# Understanding Poincaré Conjecture

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## Introduction & Statement of Poincaré Conjecture

In 1904, the french Mathematician Henri Poincaré posed an epoch making question in one of his papers, which asked:

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

## Explanation

The statement can be explained by considering the 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. [for illustrations Watch these videos on youtube.]
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”, whereas, 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 in both sides of the plane. 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 Again.

Student... and that's it! Gaurav's life lies in the interval $(x-\varepsilon, x+\varepsilon)$ .