Poincare conjecture

[AntonioLao]

The following site discusses the important of Poincare conjecture and its use in topology of manifold. http://mathworld.wolfram.com/PoincareConjecture.html

[Guille]

Actually this conjecture is only a special case or part of the geometrization conjecture. So proving the second one would prove automatically the first, and proving the first, would give some hints to proof the second.

[AntonioLao]

the geometrization conjecture.

Thanks for this information. The site at http://en.wikipedia.org/wiki/Thurston's_geometrization_conjecture
discusses more about it. According to this site both conjectures have been solved in 2002 by Grigori Perelman.

[Guille]

Thanks for this information. The site at http://en.wikipedia.org/wiki/Thurston's_geometrization_conjecture
discusses more about it. According to this site both conjectures have been solved in 2002 by Grigori Perelman.

True, although, quoting the site you give:

"but his work will need to survive two years of systematic scrutiny after publication, before the Clay Mathematics Institute can deem the conjecture to have been solved."

If his proof was in 2003, and we are in 2005, this scrutiny shoudl ahve been finished by now. Maybe the site isn't updated, maybe look in the oficial web page of the prizes we can see if it's accepted to be solved.

[Guille]

By the way, I've searched alla round and I have found the original papers of the proof by Perelman in arxim. The link is this:
http://arxiv.org/find/math/1/au:+Per...c26d61a3f27d1c

I don't really understand it, I have to increase my knowledge of topology.

[AntonioLao]

Thanks for this site. I'll try to find time to read these articles beginning from the one dated November 2002.

[protheory]

Hi, my name is PRO and I have recently published my own ideas about the Poincare Conjecture online.

Here's the link http://www.protheory.com/Poincare%20conjecture.htm

I would really appreciate any feedback you could give me.

Thanks everybody, and I'm sorry to post in so many topics in such a short time

PRO.

[protheory]

5. The Poincaré Conjecture

Introduction

Manifold

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

Relation

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 Problem

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.

The Answer

Conjecture

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.

Dimensional

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.

Restricted

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.

Combination

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.

Answers

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

Simply

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

Simultaneously.

Conjecture

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.

My point is literal.

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

This is the rough version of my answers, this page is intended only to start the conversation

I'm still working on how best to communicate my theory as postitively as I can

My other Millennium Problems answers can be found here... http://www.protheory.com/Millenium%20Problems.htm

Thanks everybody.

PRO.

[bsaucer]

Hypothetically speaking, if the poincare conjecture had been proven to be false, what would this have done to the geometrization conjecture? Would there have been another geometry to be discovered, or would some manifolds exist which couldn't be decomposed into geometric pieces?

Also, wasn't there another part of the geometrization conjecture that needed to be proven? I think it had something to do with the hyperbolic manifolds...

[AntonioLao]

Whoever proved it will be reward $1,000,000 by the Clay Mathematics Institute. See http://www.claymath.org/ and http://en.wikipedia.org/wiki/Clay_Mathematics_Institute