Démonstration du résultat principal et comparaison à l`homologie de

n= 2
n
n
n n
n= 2
∆epi
∆n∈N∗Epin

n n −1
∆epi [rn]fn −→ [rn−1]fn−1 −−−→ . . . f2 −→ [r1] [m] = {0<
1<· · · < m}
Epin
k
2
EpinEpink F
Cn
(rn,...,r1)(F) = M
a=[rn]fn −→... f2 −→[r1]∈Epin
F(a)
Cn(F) = M
r1,...,rn
Cn
(rn,...,r1)(F),
n
P
i=1
ri
∂j:Cn
(rn,...,r1)→Cn (rn,...rj−1,...,r1)
n
CnH[n]
Epink Cn
n
k A Epin
Ln(A) Epin
[rn]
k E k A
E
bnEpink
k[0] → · · · → [0] 0
H[n]TorEpin(bn,−)
H[n]
i(Epia
n)
i > 0n a Epia
n
k[Epin(a, −)] a
H[n]
i(Epia
n)i > 0
Epia
nH[n]

H[n](F)F([1] →[0] → · · · → [0]) →F([0] →
· · · → [0]) ([1] →[0] → · · · → [0]) →([0] →
· · · → [0])
bnepI[1]→[0]→···→[0]
n→epI[0]→···→[0] n
epIa
nk[Epin[(−, a)]
n= 1
n= 2
n= 2
2
C(F)FEpi2
∂2:C(r,s)(F)→C(r−1,s)(F)h(r,s)(F)
Epi2
∆epi b0, . . . , bs
< b0, . . . , bs>
∆epi [r] [s]r+ 1
bib0+· · · +bi−1b0+· · · +bi−1i
Φi(b0, . . . , bs) =< b0, . . . , bs>−1(i) = [b0+· · ·+bi−1, b0+· · ·+bi−1]
C(r,s)(F) = M
b0+···+bs=r+1
F(< b0, . . . , bs>).
Dj:F(< b0, . . . , bs>)→F(< b0, . . . , bj−1, . . . , bs>)
bj= 1
∂2Dj
[r]<b0,...,bs>//
di
[s]
id
[r−1] <b0,...,bj−1,...,bs>//[s]
i, i + 1 ∈Φj(b0, . . . , bs)
Dj
j(H(∗,s), ∂2)
(s+ 1) F(< b0, . . . , bs>)D0, . . . , Ds
a= [u]→[v] Epi2h(r,s)(Epia
2)=0
r=u=v h(r,s)(Epia
2) = k[∆epi([r],[s])]
s
[0] ∆epi

n k[Epi2(a, [n]→[0])] E(a) [u]
x≤y a(x) = a(y)y∈E x ∈E f : [u]→[v]
Epi2a[n]→[0]
(Ti(f) = f−1([0, i]))0≤i≤n
f
E(a)
Tn(f) = [n]
fEpi2
Tj(di◦f) = Tj(f)j < i
Tj(di◦f) = Tj+1(f)j≥i
(C(r,0)(Epia
2), ∂2)
E(a)
h(r,0)(Epia
2)'
b
Hr(E(a)) b H
Fj=a−1(j)cj=Card FjE E(a)
E∩FjFj
E(a)
E(a)'
v
Y
j=0
[cj].
s= 0 b H∗(A×B)'
b
H∗(A)⊗b H∗(B)A B
b
Hi([n]) = 0 i=n= 1
k u =v cj1
s
Epi2(a, [r]f −→ [s]) = a
σ∈∆epi([v],[s])
{(τ, σ)|τ: [u]→[r]}
∂2
σ: [v]→[s] Epi2
σ
E[s]aEa
E aE= (σ◦a)−1(E)¯
f
−→ σ−1(E)σE:σ−1(E)→E
σ∆epi fE:f−1(E)→
E F : [r]→[s] Epi2
E0, E1E
Epi2(a, f)σ= Epi2(aE0, fE0)σE0×Epi2(aE1, fE1)σE1.
E
m M
b
Hr(E) = HHr(E;BE)BEBE(x, y) = k
x=M y =m0BI×J=BIBJ

(C(∗,s)((Epia
2)σ), ∂2)'
s
O
j=0
(C(∗,0)(Epiaj
2), Dj'∂2).
h(r,s)((Epia
2)σ)aj
u=v k u
σ∈∆epi([v],[s])
H[2] Epia
2a
a= [u]id −→ [u]
r∈Na= [r]→[r]∈Ob Epi2
k h(r,0)(Epia
2)
cr=X
σ∈Σr+1
σ.
0≤s≤r k h(r,0)(Epia
2)
cb0,...,bs=X
(σ0,...,σs)∈Σb0×...Σbs
(σ0, . . . , σs)
< b0, . . . , bs>∈∆epi([r],[s])
Epi2(a, < b0, . . . , bs>: [r]→[s])
a=id
[r]→[r] Σr+1
Σb0×. . . Σbs
(C(∗,s)(Epia
2), ∂2)
> r
r k
1crr
n= 2
H[2] Epia
2a=id[r]
∂1h(r,s)(Epia
2)
∂1(cb0,...,bs) =
s−1
X
i=0
cb0,...,bi+bi+1,...,bs.
(h(r,∗)(Epia
2), ∂1)
∆epi Epir
1n= 1