Visualization of PC

I. Mendiaizagaz (NO RELATIONS WITH/NOT WORKING AT THE INSTITUTION, Lagos State University)

Published in matho.philica.com

Abstract
Text for the divulgative video Visualization of PC

Analyze the statement of the theorem, for a lower

dimension, in the 3D space. Every simply connected,
closed 2manifold
is homeomorphic to the 2sphere.
The
statement says: If a 2manifold
is closed, bounded and
simply connected, then, the 2manifold
is always
deformable into S². Simply connected: any two lines
connecting any two points, can be transformed one to the
other through the surface. To verify the statement is true,
deform a generic closed, bounded and simply connected
2manifold
step by step. [(1) Two points of the 2manifold.
(2) A line connecting them. (3) Deform the
line and the 2manifold.
(4) Another line connecting the
two points. (5) Deform the line and the 2manifold.
(6)
As a property of the 2manifold,
these two lines can be
transformed one to the other through the surface. Even
there are different possible directions for the
transformation, the results are equal topologically. (7)
Deform the 2manifold
until the transformation way is
represented as half sphere. (8) The previous two points of
the 2manifold.
(9) One of the previous lines connecting
them. (10) Another line connecting them. (11) Deform
the line and the 2manifold.
(12) As a property of the 2manifold,
these two lines can be transformed one to the
other through the surface. Even there are different
possible directions for the transformation, the results are
equal topologically. (13) Deform the 2manifold
until the
transformation way is represented as 3/4 sphere. (14 ∞
)
Repeat the process.] So, the only possible simply
connected, closed and bounded 2manifold
topology is
S².

The original theorem, in the 4D space. Every simply
connected, closed 3manifold
is homeomorphic to the 3sphere.
Mentally, it is easy to imagine a 4D space, but
that space does not fit into the 3D reality. It can be
visualized using 3D cuts of the 4D space. S³ radius 1,
will be formed by all the points at distance 1 from the
center (S² in each w). In 4D space, a closed, bounded and
simply connected 3manifold.
Each cut used to represent
it, will be a closed and bounded 2manifold.
If one of
the 2manifolds
were notclosed,
the 3manifold
would
be notclosed.
If one of the 2manifolds
were not
bounded, the 3manifold
would be not bounded. The
theorem says: every closed, bounded and simply
connected 3manifold
is deformable into S³. To verify the
statement is true, deform a generic closed, bounded and
simply connected 3manifold
step by step. [(1) In an
element of the w=0 cut, two points. (2) In that element of
the w=0 cut, a line connecting these two points. (3)
Deform the line and the 3manifold.
(4) Another line
connecting the two points, in the same element of the
w=0 cut.(5) Deform the line and the 3manifold.
(6) As a
property of the 3manifold,
these two lines can be
transformed one to the other through the surface. (7)
Choose one of the ways to transform the lines. (8)
Deform the 3manifold
until the transformation way is
represented in the w=0 cut. Even there are different
possible directions for the transformation, the results are
equal topologically. (9) Deform the 3manifold
until the
transformation way is represented as half sphere in the
w=0 cut. (10) The two points. (11) One of the previous
lines connecting the two points. (12) Another line in w=0
connecting the two points. (13) Deform the line and the
3manifold.
(14) As a property of the 3manifold,
these
two lines can be transformed one to the other through the
surface. (15) Choose one of the ways to transform the
lines. (16) Deform the 3manifold
until the
transformation way is represented in the w=0 cut. Even
there are different possible directions for the
transformation, the results are equal topologically. (16)
Deform the 3manifold
until the transformation way is
represented as 3/4 sphere in the w=0 cut. (17 ∞
)
Repeat the process.] Finally obtain S² in the w=0 cut. The
3manifold
is simply connected, locally smooth, so the
rest of the 3manifold
always can be deformed as a
continuation of S² w=0. This is the basis form, (common
for all closed, bounded and simply connected 3manifolds)
a continuation of S² with unknown sides.
Starting from this basis form (which is open in each
side), there are only two options to create closed and
bounded 3manifolds.
First option, solutions connecting
the two ends, but this type of 3manifolds
are not simply
connected, not our case. Second option, the only one
possible, to turn the two sides closed independently. So,
the only possible simply connected, closed and bounded
3manifold
topology is S³.

Analyze the statement of the theorem, for a higherdimension, in the 5D space. Every simply connected,
closed 4manifold
is homeomorphic to the 4sphere.
(This is not a particular case of the generalized Poincaré
Conjecture.) Closed, bounded and simply connected 4manifold.
Using the same procedure as before, the basis
form will be obtained. Starting from this basis form,
different topologies of closed and bounded 4manifolds
are possible. Including different topologies of closed,
bounded and simply connected 4manifolds.
So, not all
closed, bounded and simply connected 4manifolds
can
be deformed into S?.