Feuilletages et Groupoïdes

∗
K
∗
∗
∗
∗
∗
KK

M
p M Fp
M i :F → M
TF
M
F X (F) = {i∗(X) : X∈C∞(F, T F)}
XF
X, Y ∈ X (F) [X, Y ]X(F)
X(F)F
A
MF
MX(F) = A
F F
FM
R
α∈R−Q R2/Z2
{(a+t, b+αt) : t∈R}
(a, b)∈R2/Z2
G, G0
u:G0→G
i:G→G
i(u)u−1,
s, t :G→G0
m:G2→G G2
{(x, y)∈G|t(y) = s(x)}
t(u(x)) = s(u(x)) = x m(y, u(s(y))) = y=m(u(t(y)), y)
x∈G0y∈G
t(i(y)) = s(y)m(i(y), y) = u(s(y))
s(m(y, z)) = s(z)t(m(y, z)) = t(y) (y, z)∈G2
m(w, m(y, z)) = m(m(w, y), z) (w, y),(y, z)∈G2
G, G0
r, s, u, i, m r, s
G2r, s
i u
G0G
∗

∗
M
G=G0=M i =u=s=t=
G=M×M, G0=M, u(x)=(x, x), t(x, y) =
x, s(x, y) = y, i(x, y)=(y, x)m((z, y),(y, x)) = (z, x)
E M π :E→M
G=E, G0=M, u(x)=(x, 0), s =t=π, i(x, ξ) =
(x, −ξ)m((x, ξ),(x, η)) = (x, ξ +η)
ΓM G =
{(x, γ, y)∈M×Γ×M|x=γy}, G0=M, u(x) = (x, e, x), i(x, γ, y) =
(y, γ−1, x), s(x, γ, y) = y, t(x, γ, y) = x m((x, γ, y),(y, η, z)) = (x, γη, z)
G=M×R∗∪T M ×0, G0=M×R.
u, i, s, t m
M×R∗
T M ×0
M×R∗
T M ×0 (xn, yn, tn)M×R∗
(x, ξ, 0) ∈T M ×0tn→0, xn→x, yn→y
xn−yn
tn→ξ.
G
AG G0G
Gad =G×R∗∪AG×{0}.
5Gad G0
ad =
G0×R. s, t, i, u
G={(x, [γ], y)|x, y ∈M}[γ]
x y G0
M
W(a,[γ],b)={(x, [γxγγ−1
y], y)|x∈U, y ∈V}U, V
x y γx(γy)
U(V)x(y)a(b)
FM
{(x, y)|xRy}
FG={(y, [γ], x) : xRy, γ ∈
C1(F), γ(0) = x, γ(1) = y}[γ]γ

G
∗
∗
A∗
∗ ∗ A
kxyk≤kxk kyk,kx∗xk=kxk2.
∗ ∗
∗
∗
∗
G Cc(G)
f∗g(γ) = Zγ1γ2=γ f(γ1)g(γ2), f∗(γ) = f(i(γ)).
1kfk1=R|f|.
∗
∗
kfkmax = sup
k·k
kfk
∗kk
∗kgk≤kgk1g∈Cc(G).
k·kmax ∗
G∗G C∗(G)
Cc(G)k·kmax
∗
1.5.
Ω1
2(Tγs−1(s(γ))) ⊕Ω1 2(Tγt−1(t(γ))) Ω 1 21
2γ∈G
C∗
max(G)∗
∗C∗
r(G)

∗C0(M)
M
∗
L2(M)
∗C0(E∗)
E
∗C0(M)oΓ
C∗(G)
C0(M/Γ) ⊗ K(L2(Γ)).
2 3
0→ K(L2(M)) ⊗C0(R∗)→C∗(G)ev0 −→ C0(T∗M)→0.
0→C∗(G)⊗C0(R∗)→C∗(Gad)ev0 −→ C0(A∗G)→0.
∗
∗(π1(M)) ⊗ K(L2(M)) ∗(π1(M))
∗π1(M)
∗ ∗
K
∗
K
M
C0(M)
ξ→Γ(ξ) Γ(ξ)
ξ
K0(A)
A K1(A) := K0(C0(R)⊗A)
φ:A→B φ∗:K0(A)→K0(A)
φ∗([E]) = [E⊗φB]
K1K
K0K1
0→Aα −→ Bβ −→ C→0
∗