Understanding Poincare Conjecture

In 1904, the french Mathematician Henri Poincaré (en-US: Henri Poincare) posed an epoch-making question, which later came to be termed as Poincare Conjecture, in one of his papers, which asked:

If a three-dimensional shape is simply connected, is it homeomorphic to the three-dimensional sphere?

Henri Poincare – 1904

So what does it really mean? How can a regular math reader understand what poincare conjecture is and why does it matter?

Explanation

The ambiguous statement of Poincare conjecture can be explained by considering an analogous two-dimensional situation.

Let us think of a rubber band stretched around the spherical surface of an apple (or any other spherical body like ball) . It is easily seen that it can be shrunk to a point by moving it slowly, without tearing it and without allowing it to leave the surface.

On the other hand, let up take a doughnut, which is torus. The band can’t be shrunk in any way to a point without tearing it or the doughnut. It can be stated mathematically that the apple/ball is “simply connected”, while the doughnut is not.

In order to characterize the spherical surface of the apple/ball, we can imagine a two-dimensional disc lying in a three-dimensional plane with its boundary lifted up and tied to a single point on both sides of the plane.

Also read: The Collatz Conjecture : Unsolved but Useless

In logical words, it can be said that all the points are identified to a single point [in apple/ball].

It can be done if the two-dimensional disc is in a three-dimensional space, i.e., we have a two-dimensional sphere in a three-dimensional space.

Read the above statement again.

Therefore, Poincaré Asked,

If a two-dimensional sphere is characterized by the property of simple connectivity, a similar characteristic is valid for all closed three – dimensional objects, embedded in a four dimensional spaces, (and) which are like three-dimensional spheres.

Which is:

Every simply connected, closed 3-manifold is homeomorphic to a 3-sphere.

A space is called simply connected (or 1-connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.

Image

See: https://en.wikipedia.org/wiki/Simply_connected_space

3-manifold = a space such that every point has an open neighborhood homeomorphic to 3-dimensional Euclidean space. Euclidean space is the normal X-Y-Z space that you read now and then.

3-sphere is actually a 4-dimensional sphere in Eulidean space.

A disk is a 1-sphere and a sphere is a 2-sphere.

Although analogous results were found to be true at higher dimensions (>3D), in case of three-dimensional sphere they have proved to be the hardest. That’s why the question had been viewed as extraordinarily difficult and for long, mathematicians were struggling to find an answer to it. Even Poincaré himself pondered over the problem. Of course, these led to the creation of new vistas in mathematics and many new theorems and problems related to it solved. But Poincare Conjecture remained unsolved. This was named as one of the Millennium Prize Problems by Clay Mathematical Institute.

At last in 2002-03, Grigori Perelman succeeded in finding a solution based on the theory of Ricci Flow.

Gaurav Tiwari

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