In der Mathematik ist Serres mod-C-Theorie ein Konzept der Homotopietheorie , demzufolge sich manche Sätze der algebraischen Topologie modulo Klassen abelscher Gruppen formulieren lassen.
Sei
C
{\displaystyle {\mathcal {C}}}
abelscher Gruppen mit der Eigenschaft, dass für Gruppen in einer exakten Sequenz
0
→
A
→
B
→
C
→
0
{\displaystyle 0\to A\to B\to C\to 0}
A
∈
C
{\displaystyle A\in {\mathcal {C}}}
C
∈
C
{\displaystyle C\in {\mathcal {C}}}
B
∈
C
{\displaystyle B\in {\mathcal {C}}}
A
∈
C
{\displaystyle A\in {\mathcal {C}}}
A
⊗
B
∈
C
{\displaystyle A\otimes B\in {\mathcal {C}}}
B
{\displaystyle B}
A
∈
C
{\displaystyle A\in {\mathcal {C}}}
H
i
(
A
)
∈
C
{\displaystyle H_{i}(A)\in {\mathcal {C}}}
i
>
0
{\displaystyle i>0}
Ein Homomorphismus zwischen abelschen Gruppen
f
:
A
→
B
{\displaystyle f\colon A\to B}
C
{\displaystyle {\mathcal {C}}}
C
{\displaystyle {\mathcal {C}}}
C
{\displaystyle {\mathcal {C}}}
C
{\displaystyle {\mathcal {C}}}
C
{\displaystyle {\mathcal {C}}}
C
{\displaystyle {\mathcal {C}}}
C
{\displaystyle {\mathcal {C}}}
Der von Serre bewiesene „Satz von Hurewicz mod
C
{\displaystyle {\mathcal {C}}}
X
{\displaystyle X}
π
0
X
=
π
1
X
=
0
{\displaystyle \pi _{0}X=\pi _{1}X=0}
π
i
X
∈
C
{\displaystyle \pi _{i}X\in {\mathcal {C}}}
i
<
n
{\displaystyle i<n}
H
i
(
X
)
∈
C
{\displaystyle H_{i}(X)\in {\mathcal {C}}}
0
<
i
<
n
{\displaystyle 0<i<n}
π
n
X
→
H
n
(
X
)
{\displaystyle \pi _{n}X\to H_{n}(X)}
C
{\displaystyle {\mathcal {C}}}
C
=
{
0
}
{\displaystyle {\mathcal {C}}=\left\{0\right\}}
Satz von Hurewicz .
Der von Serre bewiesene „Satz von Whitehead mod
C
{\displaystyle {\mathcal {C}}}
X
,
Y
{\displaystyle X,Y}
π
0
X
=
π
0
Y
=
0
,
π
1
X
=
π
1
Y
=
0
{\displaystyle \pi _{0}X=\pi _{0}Y=0,\pi _{1}X=\pi _{1}Y=0}
f
:
X
→
Y
{\displaystyle f\colon X\to Y}
f
∗
:
π
2
X
→
π
2
Y
{\displaystyle f_{*}\colon \pi _{2}X\to \pi _{2}Y}
C
{\displaystyle {\mathcal {C}}}
n
{\displaystyle n}
f
∗
:
H
i
(
X
)
→
H
i
(
Y
)
{\displaystyle f_{*}\colon H_{i}(X)\to H_{i}(Y)}
C
{\displaystyle {\mathcal {C}}}
i
<
n
{\displaystyle i<n}
C
{\displaystyle {\mathcal {C}}}
i
=
n
{\displaystyle i=n}
f
∗
:
π
i
X
→
π
i
Y
{\displaystyle f_{*}\colon \pi _{i}X\to \pi _{i}Y}
C
{\displaystyle {\mathcal {C}}}
i
<
n
{\displaystyle i<n}
C
{\displaystyle {\mathcal {C}}}
i
=
n
{\displaystyle i=n}
Für
C
=
{
0
}
{\displaystyle {\mathcal {C}}=\left\{0\right\}}
J.-P. Serre: Groupes D'Homotopie Et Classes De Groupes Abelien , Ann. Math. 58, 258–294, 1953. online 
