Can anyone explain the Poincaré conjecture in simple terms?

OK

This is tough stuff, and in reality, too toush to summarise in a paragraph but I'll have a go.

In topology, a sphere with a two-dimensional surface is essentially characterized by the fact that it is simply connected (In topology, a geometrical object or space is called simply connected if it is path-connected and every path between two points can be continuously transformed into every other. Informally, an object is simply connected if it consists of one piece and doesn't have any "holes" that pass all the way through it. For example, neither a doughnut nor a coffee cup (with handle) is simply connected, but a hollow rubber ball is simply connected. In two dimensions, a circle is not simply connected, but a disk and a line are. Spaces that are not simply connected are called non–simply connected or, in a somewhat old-fashioned term, multiply connected.).. The Poincaré conjecture is that this is also true for spheres with three-dimensional surfaces. The question has been solved for all dimensions above three. Solving it for three is central to the problem of classifying 3-manifolds.

To understand this thought, you need to understand simple connection, 3 maniforlds, loops and algebraic topology. It really is a nightmare stuff.

What Is the Conjecture?

Poincaré's Conjecture deals with the branch of math called topology, which is the study of shapes, spaces, and surfaces. The Clay Institute offers this deceptively friendly-sounding doughnut-and-apple explication of the bedeviling problem:

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is ‘"simply connected,"’ but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two-dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three-dimensional sphere (the set of points in four-dimensional space at unit distance from the origin).

So, basically....this Russian mathmetician solved the three sphere theory stating that when each is pulled equally the three spheres all get smaller, thus creating no 'hole' in the center. It all has to do with topology (the study of shapes, land, etc....) This was trying to find the shape of the universe. All over my head, but at least it was finally solved! Phew...I will be able to sleep at night........LOL

the Poincaré conjecture states that every simply connected closed three-manifold is homeomorphic(Possessing similarity of form, ) to the three-sphere (in a topologist's sense) S^3, where a three-sphere is simply a generalization of the usual sphere to one dimension higher. More colloquially, the conjecture says that the three-sphere is the only type of bounded three-dimensional space possible that contains no holes. This conjecture was first proposed in 1904 by H. Poincaré Eric Weisstein's World of Biography (Poincaré 1953, pp. 486 and 498), and subsequently generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere iff it is homeomorphic to the n-sphere. The generalized statement reduces to the original conjecture for n==3.

The Poincaré conjecture has proved a thorny problem ever since it was first proposed, and its study has led not only to many false proofs, but also to a deepening in the understanding of the topology of manifolds (Milnor). One of the first incorrect proofs was due to Poincaré himself (1953, p. 370), stated four years prior to formulation of his conjecture, and to which Poincaré subsequently found a counterexample. In 1934, Whitehead (1962, pp. 21-50) proposed another incorrect proof, then discovered a counterexample (the Whitehead link) to his own theorem.

The n==1 case of the generalized conjecture is trivial, the n==2 case is classical (and was known to 19th century mathematicians), n==3 (the original conjecture) appears to have been proved by recent work by G. Perelman (although the proof has not yet been fully verified), n==4 was proved by Freedman (1982) (for which he was awarded the 1986 Fields medal), n==5 was demonstrated by Zeeman (1961), n==6 was established by Stallings (1962), and n>=7 was shown by Smale in 1961 (although Smale subsequently extended his proof to include all n>=5).

Poincare conjecture essentially says that in three dimensions you cannot transform a doughnut shape into a sphere without ripping it, although any shape without a hole can be stretched or shrunk into a sphere.

Actually, a Russian mathematician solved the century old problem. His name is Grigory Perelman, a 40-year-old native of St. Petersburg.

The assertion states that you can't take a 3-D shape that has a hole in it and remould it into a ball shape without making a tear on it's surface.

I don't understand either.... it's all too complicated for me.... infact..... you must be cleverer than me!:-o hehe! Hope you find the answer.

you are right it is complicated