23-465: (Redirected from Periodic ) [REDACTED] Look up periodicity or periodic in Wiktionary, the free dictionary. Periodicity or periodic may refer to: Mathematics [ edit ] Bott periodicity theorem , addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups Periodic function ,
46-442: A compound of iodine Principle of periodicity, a concept in generally accepted accounting principles Quasiperiodicity , property of a system that displays irregular periodicity See also [ edit ] Aperiodic (disambiguation) Cycle (disambiguation) Frequency (disambiguation) Period (disambiguation) Periodical Seasonality , periodic variation, or periodic fluctuations Topics referred to by
69-442: A compound of iodine Principle of periodicity, a concept in generally accepted accounting principles Quasiperiodicity , property of a system that displays irregular periodicity See also [ edit ] Aperiodic (disambiguation) Cycle (disambiguation) Frequency (disambiguation) Period (disambiguation) Periodical Seasonality , periodic variation, or periodic fluctuations Topics referred to by
92-450: A function whose output contains values that repeat periodically Periodic mapping Physical sciences [ edit ] Periodic table of chemical elements Periodic trends , relative characteristics of chemical elements observed Redshift periodicity, astronomical term for redshift quantization Other uses [ edit ] Fokker periodicity blocks , which mathematically relate musical intervals Periodic acid ,
115-548: A modulo-8 recurrence relation in the homotopy groups of classical groups Periodic function , a function whose output contains values that repeat periodically Periodic mapping Physical sciences [ edit ] Periodic table of chemical elements Periodic trends , relative characteristics of chemical elements observed Redshift periodicity, astronomical term for redshift quantization Other uses [ edit ] Fokker periodicity blocks , which mathematically relate musical intervals Periodic acid ,
138-432: Is Ω 2 U ≃ U . {\displaystyle \Omega ^{2}U\simeq U.} Either of these has the immediate effect of showing why (complex) topological K -theory is a 2-fold periodic theory. In the corresponding theory for the infinite orthogonal group , O , the space BO is the classifying space for stable real vector bundles . In this case, Bott periodicity states that, for
161-431: Is defined as the inductive limit of the orthogonal groups , then its homotopy groups are periodic: and the first 8 homotopy groups are as follows: The context of Bott periodicity is that the homotopy groups of spheres , which would be expected to play the basic part in algebraic topology by analogy with homology theory , have proved elusive (and the theory is complicated). The subject of stable homotopy theory
184-404: Is different from Wikidata All article disambiguation pages All disambiguation pages periodicity [REDACTED] Look up periodicity or periodic in Wiktionary, the free dictionary. Periodicity or periodic may refer to: Mathematics [ edit ] Bott periodicity theorem , addresses Bott periodicity:
207-554: Is different from Wikidata All article disambiguation pages All disambiguation pages Bott periodicity theorem In mathematics , the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups , discovered by Raoul Bott ( 1957 , 1959 ), which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles , as well as
230-422: The classifying space construction. Bott periodicity states that this double loop space is essentially BU again; more precisely, Ω 2 B U ≃ Z × B U {\displaystyle \Omega ^{2}BU\simeq \mathbb {Z} \times BU} is essentially (that is, homotopy equivalent to) the union of a countable number of copies of BU . An equivalent formulation
253-428: The stable homotopy groups of spheres . Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the unitary group . See for example topological K-theory . There are corresponding period-8 phenomena for the matching theories, ( real ) KO-theory and ( quaternionic ) KSp-theory , associated to
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#1732851115459276-402: The 8-fold loop space, Ω 8 B O ≃ Z × B O {\displaystyle \Omega ^{8}BO\simeq \mathbb {Z} \times BO} or equivalently, Ω 8 O ≃ O , {\displaystyle \Omega ^{8}O\simeq O,} which yields the consequence that KO -theory is an 8-fold periodic theory. Also, for
299-495: The complex numbers: Over the real numbers and quaternions: where the division algebras indicate "matrices over that algebra". As they are 2-periodic/8-periodic, they can be arranged in a circle, where they are called the Bott periodicity clock and Clifford algebra clock . The Bott periodicity results then refine to a sequence of homotopy equivalences : For complex K -theory: For real and quaternionic KO - and KSp-theories: The resulting spaces are homotopy equivalent to
322-430: The connection of their cohomology with characteristic classes , for which all the ( unstable ) homotopy groups could be calculated. These spaces are the (infinite, or stable ) unitary, orthogonal and symplectic groups U , O and Sp. In this context, stable refers to taking the union U (also known as the direct limit ) of the sequence of inclusions and similarly for O and Sp. Note that Bott's use of
345-741: The infinite symplectic group , Sp, the space BSp is the classifying space for stable quaternionic vector bundles , and Bott periodicity states that Ω 8 BSp ≃ Z × BSp ; {\displaystyle \Omega ^{8}\operatorname {BSp} \simeq \mathbb {Z} \times \operatorname {BSp} ;} or equivalently Ω 8 Sp ≃ Sp . {\displaystyle \Omega ^{8}\operatorname {Sp} \simeq \operatorname {Sp} .} Thus both topological real K -theory (also known as KO -theory) and topological quaternionic K -theory (also known as KSp-theory) are 8-fold periodic theories. One elegant formulation of Bott periodicity makes use of
368-474: The infinite unitary group , U , the space BU is the classifying space for stable complex vector bundles (a Grassmannian in infinite dimensions). One formulation of Bott periodicity describes the twofold loop space, Ω 2 B U {\displaystyle \Omega ^{2}BU} of BU . Here, Ω {\displaystyle \Omega } is the loop space functor, right adjoint to suspension and left adjoint to
391-542: The observation that there are natural embeddings (as closed subgroups) between the classical groups. The loop spaces in Bott periodicity are then homotopy equivalent to the symmetric spaces of successive quotients, with additional discrete factors of Z . Over the complex numbers : Over the real numbers and quaternions: These sequences corresponds to sequences in Clifford algebras – see classification of Clifford algebras ; over
414-399: The real orthogonal group and the quaternionic symplectic group , respectively. The J-homomorphism is a homomorphism from the homotopy groups of orthogonal groups to stable homotopy groups of spheres , which causes the period 8 Bott periodicity to be visible in the stable homotopy groups of spheres. Bott showed that if O ( ∞ ) {\displaystyle O(\infty )}
437-419: The same term [REDACTED] This disambiguation page lists articles associated with the title Periodicity . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Periodicity&oldid=1164559000 " Category : Disambiguation pages Hidden categories: Short description
460-419: The same term [REDACTED] This disambiguation page lists articles associated with the title Periodicity . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Periodicity&oldid=1164559000 " Category : Disambiguation pages Hidden categories: Short description
483-416: The subject of the famous Adams conjecture (1963) which was finally resolved in the affirmative by Daniel Quillen (1971). Bott's original results may be succinctly summarized in: Corollary: The (unstable) homotopy groups of the (infinite) classical groups are periodic: Note: The second and third of these isomorphisms intertwine to give the 8-fold periodicity results: For the theory associated to
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#1732851115459506-606: The word stable in the title of his seminal paper refers to these stable classical groups and not to stable homotopy groups. The important connection of Bott periodicity with the stable homotopy groups of spheres π n S {\displaystyle \pi _{n}^{S}} comes via the so-called stable J -homomorphism from the (unstable) homotopy groups of the (stable) classical groups to these stable homotopy groups π n S {\displaystyle \pi _{n}^{S}} . Originally described by George W. Whitehead , it became
529-451: Was conceived as a simplification, by introducing the suspension ( smash product with a circle ) operation, and seeing what (roughly speaking) remained of homotopy theory once one was allowed to suspend both sides of an equation as many times as one wished. The stable theory was still hard to compute with, in practice. What Bott periodicity offered was an insight into some highly non-trivial spaces, with central status in topology because of
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