib a ion
BSL(k)
--->
BGL(k)
---+
Bk*
and
he
one
co esponding
o
GL(k)
6
,
i
su ices
o
see
ha
BSL(k)
6
-->
BSL(k)
168

P
.PASCUAL
GAINZA
is
an
H,(-,
Z/p)-isomo phism
and
so
i
su ices
o
p o e
ha
H

(BSL(k)
E
,
Z/p)
=
o
.
By
he
Se e
spec al
sequence
o
he
ib a ion
BSL
n
(k)
E
BSL
n
(k)
6
BSL,,(k)
and
using
he
ac
ha
H
*
(BSL
n
(k)
6 ,
Z/p)
%
H,
(BSL
n
(k),
Z/p)
is
su jec i e
(c
.
(3
.5)),
we
can
deduce
ha
i i
o
is
he
leas
in ege
wi h
Hio
(BSL

,(k)E,
Z/p)
7É
6
hen
H¡
.
(BSL
.(k)
E
,
Z/p)
%
Hi
o
(BSL
n
(k)
6
,
Z/p)
is
injec i e
.
Bu
i
i
<
(n
-
1)/2,
we
ha e
by
he
s abili y
heo em,
hence
io
>
(n
-
1)/2
by
he
heo em
abo e
.
Now
he
esul
ollows
by
passing
o
he
limi
.
Using
now
he
second
s a men
o
he
s abili y
heo em we
can
deduce
he
isomo phism
conjec u e
o
he
g oups
GL
n
(k)
in
deg ees
<
n
Co olla y
.
The
na u al
map
induces
isomo phisms
H
;
(BSL
n
(k)
6
,
Z/m)
=
H
;
(BSL(k)
6
,
Z/p)
BGL

(k)
6

)
BGLn(k)
H
i
(BGL
n
(k)
6 ,
Z/p)

-

Hi
(BGL
n
(k),
Zlp)

i
<
n
.
Making
use
o
an
adequa e
e sion
o
he
abo e
echniques
Suslin
and
Ju -
j
ako
p o e
Theo em
.

([37,
§3])
.
Le
H
be
¡he
qua e nion
algeb a
;
hen
¡he
na u al
mo phism
BGL(H)
6
)
BGL(H)
induces
an
isomo phism
in
homology
wi h
ini e
coe cien s
.
(5
.3)

In
[16]
Ja dine
gi es
ano he
p oo
o
he
isomo phism
conjec u e
o
BGL(C)
.
This
new
p oo ,
o
an
algeb aic
la o ,
also
makes
essen ial
use
o
he
igidi y
heo em, al hough
in
his
case
he
has no
used
he
s abili y
esul s
.
is
an
isomo phism
.
COHOMOLOGY
OF
LIE
GROUPS
MADE
DISCRETE

169
We
scke ch
Ja dine's
idea
:
Ja dine
conside s
BGL
nas
a
shea
o
simpli-
cial
se s
on
he
ca ego y
o
smoo h
C-schemes
and
de ines
he
shea
BGL
=
limBGL
n
.
The
global
sec ion
unc o
*
has
a
le
adjoin ,
hence
he e
is
an
adjunc ion
map
e,
:
*BGL(C)
=
*
*
BGL
-~
BGL
co esponding
o
17
in
he
p e ious
no a ions,
and
passing
o
homology
he e
is
a
map
e *
:
H
*
( *BGL(C),
Z/p)
--~
H
*
(BGL,
Z/p)
The
ibe
o
e
*
in
a
a ionnal poin
x
o
a
smoo h
a ie y
X
is
he
map
H
*
(BGL(C),
Z/p)

H
*
(BGL(Oh),
Z/P)

,
and
his
map
is
an
isomo phism
by
he
igidi y
heo em,
hence
e
*
and
e*
a e
shea
isomo phisms,
and
so
we
ob ain
g oup
isomo phisms
e*
:
H*
(BGLC,
Z/p)
--~
H*
( *BGL(C),
Z/p)
Finally
he
p o es
ha
H*
( *BGL(C),
Z/p)
is
isomo phic
o
H*
(BGL(C),
Z/p),
concluding
he
p oo
.
One
o
he
objec i es
o
Ja dines
pape
is
o
de elop
he
me hods
o
sim-
plicial
shea es
on
a
G o hendieck
opos
o
make
sense
o
he
p og am
jus
ske ched
.
As
hese
a e
gene al
me hods he
can
apply
hem
o
he
si ua ion
desc ibed
in
§4,
so
i
k
is
an
algeb aically
closed
ield
and p
is
a
p ime
numbe
di e en
om
he
cha ac e is ic
o
k,
he
ob ains
:
Theo em
.
The
map
Hé
(BGLk,
Z/p)

+
H*
(BGL(k),
Z/p)
In ac
Ja dine's
p oo
pe mi s
o
asse
ha
he
GIC
is
ue
o
an
algeb aic
g oup
G
o e
k
i
o
e e y
a ionnal
poin
x o
e e y
smoo h
a ie y
X
o e
k
he
map
H
*
(BG(k),
Z/p)
--!-H
*
(BG(Oh),
Z/p)
is
an
isomo phism
.
One
has
o
obse e
he e ha
a
so
gene al
esul
is
unknown
e en
o
k
=
F
p
,
in
which
case
he
GIC
is
known
o
be
ue
c
.
(4
.2)
.
(5
.4)

Ka oubi
[171
has p o ed
he
s able
o m
o
he
isomo phism
conjec-
u e
o
he
g oups Sp(2n,
k)
and SO(p,q
;
k) and
SO(n,
n,
C),
using
his
ime
Vog mann
s abili y esul s,
c
.
[27]
.
170

P
.PASCUAL
GAINZA
6
.
The
Lich enbaum-Quillen
conjec u e
(6
.1)
Le
kbe a
ield
.
Quillen
[26]
associa es
o
he
opological
space
BGL(k)
6
a
new
space
BGL(k)+
and a
map
i
:
BGL(k)
6
--
i
BGL(k)+
such
ha
i) 7
l
(BGL(k)6)
->
7
i
(BGL(k)+)
co esponda
o
he p ojec ion
GL(k)
-~
GL(k)
/
[GL(k),
GL(k)]
ii)
o
any
local
coe icien
sys em
F
on
BGL(k)+
he
mo phism
i*
:
H*
(BGL(k)
6
,
F)

a

-
H*
(BGL(k)+,
F)
is
an
isomo phism,
and
hén
de ines
he
K- heo y
o
he
ield
k
by
K
;(k)
=
7
;
(BGL(k)+)

i
>
1
.
(6
.2)

I
k
is
algeb aically
closed,
Lich enbaum and
Quillen
conjec u ed
ha
he
g oups
K
;(k)
a e
di isible,
wi h
ze o
o sion
o
e en
i
and
equal
o
W(n)
he
Ta e
n
-
wis
o
he
oo s
o
uni y
o
k*
,
o
i
=
2n
-
1
.
The
conjec u e
was
known
o
ha e
a posi i e
answe
i
k
=
F
p
a e
he
de e mina ion
o
he
g oups
K
;(F
p
)
by
Quillen
[25]
.
Suslin's esul s
explained
in
§
5
gi e
a
posi i e
answe
o
k
=
C
Theo em
.
I
k
=
R,
C
and
GL
n
(k)
is
he
co esponding
Lie
g oup,
he
na u al
map
BGL(k)+

)
BGL(k)
induces
an
isomo phism
o
homo opy
wi h
ini e
coe cien s
.
As
was
he
case
in (5
.2)
i
su ices
o
p o e
he
heo em
o
SL(k),
bu
in
his
case
he
map
BSL(k)+
:
BSL(k)
induces
isomo phism
in
homology
wi h
ini e
coe iicien s
by
he
esul s
o
§
5
and
p ope y
ii)
o
he
- --cons uc ion
.
Now
BSL(k)+
and
BSL(k)
being
simply-connec ed
he
esul
ollows
(c
.
[23])
.
BGL(C)
has
he
homo opy
ype
o
BU
hence
Z/p
i i
odd
0
i i
e en
Using
a
esul
o
Weibel,
Suslin
can
deduce
Theo em
.
Modulo
uniquely
di isible
g oups
he
K- heo y
o
R
and
C
is
displayed
in
he
ollowing
able
(i
>
0)
:
(6
.2)

Suslin
also
p o es
in
[35]
ha
hese
wo
cases o
he conjec u e
pe -
mi s
o
esol e
i
in
gene al
:
Theo em
.
([351)
The
g oups
Ki(k,
Z/n)
and
he
n- o sion
g oups
n
K
;(k),
only
depend
in
he
cha ac e is ic o
he
ield
.
In
ac ,
i
ko
C
k
is
an
ex ension
o
algeb aically
closed
ields
we
may
w i e
whe e
A
uns
o e
he
ini elly
gene a ed
ko-algeb as
in k
.
As k
o
is
algeb aically
closed
he e
is
a
mo phism
o
k
o
-algeb as
spli ing
he
map
K¡
(k
o
,
Z/n)
-~
K
;
(A,
Z/n)
and
hence
he
map
COHOMOLOGY
OF
LIE
GROUPS
MADE
DISCRETE

171
i
mod
8
K
;
(R)
K
;
(C)

0

Q/Z

o

Q/Z

o

Q/Z

o

Q/Z
k=limA
,
A

)
ko
K
(k
o
,
Z/n)
-
K
;
(k,
Z/n)
is
injec i e
.
The
idigi y
heo em
pe mi s
now
assu e
ha
he
mo phism
in-
duced by
A
-->
k
is
he
same
ha
he
one
induced
by
(c
.
loc
.
ci
.)
A

)k
o

+
k
hence
su jec i i y
ollows
.
Rema k
.
As an
easy
co olla y o
he
echniques
abo e
one
may
deduce
ha
he
g oups
H
;
(GL
n
(k),
Z
,
i
_<
n,
only
depend
on
he
cha ac e is ic o
k
.
Suslin
p o es
e en
mo e
:
any
one
o
he
cases
F
p
,
C
implies
he
u h
o
he
conjec u e,
because
using
he
igidi y
heo em and
he
homo opy
uni e sal
cons uc ions
o (5
.2),
he
s ablishes
:
Theo em
.
(c
.
[36, 3
.12])
.
Le k
be
an
algeb aically
closed
ield
o
posi i e
cha ac e is ic
p
>
0
and
L
¡he
algebaic
closu e
o
he
quo ien
ield
L
o
o
he
ing o
Wi
ec o s
o
k,
W(k)
.
I
n
is
a
p ime
di e en
om
p
,
he e
is
a
canonical
isomo phisms
K,
(k,
Z/n)
-
K,
(L,
Z/n)
0
1
2
3
4 5 6
7
0Z/2
Z/2
Q/Z
00
0
Q/Z
0
J
.
0
mul
.2
0
0 0
11

172

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GAINZA
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173
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D
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Depa amen
de
Ma emá ica
aplicada
I
ETSEIB
Uni e si a
Poli écnica
de
Ca alunya
Diagonal,
647
-
08028
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SPAIN
Rebu
el
2
de
Juny
de
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