"Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche" (1931)
E679325
"Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche" (1931) is Heinz Hopf’s seminal paper in which he introduced the Hopf fibration, a foundational concept in algebraic topology describing a nontrivial mapping from the 3-sphere to the 2-sphere.
All labels observed (1)
| Label | Occurrences |
|---|---|
| "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche" (1931) canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7648336 — 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.
Target entity: "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche" (1931) Context triple: [Heinz Hopf, notablePublication, "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche" (1931)]
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A.
Topologie (with Heinz Hopf)
"Topologie" is a foundational 1935 textbook on general topology co-authored by Pavel Alexandrov and Heinz Hopf that helped shape the modern development of the field.
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B.
Neue Geometrie des Raumes
Neue Geometrie des Raumes is a foundational 19th-century mathematical work by Julius Plücker that develops projective and line geometry in three-dimensional space.
-
C.
Theorie der Transformationsgruppen
Theorie der Transformationsgruppen is Sophus Lie’s foundational multi-volume work that established the theory of continuous transformation groups, now known as Lie groups, and their applications to differential equations and geometry.
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D.
“Solutio problematis ad geometriam situs pertinentis”
“Solutio problematis ad geometriam situs pertinentis” is Leonhard Euler’s 1736 Latin paper that founded graph theory and topology by solving the Seven Bridges of Königsberg problem.
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E.
On the Curvature of Space
"On the Curvature of Space" is a pioneering 1922 paper by Alexander Friedmann that introduced non-static, expanding-universe solutions to Einstein’s field equations, laying the theoretical foundation for modern cosmology.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
Target entity: "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche" (1931) Target entity description: "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche" (1931) is Heinz Hopf’s seminal paper in which he introduced the Hopf fibration, a foundational concept in algebraic topology describing a nontrivial mapping from the 3-sphere to the 2-sphere.
-
A.
Topologie (with Heinz Hopf)
"Topologie" is a foundational 1935 textbook on general topology co-authored by Pavel Alexandrov and Heinz Hopf that helped shape the modern development of the field.
-
B.
Neue Geometrie des Raumes
Neue Geometrie des Raumes is a foundational 19th-century mathematical work by Julius Plücker that develops projective and line geometry in three-dimensional space.
-
C.
Theorie der Transformationsgruppen
Theorie der Transformationsgruppen is Sophus Lie’s foundational multi-volume work that established the theory of continuous transformation groups, now known as Lie groups, and their applications to differential equations and geometry.
-
D.
“Solutio problematis ad geometriam situs pertinentis”
“Solutio problematis ad geometriam situs pertinentis” is Leonhard Euler’s 1736 Latin paper that founded graph theory and topology by solving the Seven Bridges of Königsberg problem.
-
E.
On the Curvature of Space
"On the Curvature of Space" is a pioneering 1922 paper by Alexander Friedmann that introduced non-static, expanding-universe solutions to Einstein’s field equations, laying the theoretical foundation for modern cosmology.
- F. None of above. chosen
Statements (34)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical research paper
ⓘ
scientific article ⓘ |
| author | Heinz Hopf NERFINISHED ⓘ |
| citedFor |
geometric description of linked circles in S^3
ⓘ
original definition of the Hopf fibration ⓘ |
| classification | classic paper in 20th-century mathematics ⓘ |
| codomain | 2-sphere S^2 ⓘ |
| describes | nontrivial mapping from 3-sphere to 2-sphere ⓘ |
| domain | 3-sphere S^3 ⓘ |
| fiber | circle S^1 ⓘ |
| field |
algebraic topology
ⓘ
topology ⓘ |
| hasKeyResult |
construction of a map S^3 → S^2 with all fibers circles
ⓘ
description of S^3 as union of linked circles (Hopf fibers) ⓘ example of nontrivial element in π_3(S^2) ⓘ |
| historicalSignificance |
first explicit construction of the Hopf fibration
ⓘ
foundational work in modern algebraic topology ⓘ |
| influenced |
development of fiber bundle theory
ⓘ
development of homotopy theory ⓘ |
| introduces | Hopf fibration NERFINISHED ⓘ |
| introducesConcept | linking of fibers in S^3 ⓘ |
| language | German ⓘ |
| mainTopic |
Hopf fibration
NERFINISHED
ⓘ
algebraic topology ⓘ maps between spheres ⓘ |
| maps | S^3 onto S^2 ⓘ |
| publicationYear | 1931 ⓘ |
| relatedConcept |
Hopf invariant
NERFINISHED
ⓘ
fiber bundle ⓘ homotopy group of spheres ⓘ principal circle bundle ⓘ |
| shows | existence of nontrivial fiber bundle structure on S^3 over S^2 ⓘ |
| studies | continuous maps from S^3 to S^2 ⓘ |
| titleTranslation | On the mappings of the three-dimensional sphere onto the spherical surface ⓘ |
How these facts were elicited
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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.
Subject: "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche" (1931) Description of subject: "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche" (1931) is Heinz Hopf’s seminal paper in which he introduced the Hopf fibration, a foundational concept in algebraic topology describing a nontrivial mapping from the 3-sphere to the 2-sphere.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.