Constructed
here is a fiber bundle over a Base Space comprised of a circle **
S**^{1} with Fiber given by a
fractal with **
Z**^{n} cyclic symmetry group ( **n** = degree of cyclical symmetry ) and
produced by an
Iterative Function System, IFS. The Total Space **E **is comprised of the fractal parametrically winding or
twisting about it's local center and following the global orbit line around
a non-local axis until it matches back to the Fractal Fiber's starting position.

The angle of local Twist**T **can be in
integral multiples of **N*(360/n), **where {
**N **= 0, 1, 2 ...} and { **n** =
degree of cyclic symmetry}. A Fiber Bundle is said to
have a Twist**T **
which can determine the orientability of the Total Space. The Twist Group **T , **invariants, soliton, abelian,
non-abelian, trivial, non-trivial, Gauss-Bonnet, symplectic, Gauge Group,

Fiber Bundles are spaces which are locally Products but not
necessarily globally. A discrete fiber
**space **produces a fibration that is a covering
**space**, like a helix covering a circle. A vector fiber
**space **creates a vector bundle and is classified either
algebraically by characteristic classes which are elements of homology
groups OR geometrically by maps to classifying
spaces. Also, a loop **space ** produces the space
of loops in a given space.

These invariants are called
characteristic classes and are viewed formally as elements in the cohomology
groups of the manifold, the duals of the homology groups. Characteristic classes
can also be defined purely topologically by investigating the global motion of
the tangent space as the manifold is traversed. They measure to some extent the
failure of the manifold to possess a globally defined notion of **parallelism****: **for example, a global
parallelism exists on the torus in which Euler-Poincaré characteristic = 0, but not on the two-dimensional sphere in which
Euler-Poincaré characteristic =
2.

(... the Euler-Poincaré characteristic number
is related to the Euler graph number for the graph that is related to the
surface ? ...)

Fibers are objects in some
Category. All fibers must be isomorphic, homeomorphic, or
self-similar. Every Category is built with a notion of
isomorphism. Vector Spaces, Groups, Topological Spaces, Manifolds,
G-sets, and **Fractals**. Two Fibers are homeomorphic if
one Fiber can be deformed into another by a one-to-one mapping and a continuous
function. Homeomorphism is too weak to preserve the **fractal
**nature of objects.

For
example, the Koch curve is actually homeomorphic to the plain old non-fractal line
segment of length 1.

People usually talk about self-similar fractals. A similarity is stronger than a homeomorphism. A one-to-one and onto mapping is called a similarity if there is a constant c (called the similarity factor) such that for any

two points x1 and x2 in X their images y1 and y2 in Y have the property that the distance from y1 to y2 is c times the distance from x1 to x2.

A fractal is called
self-similar if it it can bew divided into finitely many pieces
:

X = X1 union X2 union ....

such that there is a similarity f1 from X onto X1, a similarity f2 from X onto X2 and so on.

In that sense the Koch curve is self-similar because it can be divided into four pieces, each of which is 1/3 the size of the original Koch curve.

X = X1 union X2 union ....

such that there is a similarity f1 from X onto X1, a similarity f2 from X onto X2 and so on.

In that sense the Koch curve is self-similar because it can be divided into four pieces, each of which is 1/3 the size of the original Koch curve.

A Fiber Bundle can be a Fibration, but not necessarily
everytime.

( ... something
about the parallel fibres .... in the special case of the affine space where the
connection ..., parallelism of *A*_{j}^{i}
means the constancy of all the components. Properties of the connections
imply certain types of manifolds falling in certain types of characteristic
classes ... parallel transport ...

Connection
A <> Curvature dA <> Torsion A ^ dA <> Hypervolume dA ^ dA
...

Gauge Theory <> Connection
Manifold <> Choice of A - Gauge Freedom)

Geodesics, Foliations with Leaves,
(non) Anosov Flow > Ergodicity > Topo Entropy, diffeomorphism, orbits,
orbitfolds, orientifolds, Laminations

Each Fiber **F** in the Total Space **E
**
projects to exactly one point in the Base Space **B**. The Product of a Fiber **F** and a local Neighborhood **U **of the Base Space **B** creates the Total Space **E**. The Product can be denoted as

The
inverse image of a point in the Base Space **B** is the Fiber **F**
.

A Fiber Bundle with a Fractal as a
Fiber *F *is a Map **f : E -> B **where **E **is called the Total Space of the
fiber bundle and **B **the
Base Space of the fiber bundle. Showing that for every point in the Base
Space ( b in **B **) has a Neighborhood
**U **so that ** f**^{1}(U)
is Homeomorphic to *U x F
*by some continuous function and one-to-one
mapping.

Troy Christensen

Email