Complete Solution of Poincare Conjecture

In summary, this paper proves that the Poincare and geometric conjecture are both true. Differential Geometry meets Geometric Surgery on three-manifolds. Perelman clarified and (perhaps) corrected.
  • #1
selfAdjoint
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Announced in http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2-165-492.pdf" . Differential Geometry meets Geometric Surgery on three-manifolds; Perelman clarified and (perhaps) corrected.


A COMPLETE PROOF OF THE POINCAR´E AND GEOMETRIZATION CONJECTURES – APPLICATION OF THE HAMILTON-PERELMAN THEORY OF THE RICCI FLOW

HUAI-DONG CAO† AND XI-PING ZHU‡

Abstract. "In this paper, we give a complete proof of the Poincar´e and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow. "

The first sections give a clear history of the recent approaches to the Poincare Conjecture and Thurman's Geometric Conjecture, which are joined at the hip. The guy who I feel sorry for is Hamilton, who did fantastic things to lay almost all of the groundwork for the solution but, like Moses, was not able to enter the promised land.
 
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  • #3
neutrino said:
I don't understand an iota of all this, but here's some related discussion at NEW
http://www.math.columbia.edu/~woit/wordpress/?p=434


Right. I should have made it clear that's where I got the link to the Cao and Zhu paper from.

While I fully expect the actual proof to be over my head, and I can't hope to maike it through all 200 + pages, the historical account and the general idea of what they're doing is pretty clear to me.
 
  • #4
selfAdjoint said:
the historical account and the general idea of what they're doing is pretty clear to me.
Even that's waaay over my head. :biggrin:
 
  • #7
Why does it start at page 165? What are on the previous pages?
 
  • #8
That's because it's been taken from a journal.
 
  • #9
Last two sentences.

"Hence in case (2), M is diffeomorphic to a flat manifold and then it is also geometrizable.

Therefore we completed the proof of the theorem. "


That's a million dollar conclusion, heh.
 
  • #10
waht said:
That's a million dollar conclusion, heh.

Welcome to modern mathematics.
 

What is the Poincare Conjecture?

The Poincare Conjecture is a mathematical problem first proposed by Henri Poincare in 1904. It states that any closed, simply connected 3-dimensional manifold (a mathematical object that looks like a 3-dimensional space in every small region) is topologically equivalent to a 3-dimensional sphere.

Why is the Poincare Conjecture significant?

The Poincare Conjecture is significant because it is one of the most famous and long-standing unsolved problems in mathematics. Its solution would have implications in many areas of mathematics and physics, including topology, geometry, and cosmology.

Who solved the Poincare Conjecture?

The Poincare Conjecture was solved by Russian mathematician Grigori Perelman in 2003. His proof was verified and accepted by the mathematical community in 2006, and he was awarded the prestigious Fields Medal for his work.

What is the complete solution of the Poincare Conjecture?

The complete solution of the Poincare Conjecture is a mathematical proof that shows that any closed, simply connected 3-dimensional manifold is topologically equivalent to a 3-dimensional sphere. This means that any shape that looks like a 3-dimensional space in every small region can be deformed into a sphere without any tearing or holes.

What are some applications of the Poincare Conjecture?

The Poincare Conjecture has applications in many areas of mathematics and physics, including knot theory, differential geometry, and string theory. It has also led to advancements in understanding the topology of higher-dimensional spaces and the structure of the universe.

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