4-sphere S^4
E911210
Riemannian manifold
closed manifold
compact manifold
connected manifold
differentiable manifold
homology sphere
n-sphere
oriented manifold
simply connected manifold
smooth manifold
topological space
The 4-sphere S⁴ is the four-dimensional analogue of the ordinary sphere, a compact, smooth, simply connected manifold that serves as a fundamental object in topology and differential geometry.
All labels observed (1)
| Label | Occurrences |
|---|---|
| 4-sphere S^4 canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11205506 — 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: 4-sphere S^4 Context triple: [Yang monopole, baseSpace, 4-sphere S^4]
-
A.
Hopf fibration
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.
-
B.
SPHERE
SPHERE is a high-contrast imaging instrument on the Very Large Telescope designed primarily for directly observing and characterizing exoplanets and circumstellar disks.
-
C.
Riemann sphere
The Riemann sphere is the complex plane plus a point at infinity, forming a one-dimensional complex manifold topologically equivalent to a sphere and used to study meromorphic functions and complex analysis.
-
D.
Sphere
"Sphere" is a 1998 science fiction thriller film directed by Barry Levinson, based on Michael Crichton's novel about a team of scientists investigating a mysterious spacecraft on the ocean floor.
-
E.
Sphere
Sphere is a British publishing imprint known for releasing a wide range of commercial fiction and non-fiction titles.
- 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: 4-sphere S^4 Target entity description: The 4-sphere S⁴ is the four-dimensional analogue of the ordinary sphere, a compact, smooth, simply connected manifold that serves as a fundamental object in topology and differential geometry.
-
A.
Hopf fibration
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.
-
B.
SPHERE
SPHERE is a high-contrast imaging instrument on the Very Large Telescope designed primarily for directly observing and characterizing exoplanets and circumstellar disks.
-
C.
Riemann sphere
The Riemann sphere is the complex plane plus a point at infinity, forming a one-dimensional complex manifold topologically equivalent to a sphere and used to study meromorphic functions and complex analysis.
-
D.
Sphere
"Sphere" is a 1998 science fiction thriller film directed by Barry Levinson, based on Michael Crichton's novel about a team of scientists investigating a mysterious spacecraft on the ocean floor.
-
E.
Sphere
Sphere is a British publishing imprint known for releasing a wide range of commercial fiction and non-fiction titles.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
Riemannian manifold
ⓘ
closed manifold ⓘ compact manifold ⓘ connected manifold ⓘ differentiable manifold ⓘ homology sphere ⓘ n-sphere ⓘ oriented manifold ⓘ simply connected manifold ⓘ smooth manifold ⓘ topological space ⓘ |
| admitsRoundMetric | true ⓘ |
| boundaryOf | 5-ball B^5 ⓘ |
| canBeDescribedAs | SO(5)/SO(4) ⓘ |
| cohomologyRing | Z[α]/(α^2) with deg(α)=4 ⓘ |
| constantSectionalCurvature | 1 ⓘ |
| definedAs | {x in R^5 : ||x|| = 1} ⓘ |
| dimension | 4 ⓘ |
| embeddedIn | R^5 ⓘ |
| EulerCharacteristic | 2 ⓘ |
| fundamentalGroup | trivial group ⓘ |
| generalizes |
2-sphere S^2
ⓘ
3-sphere S^3 ⓘ |
| hasCanonicalRiemannianMetric | round metric ⓘ |
| homologyGroupH_0 | Z ⓘ |
| homologyGroupH_1 | 0 ⓘ |
| homologyGroupH_2 | 0 ⓘ |
| homologyGroupH_3 | 0 ⓘ |
| homologyGroupH_4 | Z ⓘ |
| isBoundary | true ⓘ |
| isCompact | true ⓘ |
| isConnected | true ⓘ |
| isModelSpaceFor | constant positive curvature geometry in dimension 4 ⓘ |
| isOrientable | true ⓘ |
| isParallelizable | false ⓘ |
| isPrototypeOf | compact simply connected 4-manifold ⓘ |
| isSimplyConnected | true ⓘ |
| isSimplyConnectedAtInfinity | true ⓘ |
| isSimplyConnectedHomologySphere | true ⓘ |
| isSymmetricSpace | true ⓘ |
| notation | S^4 ⓘ |
| pi_1 | 0 ⓘ |
| pi_2 | 0 ⓘ |
| pi_3 | Z ⓘ |
| pi_4 | Z_2 ⓘ |
| usedIn |
differential geometry
ⓘ
gauge theory ⓘ general relativity ⓘ homotopy theory ⓘ 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: 4-sphere S^4 Description of subject: The 4-sphere S⁴ is the four-dimensional analogue of the ordinary sphere, a compact, smooth, simply connected manifold that serves as a fundamental object in topology and differential geometry.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.