Seifert fibered spaces
E911225
Seifert fibered spaces are three-dimensional manifolds that can be decomposed into a disjoint union of circles arranged in a highly structured, fibered way over a two-dimensional orbifold.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Seifert fibered space | 0 |
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
3-manifold
ⓘ
fiber bundle (up to singular fibers) ⓘ mathematical concept ⓘ |
| appearsIn |
Thurston eight geometries
NERFINISHED
ⓘ
Thurston geometrization conjecture NERFINISHED ⓘ |
| baseDimension | 2 ⓘ |
| baseOrbifoldProperty |
base orbifold may have cone points
ⓘ
base orbifold may have reflector curves ⓘ |
| baseSpace | 2-dimensional orbifold ⓘ |
| classifiedBy |
Euler number of the fibration
ⓘ
Seifert invariants NERFINISHED ⓘ base orbifold ⓘ multiplicities of singular fibers ⓘ |
| constructionMethod | Dehn filling on a circle bundle over a surface ⓘ |
| covers | circle bundle over a surface (in some cases) ⓘ |
| dimension | 3 ⓘ |
| fiber |
S^1
ⓘ
circle ⓘ |
| field |
3-manifold theory
ⓘ
geometric topology ⓘ |
| fundamentalGroupProperty | contains infinite cyclic normal subgroup generated by a regular fiber ⓘ |
| generalizes | circle bundles over closed surfaces ⓘ |
| hasComponent |
regular fibers
ⓘ
singular fibers ⓘ |
| hasExample |
3-sphere with Hopf fibration
ⓘ
S^2 × S^1 ⓘ lens spaces ⓘ torus bundles over S^1 ⓘ |
| hasFundamentalGroup | group fitting into short exact sequence with Z as normal subgroup ⓘ |
| hasInvariant |
Seifert volume (in geometric cases)
ⓘ
orbifold Euler characteristic of the base ⓘ |
| hasStructure |
decomposition into disjoint union of circles
ⓘ
locally trivial circle fibration away from singular fibers ⓘ |
| introducedBy | Herbert Seifert NERFINISHED ⓘ |
| introducedInYear | 1933 ⓘ |
| isSubsetOf |
Haken 3-manifolds (in many cases)
ⓘ
prime 3-manifolds ⓘ |
| localModel |
S^1-bundle over a 2-dimensional disk for regular fibers
ⓘ
fibered solid torus with nontrivial gluing for singular fibers ⓘ |
| namedAfter | Herbert Seifert NERFINISHED ⓘ |
| orientationProperty | can be orientable or non-orientable ⓘ |
| specialCaseOf | graph manifold ⓘ |
| supportsGeometry |
ilde{SL_2(R)}-geometry (in some cases)
GENERATED
ⓘ
E^3-geometry (in some cases) GENERATED ⓘ Nil-geometry (in some cases) GENERATED ⓘ S^2 × R-geometry (in some cases) GENERATED ⓘ S^3-geometry (in some cases) GENERATED ⓘ |
| usedIn |
JSJ decomposition of 3-manifolds
ⓘ
classification of closed orientable 3-manifolds with infinite fundamental group ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.