Un fibrado de Seifert es una 3-variedad que se obtiene construyendo un fibrado del tipo
S
1
⊂
E
→
Σ
{\displaystyle S^{1}\subset E\to \Sigma }
donde
Σ
{\displaystyle \Sigma }
orbifold conos pero no líneas reflectoras (reflector lines ).
Esto último significa que
E
{\displaystyle E}
U
×
S
1
{\displaystyle U\times S^{1}}
U
{\displaystyle U}
Σ
{\displaystyle \Sigma }
p
1
,
p
2
,
.
.
.
,
p
k
{\displaystyle p_{1},p_{2},...,p_{k}}
D
1
,
D
2
,
.
.
.
,
D
k
{\displaystyle D_{1},D_{2},...,D_{k}}
p
i
{\displaystyle p_{i}}
S
1
{\displaystyle S^{1}}
D
×
S
1
{\displaystyle D\times S^{1}}
toros sólidos ).
Para obtener una fibración no trivial en un toro sólido , primero cortamos este
en un disco meridional. Luego en este cilindro sólido damos un giro de
2
π
a
b
{\displaystyle 2\pi {a \over b}}
b
{\displaystyle b}
La siguiente tabla es un diccionario bilingüe entre la primera clasificación original de H. Seifert en 1933 y la 1968-moderna de P. Orlik -F. Raymond
o
1
=
O
o
{\displaystyle o_{1}=Oo}
n
2
=
O
n
{\displaystyle n_{2}=On}
o
2
=
N
o
{\displaystyle o_{2}=No}
n
1
=
N
n
I
{\displaystyle n_{1}=NnI}
n
3
=
N
n
I
I
{\displaystyle n_{3}=NnII}
n
4
=
N
n
I
I
I
{\displaystyle n_{4}=NnIII}
He aquí los once primeros SFS cuya caractéristica de Euler del orbifold es χ>0:
(
O
o
,
0
|
b
)
{\displaystyle (Oo,0|b)\,}
S
2
×
S
1
{\displaystyle S^{2}\times S^{1}}
S
3
{\displaystyle S^{3}}
espacios lentes no triviales.
(
O
o
,
0
|
b
:
(
a
1
,
b
1
)
)
=
L
(
b
a
1
+
b
1
,
a
1
)
,
{\displaystyle (Oo,0|b:(a_{1},b_{1}))=L(ba_{1}+b_{1},a_{1}),\,}
(
O
o
,
0
|
b
:
(
a
1
,
b
1
)
,
(
a
2
,
b
2
)
)
{\displaystyle (Oo,0|b:(a_{1},b_{1}),(a_{2},b_{2}))\,}
(
O
o
,
0
|
b
:
(
2
,
1
)
,
(
2
,
1
)
,
(
a
3
,
b
3
)
)
{\displaystyle (Oo,0|b:(2,1),(2,1),(a_{3},b_{3}))\,}
(
O
o
,
0
|
b
:
(
2
,
1
)
,
(
3
,
b
2
)
,
(
3
,
b
3
)
)
{\displaystyle (Oo,0|b:(2,1),(3,b_{2}),(3,b_{3}))\,}
(
O
o
,
0
|
b
:
(
2
,
1
)
,
(
3
,
b
2
)
,
(
4
,
b
3
)
)
{\displaystyle (Oo,0|b:(2,1),(3,b_{2}),(4,b_{3}))\,}
(
O
o
,
0
|
b
:
(
2
,
1
)
,
(
3
,
b
2
)
,
(
5
,
b
3
)
)
=
(
O
o
,
0
|
−
1
:
(
2
,
1
)
,
(
3
,
1
)
,
(
5
,
1
)
)
{\displaystyle (Oo,0|b:(2,1),(3,b_{2}),(5,b_{3}))=(Oo,0|-1:(2,1),(3,1),(5,1))\,}
esfera de Poincaré
(
N
n
I
,
1
|
b
)
{\displaystyle (NnI,1|b)\,}
R
P
2
×
S
1
{\displaystyle \mathbb {RP} ^{2}\times S^{1}}
fibrado por la esfera
S
2
{\displaystyle S^{2}}
S
2
⊗
S
1
{\displaystyle S^{2}\otimes S^{1}}
(
N
n
I
,
1
|
b
:
(
a
1
,
b
1
)
)
{\displaystyle (NnI,1|b:(a_{1},b_{1}))\,}
R
P
2
×
S
1
{\displaystyle \mathbb {RP} ^{2}\times S^{1}}
b
a
1
+
b
1
{\displaystyle ba_{1}+b_{1}}
S
2
⊗
S
1
{\displaystyle S^{2}\otimes S^{1}}
b
a
1
+
b
1
{\displaystyle ba_{1}+b_{1}}
(
O
n
,
1
|
b
:
)
{\displaystyle (On,1|b:)\,}
prisma-variedad .
(
O
o
,
0
|
b
:
(
a
1
,
b
1
)
)
{\displaystyle (Oo,0|b:(a_{1},b_{1}))\,}
Ahora los siguientes 11 que cumplen χ=0:
(
O
o
,
0
|
b
:
(
3
,
b
1
)
,
(
3
,
b
2
)
,
(
3
,
b
3
,
)
)
{\displaystyle (Oo,0|b:(3,b_{1}),(3,b_{2}),(3,b_{3},))\,}
(
O
o
,
0
|
b
:
(
2
,
1
)
,
(
4
,
1
)
,
(
4
,
b
3
)
)
{\displaystyle (Oo,0|b:(2,1),(4,1),(4,b_{3}))\,}
(
O
o
,
0
|
b
:
(
2
,
1
)
,
(
3
,
b
2
)
,
(
6
,
b
3
)
)
{\displaystyle (Oo,0|b:(2,1),(3,b_{2}),(6,b_{3}))\,}
(
O
o
,
0
|
b
:
(
2
,
1
)
,
(
2
,
1
)
(
2
,
1
)
,
(
2
,
1
)
,
(
2
,
1
)
)
{\displaystyle (Oo,0|b:(2,1),(2,1)(2,1),(2,1),(2,1))\,}
(
O
o
,
1
|
b
)
{\displaystyle (Oo,1|b)\,}
T
×
S
1
{\displaystyle T\times S^{1}}
(
N
o
,
1
|
b
)
)
{\displaystyle (No,1|b))\,}
(
N
n
I
,
1
|
0
:
(
2
,
1
)
,
(
2
,
1
)
)
{\displaystyle (NnI,1|0:(2,1),(2,1))\,}
(
N
n
I
,
2
|
b
)
{\displaystyle (NnI,2|b)\,}
K -fibrados sobre el círculo. Para b=0 es
K
×
S
1
{\displaystyle K\times S^{1}}
K
×
t
S
1
{\displaystyle K\times _{t}S^{1}}
giro de Dehn de K .
(
O
n
,
1
|
b
:
(
2
,
1
)
,
(
2
,
1
)
)
{\displaystyle (On,1|b:(2,1),(2,1))\,}
(
O
n
,
2
|
b
)
{\displaystyle (On,2|b)\,}
(
N
n
I
I
,
2
|
b
)
{\displaystyle (NnII,2|b)\,}
K -fibrados sobre el círculo
K
×
f
S
1
{\displaystyle K\times _{f}S^{1}}
f
{\displaystyle f}
y-homeomorfismo y el y-homeomorfismo compuesto con el único giro de Dehn en la botella de Klein K . 
Datos: Q1935541