Introduction E9F5FC

Questions FFFFC0

• View
• Edit
• History
• Print

博特周期性定理

Videos

• Ravi Vakil "Bott periodicity, algebro-geometrically", part 1
• Ravi Vakil "Bott periodicity, algebro-geometrically", part 2
• Ravi Vakil slides
• Peter May. An excellent overview. Text
  • Look at May's first lecture introducing algebraic topology and find where he defines the adjunction relating loop space and suspension.
• Peter May. History of algebraic topology in the 1950s.
  • Stable algebraic topology, 1945-1966
• Algebraic Topology July 2016 University of Chicago
• Max Karoubi. Algebraic maps between spheres and Bott periodicity.
• Max Karoubi. Quadratic forms and Bott periodicity.
• Raoul Bott. What Morse missed by not talking to Weyl

Statement

• Wikipedia: Bott periodicity theorem

Expositions

• 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
• Todd Trimble. The Super Brauer Group and Super Division Algebras.
  • Brauer-Wall group Super Brauer Group
  • Brauer group
• John Baez. Bott periodicity.
• John Baez. Octonions.
  • John Baez. Octonion projective geometry.
  • John Baez. Projective lines.
  • 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.
  • John Baez. The Magic Square.
• John Milnor. Morse Theory. (AM-51), Volume 51 Contains a proof of the Bott periodicity theorem.
• Eschenbur. Geometry of Octonions
• Arkadii Slinko. 1, 2, 4, 8,... What comes next?
• Dubrovin, Fomenko, Novikov. Modern Geometry - Methods and Applications. Part III. Introduction to Homology Theory. Includes Bott Periodicity.
• 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 operatordoes for complex (or real) periodicity.
• M.A. Aguilar, Carlos Prieto. Quasifibrations and Bott periodicity. May says this is a modern, simplest proof in terms of homotopy theory.
• Mark J. Behrens. A new proof of the Bott periodicity theorem. A simpler proof based on an appropriate restriction of the exponential map to construct an explicit quasifibration with base space U and contractible total space.
• Raoul Bott. The Periodicity Theorem For The Classical Groups And Some Of Its Applications
• 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.
• https://math.columbia.edu/~calebji/Bott-periodicity.pdf Caleb Ji. Various Statements of Bott Periodicity.
• The Larson Research Center. Bott Periodicity Theorem.

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

Proofs

• Peter May's notes for his overview talk.
• Bosman Honor's Thesis: Bott Periodicity with sketch of proof in terms of Morse theory
• Proofs of Bott-periodicity

Bott-periodicity also arises in string theory.

Related concepts

• 维基百科: Classifying space
• 维基百科: Classifying space for U(n)
• 维基百科: Classifying space for O(n)
• 维基百科: Orthogonal group discusses Bott periodicity.
• 维基百科: Symmetric space
• 维基百科: Morita equivalence
• 维基百科: Random matrix
• Dyson's threefold way
• 维基百科: Hopf fibration
• 维基百科: Hopf invariant and Adam's theorem.
• 维基百科: Homotopy group of spheres
• A Survey of Elliptic Cohomology by Jacob Lurie, mentions the Bott element, whose inversion is perhaps related to the period 2.
• 书: D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, modules, and algebras in stable homotopy theory.

Math facts

• Two types of results
  • Periodicity of the homotopy groups of the unitary groups.
  • Periodic theorem of vector bundles.

Ideas

• The relevant Lie groups are all rotations about a fixed origin. That fixed origin represents a universal, absolute perspective, God's perspective upon everything, God's knowledge of everything.
• Divisions of everything are perhaps chopping up a sphere where the sphere is everything also circle folding
• Bott periodicity should be related to the collapse of the eightsome into the nullsome, and thus the definition of contradiction
• Perspective arises because of base point - there is a fixed point for the isometries. We are that fixed point.
• Understand the dimensions of a Lie group as perspectives. And look at Lie groups as rotations of a sphere.
• Max Karoubi mentioned in his video that loop equations for rings with R,C,H,H' and {$\epsilon = +/-1$} yields 10 homotopy equivalences.

K-theory

• Peter May. A Concise Course in Algebraic Topology. Chapter 24. An Introduction to K-Theory. Has a section on the complex Bott periodicity.
• Allen Hatcher. Vector Bundles & K-Theory. Welcome.
  • Table of Contents
  • Allen Hatcher. Vector Bundles & K-Theory. Version 2.2 (Half-written).
  • Explanation
• Charles Weibel. The K-book: an introduction to algebraic K-theory.
• Zachary Halladay. Bott periodicity and K-theory
• Max Karoubi. Bott Periodicity in Topological, Algebraic and Hermitian K-Theory
• Qayum Khan. Classical introduction to K-theory.
• Andrew Blumberg. Introduction to Algebraic K-Theory.
• Clifford Modules by Atiyah, Bott, Schapiro, 1963 and their isomorphism of topological K-theory and quotients of Clifford modules.
• Attiyah. K-theory and Reality
• Inna Zakharevich. Algebraic K-theory, combinatorial K-theory and geometry
• https://physics.stackexchange.com/questions/104258/topological-insulators-why-k-theory-classification-rather-than-homotopy-classif
• K-theory = how dimensions are related
• K-theory. Cohomology - sequence of groups - which may not exist
• Cameron Krulewski. The K-Theoretic Classification of Topological Materials
• https://sites.google.com/view/dmonaco/teaching/k-theory-in-condensed-matter-physics
• Guo Chuan Thiang. Lecture notes on symmetries, topological phases and K-theory. A broad and helpful overview of the math and physics, including the tenfold way, Clifford algebras.
• Matthew Magill. Topological K-theory and Bott Periodicity
• When you move from K-theory to KO-theory, you add charge conjugation operator J.

Complex vector bundles

• Circle (topology) = Complex number (rotation) (algebra)
• Edward Witten. D-Branes And K-Theory.

K-theory and physics

• Atiyah: K-theory is related to quantum theory, cohomology is related to classical theory.
• Atiyah: Cohomology (graded by dimension, de Rham complex)(corresponds to classical theory) is related by the AH spectral sequence to K-theory (and Clifford algebras) (corresponds to quantum theory)
• Atiyah: Dirac operator is related to Bott periodicity and K-theory.
• Dirac operator is the formal square root of the Laplace operator.
• The spin-Dirac operator (for a spin-{$\frac{1}{2}$} particle confined in the plane) can be written in terms of the Pauli matrices {$D=-i\sigma_x\partial_x - i\sigma_y\partial_y$}. What is this raised to the exponential?
• Laplace operator is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space.
• This suggests that the curl (between the divergence and the gradient) is trivial. What is a trivial curl?
• Physically, the Laplacian describes diffusion.
• Laplace's equation is when the Laplacian is zero. Solutions are harmonic equations and are given by potential theory. They describe states of equilibrium, independent of time.
• The Laplacian is the simplest elliptic operator.
• Elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions.
• Dirac operator is the basis for spinors and for Bott periodicity.
• Dirac operator is, conceptually, the square root of the Laplacian. In what sense is a tableaux the square root of a matrix?

Alain Connes about Bott periodicity and CTP. "Why Four Dimensions and the Standard Model Coupled to Gravity...

• Video 39:00
• Ali H. Chamseddine, Alain Connes. Why the Standard Model
• Connes. Why Four Dimensions and the Standard ModelCoupled to Gravity - A Tentative Explanation Fromthe New Geometric Paradigm of NCG
• Alain Connes. Noncommutative Geometry and the Standard Model with Neutrino Mixing.]

Notes

Bott Periodicity

• 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.
• How can the fact that {$\Gamma(\frac{1}{2})=\sqrt{\pi}$} relate {$\pi$} and {$e$}? Note that {$\Gamma(1)=0!$} and {$\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{2}$}. Consider the integral definition.
• 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.
• The Elliptic Umbilic Diffraction Catastrophe. Optics, Bott periodicity?
• Study orthogonal groups and Bott periodicity.
• Allen Hatcher. Algebraic topology. Explanation Homotopy, homology, cohomology. We will show in Theorem 3.21 that a finite-dimensional division algebra over R must have dimension a power of 2. The fact that the dimension can be at most 8 is a famous theorem of [Bott & Milnor 1958] and [Kervaire 1958]. Example 4.55: Bott Periodicity.
• Bott periodicity exhibits self-folding. Note the duality with the pseudoscalar. Consider the formula n(n+1)/2 does that relate to the entries of a matrix?
• Bott periodicity is the basis for 8-fold folding and unfolding.
• What is the connection between Bott periodicity and spinors? See John Baez, The Octonions.
• Orthogonal add a perspective (Father), symplectic subtract a perspective (Son).
• My dream: Sartre wrote a book "Space as World" where he has a formula that expresses Bott periodicity / my eightfold wheel of divisions.
• A question about the topological proofs of Bott periodicity Answer by Peter May.
• Daisuke Kishimoto. Topological proof of Bott periodicity and characterization of BR.
• Dyer, Lashof. A topological proof of the bott periodicity theorems.
• ketverybė: H, nulybė: R
• 2-periodicity: Dievas - gerumas - Dievas
• sukeisti buvimą ir nebuvimą, tapimą ir netapimą
• 10 = 4+6 = 4+(4+2) = 8 + 2 (2 tai laisvumo atvaizdai)
• Z 0 Z 0 Z 0
• Z 0 (replace Z 0 with Z2 Z2 particle) Z 0 (replace Z 0 with 0 0 hole)
• 8 external dimensions (adding 4 dimensions or +2 or +3)
• internally expressed dimensions - adding integrity
• same pattern reversed internally
• syllabus by Roberto Rubio relate to Bott periodicity.
• 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 eignevalues. There are three families of double coverings.
• Bott periodicity 8=0 could be the basis for reproduction.
• Atiyah: What does octonionification mean for the magic square?
• 6+4=2 modulo 8
• https://en.wikipedia.org/wiki/Del_Pezzo_surface see especially from degree 2 to 9, is this an eightfold phenomenon? How is it related to https://en.wikipedia.org/wiki/M-theory ? How is it related to https://en.wikipedia.org/wiki/Veronese_surface and the conics?
• Related to Adams's work on vector fields on spheres.
• The odd and even dimensional orthogonal groups are related to odd and even dimensional (crosspolytope-hypercube) choice frameworks. How does the duality of center and totality that distinguishes these two choice frameworks add an additional distinction that makes for eightfold Bott periodicity?
• What is the significance (for Bott periodicity) of {$S^{n-1}=SO(n)/SO(n-1)$} ?

Shintaro Fushida-Hardy. Notes for MATH 282B Homotopy Theory.

• Chapter 2 on Fibre Sequences contains examples in Sections 2.5 and 2.6.
• Seems relevant for mastering Bott periodicity. Mentions the fibration for orthogonal groups.

Is Bott periodicity related to the triality of SO(8) ? and thus the unit octonions? See John Baez.

• OP1 and Bott Periodicity

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
• So(8) and Bott?
• Could Feynman diagrams for four fields express the foursome? How goes to What in one direction and Why goes to Whether in the opposite direction. And could Feynman diagrams for N fields express the division of everything into N perspectives? And does Bott periodicity apply?
• In my understanding of Bott periodicity, is it the + signs that are growing or the - signs?
• 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.
• Krulewski bott proof
• J. F. Adams, Infinite Loop Spaces, Ann. of Math. Studies 90, 1978.
  • 4.66. If F→E→B is a fibration or fiber bundle with E contractible, then there is a weak homotopy equivalence F→ΩB .
  • For each topological group G there is a fiber bundle G→EG→BG with EG contractible, hence by the proposition there is a weak equivalence G ≃ ΩBG. There is also a converse statement
• Combining this duality between Σ and Ω with the duality between fibers and cofibers, we see a duality relationship between the fibration and cofibration sequences of §4.3:
• One formulation of Bott periodicity describes the twofold loop space, {$\Omega ^{2}BU$} of {$BU$}. Here, {$\Omega$} is the loop space functor, right adjoint to suspension and left adjoint to the classifying space construction. Bott periodicity states that this double loop space is essentially {$BU$} again.
• 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 the 8-fold loop space,
• https://en.m.wikipedia.org/wiki/Periodic_table_of_topological_invariants
• There are ten discrete symmetry classes of topological insulators and superconductors, corresponding to the ten Altland–Zirnbauer classes of random matrices.
• https://en.m.wikipedia.org/wiki/Spin_representation

Morita equivalence

• If R is Morita equivalent to S, then their centers C(R) and C(S) are isomorphic.
• Morita equivalence is not interesting in the case of commutative rings because then they are isomorphic.
• The ring {$M_n(R)$} of n-by-n matrices in R is Morita-equivalent to R for any n>0. Work out the functors here.
• Morita equivalence could describe a noncontradictory model of contradiction.

Morita equivalence is an additive equivalence

• Presumably, an additive equivalence is one in which the equivalence is by way of an additive functor.
• Additive category An additive category is one such that all of its hom-sets are abelian groups, composition of morphisms is bilinear, and it admits all finitary biproducts.
• Additive functors A functor F: C → D between preadditive categories is additive if it is an abelian group homomorphism on each hom-set in C. If the categories are additive, then a functor is additive if and only if it preserves all biproduct diagrams.
• Additivity of adjoint functors All adjoint functors between additive categories must be additive functors.

Loop spaces

• Think of loopspace as expressing a space for learning.
• Loop - open arc - perspective - defines inside vs. outside - the ambiguity of a point.

Divisions of everything

• Bott Periodicity of states of mind.
• The widgets at the end of the Dynkin chain allow for turning around and thus for ambiguity in the understanding of the perspective of the chain and thus they can be understood as divisions of everything. And this should relate to Bott periodicity.

Symmetric spaces

• https://johncarlosbaez.wordpress.com/2023/05/10/symmetric-spaces-and-the-tenfold-way/
• Symmetric spaces - inversion is the symmetry of choice - like the symmetry in the binomial theorem.
• Symmetric spaces - those that undo any perturbation, thus make for stability.

Grassmannian

• Grassmannian can be thought as defined on center at x as it moves in a manifold and yields tangent spaces.
• https://en.wikipedia.org/wiki/Freudenthal_magic_square organizes not just Lie groups but also symmetric spaces. http://www.ims.cuhk.edu.hk/~leung/PhD%20students/Thesis%20Yong%20Dong%20Huang.pdf

24-cell

John Baez, 8 and 24

• https://johncarlosbaez.wordpress.com/2023/08/14/8-and-24/ John Baez's talk
• http://math.ucr.edu/home/baez/8_and_24/8_and_24_web.pdf slides
• YouTube

Notes

Equivalence up to isomorphism is relevant for "user requirements" as opposed to "material implementation". This is relevant for Bott periodicity.

Bott periodicity. Sulfur as an eight-cycle. The origin of life.

• https://en.wikipedia.org/wiki/Octasulfur
• https://en.wikipedia.org/wiki/Iron%E2%80%93sulfur_world_hypothesis

In what sense is Bott periodicity the paradigm for recurring activity in Christopher Alexander's patterns? Is it related to God going beyond himself in four steps and human going beyond himself in four steps back to God, on the complementary side of the divisions of everything?

Bott periodicity provides order to the homotopy groups of spheres and that reflects that we are left with Consciousness when we unplug the Unconscious and likewise unplug the Conscious. So the relation between n and m is the relation between the Unconscious and the Conscious and their content.

Bott periodicity expresses the symmetry of mathematics

• compare this with how Lie algebra root systems express the symmetry in counting forwards and backwards, for example, in one direction {$e_i-e_j$} and in the opposite direction {$e_j-e_i$}.

In Bott Periodicity, the matrix representations are related:

• {$\mathbb{Z}_2$} as a product (limit) expressing external relationships {$\begin{pmatrix}A & \\ & B \end{pmatrix}$}
• {$\mathbb{Z}$} as a sum (colimit) {$A+B$} and {$A-B$} in terms of internal structure
• {$0$} representations are the same size. What is the significance?

In Bott periodicity, do the real numbers express God (the nullsome)? and the quaternions express knowledge (the foursome)?

Cohl Furey. Bott Periodicity Particle Physics

Try to undersand Morita equivalence in terms of irreducible representations.

https://math.stackexchange.com/questions/154909/8-periodicity-clifford-clock-bott-periodicity-ko-dimension-in-noncommutative

https://mathoverflow.net/questions/8800/proofs-of-bott-periodicity

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).

Complex Bott periodicity, which relates U(n) and U(n+m)/U(n)xU(m), and goes back and forth between them, suggests that real Bott periodicity goes back and forth between O(n) and O(n+m)/O(n)xO(m). In one direction there is one step but in the other direction there are seven steps.

In Bott periodicity, {$\mathbb{Z}_2$} may count the unconscious and the conscious. They are the two pieces of {$O(\infty)$}. The homotopy groups explain the different ways of relating them.

A possible interpretation of the sequence. {$\mathbb{Z}$} expresses a hole as with the threesome and the sevensome. {$\mathbb{Z}_2$} describes the nullsome and the onesome.

Tyler Marshall Goldstein

• https://www.instagram.com/p/CPTVrophgua/ Stages of sentience
• https://www.linkedin.com/in/tmg77/

Minimal geodesics (from pode to antipode) parametrize the next space (in the next dimension). In low dimensions, minimal geodesics adequately model loop spaces.

• Three minds are three shifts adding 1, 2, 3 dimensions in the Bott clock shift.
• John Baez and James Dolan on Bott Periodicity
• How is the Yates Index Set Theorem {$+3$} related to Bott periodicity?
• In what sense is {$SO(n)$} not simply connected? And what is the relationship between its covering group {$\textrm{Spin}(n)$} and the special unitary group?
• In the Standard Model, fermions are not their own antiparticles, but in some theories they can be. Among other things, this involves the question of whether the relevant spinor representations of the groups Spin(p,q) are complex, real (‘Majorana spinors’) or quaternionic (‘pseudo-Majorana spinors’). The options are well-understood, and follow a nice pattern depending on the dimension and signature of spacetime modulo 8.
• Bosons - real representations, fermions - quaternionic representations.

Topological Analysis of Networks of Neurons

• Directed simplices. Streamling information.
• Directed flag complex.
• n-dimensional cavities
• cohomology mod 2, homology nonzero
• if multiplication of classes is meaningful
• Steiner operations
• representations
• use tools of Hodge theory