Bott periodicity
E886930
Bott periodicity is a fundamental theorem in homotopy theory and K-theory that reveals a repeating pattern in the homotopy groups of classical groups, leading to the periodic structure of topological K-theory.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Bott periodicity canonical | 1 |
| Bott periodicity theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10829434 — 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: Bott periodicity Context triple: [K-Theory, aboutConcept, Bott periodicity]
-
A.
K-theory
K-theory is a branch of algebraic topology and algebraic geometry that studies vector bundles and generalized cohomology theories using algebraic and categorical methods.
-
B.
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.
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C.
Eilenberg–MacLane spaces
Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
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D.
Hopf invariant
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.
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E.
Pontryagin classes
Pontryagin classes are characteristic classes associated with real vector bundles that capture topological information about the bundle’s curvature and play a central role in differential topology and geometry.
- 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: Bott periodicity Target entity description: Bott periodicity is a fundamental theorem in homotopy theory and K-theory that reveals a repeating pattern in the homotopy groups of classical groups, leading to the periodic structure of topological K-theory.
-
A.
K-theory
K-theory is a branch of algebraic topology and algebraic geometry that studies vector bundles and generalized cohomology theories using algebraic and categorical methods.
-
B.
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.
-
C.
Eilenberg–MacLane spaces
Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
-
D.
Hopf invariant
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.
-
E.
Pontryagin classes
Pontryagin classes are characteristic classes associated with real vector bundles that capture topological information about the bundle’s curvature and play a central role in differential topology and geometry.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in K-theory ⓘ result in algebraic topology ⓘ result in homotopy theory ⓘ |
| appliesTo |
orthogonal groups
ⓘ
stable homotopy groups of classical Lie groups ⓘ symplectic groups ⓘ unitary groups ⓘ |
| describes |
periodic behavior of homotopy groups of classical groups
ⓘ
periodic structure of topological K-theory ⓘ |
| field |
algebraic K-theory
ⓘ
homotopy theory ⓘ operator K-theory ⓘ topological K-theory ⓘ topology ⓘ |
| generalizedBy |
Bott periodicity for C*-algebras
NERFINISHED
ⓘ
Bott periodicity in operator K-theory ⓘ |
| hasConsequence |
classification of complex vector bundles via K-theory
ⓘ
classification of real vector bundles via KO-theory ⓘ computation of homotopy groups of classical Lie groups ⓘ periodic table of topological insulators and superconductors ⓘ structure of generalized cohomology theory K ⓘ |
| implies |
2-fold periodicity in complex K-theory
ⓘ
8-fold periodicity in real K-theory ⓘ |
| influenced |
applications of topology in mathematical physics
ⓘ
development of index theory ⓘ development of topological K-theory ⓘ |
| namedAfter | Raoul Bott NERFINISHED ⓘ |
| period |
2 for complex K-theory K
ⓘ
8 for real K-theory KO ⓘ |
| proofUses |
Morse theory on loop spaces
ⓘ
geometry of symmetric spaces ⓘ |
| provedBy | Raoul Bott NERFINISHED ⓘ |
| relatedTo |
Atiyah–Singer index theorem
NERFINISHED
ⓘ
C*-algebra K-theory NERFINISHED ⓘ Clifford algebras NERFINISHED ⓘ Kasparov KK-theory NERFINISHED ⓘ spin geometry ⓘ |
| states | stable homotopy groups of classical groups are periodic ⓘ |
| timePeriodOfDiscovery | mid 20th century ⓘ |
| usesConcept |
Bott element
ⓘ
classifying space ⓘ loop space ⓘ stable homotopy ⓘ |
| yieldsIsomorphism |
KO^{n}(X) ≅ KO^{n+8}(X) for real K-theory
ⓘ
K^{n}(X) ≅ K^{n+2}(X) for complex K-theory ⓘ π_{k}(O) ≅ π_{k+8}(O) ⓘ π_{k}(U) ≅ π_{k+2}(U) ⓘ |
How these facts were elicited
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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: Bott periodicity Description of subject: Bott periodicity is a fundamental theorem in homotopy theory and K-theory that reveals a repeating pattern in the homotopy groups of classical groups, leading to the periodic structure of topological K-theory.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
K-Theory
this entity surface form:
Bott periodicity theorem