TD : feuille n°2 Homotopie

X Y
f, g :X→Y f g
Y
f, g :X→Rn
f, g :X→Rn\{0}x∈X
|f(x)−g(x)|<|f(x)|f g
p q r > 0
R > r
{z∈C,|z|=R} → C\{0}
z7→ p(z)z7→ q(z)
n≥1X→Sn
n≥1f, g :X→Sn
x∈X|f(x)−g(x)|<2f g
f:Sn→Snx7→ −x
M[0,1] ×[0,1]
(x, 0) ∼(1 −x, 1) M M S1
On(R)GLn(R)
X X0Y Y 0
X×Y X0×Y0
E k Rnk < n Rn\E
Sn−k−1
CRnRn\C
Sn−1
X A B X X
A B X \A X \B
CX X
X× {0}X×[0,1] q:X×[0,1] →CX ιX
ιX:X→CX
x7→ q(x, 1) .

X CX
f:X→Y f
F:CX →Y F ◦ιX=f
n∈NCSn¯ Dn+1
X Y f :X→Y f
CfCX Y f
CX ∪Y ιX(x)∼f(x)x∈X
n X =SnY
n Cf
n+ 1
f, g :X→Y CfCg
X r :X→A X A ⊂X
r◦i= idAi:A→X A
X
X X X
X
I[0,1] P⊂R2
P=I× {0} ∪ {0} ∪ 1
n, n ∈N∗×I.
P
P
X Y X C(X, Y )
X Y
f, g C(X, Y )
X X γ : [0,1] →X
X γ γ(0) = γ(1)
n≥1γ Sn
γ
n≥1Sn
n≥2γ Sn
0 = a0< a1< . . . < ak= 1
k≥1 [0,1] i∈ {0, . . . , k −1}γ|[ai,ai+1]
aiai+1 Sn
n≥2Sn
Sn
n= 1