Cent ans de topologie algébrique

e
Ea
x2+y2/a2= 1,
a > 0a6=a0EaEa0
Ea
f f(x,y)=(x,a0y/a)
EaEa0(x,y)7→ (x,ay/a0)
a
1
x
y
Ea
(5)
(4)
°

Kassel
e
e
(16)
e e
(10)
(18)
s
a f
s−a+f= 2.(1)

Cent ans de topologie alg brique
s a f
s a f
s−a+f= 2
f
a s
s−a+f
s−a+f
s−a+f
Σ Σ0
Σ Σ0
Σ Σ0s−a+f
Σ Σ0
S2S2
χ(S2)χ(S2) = 2
S χ(S)
s−a+f S
χ(S) = s−a+f. (2)
S S0
S S0
χ(S) = χ(S0)
R3
S
S
S0

Kassel
χ(S0) = χ(S)−2.(3)
Sg
g
χ(Sg) = 2 −2g, (4)
SgSg0g6=g0g= 1
T=S1
χ(T) = 0
S2
f,f0:X→Y
ht:X→Y

Cent ans de topologie alg brique
t∈[0,1] t= 0 t= 1 h0=
f h1=f0ht
RnRnx∈Rnt∈[0,1]
ht(x) = (1 −t)xRn
X Y
f:X→Y g :Y→X
g◦f:X→X f ◦g:Y→Y
RnRn
Sg
Sg0g=g0
S β0(S)β1(S)β2(S)
χ(S) = β0(S)−β1(S) + β2(S).(5)
β0(S)
β1(S)β2(S)
ΣS C0(Σ) C1(Σ)
C2(Σ)
ΣC0(Σ)
λ1v1+· · · +λkvkv1, . . . ,vkΣ
λ1, . . . ,λk
C1(Σ) C2(Σ)
Σ
dim C0(Σ) = s, dim C1(Σ) = a, dim C2(Σ) = f,
s a f Σ
d1:C1(Σ) →C0(Σ)