sometime around a year ago, i read in several nonscientific news sources that the poincaré conjecture had been successfully proven by some englishman, who put his proof on his website. i talked to several mathematicians who were skeptical, this not having been the first claim of proof, and were waiting for it to undergo peer review and get published in a reputed mathematical journal. i have seen nothing in the news since then. does anyone know whether it was established?
nothing to forgive. the poincaré conjecture is the most important unsolved problem in algebraic topology (unless it is in fact solved). it is a claymath problem, which means the claymath organization will pay you USD 1 000 000 if you solve it. also, you will certainly get a fields medal. you can read their description of the problem here. also, check out what they have to say about it on mathworld. you can also read my 5 minute crash course on topology on these boards here. it contains a description of the poincaré problem as well. briefly stated, the poincaré conjecture is: any simply connected closed 3-manifold is homeomorphic to a 3-sphere. basically that means that if you can draw closed any loop string in a three dimensional closed space, then it has the same basic structure (meaning you can bend it, but not punch holes in it) as a 3-sphere. a 3-sphere, by the way, is the set of points in four dimensional space that are equidistant to the origin. it s hard to imagine what it looks like. just think of it like a higher dimensional analog to a regular (2-)sphere. any questions?
Poincaré's conjecture is NOT proven. There was one major step in the proof that was, even when published, left unproven. The author hasn't reconciled it. It's not exactly wrong, just incomplete. (The guy isn't a crank.) Here's the paper: A Proof of Poincare's Conjecture? The question mark is in the actual paper title itself, giving a hint as to its, well, questionableness. ----------------------- [Editor's Note: Sorry, looks like Lethe beat me to the punch--and with better footwork. I figure I'll leave it if for no other reason than it took a few minutes to write.] The Poincaré conjecture is that every simply connected and closed 3-manifold is homeomorphic to the 3-sphere. (This is the simple form.) Somewhat less than rigorously: Simply connected means that it doesn't have any holes (simply) and it is all one piece (connected). An example of not all one piece is two spheres not touching. Considered as one manifold, it isn't connected. Closed means it doesn't have a boundary. (Think of that as an edge.) Also, it has to be compact, which is a little more difficult to describe. Think of compact as being "nice." Most manifolds you generally think of are compact. Two manifolds are homeomorphic if they can be deformed into each other without ripping. Here's the idea: our normal sphere is a 2-sphere. Any nice 2-manifold without holes is basically equivalent to a sphere. Any nice 2-manifold with one hole is equivalent to a donut. Etc., Etc. So Poincaré just stated that this easy classification system will also work for the simplest case (no holes) in 4 dimensions. The conjecture has been generalized to higher dimensions using homotopy, but that's a bit out of our realm, I think. (I welcome any clarifications or corrections. I can't find Munkres at the moment.)
thanks for the reference tarrou. i ll print it out tomorrow and give it a closer inspection, but scanning it briefly, it looks over my head. but this is what i was looking for. so is he (dunwoody) working on patching up the proof? PS munkres is a good book.
As far as know, he still is. His research interests still include 3-manifolds, but I really don't know for sure. From what I hear from some topologists friends of mine, the fix won't be easy. I also found this mathworld news article after I wrote the other note. P.S. Yea, Munkres was my advisor, so I always put in a plug for him when I can.
did his proof make any new headway? or was it just a false alarm all around. have you read his proof? how hard would is it to get through and understand? i think it s over my head, but i m still schoolin. and how was working for munkres? is he still active in research? what does he do? how did you like him? does he have other books (i only know his intro book)
I think a less abstract description might be useful here. You can draw a simple closed curve (EG: a circle) on the surface of an ordinary sphere and shrink it to a point without cutting it or the sphere (pretend the circle is a rubber band). Some simple closed curves on the surface of a 3D torus (doughnut or bagel) cannot be shrunk to a point without cutting the torus or cutting the curve. Due to the above, mathematicians call the surface of a 3D sphere simply connected, while they say that the surrace of a 3D torus is not simply connected. The Poincare conjecture states that a 4D hypersphere is simply connected. A 4D Hypersphere is the locus of all points in a 4D space which are equadistant from a central point. There is probably more to it than the above, but that is the basic idea.
Lethe: I'm not completely sure if Dunwoody's proof is worthwhile if it ends up not working out. By that, I don't think any new math is there (e.g. a weaker but important proposition is proved). However, I do know that this is considered a very valiant effort and that even if it doesn't work, gives the map for an approach that might. I did read the proof on your suggestion and, as far as I can tell, it is understandable by someone with one or two graduate topology courses under their belt. Everything in it I recognized and vaguely followed, but by no means do I claim understanding. It's just that for every step I felt that if I refreshed myself a bit and sat down with for a while, I'd be able to grasp it. I may be wrong though. Anyhow, if you want to tackle it, you probably can--assuming you have a good differential topology book and lots of time. To be honest, though, I'd probably wait until I felt familiar with what simplexes are and how powerful they can be. Further, because this is just a sketch of a proof, Dunwoody is expecting you to fill a lot of the details in, which could be very hard. Munkres is a very neat guy--one of the three most popular math professors while I was in school. (The other two were the late Gian-Carlo Rota and Michael Artin whose Algebra book you may have run into.) He's also about as old as dirt. (No one--including the administrative staff--actually knew how old he was; guesses ranged from 60 to topping a century.) He left not so long after I did (1999), I think, and now spends most of his time in his garden. Actually, by the time I was senior, he was spending most of his time in his garden. I took his famous "Calculus with Theory" class my freshman year and fell in love with the man. He knew a story for just about every mathematician you could think of and worked them seamlessly into evey lecture. His wit was fairly dry and slick. He often understated everything right up until he suddenely exploded with the punchline--at which point he would then release his impish grin, highlighted by his less than artful dental work. I'm sure the classes to either side of use thought us peculiar when all 30 of us would regularly go into hysterics. He also made math seem vital and effectively destroyed any suspicions I had that math was just the drugery they teach us in high school. He loved the exception to the rule and had an example to illustrate every concept. He generally refrained from practical examples, however, which I'm sure just deepened our appreciation for the beauty of mathematics in and of itself. As an advisor, he was active, fair and honest and I could not wish for any better a man. He's written several books--but his other classic (the first being the topology intro) is called "Elementary Differential Topology" which you might find useful should you attack the Dunwoody's proof. Well, that was probably more than you cared to hear, but alas, I'm wordy.
