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”, 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 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.

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.

Watch these YouTube Videos for a clear illustration:

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 Poincaré Conjecture still 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.

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

You May Also Like

A Problem (and Solution) from Bhaskaracharya’s Lilavati

I was reading a book on ancient mathematics problems from Indian mathematicians. Here I wish to share one problem from Bhaskaracharya‘s famous creation Lilavati. Problem A beautiful maiden , with beaming eyes, asks of which is the number that multiplied by 3 , then increased by three-fourths of the product, divided by 7, diminished by one-third of the quotient, multiplied…

Irrational Numbers and The Proofs of their Irrationality

“Irrational numbers are those real numbers which are not rational numbers!” Def.1: Rational Number A rational number is a real number which can be expressed in the form of where $ a$ and $ b$ are both integers relatively prime to each other and $ b$ being non-zero. Following two statements are equivalent to the definition 1. 1. $ x=\frac{a}{b}$…

Getting Started with Measure Theory

Last year, I managed to successfully finish Metric Spaces, Basic Topology and other Analysis topics. Starting from the next semester I’ll be learning more pure mathematical topics, like Functional Analysis, Combinatorics and more. The plan is to lead myself to Combinatorics by majoring Functional Analysis and Topology. But before all those, I’ll be studying measure theory and probability this July – August. Probability…

The ‘new’ largest known Prime Number

Great Internet Mersenne Prime Search (GIMPS) group has reported an all new Mersenne Prime Number (a prime number of type $2^P-1$) which is, now officially the largest prime number ever discovered. This number is valued to a whopping $2^{74207281}-1$ and contains 22,338,618 digits. It is quoted as M747207281 and is almost 5 million digits longer than the previous record holding prime number…

Numbers – The Basic Introduction

If mathematics was a language, logic was the grammar, numbers should have been the alphabet. There are many types of numbers we use in mathematics, but at a broader aspect we may categorize them in two categories: 1. Countable Numbers 2. Uncountable Numbers The numbers which can be counted in nature are called Countable Numbers and the numbers which can…