Has The Poincare Conjecture Been Solved?

Zack Coburn writes "An article in the Boston Globe alludes to the Poincare Conjecture being solved, possibly. For those who are unfamiliar with the conjecture, the article gives a brief description: "To solve it, one would have to prove something that no one seriously doubts: that, just as there is only one way to bend a two-dimensional plane into a shape without holes -- the sphere -- there is likewise only one way to bend three-dimensional space into a shape that has no holes. Though abstract, the conjecture has powerful practical implications: Solve it and you may be able to describe the shape of the universe." Apparently Grigory Perelman may have proved it, which would mean a $1 million award from the Clay Mathematics Institute." We've previously discussed other possible Poincare proofs.

Comments

I'm confused...

[Magic5Ball]

"To solve it, one would have to prove something that no one seriously doubts: that, just as there is only one way to bend a two-dimensional plane into a shape without holes -- the sphere -- there is likewise only one way to bend three-dimensional space into a shape that has no holes ... And while the equivalent of the Poincare conjecture has already been proven for dimensions four and up..."

Being a non-math person, it seems to me if it has been solved for two dimensions (has it?) and four and up, wouldn't three dimensions just be a special case of the many (four and up) dimensions proof? Or is there something special about that proof that limits it to four and up? Or perhaps something in a form like the two dimension proof?

Perhaps my simple understanding of proofs in euclidian geometry doesn't scale up like this :-)

Re:I'm confused...

[sam_nead]

Indeed, the Poincare Conjecture (that every n-manifold with the homotopy groups of an n-sphere is homeomorphic to an n-sphere) has been solved in dimensions n = 1, 2, 4, 5, 6, ... The only missing case is n = 3, which is the case originally conjectured (well, really "asked about") by Poincare.

The cases n = 1, 2 are not so hard and may be explained to undergraduates. n = 5 and above are not easy but not impossible to explain, either -- Smale got a Fields medal for his work in this area. It can now be covered in a single graduate level mathematics course. The idea (if I remember correctly) basically boils down to "in high enough dimensions, there is enough elbow room". To give a better analogy, generically straight lines in two dimensions meet but in three dimensions they do not. (And to really say what is going on "Two-dimensional surfaces generically do not meet each other if embedded in a five-dimensional space")

The case n = 4 was handled by Michael Freedman using very subtle techniques (at least to me!) but again relying on "having enough space to move around in".

I don't understand the n = 3 case at all, really -- no one has given a simple "These techniques should work because x, y, znd z" sort of explaination, yet. The closest they come is to mutter uncomprehensible things about the heat equation... Suffice to say -- in dimension three there is not enough room to move around in. So it is not a complete surprise that the proof for n = 3 is rather different from higher n.

formalize the proof

[penguin7of9]

Proofs have reached such a level of complexity that I really have my doubts that mathematicians can verify them reliably.

It's rather like writing a 50000 line program from scratch, without ever running it through a compiler, and then having a dozen people look it over for whether it would compile. Do you really believe that a dozen people looking at a 50000 line program would be able to find all the syntax and type errors contained in it just by eye? And, if anything, mathematical proofs are more complex and subtle. With type checking and syntax, there is at least something where people have years of experience with an unforgiving "proof checker", whereas (most) mathematicians have never had to face the rigor of a formal, automated, unforgiving proof checker.

For any proof of this complexity, I think the proof needs to be formalized and the checked by computer. Even then, there is a big risk that there is some bug in the formalization of the proof.