I- Qu`est-ce que le mapping class group? A.) Définition. B.) Propriété

o
o
o
[f] [f] = [g]
[f] = [g]⇔fH =gH H =o
⇔g−1fH =H⇔g−1f∈H
⇔ ∃G∈F(X×[0; 1], X)G(x, o) = g−1f(x), G(x, 1) = Id(x)
G(x, s)
As(x) = G(f−1(x), s)
s
F(x, s) = As(x)−1
Bs(x) =
F(x, s)G(x, s) = Bs(f−1(x))−1.
g−1f

D2
F: (x, t)7→ ((1 −t)φ(x
1−t) 0 ≤ |x|<1
x1−t≤ |x| ≤ 1
φ
R
F(x, t) = t∗f(x) + (1 −t)∗g(x)
R Z Z
R
− ∝ +∝
F(x, t) = t∗f(x) + (1 −t)∗g(x)
R
RA B
A
R
A A
B[0,1]
f∈ A IdR∈ B

Π2
MCG(Π2)GL2(Z)
x0
x1[0,1] x0x1
x0x0x1x0=x1
γ1γ2
γ(t)7→ (γ1(2u) 0 ≤u≤1
2
γ2(2u−1) 1
2≤u≤1
γ1γ2
([0; 1]×[0; 1], X)
γ1γ2x0
[γ]
[γ1][γ2] = [γ1γ2]
γ1
τ1γ2τ2H1H2
H(t, x)7→ (H1(2u, t) 0 ≤u≤1
2
H2(2u−1, t)1
2≤u≤1
γ1γ2τ1τ2
x0= (1,1)
[γ] [γ−1]
u→1−u

γ(0) = γ(1) = (1,1)
γ γ = (exp(iθ1(u)),exp(iθ2(u)))
k1k2θ1(1) = 2k1π;θ2(1) = 2k2π
A
A
t= 1/2
A(γ2)γ1
A(γ1γ2) = A(γ1) + A(γ2)
A(γ1) = A(γ2)
F F
η|(u1, t1)−
(u2, t2)| ≤ η⇒ |F(u1, t1)−F(u2, t2)| ≤ 1n
η fi
F(u, ti)ti
A(fi) = A(fi+1)|fi(u)−fi+1(u)|=|F(u, ti)−
F(u, ti+1)| ≤ 1fi(u) = (exp(iθi1(t)),exp(iθi2(t)))
fi+1 |exp(iθi1(u))−exp(iθ(i+1)1(u))| ≤ 1⇒ | sin(θi1(u)−θ(i+1)1(u)
2| ≤
1/2
A(fi)6=A(fi+1)π
F
A(f0) = A(fn)
A(γ1) = A(γ2)
A(γ1) = A(γ2)γ1= (exp(iθ1),exp(iτ1))); γ2=
(exp(iθ2),exp(iτ2))) H(u, t) = (exp(i(tθ1(u))+(1−t)θ2(u)),exp(i(tτ1(u))+
(1 −t)τ2(u)))
b
A[γ]A(γ)
b
A
(exp(u∗2iπk1),exp(u∗
2iπk2)) (k1, k2)b A([γ1][γ2]) = b A([γ1γ2]) =
A(γ1γ2) = A(γ1) + A(γ2) = b A([γ1]) + b A([γ2])
γ γ γ

M CG(Π2)
GL2(Z)
MCG(Π2)
f−1
f([γ1][γ2]) = f([γ1γ2]) = [f(γ1γ2] = [f(γ1)f(γ2)] =
[f(γ1)][f(γ2)]
fZ2Z2
f((a, b)) = b A◦f◦b A−1(a, b)
Φf
f L(Z2)
b
A f((a1, b1)+(a2, b2)) =
f(b A(γ1)+ b A(γ2)) = f(b A([γ1γ2])) = f((a1+a2, b1+b2)))
b
Ab A
f
f◦g(b A([γ])) = b A([f(g(γ))]) = f(b A([g(γ)])) =
f◦g(b A([γ])
f=g
ΦMCG(Π2)
f
ΦMCG(Π2)GL2(Z)
M=a b
c d f(exp(iτ),exp(iθ)) =
(exp(2iπ(aτ +bθ)),exp(2iπ(cτ +dθ)))
f=g
γ f(γ)g(γ) Φ f=g
b
A