n-types d`homotopie

n
X
(πn(X))n≥1
X Y
X Y X Y
f:X→Y
X n Y
f n
X
n n = 0
X n = 1
n= 2
2
n n
n≥3
I[0,1]

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f:X→Y g :X→Y f g
F:X×I→Y f =F i0g=F i1
i0:X→X×I x (x, 0) i1:X→X×I x
(x, 1)
Ft:X→Y Ft(x) = F(x, t)F0=f F1=g
Ftf g
Toph
X Y
X Y
Rn
n0
Snn Bnn
X
(Xn)n≥0n
αn(φα:Sn→Xn)α∈αn
X XnX
XnO X
Xn
X0
n≥0Xn+1 Xnn
(φα)α∈αnXn+1
Xnq(qα∈αnBn+1
α)x Bn+1 αφα(x)X
x Sn+1 Bn+1

CW CWh
X Y f :X→Y
i f(Xi)⊂Yi
f:X→Y
f
(X, x0)
π1(X, x0)
x0γ I X
γ(0) = γ(1) = x0γ1γ2
γ2γ1[0,2] X
I X γ1γ2
π1(X, x0)x0
π1(X, x0)
x0γ
γ
π1
f: (X, x)→(Y, y)γ
Y fγ X
X
π1(X, x)
x0y0γ π1(X, y0) = γπ1(X, x0)γ−1
X π1(X)

X X
x0x0
Π(X)
Π1(X)X x y
X x y
x y HomΠ(X)(x, x) = π(X, x)
1X
π1(X)X22
1 1
π1
π1π1(Rn) = 0 π1(S1) = Z
π1(Sn) = 0 n≥2π1(Rn\{x0, ..., xk})k+ 1
n≥2
(X, x0)
n≥1
n f :In→X Inx0
f: (In, ∂In)→(X, x0)f1f2
(f1f2)(x) = (f2(x1/2, x2, . . . , xn)x1≤1/2
f1(1 −x1/2, x2, . . . , xn)x1≥1/2
πn(X, x0)
πn(X, x0)
n= 1
f1+f2f1f2
n= 1
n

n
πnn X
(Xi)πn(X)
Xn+1
πn
f:X→Y f
f
n≥1x X πn(f, x)
πn(X, x)πn(Y, f(x))
f:X→Y
f
X
Y X Y
C A C
F:C → C0C0A