Table des matières 11 L`homologie singulière et ses outils de ca

(X, A)X
A X
(X, ∅)
R X A ⊂X ι :
A →X
ι∗:C∗(A, R)→C∗(X, R).

C∗(A, R)C∗(X, R)
C∗(X, A, R)
C∗(X, R)C∗(A, R)
C∗(X, A, R) (X, A)
Hn(X, A, R)
C∗(X, A)Hn(X, A)
R
f:
(X, A)→(Y, B)
f∗:C∗(X, A)→C∗(Y, B)
[Pλiσi]7→ [Pλif◦σi],
Hn(f) : Hn(X, A)→Hn(Y, B)
Hn:Top2→R .
C∗(X) = C∗(X, ∅)
Hn(X) = Hn(X, ∅)
Hn({pt}) = (R n = 0
0n > 0
X Xα
Hn(X)'M
α
Hn(Xα).
f, g : (X, A)→(Y, B)
H:X×I→Y f g
H:A×I→B f|B
|Ag|B |A
f, g : (X, A)→(Y, B)

(X, A)
. . . δn+1 −−−→ Hn(A)→Hn(X)→Hn(X, A)δn −→ Hn−1(A)→. . .
→H0(A)→H0(X)→H0(X, A)→0.
f: (X, A)→(Y, B)
. . .δn+1
//Hn(A)//
Hn(f)
Hn(X)//
Hn(f)
Hn(X, A)δn//
Hn(f)
Hn−1(A)//
Hn−1(f)
. . .
. . .δn+1
//Hn(B)//Hn(Y)//Hn(Y, B)δn//Hn−1(B)//. . .
.
(X, A)B⊂A
B⊂◦
A(X\B, A \B)→(X, A)
n
Hn(X\B, A \B)' −→ Hn(X, A).
X A, B ⊂X
◦
A∪◦ B=X
· · · → Hn+1(X)δn+1 −−−→ Hn(A∩B)(Hn(ιA
A∩B),Hn(ιB A∩B))
−−−−−−−−−−−−−−→ Hn(A)⊕Hn(B)
Hn(ιX
A)−Hn(ιY B)
−−−−−−−−−−→ Hn(X)δn −→ Hn−1(A∩B)→. . .
A0, B0X0
f:X→X0f(A)⊂A0f(B)⊂B0
. . . δn+1//Hn(A∩B)//
Hn(f)
Hn(A)⊕Hn(B)//
Hn(f)
Hn(X)δn//
Hn(f)
Hn−1(A∩B)//
Hn−1(f)
. . .
. . .δn+1
//Hn(A0∩B0)//Hn(A0)⊕Hn(B0)//Hn(X0)δn//Hn−1(A0∩B0)//. . .
.
H0
X H0(X)R
π0(X)f:X→Y
H0(f)R π0(f)

X
X→ {pt}Hn(X)
Hn(X)→Hn({pt})
Hn(X) = Hn(X)n > 0
H0(X) = H0(X)⊕R
(X, A)
. . . δn+1 −−−→ Hn(A)→Hn(X)→Hn(X, A)δn −→ Hn−1(A)→. . .
→H0(A)→H0(X)→H0(X, A)→0.
d≥0n
Hn(Sd)'(0d6=n
R d =n
Sn
n≥2
n≥1
f:Dn→Dn
2R3
(1/n, 0,0) 1/n n > 0
Z
Hn(Sn,Z)
x7→ λx λ ∈Z

f:Sn→Snf
deg(f)Hn(f)
deg(f)
rRn+1 r|Sn
deg(r|Sn) = −1
Sn(−1)n+1
V
Hn(V) = Hn(S2)
V S2
I
SO3
A5
Σ3=SO3/I
3
S3
S3
Snn
V3
V S3
gRP2