Bott periodicity

E886930

mathematical theorem result in K-theory result in algebraic topology result in homotopy theory

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.

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All labels observed (3)

Label	Occurrences
Bott periodicity canonical	1
Bott periodicity for C*-algebras	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.
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
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 ⓘ linked to: Bott periodicity 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 ⓘ
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 ⓘ
relatedTo	Atiyah–Singer index theorem ⓘ C*-algebra K-theory ⓘ linked to: K-theory Clifford algebras ⓘ linked to: Clifford algebra Kasparov KK-theory ⓘ 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

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

K-Theory → aboutConcept → Bott periodicity ⓘ

subject linked to: “K-Theory” (book with Friedrich Hirzebruch and others)

Raoul Bott → knownFor → Bott periodicity theorem ⓘ

linked to: Bott periodicity

Bott periodicity → generalizedBy → Bott periodicity for C*-algebras ⓘ