Un théorème d`Adams

∗
n
n
Rn
SnRn+1 n
Snn= 0,1,3 7
n= 0,1,3,7Sn
n= 0,1 3 Sn
S7
∗

S0,S1,S3S7
Sn

S0 ,S1 ,S3S7
•S0
•S1
S1S1
1
•S3SU(2)
a b
−b aa b |a|2+|b|2= 1
C2S3
S3
SU(2)
su(2) SU(2)
su(2) = a b
−b−a, a ∈R, b ∈C.
e1=1 0
0−1, e2=0 1 −1 0, e3=0i i0.
ξkekξk(A) = Aek
A SU(2) Ae1, Ae2, Ae3A
A SU(2)
S3
•S7R8
O
O R 1i j k l li lj lk
·1i j k l li lj lk
1 1 i j k l li lj lk
i i −1k−j−li l −lk lj
j j −k−1i−lj lk l −li
k k j −i−1−lk −lj li l
l l li lj lk −1−i−j−k
li li −l−lk lj i −1−k j
lj lj lk −l−li j k −1−i
lk lk −lj li −l k −j i −1

OxO
x∗1
i j lk ||x|| =x.x∗
R8y/x =||x||−2y.x∗
(y/x).x =y
S7
V1VR7
(x, y)∈TS77→ (x, y/x)TS7S7×V y/x
V1
x y/x =y.x∗
1 (y.x∗)1=< y|x > < ·|· >
y x (y.x∗)1=< y|x >= 0
(x, z)∈S7×V7→ (x, z.x)
< z.x|x >= ((z.x).x∗)1=z1= 0 z V
(z.x).x∗=z
y/x
C1TS7
S7×R7S7
S3
R4H
nSn
Sn
Sn
XSn
γ: [0, π]×Sn→Sn
(t, x)7→ cos(t)x+ sin(t)X(x)
X X(x)x
γSnγ
t= 0 t=πSn
Hn(Sn)R

XK
R C
•
X
X
(ξ, π, X)ξ π :ξ→X
ξx=π−1(x)K
x∈X U x n
h:U×Kn→π−1(U)y∈U hy:v7→ h(y, v)
K
ξ
π
n x ξ
X×Rn
X=Snξx={v∈Rn+1, x⊥v}ξ=Fx∈Xξx
Sn×Rn+1 Sn
ξ s :X→ξ π◦s=IdX
x7→ 0x0xx
s1, ..., snn ξ
x∈X(s1(x), ..., sn(x)) ξx