The Poincare Conjecture, along with several other notorious famous conjectures (Riemann Hypothesis, for example), have the nasty problem of making people who try proving it to use it as an assumption somewhere within the proof. Basically, the PC is logically equivalent to several other statements. Somewhere along the proof, in many prior failed cases, the author uses one of the other statements and then somehow claims, voila! PC is proved. Then, someone shows where the mistake by showing where the equivalent statement is used as an assumption.
Other conjectures, like the twin prime and such (most are number theory based) are so enigmatic that no one has even a slightest inkling on how to proceed.
Let's look at 2-dimensional objects. There is a classification of 2-dim objects (done mostly by Euler in the late 1700s). Assume they're bounded (that is, doesn't go to infinity). There are the boundaried ones, like a sheet of paper: there's an edge to it. Then there are those without boundaries, like a sphere, or the surface of a donut, or the surface of a pretzel (three-holed). There are also orientable and non-orientable surfaces. A regular sphere is orientable: there's an outside and inside direction (if you're on the surface of the earth looking at the stars, no matter how your walk and travel, you can get back to where you were and look at the stars by looking in the same direction). A non-orientable surface would be a mobius strip. If you walk once around the strip, then looking at the stars in one direction will get to looking at a different direction. There is a non-orientable version of a sphere. The non-orientable version of a torus (the donut's surface) is called a Klein-bottle.
In any case, the classification of 2-dim surfaces, orientable, compact, connected, no boundaries are basically the number of holes. Zero holes is the sphere. One hole is the torus, 2 holes is the two-torus (take two pairs of pants, sew the waists together. Then, sew the two leg holes of each pant together), 3-torus, 4-torus, etc. They're it, topologically. They're all equivalent (and equivalent here means homomorphic to each other; homeomorphic requires some additional geometric structure to stay the same). The genus of the surface is the number of holes.
The main point for 2-dim is that any zero-genus surface is homomorphic to the sphere. The generalized conjecture is that any simply connected compact orientable n-dim manifold (fancy word for higher dimensional geometric object) is homomorphic to the n-sphere. Apparently, it wasn't very hard to prove it for very high dimensions. Supposedly, the constraints on the higher dimensions forced the case. The lower dimensions were rather difficult and the fifth and fourth dimension cases were solved only rather recently (in the late 80s?). The poincare conjecture was left to the 3-dimensional case, which apparently is now solved, if correct.
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The Poincare Conjecture, along with several other notorious famous conjectures (Riemann Hypothesis, for example), have the nasty problem of making people who try proving it to use it as an assumption somewhere within the proof. Basically, the PC is logically equivalent to several other statements. Somewhere along the proof, in many prior failed cases, the author uses one of the other statements and then somehow claims, voila! PC is proved. Then, someone shows where the mistake by showing where the equivalent statement is used as an assumption.
Other conjectures, like the twin prime and such (most are number theory based) are so enigmatic that no one has even a slightest inkling on how to proceed.
Let's look at 2-dimensional objects. There is a classification of 2-dim objects (done mostly by Euler in the late 1700s). Assume they're bounded (that is, doesn't go to infinity). There are the boundaried ones, like a sheet of paper: there's an edge to it. Then there are those without boundaries, like a sphere, or the surface of a donut, or the surface of a pretzel (three-holed). There are also orientable and non-orientable surfaces. A regular sphere is orientable: there's an outside and inside direction (if you're on the surface of the earth looking at the stars, no matter how your walk and travel, you can get back to where you were and look at the stars by looking in the same direction). A non-orientable surface would be a mobius strip. If you walk once around the strip, then looking at the stars in one direction will get to looking at a different direction. There is a non-orientable version of a sphere. The non-orientable version of a torus (the donut's surface) is called a Klein-bottle. In any case, the classification of 2-dim surfaces, orientable, compact, connected, no boundaries are basically the number of holes. Zero holes is the sphere. One hole is the torus, 2 holes is the two-torus (take two pairs of pants, sew the waists together. Then, sew the two leg holes of each pant together), 3-torus, 4-torus, etc. They're it, topologically. They're all equivalent (and equivalent here means homomorphic to each other; homeomorphic requires some additional geometric structure to stay the same). The genus of the surface is the number of holes. The main point for 2-dim is that any zero-genus surface is homomorphic to the sphere. The generalized conjecture is that any simply connected compact orientable n-dim manifold (fancy word for higher dimensional geometric object) is homomorphic to the n-sphere. Apparently, it wasn't very hard to prove it for very high dimensions. Supposedly, the constraints on the higher dimensions forced the case. The lower dimensions were rather difficult and the fifth and fourth dimension cases were solved only rather recently (in the late 80s?). The poincare conjecture was left to the 3-dimensional case, which apparently is now solved, if correct.