Théorème du point fixe de Lefschetz

K
Cp∂p:Cp→Cp−1
Hp
K

f:|K| → |K| |K|
K
f
f
f f
Rnkxk:= Px2
i1
2
x= (x1, ..., xn)n
Bn={x∈Rn| kxk ≤ 1}
n−1
Sn−1={x∈Rn| kxk= 1}
B0B1[−1,1]
S0={−1,1}
X
A⊂X A◦A A
{a0, a1, ..., an} ⊂ RN
∀t1, ..., tn∈R
n
X
i=0
ti= 0
n
X
i=0
tiai= 0
ti= 0 ∀0≤i≤n
a0, a1, ..., an
a1−a0a2−a0, ..., an−a0
n
X
i=1
si(a0−ai)=0
⇐⇒

n
X
i=1
si!a0−
n
X
i=1
siai= 0
⇐⇒
t0a0+
n
X
i=1
tiai= 0
n
X
i=0
ti= 0
n
X
i=1
si(a0−ai)=0⇒si= 0 ∀i
n
X
i=0
ti= 0
n
X
i=0
tiai= 0 ⇒ti= 0 ∀i
n P
{a0, ..., an}
x=
n
X
i=0
tiai{t0, ..., tn} ⊂ R
P
s1, ..., sn∈R
x=a0+
n
X
i=1
si(ai−a0)
tisi
x∈P w ∈RN
{w, a0, ..., an}
{a0, ..., an}
RNσ n {a0, ..., an}
x∈RN
x=
n
X
i=0
tiai
n
X
i=0
ti= 1 ti≥0∀i
tix∈σ
ti(x)x
a0, ..., an