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博特周期性定理

Overviews

• Wikipedia: Bott periodicity theorem
• Raoul Bott. The Periodicity Theorem For The Classical Groups And Some Of Its Applications In 1970, Bott looked back at his original proof, reconsidered it in terms of functors, vector bundles and K-theory, and considered applications, such as parallelizability.
• Math Overflow. Overview of proofs of Bott-periodicity Emphasizes the role of Thom isomorphisms. Asking for the gist of why this should be periodic.

Videos

• Ravi Vakil slides Symmetric spaces and loop space iteration. Complex and real. Focuses on double loop space iteration, using short exact sequences to show that mapping spheres into one space is the same as mapping points into another space.
  • Ravi Vakil "Bott periodicity, algebro-geometrically", part 1
  • Ravi Vakil "Bott periodicity, algebro-geometrically", part 2

Homotopy groups

• John Baez. Bott periodicity. Considering how {$\pi_0(O(\infty))$} relates to {$\mathbb{R}$}, {$\pi_1(O(\infty))$} relates to {$\mathbb{C}$}, {$\pi_3(O(\infty))$} relates to {$\mathbb{H}$} and {$\pi_7(O(\infty))=\pi_{-1}(O(\infty))$} relates to {$\mathbb{O}$} and the Grassmannian.
• Caleb Ji. Various Statements of Bott Periodicity. A short overview of statements in homotopy theory and K theory.
• Jonathan Block. The Bott Periodicity Theorem. First, periodicity allows one to deloop classifying spaces and thus define cohomology theories. Second, using periodicity, “wrong way” functoriality maps can be defined and these are of integral importance in the index theorem. Siebenmann periodicity states that for M a closed manifold {$S(M)∼=S(M×I4,M×∂I4)$}. A map {$S(M)→KO[12]∗(M)$} which intertwines the two periodicities. It should be noted that {$KO[1/2]$} is four periodic with the signature operator playing the same role as the inverse of the Bott element as the Dirac operator does for complex (or real) periodicity.

Loop spaces and suspension

• Tai-Danae Bradley. "One-Line" Proof: Fundamental Group of the Circle.
• Tai-Danae Bradley, Bryson, Terilla. Chapter 6. Paths, Loops, Cylinders, Suspensions... Suspension-loop adjunction.
• Algebraic Topology July 2016 University of Chicago
• Peter May. Introduction to Algebraic Topology 10:30 Freudenthal Suspension theorem. {$X\wedge Y=X\times Y/X\cup Y$}, {$\Sigma X = X\wedge S^1$}, {$S^n\wedge S^1=S^{n+1}$}
• Peter May. An excellent overview of Bott periodicity. Text
  • He goes through Bott's approach. Given symmetric space {$M$} and a triple {$ν = (P, Q; h)$} where {$h$} is the homotopy class of curves joining point {$P$} to {$Q$} in {$M$}. Define {$M^{\nu}$} as the set of all geodesics of minimal length which join {$P$} to {$Q$} and are in the homotopy class {$h$}. Bott shows that {$M^{\nu}$} is a symmetric space. {$\pi_k(M^{\nu})=\pi_{k+1}(M)$} for {$0<k<|\nu |-1$}. Here {$|\nu |$} is the first positive integer which occurs as the index of some geodesic from {$P$} to {$Q$} in the class {$h$}. The index is given by the Morse index theorem.
  • He also considers Hopf algebras as relevant for complex Bott periodicity.
• Stable algebraic topology, 1945-1966 History of algebraic topology in the 1950s. Includes a history of the proofs of Bott periodicity.
• Raoul Bott. What Morse missed by not talking to Weyl Includes Morse theory and loop spaces.
• M.A. Aguilar, Carlos Prieto. Quasifibrations and Bott periodicity. May says this is a modern, simplest proof in terms of homotopy theory. It deals with the complex case.
• Mark J. Behrens. A new proof of the Bott periodicity theorem. A simpler proof of the loop space result, both for complex and real, simplifying and extending ideas of Aguilar and Prieto, based on an appropriate restriction of the exponential map to construct an explicit quasifibration with base space U and contractible total space.

Homology, cohomology

• A question about the topological proofs of Bott periodicity Answer by Peter May. Emphasizes the duality of Hopf algebras.

Symmetric spaces

• John Milnor. Morse Theory. (AM-51), Volume 51 Contains a helpful proof of the Bott periodicity theorem, both real and complex, in terms of Morse theory, minimal geodesics and symmetric spaces. Relates the Lie groups to the symmetric spaces and loop spaces.

Lie group embeddings

• Dyer, Lashof. A topological proof of the Bott periodicity theorems. Loop spaces.

Symplectic structures

• Max Karoubi. Quadratic forms and Bott periodicity. Presents in terms of quadratic forms and symplectic structures.

Morse theory and geodesics

• Loring Tu. The life and works of Raoul Bott.
  • Morse theory
  • Lie groups and homogeneous spaces
  • Index of a closed geodesic Bott considered the critical points of the energy function, which are precisely the geodesics from {$p$} to {$q$}.
  • Homogeneous vector bundles
  • The periodicity theorem
  • Clifford algebras
• Dubrovin, Fomenko, Novikov. Modern Geometry - Methods and Applications. Part III. Introduction to Homology Theory. Includes Bott Periodicity, real and complex, pages 270 to 325. Based on Morse theory, calculus of variations. Also considers the seven dimensional sphere.
• Bosman Honor's Thesis: Bott Periodicity with sketch of proof in terms of Morse theory
• Minimal geodesics (from pode to antipode) parametrize the next space (in the next dimension). In low dimensions, minimal geodesics adequately model loop spaces.
• Langlands program related to Bott-Atiyah. The Yang Mills Equations Over Riemann Surfaces, Morse theory - calculus of variations, thus to Bott periodicity. And Langlands is related to conjugacy classes on GL(2), which are eigenvalues. There are three families of double coverings.

Real, Complex, Quaternions, Octonions

• Eschenburg. Geometry of Octonions Relates the representations of Clifford algebras to the normed division algebras. Identifies the octonions with the representations for {$Cl_{0,4}$}, {$Cl_{0,4}$}, {$Cl_{0,4}$}, {$Cl_{0,4}$} and identifies {$M_2(\mathbb{O})$} with the representations for {$Cl_{0,8}$}. Gives a proof of Bott periodicity in terms of symmetric spaces, symmetric subspaces, poles and centrioles.
• Arkadii Slinko. 1, 2, 4, 8,... What comes next? Parallelizability of spheres.

Octonionic line bundles

• John Baez. OP1 and Bott Periodicity. Canonical line bundles {$L_\mathbb{R}$}, {$L_\mathbb{C}$}, {$L_\mathbb{H}$}, {$L_\mathbb{O}$} give elements {$ [L_\mathbb{R}]$}, {$ [L_\mathbb{C}]$}, {$ [L_\mathbb{H}]$}, {$ [L_\mathbb{O}]$} that generate, respectively, {$\widetilde{KO}(S^1)\cong\mathbb{Z}_2,\widetilde{KO}(S^2)\cong\mathbb{Z}_2,\widetilde{KO}(S^4)\cong\mathbb{Z},\widetilde{KO}(S^8)\cong\mathbb{Z}$}. The isomorphism {$\widetilde{KO}(S^n)\rightarrow \widetilde{KO}(S^{n+8})$} given by {$x\rightarrow [L_\mathbb{O}]$} yields Bott periodicity. The canonical octonionic line bundle over {$\mathbb{O}P^1$} generates Bott periodicity.

Clifford modules

• Dexter Chua. Clifford Algebras and Bott Periodicity.
  • Bott Periodicity Seminar
  • Deke Zhao. Graded Morita Equivalence of Clifford Superalgebras explains how to calculate the graded representations and the Grothendieck groups.
    • Math Stack Exchange. Tensor product of graded algebras.
  • M.F.Atiyah, R.Bott, A.Shapiro | Clifford Modules.
  • H. Blaine Lawson, Marie-Louise Michelsohn. Spin geometry.
• Yuqin Kewang. Clifford Algebras and Bott Periodicity.
• Andre Henriques. A proof of Bott periodicity via Clifford algebras
• Baez: 2) H. Blaine Lawson, Jr. and Marie-Louise Michelson, "Spin Geometry", Princeton U. Press, Princeton, 1989. or 3) Dale Husemoller, "Fibre Bundles", Springer-Verlag, Berlin, 1994. These books also describe some of the amazing consequences of this periodicity phenomenon. The topology of n-dimensional manifolds is very similar to the topology of (n+8)-dimensional manifolds in some subtle but important ways!'' Physics of fermions.
• Dale Husemoller's book is detailed and helpful.
• Roberto Rubio. Clifford algebras, spinors, and applications. Classification of Clifford algebras.

Representations

• Bosons - real representations, fermions - quaternionic representations.
• Spin representation

Spinors

• Cameron Krulewski. K-Theory, Bott periodicity, and elliptic operators Explains Attiyah's proof which uses the index of elliptic operators. The real case is the spinor case and mentions Dirac operators.

Morita equivalences as quantum Hamiltonian reductions

• Theo Johnson-Freyd. The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions.
• We show that the Morita equivalences Cliff(4) ' H, Cliff(7) ' Cliff(−1), and Cliff(8) ' R arise from quantizing the Hamiltonian reductions R0|4//Spin(3), R0|7//G2, and R0|8//Spin(7), respectively.
• The Morita equivalence {$\textrm{Cliff}(7) \simeq \textrm{Cliff}(−1)$} arises from the Hamiltonian reduction {$\mathbb{R}^{0|7}//G_2$}, where {$G_2 ⊆ SO(7)$} is the exceptional Lie group of automorphisms of the octonion algebra {$\mathbb{O}$}.
• Hamiltonian Reduction by Stages

CPT symmetry

• Andrzej Trautman. On Complex Structures in Physics. Baez discusses this. Trautman relates Clifford algebra clock, spinors and charge conjugation.

Unit spheres

• John Baez. Octonions, normed division algebras and Bott periodicity. Clifford algebras {$Cl_{0,n}$}, whether they admit a representation on {$k$}-dimensional vector space, which is if and only if the unit sphere in that vector space {$S^{k-1}$}, admits {$n$} linearly independent smooth vector fields.
• Max Karoubi. Algebraic maps between spheres and Bott periodicity.
• Related to Adams's work on vector fields on spheres.
• 6+4=2 modulo 8

24-cell?

• John Baez's talk on the 24-cell, video, slides

Extensions

• Jim Bryan, Marc Sanders. Instantons on {$S^4$} and {$\cpbar$}, rank stabilization, and Bott periodicity
• Math StackExchange: 8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries
• Daisuke Kishimoto. Topological proof of Bott periodicity and characterization of BR. (1,) periodicity

Ideas

• Max Karoubi mentioned in his video that loop equations for rings with R,C,H,H' and {$\epsilon = +/-1$} yields 10 homotopy equivalences.
• Orthogonal add a perspective (Father), symplectic subtract a perspective (Son).
• John Baez's comment about 4 dimensions being most interesting because half-way between 0 and 8. The Son turns around and reverses the Father.
• In Bott periodicity, the going beyond oneself (by adding perspectives) and the going outside of oneself (by subtracting perspectives) is, in mathematics, extended to all possible integers, and thus the two directions are related in the four ways as given by the four classical Lie algebras, including gluing, fusing, folding.
• In the eight-cycle of divisions, the going beyond oneself and the going back outside oneself are reminiscent of spinors like electrons.
• The flip side of going beyond yourself - if you can add perspectives, then you can substract perspectives - on the flip side. Thus nullsome and foursome model the extremes: nullsome models the void of everything, foursome models the point of nothing. Thus adding a perspective takes us from the indefinite to the definite and back again. The three operations +1,+2,+3 make for a three-fold braid.
• Goedel's incompleteness theorem. There is irrelevant, inaccessible knowledge, such as that which describe different implementation of equivalent rings (for Bott periodicity) where the equivalence means that they have the same (isomorphically, structurally) representations.
• Bott periodicity exhibits self-folding. Note the duality of 1 with the pseudoscalar.

Raoul Bott

• Raoul Bott mirė Carslbad, Kalifornijoje. Attiyah apie Bott gyvenimą. Nekrologas New York Times Dukra Jocelyn Scott gyveno Rancho Santa Fe, dirbo C.B.S. Scientific. Jisai gyveno: assisted living - Sunrise at La Costa, 7020 Manzanita St, Carlsbad, CA 92011-5123 Studied geodesics on {$SU(2)$}.

People who may care

• Tyler Marshall Goldstein. Stages of sentience (Bott periodicity)
  • Tyler Goldstein at Linked In I contacted him.
• The Larson Research Center. Bott Periodicity Theorem.