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Investigating Bott periodicity

Goals

If possible, explain how Bott periodicity models

• divisions of everything into n perspectives, n=0,1,2,...,7, including the relevant shifts in perspective
• three minds as three operations on the divisions of everything, adding 1,2,3 perspectives

Specificially, model:

• a perspective - as a loop {$S^1$}, a Clifford algebra generator {$e_i$} such that {$e_i^2=-1$}, a linear complex structure {$J_i$}
• a shift in perspective - relating a loop of loops {$S^2$} with a loop {$S^1$}
• the operation +1, adding a perspective - the loopspace functor, adding a Clifford algebra generator, adding a mutually anticommuting linear complex structure
• the collapse of the eightsome - the Hopf fibration for the octonionic projective line
• three minds - spheres {$S^1$}, {$S^2$}, {$S^3$} related by the Hopf fibration, the complex projective line

Model a Shift in Perspective

• A shift in perspective is the product of perspectives {$J_{i-1}J_i$} but only in one direction, understood from the observer not the observed, thus modeling the equation of life. This is presumably why it ultimately collapses. For a product of perspectives can be understood in both directions, left to right and right to left, which are related by a minus sign, as they anticommute. {$J_{i-1}J_i=-J_iJ_{i-1}$}
• In Stone, Chiu, Roy, compare the matrices for the linear complex structure {$J_1$} and for the second, mutually anticommuting, linear complex structure {$J_2$}

Look for {$M$} such that {$J_1M=J_2$}. {$M=J_1^{-1}J_2$}. Consequently, {$M=-J_1J_2$}.

A shift in perspective is related to {$J_{i+1}J_i$}.

Note that {$\begin{pmatrix}0 & -1 \\1 & 0 \\ \end{pmatrix}$} functions like the scalar {$i$} where it commutes with {$2\times 2$} blocks.

How do {$J_1$} and {$J_2$} give rise to a sphere {$S^2$}?

Do the {$J_1$} and {$J_2$} describe linked circles as with the Hopf fibration?

Consider iterated loop spaces: Wikipedia: Bott periodicity theorem

Husemoller. Fibre Bundles

John Baez. Projective lines.

Bott periodicity for Clifford algebra maniacs

Model the Three-Cycle

Understand how applying the loop functor three times lets a particular point to loop.

Analyze how the geodesics are described in Stone, Chiu, Roy.

Understand the homology of a three-cycle and how that relates to the winding number {$\mathbb{Z}$}.

Sets of Mutually Anticommuting Linear Complex Structures

• I have identified perspectives with linear complex structures.
• I can identify adding a perspective, as the operation +1, or going in the opposite direction around the Bott clock.
• I want to understand what is a shift in perspective?
  • I need to understand how the twosome relates two perspectives, a first and a second.
• I need to understand symmetric spaces.
• I have cracked the code and need to make sense of it: {$J_1, J_2, J_1J_2, J_4, J_4J_1, J_4J_2, J_4J_1J_2, ?$}
  • Where {$J_3=J_1J_2$} generates the three-cycle.
  • Note that products have zero homotopy. Whereas {$J_1, J_2$} yield {$\mathbb{Z_2}$} and {$1,J_4$} yield {$\mathbb{Z}$}.
  • Note that {$J_1$} satisfies the commutation relation for {$Sp(n,R)$} and {$J_2$} satisfies the commutation relation for {$Sp(n,C)$}. What about {$J_3$} and {$Sp(n,H)$}?
• I need to understand how divisions of everything grow by adding one perspective at a time.
• I need to understand how divisions of everything are modeled by chain complexes and exact sequences.
• I need to look for such chain complexes / exact sequences in the Lie group embeddings.
• I need to understand how the eight perspectives collapse into one.
• I need to understand how that happens with the two kinds of Clifford algebras.
• I need to consider how that relates to the field with one element.
• I need to understand symmetric spaces in terms of various kinds of subspaces of spaces.
• Understand the products of mutually anticommuting linear complex structures {$J_1J_2\cdots J_k$}.
  • Calculate {$J_1J_2\cdots J_k$} as given in this table.
  • Study Stone, Chiu, Roy
  • Figure out what is a shift in perspective.
  • Study how a symmetric space describes the ways of adding a new anticommuting linear complex structure, thus a new perspective.
    • Study Eschenburg
• Draw intuition from the quantum symmetries, how they relate to the process of adding mutually anticommuting linear complex structures.
  • Moore. Quantum Symmetries and Compatible Hamiltonians
  • Bott Periodicity For Clifford Algebra Maniacs

The Three Minds

Complex Bott periodicity is based on the Hopf fibration, which models the three minds. Also, the proof by Aguilar and Prieto is based on the three-cycle (from a long exact sequence of fiber, total space and base space) where one perspective is set to zero, equating the other two. So I need to understand the connection between the three minds and the three-cycle.

Hopf Fibration

• Study John Baez. Projective lines.

Suspension - Loop Space - Classifying Space Adjunction

Think of smash product (with a circle) as composition of perspectives. {$S^1$} perspective, {$S^2$} perspective on a perspective, {$S^3$} perspective on a perspective on a perspective.

The base point of a pointed space can be thought of as a zero. In the case of a wedge sum {$X\vee Y$} the base points behave as an additive zero and so we get a disjoint union of {$X$} and {$Y$} with a fused base point. In the case of the smash product {$X\wedge Y$} the base points behave as a multiplicative zero and so we get nonzero combinations {$(x,y)$} and anything with zero (anything with a base point) collapses to zero (to the resulting base point). Note the importance of zero in building up the concept of a vector space but also, upon removing it, the concept of a projective space. Note also that the two-point pointed space {$S^0$} can be thought of as zero and one, the field {$F_2$}, and the zero by itself can be thought of as the center of a polytope and as the field with one element {$F_1$}.

The Collapse

Note that we can write out all of the {$J_i$} in terms of {$J_1,J_2,L,I_1,I_2,I_3,I_4$} and with {$J_7$} (noncontradiction) we have no dependence on {$J_1$} or {$J_2$} (which are evidently the sources of contradiction). So the initial state is contradiction but {$J_i$} are expressions of that contradiction. Note also that the collapse is given by {$J_8$} depending on {$L$} and {$L_2$} so that is the heart of contradiction.

Symmetric Spaces

• Understand how symmetric spaces variously express subspaces of vector spaces.
  • Moore. Quantum Symmetries and K-Theory.
  • Study how real structures (where {$T^2=+1$} or {$C^2=+1$}) and quaternionic structures (where {$T^2=-1$} or {$C^2=-1$}) are defined.
  • See the articles by Eschenburg.

Understand what is an isotropy group

• Understand what can be said about an isotropy group of an isotropy group.
• Specifically consider the case of {$O(4n)$}, {$U(2n)$} and {$Sp(n)$}.

Vector Bundles on Unit Spheres

Understand Bott periodicity in terms of vector bundles on unit sheres and how that relates to Clifford modules.

• Study Jost Eschenburg, Bernhard Hanke. Bott Periodicity, Submanifolds, and Vector Bundles
• Understand the clutching construction.

Octonions

Understand how the octonions organize Bott periodicity.

• Jost Eschenburg. Geometry of Octonions.
• John Baez. {$\mathbb{O}P^1$} 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.

Understand the octonions as the projective plane of {$F_2$}.

• Relate the 2x2 switches for quantum symmetries and the circle of recurring activity (a third switch) with this cube.
• How do the cube and the octonions encode the 8 divisions, 6 conceptions and 12 circumstances?
• What is the projective plane of {$F_1$}?

Krebs Cycle

• Understand the Krebs cycle and look for connections.
  • Compare the eightfold Nanorooms diagram with the ninefold Nick Lane diagram.
  • Consider how glycolysis may act through eight layers (as with geometry, deriving noncontradiction out of contradiction) and establish the eight-cycle.

Other mathematical expressions of the divisions of everything

Study how chain complexes and exact sequences carve up mental space.

• Threesome: Fiber bundle
• Foursome: Yoneda Lemma
• Fivesome: Helmholtz decomposition
• Sevensome: Snake Lemma

Gain intuition regarding the field with one element.

• Combinatorial interpretations.
• Buildings

Divisions of everything

• Systematize the examples of the three minds in the Theory Translator.

• The nullsome is the lack of context (for the three minds: the nullsome, onesome, twosome, threesome). Foursome establishes arbitary context (for the nullsome, onesome, twosome, threesome). The eightsome establishes arbitrary context within arbitrary context, which is no context, thus collapses back to the nullsome.

Notes

A section on a vector bundle involves: a point {$p$} on a manifold {$X$}, a vector space {$V_p$} related to each point, a vector {$v_p$} within that vector space. Compare that to the foursome or fivesome.