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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Observed surface forms (1)

Surface form	Occurrences
Bott periodicity theorem	1

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) ⓘ

Referenced by (2)

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

“K-Theory” (book with Friedrich Hirzebruch and others) → aboutConcept → Bott periodicity ⓘ

subject surface form: K-Theory

Raoul Bott → knownFor → Bott periodicity ⓘ

this entity surface form: Bott periodicity theorem