Variété des représentations du groupe fondamental d`une variété

4
C
CX k
k X(k)G
Hom(Γ, G)
XΓ = π1(X, x)
GRρ: Γ →G(R)
G(R)ρ
Hom(Γ, G)
XΓ = π1(X, x)
GRρ: Γ →G(R)
ρHom(Γ, G)
1,2 2,3,4
X G
GF
X x ∈X U x F|U
G
F|U→ F|VV⊂UAut(G)Fx
G

G p :E→X G
Aut(G)
U⊂X p−1(U)'U×G U, V
(U∩V)×G→(U∩V)×G(x, g)7→ (x, ϕ(g)) ϕ
U∩V→Aut(G)E
G
F
E
(Ui)
X giG Uiϕij ∈Aut(G)
Ui∩Ujgi=ϕij gjE ϕij
Ui∩UjAut(G)F
E
X G
X
X
e
X x ∈X π1(X, x)
p:E→X Exx
˜x∈Exγ: [0,1] →X
γ(0) = x˜γ: [0,1] →E˜γ(0) = ˜x γ, γ0
X˜γ, ˜γ0
π1(X, x)×Ex−→ Ex
(γ, ˜x)7−→ ˜γ(1),˜γ(0) = ˜x
γ Ex
π1(X, x)
G
π1(X, x)Ex
x, y X Ex, Ey
π1(X, x), π1(X, y)x y
π1(X, x) ( e X, ˜x)
(e X, ˜x)
γ: [0,1] →X γ(0) = x˜x x
π: ( e X, ˜x)→(X, x)γ7→ γ(1) π1(X, x)
π1(X, x)G
G X
E X e Xe X×G
Ee X×G(˜x.γ, g)∼(˜x, γ.g)
π1(X, x)
(X, x)
m E
GL(m, R)
π1(X, x)Rm
π1(X, x)GL(m, R)

XAk
X
k
m p :E→X E E
C∞E F
U p−1(U)'U×Rm
C∞f:U→p−1(U)Rmp◦f= idU
F k Ak
X(F) :=
Ak
X⊗F∇:Ak X(F)→ Ak+1
X(F)
∇(Xαi⊗fi) = Xdαi⊗fi
d
F
∇◦∇= 0 s E
∇(s)=0
E
F∇
F s ∇(s)=0
(A∗(X, F ),∇)
F
(Ω∗(X, F ),∇)FΩX
X∇
GgGg
Ad ρ:π1(X, x)→G
π1(X, x)GL(m, R) Ad ◦ρ π1(X, x)g
Ad(ρ)∇
(A∗(X, Ad(ρ)),∇) Ad(ρ)
[α⊗u, β ⊗v]=(α∧β)⊗[u, v]
α, β u, v Ad(ρ)
G
G M G
Z[G]
G G M
G MGx∈M∀g∈G, g.x =x
M7→ MG
G MGHomZ[G](Z, M)ZG

G M
Hn(G, M) := Extn
Z[G](Z, M).
M7→ MG
0−→ M−→ M0−→ M1−→ · · ·
M G
Extn(−, M) Hom(−, M)
Extn(Z, M)
0←− Z←− P0←− P1←− · · ·
Z Z[G]
Hom(−, M)
M7→ Hn(G, M)G
H0M7→ MG
0→L→M→N G
0−→ LG−→ MG−→ NG−→ H1(G, L)−→ H1(G, M )−→ H1(G, N)−→ H2(G, M )−→ · · ·
G
G
Hn(G, M)
GZ
C∗(G, M)H∗(G, M)d
Zn(G, M)Bn(G, M)
n Cn(G, M)
f:Gn+1 →M
∀g∈G, f(g.x0, . . . , g.xn) = g.f(x0, . . . , xn)
d:Cn(G, M)→Cn+1(G, M)
df (x0, . . . , xn) =
n+1
X
i=0
(−1)if(x0,..., bxi, . . . , xn+1).
(y1|. . . |yn) := (1, y1, y1y2, . . . , y1y2· · · yn)
C∗(G, M) (x0, . . . , xn)
(y1|. . . |yn)
n f :Gn→M
df (y1|. . . |yn+1) = y1.f(y2|. . . |yn+1)
+
n
X
i=1
(−1)if(y1|. . . |yi−1|yiyi+1|yi+2|. . . |yn+1)
+ (−1)n+1f(y1|. . . |yn).