Investigating Bott periodicity
- Understand the adjunction: suspension - loop space - classifying space, and relate it to the tensor-hom adjunction.
- Consider in the suspension (tensor) - loop space (hom) adjunction, how a {$k$}-fold (hom) suspension yields a division of everything (tensor) into {$k$} perspectives.
- What is the role of the classifying space?
- How is recurring activity related to a perspective?
- How can I think of a linear complex structure (based on {$i$}) as a generator for disjoint circles? Is it related to Lie algebras?
- How can I think of the swapping structure (with ones on the anti-diagonal) as a generator for hyperbolas? (hyperbolic functions)
- What is the geometry, in terms of circles, of sets of mutually anticommuting linear complex structures?
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
- In Stone, Chiu, Roy, compare the matrices for the linear complex structure {$J_1$} and for the second, mutually anticommuting, linear complex structure {$J_2$}
{$J_1$} | {$J_2$} | {$J_1J_2$} |
{$\begin{pmatrix}0 & -1 & 0 & 0 \\ 1& 0 & 0 & 0 \\ 0& 0 & 0 & -1 \\ 0& 0 & 1 & 0 \\ \end{pmatrix}$} | {$\begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0& 1 & 0 & 0 \\ \end{pmatrix}$} | {$\begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ \end{pmatrix}$} |
Look for {$M$} such that {$J_1M=J_2$}. {$M=J_1^{-1}J_2$}. Consequently, {$M=-J_1J_2$}.
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
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 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.
- Draw intuition from the quantum symmetries, how they relate to the process of adding mutually anticommuting linear complex structures.
Hopf Fibration
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 {$F_1$}.
Symmetric Spaces
- Understand how symmetric spaces variously express subspaces of vector spaces.
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.
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.
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