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Edited on Tue Sep-07-04 02:30 AM by Lucky Luciano
The Poincare Conjecture states the following. Any simply connected compact 3 dimensional manifold without boundary with the same homotopy type as a 3-Sphere is actually homeomorphic to a 3-Sphere.
Partial Explanation for Laypeople:
A n-manifold may be thought of as a gemoetric object that 'locally' looks like n-dimensional Euclidean Space. For simplicity, you may think of it as being described locally by n coordinates (X1, X2, X3,..., Xn). An example of a one dimensional manifold is a circle (A one dimensional sphere) because around any point you may take a small enough neighborhood that looks like an open interval of the Real Line (of course you would have to allow for bending it into shape, but that is ok in topology). A two dimensional manifold might be the surface of a doughnut - called a torus. You can take any point on the torus and find a small enough neighborhood such that when you take that neighborhood and flatten it out, it looks like a piece of a two dimensional plane. A 2-Sphere is another two dimensional manifold and is described by an equation in 3-dimensional Euclidean space. The equation is x^2 + y^2 + z^2 = 1 - in other words, the points of distance 1 from the origin in 3-space. Some two dimensional manifolds require 4 dimensions in order to realize them. A projective plane can be described by taking a two sphere and gluing antipodal points together - tit is impossible in three dimensions to realize this, but you can realize it in 4 dimensions. Another 2 dimensional manifold is a Klein Bottle and this requires 4 dimensions to see it as well. A 3-sphere is an example of a 3 dimensional manifold. It can be realized in 4 dimensional Euclidean Space by considering the requation w^2 + x^2 + y^2 + z^2 = 1 - in other words, the set of all points (w,x,y,z) in 4 dimensional space of distance one from the point (0,0,0,0). So this may give an idea of what a manifold is.
Now to describe what simply connected is. Very loosely speaking, simply connected means that the geometric object is 'path connected' - just use your intuition for this notion (This requires a very precise definition that is obvious in an intuitiuve way, but we can't get into the precise definition) and that any loop drawn on the object can be continuously contracted to a point. This second part needs more explanation. Suppose we draw a loop of any shape based at some point on a plane. If you think of the drawing as a string, then you could imagine pulling the string at the chosen base point until the string passes all the way back to the base point. This is simply connected. Now consider the same plane with the origin (0,0) removed. Draw a loop that goes around the origin and base it at some point like (1,0). If you were to pull on that string to try to bring it back through the base point in its entirety, it woulg get blocked at the origin because somewhere along the way, some point on the string will have to pass through the origin - but we removed that point! Think of the string analogy and place a nail at the origin - tring to pull the string entirley to (1,0) requires that we pull the string through the nail - this would break the string and prevent us from having a "continuous deformation" of the loop to the base point. This is NOT simply connected. A 3-Sphere is simply connected and this is not hard to prove, but I will pass on this.
I will avoid discussion of "compact" and "boundary" because I have to describe some very elemtary areas of topology that will make this post too long (it is already too long). I would like to describe homeomorphic and homotopy type in greater detail, but this requires significant discussions and understand of what a continuous function is - and the definition of being continuous hinges on the concepts of open and closed sets of topological spaces. But, I will say this. In topology, people want to classify all geometric objects. I will make some definitions that will make no sense for the layperson, but read them anyway and look at the end of this paragraph if you are interested. Two geometric objects will be considered homeomoprhic (Or exactly the same from the topologist's standpoint even if they do not look exactly alike - example - a square and a circle are homeomorphic so a topologist considers them the same) if there exists a one to one and onto continuous function from one to the other with an inverse function that is also continuous. Now, we want to describe what it means to be of the same homotopy type. Two spaces X and Y are of the same homotopy type if there is a continuous function f: X---> Y and a continuous function g:Y---> X such that g composed with f is homotopic to the identity function on X and f composed with g is homotopic to the identity function on Y - I will not explain homotopic, but it relates the concept spoken of above with regard to simple connectedness - it relates to the ideas of "continuously deforming" objects or functions. If anyone were to study this stuff, it would be readily apparent and completely trivial that if two geometric objects are homeomorphic, then they are of the same homotopy type, but it is usually not true that if two spaces are of the same homotopy type, then they are homeomorphic. A simple counterexample would be that the real line (or any Euclidean space for that matter) is of the same homotopy type as a point - just squashing the real line down to a point and including the point into the real line would satisfy the requirements in the definition - they are clearly not homeomorphic since you cannot map the real line to the point in a one to one fashion - All points go to the same point in any function from the real line to that point!
The Poincare Conjecture is simply stating that if you have any 3 dimensional manifold that is compact without boundary with the same homotopy type as a 3-sphere, then it is indeed homeomorphic to the 3-sphere, so that in this case being of the same homotopy type and homeomorphic are the same. This was easy to prove for the one dimensional case and the two dimensional case. Then it was proven for dimensions 6 and higher. Then someone proved it in the fifth dimension. Then Stephen Smale won a Fields Medal (The highest mathematical honor in the world - offered every four years) for prtoving the Poincare Conjecture in the fourth dimension - the one that eluded us until now (If Perlemann is correct) has been the third dimension - the proof, again if correct, is astonishingly difficult to understand - the most brilliant people in the field will need a year to digest this....even if it fails, they will have learned a great deal.
If anyone understands any of this, then great! If not, then oh well! If anyone wants me to say something about the Riemann Hypothesis, then I will. It might be easier to explain that problem, but I will not attempt to explain the ramifications since that requires a great deal of mathematical training.
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