Hopf invariant
E679320
The Hopf invariant is a topological integer-valued invariant that classifies certain continuous maps between spheres, playing a central role in homotopy theory and the study of higher-dimensional linking.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hopf degree theorem | 1 |
| Hopf invariant canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7648309 — 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 invariant Context triple: [Heinz Hopf, notableWork, Hopf invariant]
-
A.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
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B.
h-cobordism theorem
The h-cobordism theorem is a fundamental result in differential topology that classifies when two high-dimensional manifolds are diffeomorphic by analyzing the structure of a cobordism between them.
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C.
Hirzebruch signature theorem
The Hirzebruch signature theorem is a fundamental result in differential topology that expresses the signature of a smooth, compact, oriented 4k-dimensional manifold as a polynomial in its Pontryagin classes.
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D.
Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
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E.
Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
- 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 invariant Target entity description: The Hopf invariant is a topological integer-valued invariant that classifies certain continuous maps between spheres, playing a central role in homotopy theory and the study of higher-dimensional linking.
-
A.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
-
B.
h-cobordism theorem
The h-cobordism theorem is a fundamental result in differential topology that classifies when two high-dimensional manifolds are diffeomorphic by analyzing the structure of a cobordism between them.
-
C.
Hirzebruch signature theorem
The Hirzebruch signature theorem is a fundamental result in differential topology that expresses the signature of a smooth, compact, oriented 4k-dimensional manifold as a polynomial in its Pontryagin classes.
-
D.
Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
-
E.
Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
homotopy invariant
ⓘ
integer-valued invariant ⓘ topological invariant ⓘ |
| appearsIn |
Adams’s solution of the Hopf invariant one problem
ⓘ
obstruction theory ⓘ rational homotopy theory ⓘ study of H-spaces ⓘ |
| appliesTo |
continuous maps between spheres
ⓘ
maps S^{2n-1} → S^n ⓘ |
| centralIn | Hopf invariant one problem NERFINISHED ⓘ |
| constraint | Hopf invariant one maps exist only in dimensions 1, 2, 4, and 8 ⓘ |
| context |
CW-complex mapping cones
ⓘ
maps between spheres of odd dimension domain ⓘ |
| definedOn | homotopy classes of maps ⓘ |
| definedUsing |
cellular decomposition of mapping cone
ⓘ
cohomology operations ⓘ cup product in cohomology ⓘ |
| field |
algebraic topology
ⓘ
homotopy theory ⓘ |
| generalization |
Massey products
NERFINISHED
ⓘ
secondary cohomology operations ⓘ |
| hasSpecialCase |
Hopf invariant of the Hopf fibration S^7 → S^4 equals 1
ⓘ
Hopf invariant of the Hopf fibration S^{15} → S^8 equals 1 ⓘ Hopf invariant of the Hopf map S^3 → S^2 equals 1 ⓘ |
| hasVariant |
mod p Hopf invariant
NERFINISHED
ⓘ
reduced Hopf invariant ⓘ stable Hopf invariant ⓘ |
| implies | existence of higher-dimensional linking phenomena ⓘ |
| namedAfter | Heinz Hopf NERFINISHED ⓘ |
| property |
additive under composition in certain contexts
ⓘ
homotopy invariant of maps ⓘ |
| relatedTo |
Adams spectral sequence
NERFINISHED
ⓘ
Hopf fibration NERFINISHED ⓘ J-homomorphism NERFINISHED ⓘ Whitehead product NERFINISHED ⓘ cohomology cup product ⓘ complex numbers ⓘ linking number ⓘ normed division algebras ⓘ octonions ⓘ quaternions ⓘ real numbers ⓘ stable homotopy groups of spheres ⓘ |
| usedFor |
classification of certain homotopy classes of maps between spheres
ⓘ
distinguishing non-homotopic maps with same degree ⓘ study of higher-dimensional linking ⓘ |
| usedToProve | nontriviality of certain homotopy groups of spheres ⓘ |
| valueType | integer ⓘ |
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 invariant Description of subject: The Hopf invariant is a topological integer-valued invariant that classifies certain continuous maps between spheres, playing a central role in homotopy theory and the study of higher-dimensional linking.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hopf degree theorem