"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.
Statements (34)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical research paper
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scientific article ⓘ |
| author | Heinz Hopf NERFINISHED ⓘ |
| citedFor |
geometric description of linked circles in S^3
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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
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topology ⓘ |
| hasKeyResult |
construction of a map S^3 → S^2 with all fibers circles
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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
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foundational work in modern algebraic topology ⓘ |
| influenced |
development of fiber bundle theory
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development of homotopy theory ⓘ |
| introduces | Hopf fibration NERFINISHED ⓘ |
| introducesConcept | linking of fibers in S^3 ⓘ |
| language | German ⓘ |
| mainTopic |
Hopf fibration
NERFINISHED
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algebraic topology ⓘ maps between spheres ⓘ |
| maps | S^3 onto S^2 ⓘ |
| publicationYear | 1931 ⓘ |
| relatedConcept |
Hopf invariant
NERFINISHED
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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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
Heinz Hopf
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notablePublication
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"Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche" (1931)
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