La dimension cohomologique des variétés topologiques

X
X n FX
Hn+1
c(X, F)=0
Hi
c(X, F) =
0i > n i
0−→ F −→ I −→ G −→ 0
I
0 = Hi
c(X, G)−→ Hi+1 c(X, F)−→ Hi+1 c(X, I) = 0
Hi+1
c(X, F)
X n −1X
n
n
n
X n
n−1j:Rn−→ X
Hn
c(X, j!R) = Hn c(Rn,R) = R6= 0
n n
(Fi)i∈I
X n
lim
−→ Hn
c(X, Fi)−→ Hn c(X, lim
−→ Fi)
n
Rn

FX A B
X
· · · −→ Hn
c(A∩B, F)−→ Hn+1 c(A∪B, F)−→ Hn+1 c(A, F)⊕Hn+1 c(B, F)−→ Hn+1 c(A∩B, F)−→ · · ·
n U
n ∂U =¯ U\U n−1
X
n X
n
FX α
Hn+1
c(X, F)
S={Z⊆X|Z X ι∗
Zα6= 0}
ιZZ−→ X X S
S T S Z
T
lim
−→
Z0∈T
(ιZ0)∗ι∗
Z0F −→ (ιZ)∗ι∗
ZF
lim
−→
Z0∈T
Hn+1
c(Z0,F) = lim
−→
Z0∈T
Hn+1
c(X, (ιZ0)∗ι∗
Z0F)−→ Hn+1
c(X, (ιZ)∗ι∗
ZF) = Hn+1 c(Z, F)
ι∗
Z0α α Hn+1
c(Z0,F)Z0
Tι∗
Zα α Hn+1
c(Z, F)Z
T S S
ZS
Hn+1
c(Z, F)Z
p q Z X
U0V0p q
n U p U0
A=Z∩¯ U B =Z∩(X\U)
Hn
c(A∩B, F)−→ Hn+1 c(A∪B, F)−→ Hn+1 c(A, F)⊕Hn+1 c(B, F)
Hn
c(∂U, ι∗ι∗
A∩BF)ι A ∩B=
Z∩∂U ∂U U n ∂U U
n−1
Hn+1
c(Z, F) = Hn+1 c(A∪B, F)−→ Hn+1 c(A, F)⊕Hn+1 c(B, F)

p Z B q Z
A A B Z Z
ι∗
Aα= 0 ι∗ Bα= 0
ι∗
Zα
Hn+1
c(X, F) = 0
X n
X n n
. n = 0 X
. n 1n−1
n−1n−1
Rnn X
RnX
X n