The Poincare Conjecture
Well, after a year of looking at the background mathematics, we're finally ready to state the Poincaré Conjecture:
Every simply-connected, closed 3-manifold is homeomorphic to the 3-sphere.
This isn't exactly how Poincaré put it (for a start, he said it in French) but this is the best way to express it given the terminology we've used up until this point.
Poincaré's conjecture has to do with three-dimensional manifolds but it might be easiest to start off thinking about the two-dimensional version:
Every simply-connected, closed 2-manifold is homeomorphic to the 2-sphere.
Consider the surface of a ball and a torus. Both are closed 2-manifolds. But only one is simply-connected. A torus isn't simply-connected so it can't be homeomorphic to the 2-sphere. The surface of a ball is simply-connected and it is homeomorphic to the 2-sphere.
The big question is: is it possible to find a closed 2-manifold that is simply-connected but is not homeomorphic to the 2-sphere? In a nutshell: no, it's not. If it's a closed 2-manifold and it's simply-connected then there isn't any topological property that will distinguish it from the 2-sphere.
The Poincaré Conjecture is that this is true for 3-manifolds as well.
Of course, this is really just the beginning of our journey. Mathematicians have spent the last century trying to prove this so we still have a lot to cover.
Interestingly, it's already been proven that it's true for dimensions greater than 3 (as well as for 1 and 2 dimensions). Stephen Smale proved it in 1960 for dimensions greater than 6 and then extended his proof to cover dimensions greater than 4. In 1966, he was awarded the highest prize in mathematics, the Fields Medal, for this proof. Michael Freedman then proved in 1982 that it's true for 4 dimensions which won him the Fields Medal in 1986.
UPDATE: next post
Comments (2)
Peter Mott on Nov. 19, 2007:
In n-dimensions, if a collection X of 'spheres' is such that any n+1 overlap
(have an area in common) then all of X have an area in common.
For n=1 a 'sphere' is just an interval of the line. Let X be a collection of intervals such that
any x,y overlap. Let L be the largest l of (l,u) in X and U be the smallest u of (l,u) in X.
Then (L,U) is an interval, for if not then L>U and so the intervals (L,u) (l,U) are a pair that don't overlap. (L,U) is in fact the common interval. qed
Thus encouraged I tried n=2 but never managed to prove it nor find anywhere a proof. Maybe your
explorations in much harder stuff provides a clue. Russell wrote the book in 1927 and discussed it
with a Cambridge topologist called M. Newman. It will follow from some properties of metric spaces I think. You are allowed to make assumptions as required. Russell was sloppy!
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Last Modified: Aug. 31, 2005
Author: jtauber
sunbin on June 5, 2006:
http://sun-bin.blogspot.com/2006/06/poincare-conjecture-proved-by-chinese.html