The Poincaré Conjecture

This problem concerns the equations that are thought to govern a three dimensional sphere, and the mathematical properties that define its characteristics.

The Poincaré conjecture is a problem from the field of topology.

Topology is the study of the properties of a given object after certain applied events.

These events may be mathematically twisting, stretching, or otherwise deforming a given object, though tearing is not allowed.

If you imagine stretching a rubber band around an apple, it is theoretically possible to slowly shrink it to a point without breaking it or the apple.

The opposite example is the same idea but using a doughnut instead of an apple.

The previous result is not possible so the opposite occurs.

This principle of opposite properties became known as simple connectivity.

The surface of the apple is said to be simply connected, and the surface of the doughnut is not.

Topologically speaking, a two dimensional sphere is thought to be governed by the property of simple connectivity.

Poincaré asked whether or not the three dimensional sphere is characterized as the unique simply connected three manifold.

He wanted to solve the equations that define the same mathematical property in relation to a three dimensional sphere, but was unable to provide a singular answer.

Is a three dimensional sphere simply connected or not?

Poincaré assumed that it was at first but he was unable to prove it.

The Millennium problem is to prove whether or not the Poincaré conjecture is correct, and to fully explain this principle of simple connectivity in relation to a three dimensional sphere.

For the exact problem description please refer to Claymath.org

The Answer

The Poincaré conjecture presents us with two options, either a three dimensional sphere is simply connected or it is not, and asks us to choose one of them.

The concept of simple connectivity seems to work for a two dimensional example because the outcome is restricted to two possible opposites.

1. Simply connected.

2. Not simply connected.

The rules change when considering three dimensions because instead of being restricted to only two potential answers, we are now restricted to three potential answers, no more and no less.

A three dimensional sphere may still be simply connected or not, as it was in the two dimensional answer, but now it may also be a simultaneous combination of the two.

Three simultaneous dimensions require three simultaneous answers in the same way that all other answers to all other questions do.

A three dimensional sphere may be simply connected, not simply connected, and neutral.

Simultaneously.

1. The Poincaré conjecture is true.

2. The Poincaré conjecture is false.

3. The Poincaré conjecture is neutral.

Simultaneously.

Am I wrong?

I simultaneously oppose, agree with, and neutralise all criticism ad infinitum.

There is no point creating a theory of everything that doesn’t work.