Morphismes de Hopf

X
Γ = π1(X)π:e X−→ X
ΓX
k≥0
hk:Hk(Γ,Z)−→ Hk(X, Z),
hk
π∗
Hk(Γ,Z)hk −→ Hk(X, Z)π∗ −→ Hk(e X, Z).
h0h1h2
0−→ H2(Γ,Z)h2 −→ H2(X, Z)π∗ −→ H2(e X, Z)
G M G F :M7→ MG
G G
G

G M
F
∀k≥0, Hk(G, M)=RkF(M).
(C•(G, M), d)Ck(G, M)
GkMC0(G, M) = M
df(g1, . . . , gk+1) = g1·f(g2, . . . , gk+1)+
k
X
j=1
(−1)jf(g1, . . . , gj−1, gjgj+1, . . . )
+ (−1)k+1f(g1, . . . , gk).
M=ZG
H0(G, Z) = ZH1(G, Z) = Hom(G, Z).
c:G×G−→ Z
c(g, h) = c(h, k)−c(gh, k) + c(g, hk)
(g, h, k)∈G
GZ
1−→ Z−→ E−→ G−→ 1
H2(G, Z)
•1−→ Z−→ E−→ G−→ 1
s:G−→ E
s(gh)s(g)s(h)E
GZf(g, h)s(gh) =
f(g, h)s(g)s(h)s((gh)k) = s(g(hk))
f G
s f
•f f(g, 1) = f(1, g)=0
Z×G
(n, g)·(m, h)=(n+m+f(g, h), gh).
Z×G
f(n, g)−1= (−n−
f(g, g−1), g−1)Z=Z× {1}

Γ
Xe XΓe X
e
X
ΓEq(e X)q
e
XE0(e X) = C∞(e X)
ΓEq(e X)
Hk(Γ,Eq(e X)) = 0 ∀k≥1,∀q≥0.
k= 0
H0(Γ,Eq(e X)) = Eq(e X)Γ=Eq(X)
X
(Vj)j∈Jπ:e X−→ X
Vjj Uje X π
UjVj(ϕj)j∈J
(Vj)j
χj= (π|Uj)∗(ϕj) = ϕj◦π−1
|Uj.
Uj
(g(Uj))j∈J, g∈Γ
χj,g =g∗(χj) = χj◦g−1.
∀g, h ∈Γ,∀j∈J, χj,gh =g∗χj,h.
P:Ck+1(Γ,Eq(e X)) −→ Ck(Γ,Eq(e X))
P(f)(g1, . . . , gk) = (−1)k+1 X
(j,g)∈J×Γ
χj,gf(g1, . . . , gk, g−1
kg−1 k−1· · · g−1
1g).

d(P(f))
d((−1)k+1P(f))(g1, . . . , gk+1) = g∗
1[P(f)(g2, . . . , gk+1)] +
k
X
i=1
(−1)iP(f)(g1, . . . , gigi+1, . . . , gk+1)+(−1)k+1P(f)(g1, . . . , gk)
=X
j,g
g∗
1(χj,g)g∗ 1f(g2, . . . , gk, g−1
k+1g−1 k· · · g−1
2g)+
k
X
i=1
(−1)iX
j,g
χj,gf(g1, . . . , gigi+1, . . . , gk+1, g−1
k+1g−1 k· · · g−1
1g)+
(−1)k+1 X
j,g
χj,gf(g1, . . . , gk, g−1
kg−1 k−1· · · g−1
1g)
=X
j,g
(χj,g)g∗
1f(g2, . . . , gk, g−1
k+1g−1 k· · · g−1
1g)+
k
X
i=1
(−1)if(g1, . . . , gigi+1, . . . , gk+1, g−1
k+1g−1 k· · · g−1
1g)+
(−1)k+1f(g1, . . . , gk, gk+1g−1
k+1g−1 kg−1 k−1· · · g−1
1g)
=X
j,g
(χj,g)"g∗
1f(g2, . . . , γg
k+2)+
k+1
X
i=1
(−1)if(g1, . . . , gigi+1, . . . , γg
k+2)#
χj,g
γg
k+2 =g−1 k+1g−1 k· · · g−1
1g.
df(g1, . . . , gk+1, γg
k+2)−(−1)k+2f(g1, . . . , gk+1).
χj,g
d(P(f)) = f−(−1)kP(df))
f f
d(P(f)) = f.

e
X
Ck(Γ,Eq(e X)).
E0(e X)Γd//
_
E1(e X)Γd//
_
E2(e X)Γd//
_
. . .
R//
dΓ
E0(e X)d//
dΓ
E1(e X)d//
dΓ
E2(e X)d//
dΓ
. . .
C1(Γ,R)//
dΓ
C1(Γ,E0(e X)) d//
dΓ
C1(Γ,E1(e X)) d//
dΓ
C1(Γ,E2(e X)) d//
. . .
C2(Γ,R)//
dΓ
C2(Γ,E0(e X)) d//
dΓ
C2(Γ,E1(e X)) d//
dΓ
C2(Γ,E2(e X)) d//
dΓ
. . .
. . . . . . . . . . . .
d
dΓ
Γ
hk:Hk(Γ,R)−→ Hk(X, R)
k= 2 f
Γ
E0(e X)Γ//_____
E1(e
X)Γ//_____
E2(e
X)Γ//____ . . .
R//_______
E0(e
X)//______
E1(e
X)d//E2(e X)//____
OO
. . .
C1(Γ,R)//____
C1(Γ,E0(e
X)) d//C1(Γ,E1(e X)) //___
P
OO
C1(Γ,E2(e X)) //___
. . .
f∈C2(Γ,R)//
C2(Γ,E0(e
X)) //___
P
OO
C2(Γ,E1(e X)) //___
C2(Γ,E2(e
X)) //___
. . .
. . . . . . . . . . . .
α2(f)∈ E2(e X) Γ