K-théorie et opérateurs de Fredholm

K
K
0
K
0
⊕ ⊗ Hom
X
X
X

η X ξ
X η ⊕ξ
X
R
XR
K0(X)X
X
f, g :X→Y X
η Y f g
f∗η g∗η X
K0(X)
n∈NOn(R)
GLn(R)
K0Knn∈Z
G
G G
G X p :E→X E
G p g ∈G x ∈X G
p−1(x)→p−1(g.x)K
G K0
G(X)G
G
K(X)

K
X
X
X
K
X
λ µ X ξ
X×[0,1] i∗
0ξ'λ i∗ 1ξ'µ
X
λ µ X τ
X λ ⊕τ µ ⊕τ
k(X)X
K0(X)X
K0(X)
XF
H→H
H
End(H)
k(X)→K(X)
λ[Ker λ]−
[Coker λ]K0(X)
λ
T∈ F V
H V ∩Ker T = 0 U
T End(H)V∩Ker S = 0 S∈U
U
φ:V× F →H× F → H× F
T:X→ F X
V H V ∩KerT (x)=0

x∈X X
V×X →H×X→H×X
T
X
T
X H/V K0(X)
V
[X, F]→K0(X)
G G
G
H
G H
K G
G
X
X f =l+c:E0→E X E E0
X
X f l
c X
E0E
E
X
H H+
˜
E X
E
T h(E)
E
X
f=l+c:E0→E
X E 'X×H f
H T h(E0)→H+

X
Y X∧Y X×Y
X× {∗Y} ∪ {∗X} × Y
V+∧W+'(V×W)+
G V +V
G G
(X(V)) G V
H G W 0+∧X(V)→
X(W)V⊂W W 0V W
X Y V
G X(V)→Y(V)
K
G X G
G u
K0
G(X)G H
G
H
G
n∈Zπn
G,H (X;u)n
G X u
u= [η]−[X×V]η G X
V G X ×V G
X V H
T h(η)G
f=l+c:E0→E l
c G G X G
H G
G
[f]∈π0
G;H(X;Ind l)