Tuesday's New York Times had this lengthy article about progress on one of the great open problems in mathematics: Poincare's conjecture. Actually, it looks increasingly likely that the problem is no longer open:
Three years ago, a Russian mathematician by the name of Grigory Perelman, a k a Grisha, in St. Petersburg, announced that he had solved a famous and intractable mathematical problem, known as the PoincarÃ© conjecture, about the nature of space.
After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Dr. Perelman disappeared back into the Russian woods in the spring of 2003, leaving the world's mathematicians to pick up the pieces and decide if he was right.
Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics and prose between them.
As a result there is a growing feeling, a cautious optimism that they have finally achieved a landmark not just of mathematics, but of human thought.
So what is the Poincare conjecture? Well, that takes a little explaining. I notice that Mark Chu-Carroll has already explained some of this at his blog. Nonetheless, I will forge ahead. Mark's essay might be a bit tough going if you have not seen these ideas before, so I will try to explain some of the things he takes for granted.
The Earth is a giant sphere, but we are not aware of this fact in day-to-day life. So long as we confine our movements to a small region, we feel as though we are walking on a flat surface. Such a surface is said to be two-dimensional, because there are effectively only two distinct directions in which one can move.
To see this, imagine that you have a rectangular sheet of paper on the table in front of you. We will play the following game: I will choose two points on the paper. You will place a pencil at one of the points. Your task is to draw a path, without lifting your pencil from the paper, that connects one of the points to the other. The catch is that every move you make with the pencil must be parallel to one of the edges of the paper. The question is: Is it possible for me to pick two points on the paper that you cannot connect in accordance with the rules?
The answer is no, and that is what we mean when we say that there are effectively only two distinct directions on a flat surface.
What if I chose one of the points to lie on the paper, while the other resides in the air two inches above the paper? In this case the person with the pencil will lose the game. No matter how clever he is, there is no sequence of pencil movements, with each one parallel to one of the sides of the paper, that will connect one point to the other. But if we impale the paper with a metal rod and allow the pencil to move parallel to this rod as well, then the pencil will win. That is what we mean by saying that ordinary space is three-dimensional.
Mathematicians refer to objects like the surface of a sphere as a “two-dimensional manifold&rdquo. Remember, the point is that the region near any particular point might as well be a flat surface. Can we think of other two-dimensional mainfolds?
Well, we might suggest things like the surface of a hamburger patty (by which I mean a small, flat cylinder). Somehow, though, that doesn't really seem all that different from a sphere. We could take the hamburger patty, and by applying a gentle, rolling motion transform it into a spherical meatball. No ripping, tearing, cutting or puncturing is required to carry out this transformation. Two surfaces that can be transformed into one another via this sort of gentle motion will be referred to as equivalent.
Thus, cubes, pyramids and footballs are all equivalent to a sphere. Are there any two-dimensional manifolds that are not equivalent to a sphere?
Yes! How about a torus (i.e. a doughnut)? The torus has a big hole in the middle, and this hole makes it impossible to transform it into a sphere without in some way cutting the surface. Apply all the gentle pressure, or stretches, or twists that you want, and that hole will still be there. So the torus is something fundamentally different from the sphere.
Any others? Well, we might imagine a two-holed torus. Tha would be like taking two doughnuts and fusing them together along one side, with the result being rather like a figure eight. Or you could have a three-holed torus, like the traditional pretzel shape. In fact you could produce a fundamentally new surface by taking a torus with some number of holes, and poking one more hole in it.
So now we have spheres, and we have tori with arbitrary numbers of holes. Any others?
Well, if you have no prior experience with this topic you may be hard-pressed to come up with another. But in fact there is another one. In transforming one surface into another we are allowed to stretch it and bend it, but we must never cut it, tear it or puncture it. Our allowable motions make it impossible to change the number of holes in the surface. But they also make it impossible to change the number of twists in the surface. Perhaps we can use that fact to manufacture more two-dimensional manifolds.
Go back to your rectangular sheet of paper. Roll the paper so that one set of parallel edges are touching each other and tape those edges together. The result is a cylinder. If you now took the cylinder and folded it up so that the circles at either end were touching, the result would be a torus. But what if you gave one of the circles a half-twist, and glued the circles together in this offset position?
The result would be a surface called a Klein bottle. It is almost impossible to picture what such a thing would actually look like. In fact, you can't even construct the darn thing in three dimensions. Not without having the bottle pass through itself in some way. Suffice it to say, however, that the Klein bottle is a two-dimensional manifold fundamentally different from any we have considered so far. Like the torus, it has a hole. So it's definitely not equivalent to a sphere. But the half-twist guarantees that this surface is also not equivalent to the torus.
Of course, you could then take several Klein bottles and glue them together. The result would be a sort of n-holed Klein bottle.
So we have three basic types of two-dimensional manifolds. Spheres, tori with arbitrary numbers of holes, and Klein bottles with arbitrary numbers of holes. This turns out to be a complete list of all two-dimensional manifolds, up to equivalence. If I hand you some exotic, two-dimensional manifold, I promise you that by judicious use of stretches, bends and twists you will be able to transform it into one of the surfaces on the list.
This is not easy to prove. On the other hand, it is something that can be proved in an upper level undergraduate math class.
If we look at our list of two-dimensional manifolds, we might notice something curious. Imagine that you have a sphere. With a pencil, you draw a small loop on the sphere, by which I mean a curve that ends at the same place that it starts. You can imagine taking that loop and slowly contracting it to a point. In other words, you can imagine the loop getting smaller and smaller until it becomes a single point, all the while only moving the loop along the surface of the sphere.
If you try the same thing on a torus, you might find yourself stymied. Suppose I draw my loop so that it actually passes through the hole in the torus. To see what I mean, imagine grabbing a doughnut with your thumb and index finger so that the two fingers meet in the hole. Such a loop can not be contracted to a point, not if you are only allowed to slide it around on the surface of the doughnut, anyway. Likewise, it is possible to draw loops on a Klein bottle that can not be contracted to a point.
So the sphere is the only two-dimensional manifold on which every loop can be shrunk to a point. (In math speak we would say that the fundamental group of the sphere is trivial, but that's a different post).
Is that true in higher dimensions as well? In the early 1900's, Poincare conjectured that the only three-dimensional manifold on which every loop can be contracted to a point is the three-dimensional analog of the sphere. It is that conjecture that defeated everyone who attempted to solve it for close to a century. And it is that conjecture that now appears to have fallen.
A three-dimensional manifold would be one where the region around any point looks an awful lot like familiar three-dimensional space. It is difficult to picture such thing. In analogy with the two dimensional case, we might say a three-dimensional manifold would be the surface of some four-dimensional object, but I don't think that really helps much.
But there's no reason why such things can't be defined abstractly, and there's no reason to stop at three dimensions. Such objects can be defined precisely, and we can start investigating what happens when we draw loops on these surfaces.
In 1960 Stephen Smale proved that in five dimensions or higher, it remains true that being able to contract any loop to a point characterizes the spheres of that dimension. Smale won a Field's Medal (the mathematical equivalent of the Nobel Prize) for his trouble.
In 1981, Michael Freedman polished off the four dimensional case. He duly received a Field's Medal as well.
This raises an interesting question. The proofs of Smale and Freedman are towering achievements in twentieth century mathematics, and both gentlemen richly deserved their Field's medals. But doesn't it seem odd that the problem for more than three dimensions was solved before the three dimensional case? I mean, wouldn't a smaller number of dimensions make the problem simpler?
The answer is no. You see, all of those extra dimensions provide a lot of freedom of movement. Remember that the number of dimensions corresponds roughly to the number of directions in which things can be moved. So if you are trying to prove that one manifold is equivalent to another, superficially different, manifold, it can be very convenient to have lots of degrees of freedom when you start moving things around. So even though high-dimensional objects are impossible to picture in your mind, it is an amusing fact that it is often easier to prove theorems about them than for lower-dimensional objects.
So there you go. Personally, I find this sort of thing to be a bit mind-numbing. That's why I specialize in combinatorics and number theory, not topology!
"If I hand you some exotic, two-dimensional manifold, I promise you that by judicious use of stretches, bends and twists you will be able to transform it into one of the surfaces on the list."
What about a rope that's been knotted and then the ends have been stuck together. I can't see how this can be transformed into either a sphere or torus, and a klein botle type thing is clearly impossible.
Strabo: Are you pretending that the rope really has no thickness, or not? If you are, then it's really a one-dimensional manifold. If the rope has thickness, then you are really thinking just about the *surface* on the rope. In this case, it's a torus.
But what about the knots? Well, imagine that you're an ant walking on the rope, and that your eyesight is very poor. Then you can only see immediately around you, and you can't see "away" from the surface of the rope at all. Then it would be impossible for you to tell that the rope was knotted, so you couldn't tell that you weren't on a torus. This, mathematically, is what we mean.
To be technical, the knots are an artifact of the particular embedding into three-dimensional space that you've chosen. When Jason says "stretches, bends and twists" he really means "in any number of dimensions". Confusingly, one can pull apart a knot in the rope, without breaking it, in four or more dimensions. This is related to the fact that a Klein bottle can't even be properly visualised in three dimensions: it always cuts itself, which it's not meant to do.
The result would be a surface called a Klein bottle. It is almost impossible to picture what such a thing would actually look like. In fact, you can't even construct the darn thing in three dimensions.
Note my next sentence:
Not without having the bottle pass through itself in some way.
If you had a fourth dimension to work with the Klein bottle could be constructed without having it pass through itself.
Actually I was just looking for an excuse to post that link. I've always wanted to own a closed, non-orientable, boundary-free manifold...
That's why I specialize in combinatorics and number theory, not topology!
oh, right. because combinatorics and number theory are so much easier.
I just read that Perelman is refusing the Fields medal, saying that he really isn't part of the mathematical community and does not want to become its "figurehead." Sheesh, even if he IS head and shoulders above the rest of the mathematical community, spurning complimentary recognition from it is just plain rude.
That's why I specialize in combinatorics...
Could you recommend a good text on enumerative combinatorics? (I am just interested in counting problems, not abstract combinatorics.)
In 1960 Stephen Smale proved that in five dimensions or higher, it remains true that being able to contract any loop to a point characterizes the spheres of that dimension.
(If it were true, it would imply that, e.g., the product of a 2-sphere S^2 and a 3-sphere S^3 is homeomorphic to a 5-sphere. But there are many ways to see this is not the case.)
The Generalized Poincare conjecture for compact manifolds, which Smale proved at least for smooth manifolds of dimension >= 5 (and Freedman proved for topological 4-manifolds) is that if the homotopy groups pi_k(M) of an n-manifold M agree with those of the n-sphere S^n for all k in the range 0 <= k <= floor(n/2), then M is homeomorphic to S^n.
Oops -- crucial sentence was chopped off by this site's buggy software. To repeat:
Generalized Poincare conjcture:
If the homotopy groups pi_k(M) of an n-manifold M agree with those of the n-sphere S^n for all k in the range 0 <= k <= floor(n/2), then M is homeomorphic to the n-sphere S^n.
I suspect the perfidious software they use in these parts is chopping off your post after a less-than sign (which it interprets as the beginning of an HTML tag, probably). Try using the HTML entities < for < and > for >.
Assume all manifolds discussed are closed (compact and without boundary).
Generalized Poincare conjcture:
If condition (*) holds:
(*) The homotopy groups pi_k(M) of a closed n-manifold M agree with those of the n-sphere S^n for all k in the range k between 0 and floor(n/2), inclusive.
then M is homeomorphic to the n-sphere S^n.
Thanks to Smale, Freedman, Perelman and others, it's known that any *topological* n-manifold M of any dimension n satisfying (*) is indeed homeomorphic to the n-sphere.
The question for smooth manifolds is more complicated: If M is a *smooth* n-manifold of dimension n = 1,2,3,5, or 6 satisfying (*), then it is also *diffeomorphic* (smoothly homeomorphic with a smooth inverse) to S^n.
But if n equal to 7 or greater, then for most n there are examples of such M that are *not* diffeomorphic to the standard S^n.
And for n = 4, it is not yet known whether such an M must be diffeomorphic to the standard S^4.
(A related fact is that for all n unequal to 4, a smooth n-manifold M homeomorphic to Euclidean space R^n must be diffeomorphic to R^n.
But if n = 4 -- and this is truly bizarre -- there are *uncountably* many examples of such M's no 2 of which are diffeomorphic!)