A (Correct) Poincare Proof!?

aphyscher writes "About a year ago, there was an announcement that M.J. Dunwoody had proved the (in)famous Poincare conjecture. His paper turned out to have a slight problem, and so it remained unsolved... until perhaps now! Sergey Nikitin has posted a preprint of what may perhaps be an actual proof."

Comments

Re:Though i'm not stupid

[MattRog]

Here is an example with all sorts of definitions you can read.

http://mathworld.wolfram.com/PoincareConjecture.ht ml

Poincar� Conjecture

[taphu]

For super-geeks, here is is a more thorough discussion of the Poincaré Conjecture.

http://mathworld.wolfram.com/PoincareConjecture.ht ml

Poincare Conjecture

[jkauzlar]

Simpy put, the poincare conjecture implies that every compact n-manifold is homotopy-equivalent to the n-sphere iff it is homeomorphic to the n-sphere. (From Wolfram's MathWorld) Actually I don't know what that means, but having read and studied a bit about math, I can offer some explanation on the importance of such a proof. When a proof attempts to show that two algebraic structures are equal, as does this conjecture, it allows mathematicians the freedom to look at a problem in two ways instead of one. At last, a compact n-manifold problem can be safely regarded as an n-sphere problem and all the rules regarding n-spheres can be applied to certian n-manifolds. On another topic, these long-standing, but near-universally-believed-to-be-true conjectures are often assumed to be true in order to prove other theorems. i.e. a ground-breaking new primality testing algorithm ASSUMES the truth of the unproven Reimann Hypothesis. So, future encryption keys may rely on unstable hypotheses for their unbreakability.

Only need an answer for n = 3

[dvdeug]

I don't know which is worse; a problem like the Poincare problem, which has been definitively solved for 1-manifolds, 2-manifolds, and n-manifolds where n > 3, leaving only one little hole; or something like Femat's Last Theorem, which was solved for everything up to n equals a million billion and most numbers beyond that, before someone finally come up with a definitive proof.

Poincare Conjecture

[Listen Up]

The Poincaré Conjecture is widely considered the most important unsolved problem in topology. It was first formulated by Henri Poincaré in 1904. In 2000 the Clay Mathematics Institute selected the Poincaré conjecture as one of seven Millennium Prize Problems and offered a $1,000,000 prize for its solution.

The conjecture is that every simply connected compact 3-manifold without boundary is homeomorphic to a 3-sphere. (Loosely speaking, that every 3-dimensional object that has a set of sphere-like properties can be stretched or squeezed until it is a 3-sphere without breaking it).

Analogues of the Poincaré Conjecture in dimensions other than 3 can also be formulated. The difficulty of low-dimensional topology is highlighted by the fact these analogues have now all been proven, while the original 3-dimensional version of Poincaré's conjecture remains unsolved. Its solution is central to the problem of classifying 3-manifolds.

On April 7, 2002 there were reports that the Poincaré conjecture might have been solved by Martin Dunwoody; on April 12 Dunwoody acknowledged a gap in the proof and was attempting to fix it. In October of that year, Sergei Nikitin of Arizona State University announced that he had proved the result.

Re:Ok

[cperciva]

What he's saying is, the...er...well, he means that the, uh...

Look at it this way:
Suppose the universe doesn't have any "edges" -- you can keep on going forever in a straight line without "falling off the edge of the world". Suppose further than there aren't any "wormholes" -- that given two paths between a pair of points, you can continuously deform one into the other. Finally, suppose that the universe is finite in volume.

Now, the first and third conditions above imply that the universe "folds in on itself". Add in the "no wormholes" condition, and Poincare's conjecture/theorem, and you find that there is only one possible way that it can fold in on itself -- as a hypersphere.

At least, that's the best explanation I can provide without any formal background in topology or astrophysics.

Why Eric Weissman rules my world

[Nyarly]

What he's saying is, the...er...well, he means that the, uh...

Piece by piece:

• Consider a compact 3-dimensional manifold V without boundary.
  • Basically, V is a set of points with a whole bunch of properties. Among them are the fact that the points are "smooth," as defined by a funky neighborhoods deal, but it's roughly analogous to the definition of a continuous function. (I believe that the points that satisfy a continuous function would be a topological space - the first part of being a compact manifold).
  • Compactness is harder to grasp. Essentially you couldn't find a infinite set of open sets whose union is the set of points in the manifest, from which a finite number of sets could be taken whose union would also be the original set. Almost kinda like saying that there aren't discontinuities in the manifold - there aren't gaps.
  • Without boundary means that it doesn't include it's own boardary - like the open ball, where it's every point inside a certain radius - the surface is the sphere at that radius.
  • 3-dimensional means that you could refer to any point in the manifold with as few as three values. Unless the manifold set is a space, is exists in 4 dimensions. Think about a sphere - you can refer to any point on a sphere with 2 coordinates, like latitude and longitude, but it exists in 3 dimensions.
• Is it possible that the fundamental group of V could be trivial
  By which he means: there is one equivalence set of loops through the manifold. Every possible loop (A path that returns to its beginning point. Duh.) in the manifold belongs to one set of loops that are pretty much the same - you could push any one of them around and get any other one. A sphere has this quality - any loop you draw on the surface of a 2-sphere (the one that exists in three dimensions), but a torus doesn't - there are the loops that are equivalent to the loop around the outside of the torus, and the ones that run through the hole.
• even though V is not homeomorphic to the 3-dimensional sphere?" Trans: Even though you can't finagle necessarily finagle it into a 3-sphere, even if I let you do it in higher dimensions.

What he's asking is, is it possible there could be an object in 4-dimensions, that has some kind of tangle in it such that you can't make it into a hypersphere, but isn't so mangled that there are fundamentally different ways to "draw lines" on it. The suggestion is fairly reasonable, I think.

For instance, any loop drawn on a plane is homeomorphic to a circle. You're options in connected finite 1-d manifolds amount to line segments, or cirles. But when you upgrade to 2-d, suddenly you've got holes. You can't have holes in 1-d and still be connected, but in 2-d you get donuts and dresses and honeycombs and whatnot. But the thing about a hole is that it means that your fundamental group have more than one set in it, and without a hole, you wind up being a sphere.

So why shouldn't there be another characteristic of 3-d objects, one which allows for more than one kind of simply connected, non-contractable manifold?