Pub
.
Ma
.
UAB
Vol
.
26
Ne
3
Des
.
1982
In oduc ion
PROFINITECHERNCLASSES
FOR
GROUP
REPRESENTATIONS
Guido
Mislin
Le
p
:
G
->
GL
n
C
be
a
complex
ep esen a ion
o
he
disc e e
g oup
G
.
I
one
wishes
o
s udy
p
om
an
alge-
b aic
opologis 's
poin
o
iew,
one
o ms
he
induced
map
Bp
:
BG
->
BGL
n
C
,
o
classi ying
spaces,
whichgi es
ise
o
an
n-dimensional
complex
ec o bundle
j(p)
o e BG
=
K(G,1)

.
The
Che nclasses
o
his
ec o
bundle

U
p)

,
c
j
(P)

e

H
2j (G
;
a)

,
a e
called
he
Che nclasses
o
p
.
These
cohomology
classes
may
be
used
o ob ain
.in o ma ion
on H*(G
;a)
,
o o
s udy
he ep esen a ion
p
i sel
.
Fo
ins ance,
i
p
ac o s
h ough

GL
P
,
he
associa ed
complex
ec o bundle
o e

BG
will
be
in a ian
unde
complex
conjuga ion,
and
by
a
well
knownp ope y
o
Che nclasses
his
implies
ha
c
j
(P)
=
(-1)
i
c
j
(p)
,
ha
is,
2c
(p)
= 0
o
j
odd
.
Mo e
gene al-
ly,
he e
is an
ob ious
ac ion
o
ieldau omo phisms
o
C
on
he
ec o
bundles
o
he
o m
~(p)
,
and
i is
ou
ob-
jec i e
o
s udy
he
beha io
o Che nclassesunde
his
ac-
ion
.
Using
Sulli an's
compu a ion
o
he
"Galois
ac ion"
on
H*
(BGL
n
C
;ZZ
/m2Z)

(c
.

[10])
we
will
be
able
o
unde s and
his
ac ionon
he
Che nclasses educedmodulo
m
.
A
di -
e en app oach
is
desc ibed
in G o hendieck'spape
[6],
using p-adicChe nclasses
de ined
in an
algeb aic
geome y
se ing
(see
also
Soulé
[9])
;
esul s
on
o dina y
Che nclas-
ses
ollow henbymeanso
he
compa ison
heo em,
ela ing
he
e ale
homo opy
ype
o
a
complex
a ie y
wi h
i s
o dina-
y homo opy
ype
and
i s
p o ini e
comple ion
.
I
one
is
in-
e es ed
in
esul s
conce ning
ini e
g oups,
hen
a
mo e
di ec app oach
is
possibleby iden i ying
he
Galois
ac ion
on
he
ep esen a ion
ing
wi h
ce ain
Adamsope a ions
(see
[5J)
.
Fo
ou
app oach,
i
u ns
ou
o be
na u al
o
wo k
wi h
p o ini e
Che n
classes
^
c
j
(P)

e

H
2i
(G
;
a)
They
a e
de ined
as
he
images
o
he
o dina y
Che n
classes
c
j
(p)
unde
he
map
induced
by
he
coe icien homomo phism
Z
;
->
zz
,
2Z
=
lim
Z
;
/nZ
;
he
ing
o
p o ini ein ege s
.
Fo
a e
Gal(C/C)
a
ield
au omo phism
o
C
and
p
:
G
-
GLnC
a
ep esen a ion,one
de ines

p
a

by

a* o
p

'whe e
a*
:
GL
n
C
->
GL
n
C
is
ob ained
by
applying
a
o
he
en ies
o
a
ma ix
.
We
show
i s
ha
c
j
(p)
dependsonly
on
X
p
1
he
cha ac e
o
p
.
The e o e,
c
j
(P)
=
c
j
(p
o
)
i
a
ixes
he
alues
o
Xp
.
On
he
o he
hand,
we
show ha
Q
)

=

0
J
c
j
(p

c
j
(p)

,
whe e

an

is
a
uni
in

~n

which
is
de-
e minedby
he
ac ion
o
o
on
he
oo s
o
uni y
in
T
.
Ou
main heo em
hen
esul s
om
an
analysis
o hese
e-
la ions
.
I
in ol esce ainnumbe s
EK(j)
which
a ede-
ined
o
a
numbe ield
K
and
whichwe ein oduced
in
E
K
(j)
=
max{mlj
- 0
mod
exp(Gal(K(~m)/K))}
whe e
Cm
deno es
a
p imi i e

m- h oo
o
uni y,
and
exp(Gal(K(~
m
)/K))
is
he
exponen
o
he
Galois
g oup
o
K(~
m
)
o e
K
.
Main
Theo em
.
Le
p
:
G
->
GL
n
C
be
a
ep esen a ion
wi h
cha ac e
Xp
.
Suppose
K
C
T

is
a
numbe
ield
such ha
X
p
(g)
e
K
o
all
g
s
G
.
Then
he
ollowing
holds
:
A)

E
K
(j)
c
j
(P)
=0 e
H
2j (G
;ZZ
)

o
all

j>0
.
B)
The
bounds
EK(j)
on
he
o de s
o
c
j
(p)
a e
bes
possible
in
he
ob ioussense
.
Rema ks
.
The
numbe s
E
K
(j)
can
be
desc ibed
in
a
e y
ex-
plici
way
in
e mso
in a ian s
a ached
o
K
(c
.
[5])
.
Fo
ins ance,
i
j
is
e en
and
K =
Q
,
one
has
E~(j)
=
den(Bj/
2j)
wi h B
2
=
1/6
,
B
4
=
1/30
e c
.
he
Be noulli
numbe s
.
No e
also
ha he numbe s
EK(j)
ag ee wi h G o hendieck's
bounds
[6]
and
hey
a e
also
equal
o
he numbe s w
j
(K)
de-
ined
in
Cassou-Nogués' pape
[3]
,
(see
also
[7]
)
1
.
Rep esen a ions
and
aces
A
ep esen a ion
p
:
G
-+
GLnX
de ines
a
G-ac ion
on
C
n
.
We
w i e
V
=
V(p)
o he
co esponding
T[G]-module
.
As
usual,
we
de ine
he
complex ep esen a ion
ing
R(G) o
be
he
ing
addi i ely
gene a ed
by
isomo phism
cl~Lsses
o
ini e
dimensional
T[G]-modules, wi h ela ions
o
he
o m
[W]
=
[V]
+
[W/V]
E
R(G)
o
e e y
sho
exac
sequence
,
l ,

T
G1
-
odul
e

[171

acnn-
V
T
W
T
YV~
V

VL
111111
.
.C-U1llleliJlVliül

W
LVJ
iu uui~
.-1

L J

1
.
.V
es
he
image o
V
in R(G)
.
The
mul iplica ion
in R(G)
is
de ined
using
he
enso
p oduc
o e
T
o
T[G]-modu-
les
.
I
V =
V(p)
and
i we
choose
a
composi ion
se ies
V
1
C V
2
C
.
.
.
C V
n=
V
,
we
see ha

[V]
=
E
[V
j
/V
j-1
] s
R(G)
wi h V
/V

=V(P
.

j
) ,
p

an
i educible ep esen a ion
;
j
j
-1

J
his
means
ha
om
he
poin
o iew o
R(G)
,
e e y
e-
p esen a ion
is
semi-simple
.
The
Jo dan-HSlde Theo em
s a es
ha he i educible
ep esen a ions
Pj
a e
uniquely
de e -
mined
by
P
(up
o
equi alence
and
o de )
.
Thus
R(G)
has
an
addi i e basis
consis ing
o
he
elemen s
o
he
o m
[V
a
],
a
simple
T[G]-module
o
ini e
dimension
.
The
cha ac e
Xp
o
p
is
he
unc ion
G+T
de-
inedby

XP
(g)

=
ace(p(g))

,

g
e

G

.

O
cou se,

X
P

de-
pends
on
V(p)
only,
and
we
some imes
w i e
XV(P)
o
Xp
I
V
->
W
->
W/V
is
a
sho
exac
sequence
o
ini e
dimen-
sional
Q[G]-modules,
hen
X
w
=
X
V
+
XW/V
.
The e o e
P¡->
X
p
gi es
ise
o an
addi i e
homomo phism
in o
he
ing TG o T- alued unc ions
on
G
.
Since
x
©w
=

x
'
x
w

,

x
The
image
X(R(G))
is
deno edby
R
x(G)
and
we
call
i
he
cha ac e
ingo
G
.
Theo em
1
.

The
map

X

:

R(G)

->
R
X
(G)

is an
isomo phism
o
ings
.
X

:

R
(G)
--~
( G
ac ually
de inesa
homomo phism
o ings
.
P oo
.
Le
p
l
,P2
:
G -
GL
n
X
be
wo
comple ely
educible
ep esen a ions
.
Then
Xp
=
X
p
implies
V(p
l
)
=
V(P
2
)
as
1
2
T[G]
-
modules
:
his
is a
consequence
o
he
double
cen al-
ize
Theo em,c
.
Bou baki
[2
;
chapi e
VIII,
§
12,
P op
.
3]
.
I
x
e
R(G)

is
an
a bi a y
elemen ,
we
can
w i e
x
in
he
o m
x
=
E
[Vi] - E [W
j
]

wi h
V,
i
and
W
j
simple
C[G]-
modules
o
all
i
and
j
.
Suppose
now
ha X(x)
=
0
.
Then

EX([V
i
])
=
EX([W
j
])

and
he e o e
©
V
i
-
9
W
j
be-
cause
he
ep esen a ions

©
i
and
p
W
j
a e
semi-simple
.
We
in e
x =
E[ i]
-
EN
j
]
=
0

and
hus

X
is
injec i e
.

Since
X
is
su jec i e
by
de ini ion,
he
asse iono
he
heo em
ollows
.
2
.
Galoisac ion
Le
a e
Gal(T/Q)
be
an
au omo phism
o
0
.
By apply-
ing
a
o
he
en ies
o
a
.ma ix,
one
ob ains
an
induced
g oup
au omo phism
a*
:
GL
n
T
-~
GL
nT
.
I p
:
G
-
GL
nT
is
a
ep esen a ion,we
w i e
pa
o he
composi e
ep esen a-
ion
a
* o
p
.
As
usual,
we
deno e
he
g oupo
au omo phisms
o
T
o e
K
C
T
by
Gal(T/K)
.
Theo em
2
.
Le
p
:
G
->
GL
nT
be
a
ep esen a ionandle
deno e
hc
"b 'eld
o
T
gene
a
ed
by
he
eces
o
he
ma ices
p(g)
,
g e
G
.
I
a
e
Gal(T/Q(X
p
))
hen
[V(P)]
=
[V(P
a
)]
e
R(G)
P oo
.
No e
ha
o
a
an
au omo phism
o
T
o e
92(X
p
)
,
Xpa
(g)
=
a(X
p
(g))
=
X
p
(g)
o
all
g e
G
.
The e o e,
X([V(p)])
=
X([V(p
(y
)]

and
we in e
om
Theo em
1
ha
[V
(P)]

=

[V (P
a
)]
Rema k
.
I
p
:
G -
GL
nT
is
a
ep esen a ion
o
a
ini e
g oup
G
,
hen
i
is
well
known
ha
he
ep esen a ions
p
and

p
a
a e
ac ually
equi alen
o
e e y

a s
Gal(T/4
(Xp
))

.
Fo
an
in ini e
g oup,
his
need
no
be so
.
Fo
example,
i
p

:

a
-
GL
4
Q

is
gi enby
hen
Q(X
p
)
= C
and,
aking
a
o be
complexconjuga ion,
one
easilychecks
ha
V(p)
1
V(p
a
)
al hough
XP
=
X
a
.
P
Le

K C T
be
a
numbe ield
andle

u(T)

deno e
he
g oupo oo s
o
uni y
in
0
.
The
ollowingnumbe s
w
j
(K)
ha e
beenconside edby
Soulé
in
[9]
:
w
j
(K)
=
ca d{x
e
U(T)la
i
x = x
o all
a
e
Gal(T/K)}
We
wan
o
show ha
w
j
(K)
=
!K
(j)
,
E
K
(j)'
being
de ined
as
in
he
in oduc ion
(see
also
[5])
.
Le
u
m
C
U(Q)

deno e
he
g oup
o

m- h oo s
o
uni y
.
Then

p
m
C
K(I
m
)

whe e
i
m
deno es
a p imi i e
oo
o
uni y
in
T
.
The
ob ious
map
Gal
(T/K)
--->
Au
um
ac o s h ough
he
su jec i e
es ic ion
map

Gal(0/K)
--a
Gal(K(I
m
)/K)
.
Since Gal(K(C
m
)/K)
ac s
ai h ully
on um
,
he
asse ion
is
he e o e
equi alen
o
he
asse ion
" a
i
x = x
o
all
x
eu
m
andall
a
e
Gal(T/K)
"
" j
- 0
mod
exp(Gal(K(I
m
)/K))
"
whe e
exp(Gal(K(C
m
)/K))
deno es
he
exponen o
he
g oup
Gal(K(I
m
)/K)
.
Using
he
ac
ha
all
ini e
subg oups
o
u(C)
a e
cyclicwe in e
ha
w
j
(K)
ag ees
wi h
E
K
(j)
= max{mjj
= 0
mod
exp(Gal(K(E
m
)/K))}
o
e e ynumbe ield
K
and
e e y
j
> 0
.
Co olla y
1
.

Le
K
C
C
be
a
numbe
,
ield
.
Then
he
o -
sion
subg oup
o
he
mul iplica i e
g oup
K*
is
cyclic
o
o de

E
K
(1)

.
P oo
.
The
o sion
subg oup
o K*
is
p(C)
n
K
.
I s
o -
de_
i_s
ob iously
equal
o
he
la ges
numbe
m
such
ha
um
C

K

,
which
is
he
same
as

EK
(1)

o

w
1
(K)

.
3
.
Che n
classes
We
w i e

c(p)
= E
c
.(p) e H*(G
;2Z)

o
he
o al
Che n
7
class
o
a
ep esen a ion
p
:
G +
GL
nT
.
Clea ly,
c(p)
de-
pends
on
V =
V(p)
only,
and
we
some imesw i e
c(V)
o
c
(p)

.
Le

V -
W
->
W/V

be
a
sho
exac
sequence
o
ini e
dimensional

C
[G]
-modules
.
Then

c
(W)

=
c
(V)
"
c
(W/V)

since
e e y
sho
exac sequenceo
ec o
bundleso e
a CW-com-
plex
is
spli
.
Taking
Che n
classes
hus
de ines
a
map
c

:

R
(G)
--
H*
(G
;
2Z)
]
~---->

c
([ ])

:
=
c
( )
which
is
a
homomo phism
o
he
unde lying
abelian
g oup
o
R(G)
in o
he
mul iplica i e
g oup
o
uni s
o
he
g aded
ing
H*(G
;ZZ)
Theo em
3
.
Le
pl,p2
:
G
-+
GL
nT
be
wo ep esen a ions
wi h
Xp
=
X
p
.
Then
12
c
(P1)

= c (P2)

e
H
*
(G ;ZZ)
P oo
.
By
Theo em
l, X

=
x

implies
ha
p
l p
2
[V(P
1
)]
=
[V(p
2
)]
.
The e o e
c(p
l
)
=
c([V(p
l
)])
_
c
([V
(P
2)])

=C
(P
2
)
The
i s Che nclass
o
a
ep esen a ion
p
:
G
-
"
GLnT
can
be
desc ibed
in
a
e y
explici
way
.
Le
de
:
GL
nT -
T*
=
GL
1
T
deno e
he
de e minan
map
.
Then
de
P
is
a
one-dimensional ep esen a ion
and,
by
a
well-
knownp ope y
o
ec o
bundles,
c
1
(P)

=

c
l
(de

P)

s

H2 (G
;
?Z)
Conside
he
coe icien
sequence
0

>
%

1
Q
exp,
T*
--
:1
0
This
comple es
he
p oo
o pa
A)
o
he
Main
Theo em
.
I
emains
o
show
ha
he
bounds
EK
(j)
a e
bes possible
.
This
can
be
seen
using
he
calcula ions
pe o med
in
[5_]
.
We
ecall
(Theo em
4
.12
o
[5])
ha
E
K
(j)
is
he
bes pos-
sible
bound
o
he
o de
o
he
Che nclasses

c
J
.

o

K- e-
p esen a ions
o
ini e
g oups,
wi h
he
single
excep ion
when
j
is e en
and
K
o mally
eal
;
in
his
la e
case
he
bes possible
such
bound
is
2
EK
(j)
.
I
su ices he e-
o e
o
p o e
he
ollowing
.
Theo em
7
.

Le
K
be
a
o mally
eal
numbe ield
and

j
> 0
e en
.
Then he e
exis s
a
ini e
2-g oup
G
and
a
ep esen-
a ion

p
:
G
}
GL(T)

wi h
Q(X
p
)
C
K
and
EK
(j)
c
j
(P)
+
0
P oo
.
The
cons uc ion
o
such
a p
can
be
pe o med
in
essen ially
he
same
way
as
he
cons uc ion
o
p
in
he
cou se
o
he
p oo
o
P oposi ion
4
.11
(b)
o
[5]
.
One
hus
ob ains
a
ep esen a iono
a
gene alized
qua e nion
g oup
wi h
Q(X
p
)
C
K
and
wi hSchu indexequal
o
wo
wi h
e-
spec
o
Q(X
p
) ,
such
ha
2
EK
(j)
c
j
(P)
+
0
.
Rema k
.
I
p
:
G }
GL
nC
is
a
semi-simple
ep esen a ion
and

K
D
Q(X
P
)

a
sub ield
o

C

,

hen
he e
is
a
ini e
ex-
ensionL
o
K
in
T
such ha
p is
equi alen
o
a
ep esen a ion
de ined
o e
L
.
This
in e es ing
obse a-
124

ion
was
communica ed
o me by
P
.
Menal
.
We
plan
o
use
his
ac
in
a
la e
pape
o
show
ha
o
a
e y
gene al
p
he
ac ualChe n
classesc
j
(P)

( a he
han
1
j
(p))
a e
o
ini e
o de
bounded
by
EK
(j)
,
i
K
is
a
numbe
ield
con aining
Q(X
P
).
Thisexposi o ypape
is
based
on
lec u es
deli e ed
a
he
Uni e si a
Au ónoma
de
Ba celona
.
The
inal
o m
o
he
esul s
will
be
published
in
a
join
pape
wi h
B
.
Eckmann
.
Re e ences
[2]
-
N
.
Bou baki,
Algéb e
;
He mann,
Pa is,
1958
D]

P
.
Cassou-Nogués
:
Valeu s
aux
en ie snéga i s
des
onc ions
zé a
e
onc ions
zé a
p-adiques
;
In en-
ionesma h
.
51
(1979),
29-59
[4]

P
.
Deligne
and
D
.
Sulli an
:
Fib és
ec o iels
com-
plexes
á
g oupe
s uc u al
disc e
.
C
.R
.
Acad
.
Sc
.
Pa is,
.
281,
Sé ie
A,
(1975)
1081-1083
[6]

A
.
G o hendieck
:
Classes
de
Che n
e
ep ésen a ions
linéai es
des
g oupes
disc e s
.
Dans
:
Dix
exposés
su
la
cohomologie
des
schémas
.
Ams e dam
:
No h-Holland
1968
[8]

J
.-P
.
Se e
:
Cohomologie
des
g oupes
disc e s
.
Annals
o Ma h
.
S udies
70
(1971),
77-169
G
.
Baumslag,
E
.
Dye
and
A
.
Helle
:
The
opologyo
disc e e
g oups
.
J
.
Pu e
Appl
.
Algeb a
16
(1980)
1,
1-47
B
.
Eckmann
and
G
.
Mislin
:
Che n
classes
o
g oup
e-
.
p esen a ions
o e
a
numbe
ield
.
Composi io
Ma hema-
ica
44
(1981),
41-65
G
.
Mislin
:
Classes
ca ac é is iques
pou
les
ep ésen-
a ions
des
g oupes
disc e s;'Séminai e
Dub eil-
Mallia in,
Pa is
1981
(To
appea
in
:
Lec u e
No es
in
Ma h
.,
Sp inge -
Ve lag)
[10]

D
.
Sulli an
:
Gene ics
o
homo opy
heo y
and he
Adams
conjec u e
.
Annals
o
Ma h
.
100
(1974),
1-79
May
1982
ETH
-
Zü ich
Schweiz
C
.
Soulé
:
Classes
de
o sion
dáns
la
cohomologie
des
g oups
a i hmé iques
.
C
.R
.
Acad
.
Sc
.
Pa is,
.
284,
Sé ie
A
(1977),
1009-1011