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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.

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.

Two branches

• Neil Turok Mirror universe, big bang as a mirror, CPT symmetry.
• {$O(\infty)$} has two parts and that may be reflected in the fact that the simple roots can be considered in two groups {$x_i-x_j$} and {$x_j-x_i$} where {$i<j$}.
• Idea: The threesome is what links together the two worlds of O(infinty). The threesome equates a shift in one world with a node in another world and vice versa. And this creates a circle - the three-cycle - which moves in one direction - distinguishing what is unconscious and what is conscious and defining a hole for Z.

Loop spaces and suspension (Operator +2)

• 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. {$X\wedge Y=X\times Y/X\cup Y$}, {$\Sigma X = X\wedge S^1$}, {$S^n\wedge S^1=S^{n+1}$}
• 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
• 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 the self-duality of 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.
  • Somnath Basu. Bott periodicity and some immediate generalizations. An exposition of the proof by Aguilar and Prieto.
  • 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.

Clifford algebra

• The even subalgebra of a Clifford algebra {$Cl_{0,n+1}$} is the Clifford algebra {$Cl_{0,n}$}. The unit elements of that Clifford algebra is the spin group {$Sp(n+1)$}.

Functor on Clifford algebra representations

• In John Baez's talk on the symmetric space, the functor acts on the category of representations of Clifford algebras, which is the category relevant for Morita equivalence.

Complex Clifford algebra

• In a complex Clifford algebra, the coefficient {$i$} commutes with all of the generators {$e_k$} whereas the generators anti-commute with each other. If the coefficient {$i$} anticommuted with all of the generators, then this would simply be a real Clifford algebra with one more generator. So this is a very deep fact that distinguishes complex and real Clifford algebras.
• complex Bott periodicity models the being and real Bott periodicity models their mind (and their three minds)

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

Morita equivalence

• How to prove that a ring {$\mathbb{R}$} is Morita-equivalent to {$M_n(\mathbb{R})$}?

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

Vector bundles on unit spheres

• Jost Eschenburg, Bernhard Hanke. Bott Periodicity, Submanifolds, and Vector Bundles

Symmetry

• The fact that, for orthogonal matrices, {$O^T=O^{-1}$} and thus the inverse can be understood as reversing arrows (similarly for the unitary and compact symplectic cases) is an expression of the symmetry in math. Bott periodicity organizes these symmetries in math itself.
• Knowledge is the marking of a state in a symmetry. You have two views (two levels). Different states (known) and their symmetric transformation (no visible change, thus unknown).

Eigenvector

• In the orthogonal group, commuting means that we have some eigenvector.
• the foursome relates {$V_+$} and {$V_-$} as two twosomes. And then divide further into two. A perspective becomes a twosome. Until it all collapses.

Threefold way

• Freeman J. Dyson. The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics. J. Math. Phys. 3, 1199–1215 (1962)

Building block

• {$A_1$} is the building block for Lie theory and its Lie algebra serves both {$SU(2)$} and {$SO(3)$}, thus relates vectors and spinors, and seems relevant for Bott periodicity.

Permutations

• {$S_\infty$} = colimit of symmetric groups = permutations with finite support
• Consider all permutations of all lengths. (In considering the moments.) This is similar to {$O(\infty)$} or {$U(\infty)$} which considers all rotations. There is a theorem that relates them. And consider how to take the Fourier transform of all of that.

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
• Geoffrey Dixon. Division Algebras, Clifford Algebras, Periodicity. 24-fold symmetry of split dual Clifford algebras.

Ideas

• Max Karoubi mentioned in his video that loop equations for rings with R,C,H,H' and {$\epsilon = +/-1$} yields 10 homotopy equivalences.
• Bott periodicity exhibits self-folding. Note the duality of 1 with the pseudoscalar.

Outreach

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.

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 {$SU(2)$}.