**Investigation: Relate Bott periodicity and the eight-cycle of divisions of everything.**

博特周期性定理

- palyginti susijusias Lie grupes (ir jų ryšį su gaubliu) su požiūrių permainomis
- Max Karoubi savo video paskaitoje paminėjo loop lygtį žiedams kurioje R,C,H,H' ir epsilon = +/-1 gaunasi 10 homotopy equivalences. Kodėl 10? 8+2=10? ar 6+4=10, dešimt Dievo įsakymų?
- How might the fourfold periodicity of the sign of the pseudovector be related to the fourfold periodicity of the differentiation of sine and cosine functions?
- Does the constraint {$J^2=−I_n$} on complex structures and their anti-commutativity relate to the constraints on Clifford algebras?

Study and understand:

- Fiber bundle
- the long exact sequence

{$$\dots\rightarrow \pi_nF\rightarrow \pi_nE\rightarrow \pi_nB\rightarrow \pi_{n-1}F\rightarrow \dots$$}

- Characteristic class
- Clifford algebras, clock shifts
- Topological K-theory, infinite unitary group, classifying spaces.

Videos

- Peter May. An excellent overview.
- Peter May. History of algebraic topology in the 1950s.
- Algebraic Topology July 2016 University of Chicago
- Xuan Gottfried Yang

Statement

Expositions

- Bott periodicity John Baez
- John Baez about Clifford algebra periodicity
- Max Karoubi. Bott Periodicity in Topological, Algebraic and Hermitian K-Theory
- 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.

Extensions

- Wikipedia: Higher Clifford Algebras 24 periodicity of 3-categories

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

Related concepts

- Ortogonal group discusses Bott periodicity.
- Wikipedia: Hopf fibration
- Wikipedia: Hopf invariant and Adam's theorem.
- Wikipedia: Homotopy group of spheres
- Wikipedia: Clifford paralells and quaternions
- A Survey of Elliptic Cohomology by Jacob Lurie, mentions the Bott element, whose inversion is perhaps related to the period 2.

Math facts

- Bott periodicity is based on the fourfold periodicity of the sign of the pseudovector.
- Start from C - as the basic duality - and end up at C inverted (perhaps the conjugate) and go back. This circle is used to list the quotients of Clifford modules {$GL_{n}(\mathbb{R})/GL_{n}(\mathbb{C})$} check? etc If you multiply all the quotients together than you get the identity (?)

{$\begin{pmatrix} & & \mathbb{C}_{n} & & \\ & \mathbb{H}_{n} & & \mathbb{R}_{n} & \\ \mathbb{H}_{n} \times \mathbb{H}_{n} & & & & \mathbb{R}_{n} \times \mathbb{R}_{n} \\ & \mathbb{H}_{n} & & \mathbb{R}_{n} & \\ & & \mathbb{C}_{n} & & \end{pmatrix}$}

- C0 R
- C1 C
- C2 H
- C3 H + H
- C4 H(2)
- C5 C(4)
- C6 R(8)
- C7 R(8) + R(8)
- C8 R(16)

''C_{n+8} consists of 16 x 16 matrices with entries in Cn ! For a proof you might try

Generalized Clifford Algebra has clock-shift operators.

- Some matrices describe the 8-cycle clock (the trolley stops).
- Generalized Pauli matrices describe the 3 shifts (the trolley cars of different increments +1, +2, +3).

Complex structures

- We call {$J$} a complex structure on {$R^n$} if {$J\in O(n)$} and {$J^2=−I_n$}. Denote the space of complex structures {$Ω_1(n)⊂O(n)$}.
- Define {$Ω_k(n)$} to be the space of complex structures that anti-commute with fixed {$J_1,\dots ,J_{k-1}$}.
- {$Ω_0 \cong Ω_8$}
- Consider how complex structures relate to divisions of everything. Apparently, each {$J_i$} is a perspective. Anti-commutativity {$J_iJ_j = -J_jJ_i$} means that the composition of perspectives is inverted if the order is switched. So the matrix {$-I$} can be interpreted as an inversion of perspective, and thus, of chains of perspectives. A set of eight perspectives brings us back to no perspectives, which is to say, the default perspective at the origin.

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
- Complex case: 2-periodicity - divisions having 4 (nežinojimas) or 2 (žinojimas) representations. Real case: 8-peridocity.
- 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.

Parsiųstas iš http://www.ms.lt/sodas/Book/BottPeriodicity

Puslapis paskutinį kartą pakeistas 2019 birželio 17 d., 16:13