Pub
.
Ma
.
UAB
Vol
.
29
Ns
2-3
No
.
1985
INTRODUCTION
:
ON
A
THEOREMOF
M
.
FUJII
M
.O
.
Aje unmobi
In
1967
M
.
Fujii
[21
compu ed
he
KO
-i
-
ings
o
he
complex
p ojec i e
spaces
.
We
gi e
a
modi iedp oo
he e
using
some
esul s
by
S .G
.
Hogga
[31
.
Ou
me hod
seems
di ec
and
easie o handle
and
i
has
been
applied
o
compu e
he
KO
-i
-
g oups
o
he
complex
lag
mani olds
o
leng hs
2
and
3
[11
.
The
esul we ep o ed
is
Theo em
2
o
Fujii
[21
.
Theo em
[2,
p
.
1421
The
KO
-i
-
g oups
o
IP
n-l
(Q)

a e
as
ollow
i

n
-
2
(mod
4)

n -
0
(mod
4)

n
odd
0

(2 +1)

a+
ZZ
2
(2 )
a

( +1)
zz
n =
4
+
2

n
=
4

n =
2
+ 1
1
a
2
a
2
2

( +1)
a

+
ZL
2
( +1)
zz
+
=
[
n211
3

0

zz
2

0
4

( +1)
zz

( +1)
Z
;
+
a
2

( +1)
a
zz
2
( ) a+2
;2
6

( +1)
a

( +1)
ZZ

( )
2Z
Cohomology
o
Ip
n-1
(0)
Le
XR
be
he
eali ied
bundle
o
he
canonical
bundle,
X,
o e
IP
n-1
(Q)
.
Then
he
second
S ie el-Whi ney
class
w
2
(x
R
)

is
hemod
2
educ ion
o

C
1
(X)
.

Pu

x
=
w
2
(x
R
),
hen
an
addi i e
basis
o
H*(IP
n-l
(¢)
;
T
.L
2
)

is
gi enby
xl
subjec
o
he
condi ion
xn
= o
.
The
Poinca é
polynomial
o
IP
n-l
(0)

is
gi en
by
(1)

P(IP
n-1
(0)

.
)
=
1 +
2 +
4 +
. .
.
.
.
+
2(n-1)
KO
-i
(IP
n-1
(0»
.
F om
(1),
i is
clea
ha
he
2k h
Be i
numbe ,
0
2k
=1

o
0
<
k
<
n-1,

hus
he
anks
o
KO*(IPn-1
(0))
a e
de e mined
as
ollowsusinglemma
(2
.4)
o
[31
:
o all
alues
o
n
.
[
nn-11
2
ank
KO
o =
ank
KO-4
=

E
R
4k
=
[
n1
1
+
1
k=o
[
nn-11
2
Also,
ank
KO-2
=
ank
KO-6
=E
0
4k+2
=
in211
+
1
k=o
o
n
e en
and
[
nn-21
2
ank

KO
-2
=
ank

KO-6
=

E

R
4k+2
=
[
n
2
2
1

+
1
=
n21
k=o
o
n
odd
and
his
comple es
he
ee pa
.
Fo
he
o sion
pa ,
conside
he
A iyah-Hi zeb uch
spec alsequence
which
con e ges
o
KOp+q(IP,-1(¢)),
see
[2]
.
Conside
he
sequence
o
di e en ials
(2)
E
p-2,
q
+1
2
Fo

q=0,4(mod
8),

E2
'q

gi es
he
ee
pa
c

KOp
+
q
which
is
de e mined
.
Fo he
o sion
pa ,
we need
only
conside
q=
-l,
-2(mod
8)
.
Fo

q
=
-1
-
(mod
8)
(2)
becomes
E
p-2,8

d
2

Ep,8 -1
d2
Ep+2,8 -2
.
2

2

2
The
map
EZ'q
-
EP+2,q-1
is
ze o
o
p =
0,4
(mod
8)
(2a)

and
is
an
isomo phism
o

p =
2,6(mod
8)

i

E2
'q
70
.
Thus
(2b)2(n-2)
=
0,4(mod
8)
and
o
n
odd, 2(n-2)
-
2,6(mod
8),
hus
he
di e en ial
is an
isomo phisms
o
n
oddand
ze o
o
n
e en
using
(2a)
and
(2b)
.
Hence
d
2
Eo,-1
-
Eo,-1
3

2
and
Ep
,-1
=
0
o he wise
.
Ep'q
2
d
2
(3)

E
2(n-1),-1
-
{
0

n
odd
3

2z
2
;

n
e en
E
p
+
2,q
-1
.
2
zz
2
o -all
n
.
Fo
n
e en,
2(n-2),8 )
E
2(n-2),8

d
2

E
2(n-1),-1
2

2
--
~
oz

g
~
-2
~
-
x~~
8~)"
(2"be
:
c
:
¿
lé
S'
"
(2
.4)
in
[3]
w¿
-
'
no
o
EE
o(moa
«)'
we na e
`-~^-----
'
.' P' z~

~u~'-'
-

-

u~
_~ z,
_

z

-'-
---

---~-
'
z
~ . .
.~"^
=~~*
`/1
o,-2
~aud'
.
E
~
`
i
17
'12
1

`

un

.
Also
o
o
EE
:'
2(mod
4),
we ua e
zz
2
1

U!
?
X1

P,
s
z,1(mou

n),

p
0

o
o
;
o he w¡ae
.
mow,

o

i
~
`
~(mnd
4)
.
,

2
{o
-
l
.>
.~V

(m9d
8)'
o

o5
:1

](mod

«)
'

u(u-1)

~

4(mod

8)'
o

u
~
-
2 (mod
,
4J
.,

.

.~
z(u-a
~
2< md
x>,

aud
'
~
o
u
~
0(muu
4),
u(o-l)
~
0(mod
o)
.
Tbo9,
~
:
iog
(])
ni h
(4)
auu
uaiog
zenuna
(z
.1) in
131,
.
-
.^
^
we ba e` o
uouu,
.
no-]
=
oou
-
Daz
xo-4
~
o
-5

-§
~o

~

z
z

-
pac

mo

~
o

auu
no
-7 ~
uoz
-
naz
m"
o
~
o
oo
-5 ~
oz
o-
naz
mz-a ~
u
aod
nn
-7 ~
oz
z-
pa

mx
o
~
o
KO
-3
=
2Z
2 -
pa KO
-4
=
0
and
KO-5
=
zz2
-
pa
KO
-6
=
0
.
Now, we
show
ha
.he

E
3
- e mssu i e
o

E
,

o
q
=
-1(mod
8)
.
.
Le

E
0,-1
=
a
2Z

,

E2(n-1),-1
=

~
zz
(
n
3

2
3

2
Conside
he
di e en ial
E
0,-1
E
0,-1
2
d
2
Thus
E0'
-1
=
a
a2
.
e en)
d

E '-
,
E '-
=
0
excep
-
0,2,4(mod
8)
and
d = 0
o
=
0,
4(mod
8)

because
i
maps
a
ini e
g oup
o
a
ee
g oup
.
Thus,
we
a e
le
wi h
he
case
"=
2(mod
8)
.

In
his
case,
we
claim
ha
d =
0
o
-
2(mod
8)
.
P oo o
claim
:
I
su ices
o
show ha
d
10 = 0
.
F om
he
ze o
di e en ial
E2'
-2
,
we
see
ha
E3'
-1
=
d
(x
s
)
=
sx
s-1
d
(x),
inishing
he
claim
.
Also
o
n
e Rn,
we
conside
he
di e en ial
gene a ed
by
x
o
and
since
d
10
is
a
de i a ionwe ha e
dio(x0)
=
0
om
he
o mula
is
E
2(n-1)- , -2
d
-
,

E
2(n-1),
-1

E
2(n-1)- , -2
=
0



excep
_
0,2,6
(mod
8)
using
he
p ope y
o
KO
*
(*)
and
d
=

0
.
 o

=

0,6

(mod

8),

see

[31
.

When

=

2
(mod

8
)
E2(n-1)- , -2
i
s
a
ee
g oup
which
su i es
o
E
.
.
Thus
d
=
0
o
all
%
3
.
We
conside
he
il a ions
whe e
Re e ences
KO
1
= Fo
'
-1
D F
1
'
-2
D
. . . .
DF
n-1
'
-n
DF
n,-n-1
=0
and

KO2n-3
=
F
0,
2n-3
DF
1,
2n-4
D
. . .
D

n-1,
n-2
Do
a

0,-1

F
O
'
-1

0

2(n
-
1),
-1
_F
2(n-l)
'
-1
~G
2
=
E
m
_

/
F
1,-2,

ZL
2
=
E~

/
F2n-1,-2
Fp'q
= Ke (KOP
+q
(X)
X=
IP
n-1
(V)
1
Thus
KO
1
=
a
2
o all
n
.
Also
KO2n-3
=
KO-3
o
n =
0(mod
4)
and
KO2n-3
=
KO-7
o
n
_
2(mod
4)
inishing
he
p oo
o
he
heo em
.
KOP
l
q(Xp
-l
))
and
E
.

=
0
o
ei he
p
o
q
odd
.
(1)
AJETUNMOBI,
M
.O
.
:
Ph
.D
.
hesis
Uni e si y
o
Ibadan,
Nige ia
.
1984
.
(2)
FUJII,
-
Michikazu
:
KO-g oups
o
p ojec i e
spaces,
OsakaJou n
.
Ma h
.
4
(1967)
pp
.
141-149
.
(3)
HOGGAR,
S .G
. :
On
KO- heo y
o
G assmannian,
Qua
.
Jou n
.
Ma h
.,
Ox o d
Se
.
(2)
20(1969)
pp
.
447
-
463
.
Rebu
el
15
de 6ebnen
del
1985
Ma hema ics
Deoa men ,
Uni e si y
o
Ibadan
Ibadan
NIGERIA