Hopf fibration
E679319
The Hopf fibration is a fundamental construction in topology that describes the 3-sphere as a fiber bundle of circles over the 2-sphere, revealing deep connections between geometry, algebra, and higher-dimensional spaces.
Statements (64)
| Predicate | Object |
|---|---|
| instanceOf |
circle bundle
ⓘ
fiber bundle ⓘ map between manifolds ⓘ principal bundle ⓘ topological construction ⓘ |
| hasApplicationIn |
Berry phase
NERFINISHED
ⓘ
Skyrme models NERFINISHED ⓘ magnetic monopoles ⓘ quantum spin systems ⓘ topological solitons ⓘ |
| hasBaseSpace | 2-sphere ⓘ |
| hasBaseSpaceIdentifiedWith |
CP^1
ⓘ
complex projective line ⓘ |
| hasDimensionOfBaseSpace | 2 ⓘ |
| hasDimensionOfFiber | 1 ⓘ |
| hasDimensionOfTotalSpace | 3 ⓘ |
| hasFiber |
1-sphere
ⓘ
circle ⓘ |
| hasFiberDescribedAs | orbits of the U(1) action on S^3 ⓘ |
| hasHopfInvariant | 1 ⓘ |
| hasProperty |
admits connection with nonzero curvature
ⓘ
fibers are pairwise linked circles in S^3 ⓘ is not isomorphic to the trivial bundle S^2 × S^1 ⓘ |
| hasStructureGroup |
S^1
ⓘ
U(1) NERFINISHED ⓘ |
| hasStructureGroupIdentifiedWith | U(1) NERFINISHED ⓘ |
| hasTotalSpace | 3-sphere ⓘ |
| hasTotalSpaceIdentifiedWith |
SU(2)
NERFINISHED
ⓘ
unit sphere in C^2 ⓘ |
| hasTypicalFiber | S^1 ⓘ |
| isDenotedBy | S^3 → S^2 ⓘ |
| isExampleOf |
Seifert fibration
NERFINISHED
ⓘ
map of Hopf invariant 1 ⓘ nontrivial fiber bundle ⓘ nontrivial principal bundle ⓘ spherical fibration ⓘ |
| isGeneralizedBy |
S^7 → S^4 Hopf fibration
NERFINISHED
ⓘ
S^{15} → S^8 Hopf fibration ⓘ higher Hopf fibrations ⓘ |
| isNamedAfter | Heinz Hopf NERFINISHED ⓘ |
| isPrincipalBundleOver | 2-sphere ⓘ |
| isPrincipalBundleWithGroup | circle ⓘ |
| isProjectionOnto | CP^1 NERFINISHED ⓘ |
| isRelatedTo |
Clifford algebras
NERFINISHED
ⓘ
complex numbers ⓘ homotopy groups of spheres ⓘ quaternions NERFINISHED ⓘ π_3(S^2) ⓘ |
| isStudiedIn |
differential geometry courses
ⓘ
graduate-level topology ⓘ |
| isUsedIn |
Riemannian geometry
NERFINISHED
ⓘ
bundle theory ⓘ complex geometry ⓘ contact geometry ⓘ gauge theory ⓘ homotopy theory ⓘ quantum field theory ⓘ twistor theory NERFINISHED ⓘ |
| isUsedToShow | π_3(S^2) is nontrivial ⓘ |
| isVisualizedBy | linked circles in 3-space ⓘ |
| representsElementOf | π_3(S^2) ⓘ |
| wasIntroducedBy | Heinz Hopf NERFINISHED ⓘ |
| wasIntroducedInField |
algebraic topology
ⓘ
differential topology ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.