Bott periodicity theorem 782099 176252730 2007-12-06T23:11:46Z Rjwilmsi 203434 [[WP:AWB/T|Typo fixing]] , typos fixed: two-fold → twofold using [[Project:AutoWikiBrowser|AWB]] In [[mathematics]], the '''Bott periodicity theorem''' is a result from [[homotopy theory]] discovered by [[Raoul Bott]] during the latter part of the 1950s, which proved to be of foundational significance for much further research, in particular in [[K-theory]] of stable complex [[vector bundle]]s, as well as 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 the real [[orthogonal group]] and the quaternionic [[symplectic group]], respectively. They impact the [[stable homotopy groups of spheres]] even more tightly than complex [[K-theory]]. ==Context and significance== The context of Bott periodicity is that the [[homotopy group]]s of [[sphere]]s, 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]] was conceived as a simplification, by introducing the [[suspension (mathematics)|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 the connection of their [[cohomology]] with [[characteristic class]]es, 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 :<math>U(1)\subset U(2)\subset\cdots\subset U = \bigcup_{k=1}^\infty U(k)</math> and similarly for ''O'' and ''Sp''. Bott's (now somewhat awkward) use of the word ''stable'' in the title of his seminal paper refers to these stable [[classical groups]] and not to [[stable homotopy theory|stable homotopy]] groups. The important connection of Bott periodicity with the [[stable homotopy groups of spheres]] <math>\pi_n^S</math> comes via the so called stable [[J-homomorphism|''J''-homomorphism]] from the (unstable) homotopy groups of the (stable) classical groups to these stable homotopy groups <math>\pi_n^S</math>. Originally described by [[George W. Whitehead]], it became the subject of the famous [[Frank Adams|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: :<math>\pi_k(U)=\pi_{k+2}(U) \,\!</math> :<math>\pi_k(O)=\pi_{k+4}(Sp) \,\!</math> :<math>\pi_k(Sp)=\pi_{k+4}(O) ,\ \ k=0,1,\dots . \,\!</math> '''Note:''' The second and third of these isomorphisms intertwine to give the desired 8-fold periodicity results: :<math>\pi_k(O)=\pi_{k+8}(O) \,\!</math> :<math>\pi_k(Sp)=\pi_{k+8}(Sp) ,\ \ k=0,1,\dots . \,\!</math> ==Loop spaces and classifying spaces== For the theory associated to the infinite [[unitary group]], ''U'', the space ''BU'' is the [[classifying space]] for stable complex [[vector bundle]]s (a [[Grassmannian]] in infinite dimensions). One formulation of Bott periodicity describes the twofold loop space, &Omega;<sup>2</sup>''BU'' of ''BU''. Here, &Omega; is the [[loop space]] functor, [[right adjoint]] to [[Suspension (topology)|suspension]] and [[left adjoint]] to the [[classifying space]] construction. Bott periodicity states that this double loop space is essentially ''BU'' again; more precisely, :<math>\Omega^2BU\simeq Z\times BU\,</math> is essentially (that is, [[homotopy equivalence|homotopy equivalent]] to) the union of a countable number of copies of ''BU''. An equivalent formulation is :<math>\Omega^2U\simeq U .\,</math> 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 bundle]]s. In this case, Bott periodicity states that, for the 8-fold loop space, :<math>\Omega^8BO\simeq Z\times BO ;\,</math> or equivalently, :<math>\Omega^8O\simeq O ,\,</math> which yields the consequence that ''KO''-theory is an 8-fold periodic theory. Also, for the infinite [[symplectic group]], ''Sp'', the space ''BSp'' is the [[classifying space]] for stable quaternionic [[vector bundle]]s, and Bott periodicity states that :<math>\Omega^8BSp\simeq Z\times BSp ;\,</math> or equivalently :<math>\Omega^8Sp\simeq Sp .\,</math> 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. ==Geometric model of loop spaces== One elegant formulation of Bott periodicity makes use of 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 space]]s of successive quotients, with additional discrete factors of <math>\mathbf{Z}</math>. Over the complex numbers: :<math> U \times U \subset U \subset U \times U</math> Over the real numbers and quaternions: :<math>O \times O \subset O\subset U\subset Sp \subset Sp \times Sp \subset Sp\subset U\subset O \subset O \times O</math> These sequences corresponds to sequences in [[Clifford algebra]]s; over the complex numbers: :<math>\mathbf{C} \oplus \mathbf{C} \subset \mathbf{C} \subset \mathbf{C} \oplus \mathbf{C}</math> Over the real numbers and quaternions: :<math>\mathbf{R} \oplus \mathbf{R} \subset \mathbf{R}\subset \mathbf{C}\subset \mathbf{H} \subset \mathbf{H} \times \mathbf{H} \subset \mathbf{H} \subset \mathbf{C} \subset \mathbf{R} \subset \mathbf{R} \oplus \mathbf{R}</math> 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 equivalence]]s: For complex ''K''-theory: :<math>\begin{align} \Omega U &\simeq \mathbf{Z}\times BU = \mathbf{Z}\times U/(U \times U)\\ \Omega(Z\times BU)& \simeq U = (U \times U)/U \end{align}</math> For real and quaternionic ''KO''- and ''KSp''-theories: :<math>\begin{align} \Omega(\mathbf{Z}\times BO) &\simeq O = (O \times O)/O & \Omega(\mathbf{Z}\times BSp) &\simeq Sp = (Sp \times Sp)/Sp\\ \Omega O &\simeq O/U & \Omega Sp &\simeq Sp/U\\ \Omega(O/U) &\simeq U/Sp & \Omega(Sp/U) &\simeq U/O\\ \Omega(U/Sp)&\simeq \mathbf{Z}\times BSp = \mathbf{Z}\times Sp/(Sp \times Sp) & \Omega(U/O) &\simeq \mathbf{Z}\times BO = \mathbf{Z} \times O/(O \times O)\\ \end{align} </math> The resulting spaces are homotopy equivalent to the classical reductive [[symmetric space]]s, and are the successive quotients of the terms of the Bott periodicity clock. These equivalences immediately yield the Bott periodicity theorems. ==Proofs== Bott's original proof used [[Morse theory]]; subsequently, many different proofs have been given. ==Applications== * Bott periodicity expresses a considerable amount regarding the [[Orthogonal group#Topology|topology of the orthogonal group]], in particular the [[Orthogonal group#Homotopy groups|homotopy groups]] ==References== *Bott, R. ''The Stable Homotopy of the Classical Groups'', Ann. Math. 70, 1959, 313–337. *Giffen, C.H. ''Bott periodicity and the Q-construction'', Contemp. Math. 199(1996), 107–124. *Milnor, J. ''Morse Theory''. Princeton University Press, 1969. ISBN 0-691-08008-9. *[http://math.ucr.edu/home/baez/week105.html John Baez "This Week's Finds in Mathematical Physics" week 105] [[Category:Homotopy theory]] [[Category:Topology of Lie groups]] [[Category:Mathematical theorems]] [[zh:Bott周期性定理]]