Corrigé feuille n°3 Groupe fondamental

L(X)X
π1(X, x)→L(X)
γ
y∈X ` : [0,1] →X x y γ
`γ`−1
H(s, t) = 
`(t+ 4s(1 −t)) 0 ≤s≤1
4
γ(4s−1) 1
4≤s≤1 2
`(t+ (1 −t)(2 −2s)) 1
2≤s≤1
π1(X, x)
π1(X, x)
L(X)x
π1(X, x)
f0f1x H : [0,1] ×[0,1] →X
H(·,0) = f0H(·,1) = f1γ:t7→ H(0, t)
x f0γf1γ−1t
γ0t H(·, t)γ t 0
H(s, t) = 
H(0,4st) 0 ≤s≤1
4
H(4s−1, t)1
4≤s≤1 2
H(0, t (2 −2s)) 1
2≤s≤1
α, β αβ γ α
s∈[0,1/2] β s ∈[1/2,1]
αβ βα t H(·, t)
α t 1β α 0t
H(s, t) = αβ(s+t
2) 0 ≤s≤1−t 2
αβ s−1 + t
21−t 2≤s≤1
f γ γfγ−1f
X6=∅x0∈X γ : [0,1] →X x0
H: [0,1] ×[0,1] →X t ∈[0,1] H(0, t) = γ(t)
H(1, t) = x0x∈]0,1[ H(x, ·)X H
γ
x0

L(G, e)G e
L(G, e)×L(G, e)→L(G, e)
(α, β)7→ αβ
π1(G, e)×π1(G, e)→π1(G, e)
α α0β β0H I
α α0β β0(s, t)7→ H(s, t)I(s, t)αβ
α0β0
e e
α, β ∈L(G, e) (α∗e)(e∗β) = α∗β(e∗α)(β∗e) = β∗α
α∗e e ∗α α β ∗e e ∗β β
αβ α ∗β β ∗α
π1(G, e)×π1(G, e)→π1(G, e)
R2R
R R2
R2Rnn≥3
R2S1
RnSn−1
n≥3
2
SU2(C)a−¯
b
b¯a|a|2+|b|2= 1
S3
SU2(C)S3
SU2(C)×S1→U2(C)
(M, z)z0
0 1 M π1(U2(C)) 'π1(S1)'Z
GL2(C)U2(C)
Z
1→SU2(C)−→ U2(C)det −−→ S1→1,
z7→ z0
0 1 , U2(C)
SU2(C)S1
SU2(C)
S1

SU2(C)π1(SO2(R)) 'π1(S1)'Z
GL2(R)O2(R)
GL+
2(R)
SO2(R)π1(GL+
2(R)) 'π1(S02(R)) 'Z
1 0
0−1GL+
2(R)
GL−
2(R)π1(GL+ 2(R)) 'Z
X
X=U∪V U V U ∩V
X
ΣX=C+X∪C−X
C+X:= X×h0,1
2i/X × {0}, C−X:= X×h1 2,1i/X × {1}.
ε > 0C+
εX C− εX ε C+X
C−X C+X C−X
C+
εX C− εX
Xε:= C+
εX∩C− εX X
n n = 0 CP0
n≥0CPn+1
CPn2n+2 D U V ε
CPnCPn+1 ε
D
D S2n+1 n≥0S2n+1
t7→ p(Re2iπt)t7→ Re2iπt R
C∗p
p:C→C∗
t7→ Re2iπt Ct7→ p(Re2iπt)p
C∗
t7→ Re2iπt C∗
x7→ f(x)−x
r:D2→S1
i:S2→D2r◦i= idS1i r
π1(S1)→π1(D2)→π1(S1)
π1(S1)
π1(D2)

f:D2→D2
r:D2→S1x r(x)
∂D2=S1[x, f(x)[ r(x)
x f(x)
h:Dn→∆nf: ∆n→∆n
h−1◦f◦h:Dn→Dn
A1A2
[AB]
[A1A2]
(V, E)V E
X
v∈V
d(v) = 2|E|,
d(v)v
v v0d(v)d(v0)
v∈∆2(a, b, c)
v=aA1+bB2+cC2f(v)=(a0, b0, c0)
v
a>a0c(v) = 1
a≤a0b > b0c(v)=2
a≤a0b≤b0c > c0c(v)=3
f v
(Sk, Tk)k≥0
k
B1,kB2,kB3,k Tkc(Bi,k ) = i
i= 1,2,3B1,k B2,k
B3,k B1,kB2,kB3,k
C(a, b, c)C(a0, b0, c0)f(C)
B1,k →C a ≥a0
B2,k →C a ≤a0b≥b0
B3,k →C b ≤b0c0≥c
a=a0b=b0a+b+c=a0+b0+c0= 1 c=c0C
f

g g(−x) = −g(x)
h=g|S1
h h :I→S1h(x+1
2) = −h(x)˜
h:I→R
h h x ∈[0,1
2]
exp 2iπ˜ hx+1
2=−exp 2iπ˜
h(x)= exp 2iπ ˜ h(x) + 1
2
x7→ ˜ hx+1
2−˜
h(x)−1
2Z
m
˜
h(1) = ˜ h1
2+1 2+m=˜
h(0) + 2m+ 1
˜
h(1) −˜ h(0)
h
g g∗:π1(S2)7→ π1(S1)
α:S1→S2
S2h=g∗α g∗
F1, . . . , Fn+1 d1, . . . , dn
n
y−y
x7→ (d1(x), . . . , dn(x))
i∈ {1, . . . , n}di(y) = di(−y)=0 y−y Fi
Fn+1
n= 1 n= 2 S1
n= 2