Can anybody simplify Poincare's Conjecture?

Question

I was trying to describe this to a friend of mine, but I can't think of any simpler way to put it... the best I got is
"Is a spherical object is the only closed three-dimensional space with no holes?" But that doesn't really capture it because you could say "well... what about a cube?"

Simplifying questions... gotta love it :)

Answer 1

Topologically speaking, a sphere and a cube are equivalent...that is about all I can add to the conversation.

Sorry, well beyond my mathematical capabilities. I read the wiki article and understood about 1% of it.

Hopefully you get an answer from someone smarter than me.

**EDIT**
Just found this excerpt on cnn.com
"Poincare conjecture, which essentially says that in three dimensions, a doughnut shape cannot be transformed into a sphere without ripping it, although any shape without a hole can be stretched or shrunk into a sphere."

That may be the best way to explain it to a non-math person.

Answer 2

First a bit of terminology:

Manifold: A space is called a manifold if it locally looks like a euclidean space. For example, the surface of a sphere is locally like a plane (which is why people thought the earth is flat) which is a two dimensional Euclidean space. Similarly, the surface of a bagel (called a torus) is another two dimensional manifold. Three dimensional manifolds are harder to visualize, but there are analogs of both spheres and tori for three dimensions.

Simply connected: We say a manifold is simply connected if every loop drawn on the manifold can be shrunk to a point within the manifold. For example, a sphere is simply connected, but a torus is not. A loop that goes around the hole of the torus cannot be shrunk to a point within the torus.

Homeomorphic: This is the subtle concept here. Two manifolds are said to be homeomorphis if there is a way of deforming one into the other without tearing or gluing anything. For example, the surface of a cube and the surface of a sphere are homeomorphic because it is relatively easy to 'round off the corners' of the cube to get a sphere. One the other hand, a torus and a sphere are not homeomorphic: there is no way to deform one to the other without gluing or tearing.

Compact: A manifold is compact if (essentially) it is finite in extent. The sphere is compact, but a euclidean space is not.

The n-dimensional Poincare conjecture says that every compact, simply connected n-dimensional manifold is homeomorphic to the n-dimensional sphere.

This was long known to be true for two-dimensional manifolds: for example, the torus is not homeomorphic to the sphere, but it is also not simply connected. It was shown a few decades ago that it is true for n more than 5 and more recently for n=4. So the only open case was for three dimensional manifolds. This was then known as THE Poincare conjecture: that every compact, simply connected three-dimensional manifold is homeomorphic to the three-dimensional sphere.

BTW, Doug's definition of simple connectivity is wrong, as is his implication that compactness follows from simple connectivity. Three dimensional euclidean space is simply connected, but not compact.

Answer 3

You kinda have it, but Poincare actually conjectured that any simply connected 3-manifold is homeomorphic (structure preserving) to a 3-sphere. Simply connected means that any closed path with length C and enclosing area A on the manifold has the property that if C can go to 0 smoothly, then lim C -> 0 ==> A -> 0 (which also implies compactness on the manifold since it has to include the limit point C=0)

The best way to get out from under the cube (or anything else closed) discussion it to point out that a cube and a sphere are topologically equivalent since, if you put a sphere in the middle of a cube, every point on the cube is intersected by an extended radius of the sphere which maps every point on the sphere to 1 and only 1 point on the cube.


Doug

Answer 4

You could say a cube.

The question is whether under a certain type of transformation (Homeomorphic... but transformation or change for the layman- no need to talk about specifics) are all closed 3D sets equivalent to the spere.

I think thats basically all a layman can understand

A cube under and appropriate transformation is. You could rephrase the problem to a cube but a sphere is better basically.

Answer 5

http://www.claymath.org/millennium/Poincare_Conjecture/

http://www.inq7.net/inf/2003/jun/11/inf_24-1.htm

http://www.math.unl.edu/~mbrittenham2/ldt/poincare.html