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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hopf fibration canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7648308 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hopf fibration Context triple: [Heinz Hopf, notableWork, Hopf fibration]
-
A.
Conway sphere
The Conway sphere is a mathematical construct in knot theory used to decompose knots and links into simpler tangles, named after mathematician John Horton Conway.
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B.
Smale’s paradox
Smale’s paradox is a result in differential topology showing that a sphere can be turned inside out in three-dimensional space through smooth deformations without tearing or creasing, challenging intuitive notions of geometry.
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C.
Fubini–Study form
The Fubini–Study form is the canonical Kähler form on complex projective space, encoding its standard Hermitian and symplectic geometry.
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D.
Thom space construction
The Thom space construction is a fundamental operation in algebraic topology that associates a topological space to a vector bundle, playing a central role in cobordism theory and characteristic classes.
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E.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Hopf fibration Target entity description: 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.
-
A.
Conway sphere
The Conway sphere is a mathematical construct in knot theory used to decompose knots and links into simpler tangles, named after mathematician John Horton Conway.
-
B.
Smale’s paradox
Smale’s paradox is a result in differential topology showing that a sphere can be turned inside out in three-dimensional space through smooth deformations without tearing or creasing, challenging intuitive notions of geometry.
-
C.
Fubini–Study form
The Fubini–Study form is the canonical Kähler form on complex projective space, encoding its standard Hermitian and symplectic geometry.
-
D.
Thom space construction
The Thom space construction is a fundamental operation in algebraic topology that associates a topological space to a vector bundle, playing a central role in cobordism theory and characteristic classes.
-
E.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hopf fibration Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.