Pub
.
Ma
.
UAB
Vol
.
26
Nó
3
Des
.
1982
FORMAL
GROUPS
AND
RINGSTRUCTURES
FOR
CERTAIN
PERIODIC
COHOMOLOGY
THEORIES
The
pu pose
o
his alk
is o
epo
on
some
esul s
con-
ce ning
a
classi ica ionp oblem
o
a
special
kind
o 'coho-
mology
heo ies
.
Fo he
beginning,
howe e ,
I
would
like
o
conside
a di e en
and
pe haps
mo e
conc e e
p oblem
which
may
se eas
mo i a ion
o
he
es
.
Le
K
(-)
deno eo dina ycomplex
K- heo y
.
We conside
i
as
a
E/2-g aded
heo y
de ined
on
he
ca ego y
CW
*
o
poin-
ed
spaces
o
he
homo opy
ype
o
a
CW-complex
.
Recall ha
K0
(S
0
-
Z
,
K
1
(S0
)
= 0
and
ha
he e
is a
na u alequi alen
ce
~
:
KO
(X)
-
-K
O
(S
^
X),

he
Bo
isomo phism
.
K
*
(-)

is
usually
conside ed
as
a
mul iplica i e
heo y,
he
p oduc being
in-
ducedby
he
enso
p oduc
ope a iono complex
ec o
bund-
les
.
We
may
ask he
ollowing
Ques ion
1
:
A e
he ep oduc s
in
K
(-)
di e en om
he
o dina y
one
and,
i
so,
can one
desc ibe
he
se
P od(K)
o
all
isomo phism
classes
o
such
p oduc s
in
some
easo-
nable
way
?
0
U s
Wü gle
No e
ha
all
p oduc swe conside he e
a e
assumed
o be
wi h
uni ,
associa i e
and
commu a i e
in
he
g aded
sense
.Mo eo e
wo
p oduc s
U,U'
:
K
(X)®
K
(X)~
K
(X)
a e
calledisomo phic
i
he e
is
an
isomo phism
e
:
K
(-)-~
K
(-)
o
cohomology
heo ies
wi h
alues
in
he
ca ego yAb
o
abelian
g oups
(an
addi i e
isomo phism)
such
ha
he
ollowing
diag am
commu es
:
e®e

e
u
K
(-)®
K

K
(-)
To
answe
he
ques ion
abo e
one
cou`ld
ce ainly
y
o
cons uc elemen s
U
EK
0
(BUABU)
wi h
app op ia e
p ope ies
and
hen
de e mine
he
se
o
all
such
elemen s
.
He e,
howe e ,
we
will
adop
a
di e en
poin
o
iew
.
Le
A
be an
ung aded
commu a i e
ing
wi h
uni
.
Fo any
such
ing
A
we
conside
he
se
C(A)
o
all
isomo phism
classes
[T]
o
Z/2-g aded
mul iplica i e
cohomology
heo ies
T
(-)
wi h
coe icien
ing
T
(S0
)
o
he
o m
TOS
O
)
=
A,
T
1
(S
0
)
=
0
(Z/2
-g aded
ing
heo ies
wi h
coe icien s
A
o
sho )
.
.
Clea ly,
[K]E=
-
C(Z)
.
Using hisno a ion
we
ask
he
ollowing-
unp ecise-
ques ion
:
208
Ques ion
2
:
Gi en
a
ing
A,
can
one
desc ibe
he
se
C(A)
o
a
leas
some
in e es ing
subse s
o C(A) in
an
explici
way?
O
cou se,
his
is
jus
heclassi ica ion
p oblem
o
Z/2
-
g aded ing
heo ies
wi hcoe icien s
A
.
Now
we
ema k
ha
he e
is
a
connec ion
be ween
Ques ion
1
a id
Ques ion
2
.
Pu
A=Z
and
lé
CK
(a)
deno e
he
subse
o
CM)
whose
elemen s
a e
all
isomo phism
classes
[T]
E
C(Z)
wi h
he
p ope y
ha
T
*(-)
is
addi i elyisomo phic
o
K*(-),
.i
.e
.
*

*
T

K
(-)
as
Z/2
-g aded
cohomology
heo ies
wi h
alues
in
he
ca ego y
Ab
.

Suppose[TIC
CK
(E),

le
e
:
T
*
(X)>

K
*(X)

be an
addi i e
equi alence
and
suppose
a
:
T
*
(X)®
T
* (X)
-
'
T
*
(X)

is
a
p oduc
on
T
*
(-)
.

Then
eoa
o
(e
-1
®
e
-1
) :

K
*
(X)®
K
* (X)
--
>
-
K
*
(X)

de-
*
ines
a p oduc
on
K

Mo eo e ,
di e en
equi alences
e
and
isomo phic
p oduc s
on
T
*
(-)
p oduce
isomo phicp oduc son
K
*
(-)
andone
sees
easily ha he e
is
a bijec ion
(*)

CK
M)
-s
P od(K)
de ined
by
a
-'
eoao(e
-1
®e
-1
) .
Thisleads
us
o
s udy
he
p ob-
lem
aisedby
Ques ion
2
in
mo e
de ail
.
*
Obse e ha
any
X/2
-
g aded ing heo y
T
(-)
wi hcoe i-
cien san
ung aded
ing
A
is
au oma ically
a
complex-o ien able
heo y,
i .e
.
he
canonicalcomplex
line
bundle
n
m
o e
CP
-
is
*
T
(-)-o ien able
.
This ollows
immedia ely
om
[1],
p
.399.Le
m
:
Cp
w
XCP
-
--'
CP
.b
e
he
classi ying
map
o
he
bundle
n
.xn
.
and
le
x
ET
0
(CP
.)
be an
Eule
.
class
o
n
.
(a
C-
o ien a ion
*
o
T
(-))
.
Then,
as is
well
known,
he
o mal
powe
se ies
*
F(x
1
,x
2
)
:=
m
(x)E
AQx
1
,x
2 j
is
a
one-dimensional
commu a i e
o mal
g oup
law
on
A
(a
o mal
g oup
on
A
o
sho ),
whe e
* *
x
i=
p
i
(x)
ET
(CP
.
x
CP_)
.
Now
o malg oups
co esponding
o
di e en
C
-
o ien a ions
o
he
same
heo y
a e
isomo phic
and
isomo phic heo ieswi h
he
same
coe icien
ing p o-
duceisomo phic
o mal
g oups,
so i
we associa e
o
any
7L/2-
*
g aded ing heo y
T
(-)
wi h
coe icien s
A
i s
o mal
g oup
we
ge
a
map
(D

:

C
(A)

-

FG
(A)
whe e
FG(A)
deno es
hese
o (s ic )
isomo phism
classes
o
o mal
g oupso e
A
.
We will
use
his
map
o
ge
an answe
o
Ques ion
2
in
some
pa icula
cases
.
Suppose i s ha
A
=
k
is
a
ield
.
I
hecha ac e is ic
o
k
is
0,
classical esul simply
ha
C(k)
consis s
o
only
one
elemen ,
namely
H
(-
;k)
.
Fo
ields
o
posi i e
cha ac e
is ic,
howe e ,
he
si ua ion
changes
.
Fi s we
ha e
:
Theo em
1
:
Le
k
be
á
ieldo
cha ac e is ic
p>
2
.
Then
he
más
(D
:
C(k)
-->-
FG(k)

is
á
biiec ion
.
Rema k
:
Fo
p=2
we ha eonlypa ial esul s
.
In
his
case,
hemap
D
is
su jec i e
bu no
injec i e
.
Di icul ies
a ise
om
he
ac ha
all
elemen s
n
C(k)
di e en
om
H
**
(-
k)
a enon-commu a i e
.
Fo mal
g oupso e
ields
o
posi i e
ch ac e is ic
a e
a he wellunde s ood
(see
o
example
he
book
[3])
.
In
pa -
icula ,
he e
is an
impo an
isomo phism
in a ian
o
such
o malg oups
F,
hei heigh h
F
E
=
U{-}
.
B ie ly,
h
F
= n
n
i [p]
F
(x)
=
axp
+
e ms
o
highe
o de ,
a
q¿
0,
and
h
F=
i
[p]
F
(x)
= 0
.
Le
FG(k)
n
deno e
he
subse
o
FG(k)
o
o -
mal
g oups
o
heigh
n
andpu
C(k)
n
=
(D-1(FG(k)n)
.
Then
FG(k)=
U
n=w
FG(k
)
n
and
C(k)

=
U
n=-
C (k
)
n
.
The
nex
heo em
ells
us
ha
n=1

n=1
Z/2
-
g aded
ing heo ieswi hcoe icien s
k
and
o mal
g oups
o equalheigh
a e
e ys ongly
ela ed,
in
ac hey only
di e
by hei
mul iplica i e
s uc u e
:
*
Theo em
2
:
Le
p
be
any
p_ ime
and
suppose
T
1
(-),
T
2
(-)
a e
E/2-
g aded
in
heo ies
wi hcoe icien s
k,
a
ield
o
cha
-
*

*
ac e is ic
p
.
Then
T
1
(-)
and
T
2 (-)
a e
isomo phic
as
coho
-
mology heo ieswi h alues
in
he
ca ego y
o k-
ec o
-
spaces
i
andonl
i
hei
o malg oups
a e
o
he
same
heigh
.
Recall
om
[21,[41 ha
o any
in ege
n
he
/2-g aded e sion
*
wi hcoe icien s
k,
K(n)
(-
;k),
ep esen s
an
elemen
o
C(k)
n
.
Fo
n
=
-
we
se
K(w)
*
(-
;k)
= H
**
(-
;k)
.
No e
also ha
*

*
K(1)

(-
;&
p
)

= K

(- ;F
p
) .

Using
he
same
a gumen
which
lead
o
he
bijec ion
(*)
we
ge
om
heo ems
1
and
2
he
Co olla y
3
:
Le
k
be
á
ieldo cha ac e is ic
p>2
.
Then
o
all
nE
N
U
{-}
he e
a e
biiec ions
C(k)
n-
~
P od(K(n)*(-,k))
-*
in
log
n
(x)

=

EP
-1
x
p

E
w
X]
i>0
p ime
p
and
any
posi i e
o
he
n- h
Mo a aK- heo y
FG(k)
n
.
I
shouldbe no ed
ha
o
FG(k)
n
,
he e
a e
se e almo e
o
less
explici desc ip ionsa ailable
(see
e
.g
.
[3])
.
Le
us
ecall
e y
b ie ly
one
o
hem'
Conside
he
powe
se ies
andpu
Fn
(x,y)
=
logn
l
(log
n
(x)
+
log
n(y))
.
Fn
(x,y)
is
a
o mal
g oup
o e
Z(P)
.
F
n
(x,y),
i s
educ ion
mod
p, is
de inedo e
I
p
and
so
o e
e e y
ield
o
cha ac e is ic
p
.
Le
k
sep

be
a
sepa able
closu e
o
k
and
S
n=
Au k

(F
n
)
he
au omo phism
_

sep
g oup
o
Fn
o e
k
seP
.
A
classical esul
o
Dieudonné-Lubin
ellsus
ha
S
n
is
isomo phic
o
he
g oup
o
uni s
o
he
maximal
o de
in
he
cen aldi isionalgeb a
D
n
o
in a ian

1/n
and
ank n
2
o e
0p
.
Le
P
be
he
Galois
g oup Gal(k
sep
:k)
.
Then
P
ac s
on
S
(by
ac ing
on
he coe icien s
o
powe
n
se ies)and
he e
is
a
bijec ion
FG(k)
n
-,
H
1
(P,S
n
) .
This
bijec ion
oge he wi h he
ac ha
o mal
g oups
o
in ini e heigh
o e
a
ing
o
p ime
cha ac e is ic
a e
iso-
mo phic
o
he addi i e o mal
g oup
imply he ollowing
Co olla y
4
:
I k is
á
sepa able
closed
ield
o
odd
cha ac
e is ic and n<-
o
i
k
is
án
a bi a e
ield
ó
posi i e
cha ac e is ic and
n=-,
hen
,
ug
o
isomo nhism
,
K(n)
(-,k)
is
he
only
Z/2
-
a aded ing heo y wi h coe icien s k and
i
o mal
Q ouP o heigh
n
.
I
n
.
=
1,
S
.
1
is
isomo phic o
_
he
g oup
2
p
o
p-adic uni s
.I
k=Y
is
a
ini e
ield,
P
is
opologically
gene a ed
by
p
he
F obenius homomo phism and one ob ains H
(Pa
p
)=
7i
p
.
So
co olla y
3
implies
a
bijec ion
C(Yp
)~
P od(K*
(-,F p
)
)-,
~P
o
global
e sion
o
Theo em
2
.
Fo
mo e
gene al ings
A
we
ha e
only e y pa ial esul s
o
o e o he momen and he
ques ion
seems
o be
di icul
.
To
end
his
alk,le me
jus
desc ibe
some
esul s
o
he case
A =
E
.
This wiil be enough
o
answe
ou ini ial Ques ion
1 .
Le
P
deno e
he
se
o
all
p imes and
le F(x,y)
be
a
o mal
g oup
o en
2E
.
De ine he
heigh
unc ion
o
F,
h F
:
P
--
'
N
U{-},
by
se ing
h
F
(p)
=
heigh
o
F mod p
o e
F
p
.
I is an iso-
mo phism
in a ian
o
F
.
Using
his
no ion
we
ge
some so
Theo em
5
:
Le
T
1
(-)
and
T
2
(-)
be
Z/2
-
ci aded
ing
heo ies
wi h
coe icien s
2Z
and
o mal
g oupsF
1
es
p
.
F
2
.
Then
T
1
(-)
and
T
2
(-)
a e
addi i elyisomo phic
i
and
onlYi
h
F

(p)

=
h F2
(p)

o all
P imes
p
.
1
We do
no
know
i
he
map
(D
:
C(Z)
-
FG(Z)

is
sú jec i e
o
injec i e
in
gene alal hough
we
ha esomepa ial
esul s
which
we will
no
desc ibehe e
.Howe e ,
i
we
es ic
ou
s
a en ion
o
he
subse
CK
(Z)
o
C(Z),
we
can
be mo e
p ecise
.
Le
FG(Z)
1
be
he
se
o
all
isomo phism
classes
o
o mal
g oups
F
o e
Z
o
heigh
1
a
any
p ime,
i
.e
.
h
F
(p) =
1
o
all
p,
andle
(D J<
deno e
he
es ic ion
o
4)
o
CK
(E)
.
Theo em
6
:
The e
a e
biiec ions
4)K
1
P od(K)-}

CK
(Z)

FG(Z)

--->

II
Z
*
PEP
p
One
mayask
wha
all
he e
new
p oduc s
on
K
(-)
des-
c ibed
by
heo em
6
a e
good
o
.
I
u ns
ou
ha
he e
a e
in e es ing
connec ionsbe ween hem
andcha ac e is ic
classes
cx
E
H
(BU,Q)
associa ed
o
ce ain
in eg al
Hi zeb uch
gene a
(i
.e
.
ing
homomo phisms)
'
ZC
Q
which
can
be
desc ibed
in
e ms
o
Riemann-Rock
ela ions
.
Also,
o
any
exo ic
p oduc
on
K
(-)
hei co esponds
a
se
o
"exo icAdamsope a ions"wi hin e es ing
p ope ies
.
Re e ences
[11

Dold,A
. :
Che n
classes
in
gene alcohomology
.
Symposia
Ma hema ica
ol
.
V
(1970)
[21

Johnson,D
.C
.,
Wilson,S
. :
BP-ope a ions
and
Mo a a's
ex ao dina y
K- heo ies
.
Ma h
.
Z
.
144,
55-75
(1975)
[31

Hazewinkel,M
.
:
Fo malg oups
and
applica ions
.
Academic
P ess, 1978
[41

Wü gle ,U
. :
On p oduc s
in
a
amily
o
cohomology
heo ies
associa ed
o
he
in a ian
p ime
ideals
o
u
*
(BP)
.
Commen
.
Ma h
.
Hel
.
52,
457-481
(1977)
Ma hema isches
Ins i u
de
Uni e si á
Sidle s asse
5
CH-3012
Be n