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?. Information about this Article This Article has not yet been peerreviewed
Published on Monday 15th May, 2017 at 09:03:19.
