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Book: Divisions


Bott periodicity, Map of fixed point theorems

Describe mathematically the divisions of everything.



Fiber spaces (fiber bundles?) are understood as finite exact sequences and perhaps vice versa. This outlook is key for Bott periodicity.

{$F{\rightarrow}E\overset{\pi}{\rightarrow}B$}

A perspective is a map with a fixed point. Study fixed point theorems, such as Brower's fixed point theorem, and the inverse function theorem and implicit function theorem.

Finite exact sequences are divisions of everything

I believe that finite exact sequences can be thought of as divisions of everything. I am trying to show this and understand what this means.

Infinite exact sequences can be thought of the way that growth proceeds, extending the whole. Finite exact sequences can be thought of as infinite exact sequences which have turned in upon themselves, giving an autoassociative function.

For example, Bott periodicity describes how the infinite exact sequence folds in on itself in an eight-cycle as the relationship between a matrix and its entries.

The growth of the finite exact sequence seems to come from the middle, which keeps becoming more refined.

Matematikos žinojimo rūmuose trejybės ratas sukuria autoasociatyvas sekas - jos iš begalinių "tikslių sekų" padaro baigtines tikslias sekas, tad padalinimus.

An exact sequence is a way of intrinsically defining the concept of dimension. Each term in the sequence characterizes elements of a particular dimension. The terms are organized by increasing dimension. Each increase in dimension corresponds to the introduction of a new perspective which expands upon the previous dimension.

A simplex is imagined to be embedded into 0, Zero, that which is not there. So the holes are equated to Zero if you go around them. If you go around something that is there, A, then the sum is A rather than Zero. But then the cycle around A is a boundary of A. So we mod out by such boundaries so that they don't affect our search for holes. So homology is counting the unfilled holes. "Cycles" are the boundaries of holes (filled or not); "Boundaries" are the boundaries of filled holes; "Co-cycles" are... ; "Co-boundaries" are...

A multi-dimensional torus has holes (Betti numbers) given by the binomial theorem.

Note that a cross polytope has no totality - no volume and hence no "filling" but is always a cycle that is not a boundary.

Interpreting mathematical structures as divisions of everything

Roots of unity = divisions of everything?

Think of an exact sequence as starting from everything and ending with everything. Everything from above and from below, with the two identified.

Nullsome

Onesome

Twosome

Threesome

Foursome

Fivesome

Sixsome

Sevensome and eightsome

Finite exact sequences

Ideas

Examples

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Puslapis paskutinį kartą pakeistas 2019 rugpjūčio 10 d., 13:04