Homologie simpliciale de quelques sous

Rn

RNn
RNn+ 1 A0, . . . , An
σ
σ Sσσ
KRNK
K K
K K K
K
K L f K
L σ K f σ
L
KRN
|K|
RNf K L
|f|:|K|→|L|f
f|f| |K|
|L|
K
A K B ⊂A
B∈K K K K
KRN
Sσσ K
K L f :K→L∀σ∈K, f(σ)∈L
K N |K|
RNRNK
(es)s∈SKσ K (es)s∈Sσ
f

|f|
RN
RN
K
RN|K|
X K
|K|X
∆[n]{0, . . . , n}
n σ n K
σ K ∆[n] ∆[n]Rn+1
RnBn∆[n]
∆[n]\ {{0, . . . , n}} ∂∆[n]Sn−1
∼K
SK/∼π π(Sσ)
σ K K ∼
K/ ∼
∼G SKx∼y⇔
∃g∈G, x =g.y g ∈G g SK
K/G
G|K|
|K|/G π :K→K/G
|π|:|K|→|K/G|ψ:|K|/G → |K/G|
ψ|K|/G ψ
S2/(Z/2Z)Z/2Z
K
|K|S2Z/2ZK
|K|/(Z/2Z)K/(Z/2Z)
D
K L K K/L L
|L| |K| |K|/|L| |L|
K→K/L
|K|→|K/L|
ψ:|K|/|L|→|K/L|ψ
S2
D

K L |K|/|L|K/L
K, L, M f :K→L
g:K→M L ∪KM L tM
f(x)g(x)x∈K
f:|K|→|L|g:|K| → |M|
|L|∪|K||M|= (|L|t|M|)/∼f(x)∼g(x)x∈ |K|LtM→L∪KM
|L| t |M|→|L∪KM|
ψ:|L| ∪|K||M|→|L∪KM|ψ K
K L f, g
|K|
|L| |M|f
g|L| ∪|K||M|
σ n
{0, . . . , n}σ σ
{0, . . . , n}
(A0, . . . , An) [A0, . . . , An]
K n K Cn(K)
n
(A0, . . . , An)∼(ϕ)(Aϕ(0), . . . , Aϕ(n))ϕ{0, . . . , n}(ϕ)
Cn(K)Z
n
K
dn:Cn(K)→Cn−1(K)
dn([A0, . . . , An]) =
n
X
i=0
(−1)i[A0,..., ˆ Ai, . . . , An]
[A0,..., ˆ Ai, . . . , An]n−1Ai
dn◦dn+1 :Cn+1(K)→Cn−1(K)C−1={0}
d0= 0
dndndn+1
dn
Hn(K) = Ker(dn)/Im(dn+1)H∗(K)
(Hn(K))n≥0
f:K→L Cn(f) : Cn(K)→Cn(L)
Cn(f)([A0, . . . , An]) = [f(A0), . . . , f(An)] f(A0), . . . , f(An)
0f d dn+1 ◦Cn+1(f) = Cn(f)◦dn+1
f