# The Poincare Conjecture.

## Main Question or Discussion Point

Could someone lay down, in layman's terms, The Poincare Conjecture? Lol, is this even possible?

Related Differential Geometry News on Phys.org
In 3D, it would be somthing like : if you have a closed, simply-connected surface, then it is more or less a sphere. For lay(wo)man's vocabulary : simply-connected means that if you put a a rope and closed it on your surface, you can reduce it to a point, e.g. on a donut you quickly see there are possiblities such that you cannot tight the rope without breaking the donut, if you really want your rope to become a point like object (ideally). However, it is not known in 4 dimension if this is true for what is sometimes called a 3-sphere : the generalization of a sphere surface (you have 2 angles to parametrize everypoint on a sphere) to three dimension (three such angles, which is, i heard, quite hard to visualize or intuitiv. understand). However, I think it was proven that for higher dimensions this was true, so that for not mixing : for 2,-,4,5....dimensional varieties (number of free parameter on your object), this was true : i.e. the shape is deformable to a n-sphere if it has some properties like closedness and simple-connectedness...but Poincaré made at first wrong assumptions and corrected himself his mistake, but could solve after it...Technically it's quite complicated, with Homology and Homotopy groups, and other math. stuff..(which i personnally don't know even only the surface of those concepts)

Last edited:
mathwonk
Homework Helper
its an attempt to describe a sphere by simple properties. for instance the usual 2-sphere, x^2 + y^2 + z^2 = 1, can be described by saying it is a smooth, compact, connected surface, has no boundary, and every loop in it can be shrunk to a point on the surface.

so having settled this case, we go up one dimension to the "3 sphere", defined by the analogous equation x^2 + y^2 + z^2 + w^2 = 1, and we ask if it is the only three dimensional, compact connected, smooth 3 dimensional gadget, in which again all loops on it can be shrunk toa point.

no one knows for sure, but a solution (yes) has been propsed recently, and a conference on the topic will be held presently in france.