1 Espaces projectifs réels 2 Recollement de deux - IMJ-PRG

◦
n
RPn=Rn+1 \ {0}
∼
∼Rn+1 \ {0}x∼y⇔ ∃t∈R∗, y =tx
π:Rn+1 \ {0} −→ RPn
π(x1, . . . , xn+1)=[x1, . . . , xn+1].
RPnπ
RPn
[x1, . . . , xn+1]RPnxn+1 6= 0
πH={xn+1 = 1}
π SnSn
∼RPn
RPn
Sn
+={(x1, . . . , xn+1)∈Sn, xn+1 >0}Sn
π Sn
+
Sn
+
∼RPn
f:Dn−→ Sn
+
x7−→ x, p1− kx|2
π|Sn
+◦fRPn'Dn
x∼ −xpour x∈∂Dn
RPnRPn−1n
g:RPn−1qDn−→ RPng([x]) = [x, 0]
[x]∈RPn−1g(x)=[f(x)] x∈Dn
M N f :∂N −→ ∂M
g:N−→ N
M∪f◦g|∂N 'M∪fN
M∪fN f ∂N
N

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M=q s
p r ∈GL2(Z)
hM:S1×S1−→ S1×S1
(u, v)7−→ (uqvs, upvr)
S1×S1⊂C×C
LM=D2×S1∪hMD2×S1=D2×S1qD2×S1
(x, y)∈∂(D2×S1)∼hM(x, y)∈∂(D2×S1)
M=0 1
1 0 LM'S3
M=1 0
0 1 LM'S1×S2
D2×S1⊂C×C
t:D2×S1−→ D2×S1
(u, v)7−→ (uv, v)
M=q s
p r ∈GL2(Z)
n∈ZLM'LAnAn=q s +nq
p r +np 
LM'LM0M0=q−s
p−r∈GL2(Z)
LM(p, q)
p q L(p, q)
n∈Z
LM'LBnBn=q+np s +nr
p r 
L(p, q)q p
L(−p, −q)'L(−p, q)'L(p, q)
qq0≡ ±1[p]L(p, q)'L(p, q0)
π1(L(p, q)) 'Z/pZ