The Poincaré conjecture, II
Note: There are a few special mathematical symbols used in the following. Some versions of Microsoft's "Internet Explorer", maybe every one, for all I know, can't display them and show square boxes instead. Don't ask me why. MS IE sucks. Use Firefox, Netscape, or some other real browser.
In the previous post on this subject we looked, from an intuitive point of view, at what the conjecture (now, apparently, a theorem) is about. In a nutshell, it's really quite simple: What properties does an abstract topological object need to have in order to be "equivalent" to a sphere?
We provided a list of properties that have long been known to do the trick in the case of a special kind of sphere, the 2-sphere S2, a 2-dimensional surface (like a soccer ball) in ordinary 3-space.
It's natural, in fact inevitable (at least for a mathematician) to wonder whether a similar result applies in higher dimensions. The next dimension up involves the 3-sphere S3, a 3-dimensional object in "ordinary" 4-space.
At this point, most non-mathematicians start feeling queasy and weak in the knees because of all the traditional hype and mystery often associated with "the 4th dimension". In particular, how on Earth can anyone really understand what's going on with that, when admittedly it's barely possible, at best, to visualize what happens in a space of more than 3 dimensions?
Rest assured, most mathematicians can't really visualize it either. Instead, there are alternative ways to reason about spaces of more than 3 dimensions, and they don't depend much on "visualizing". All it takes is some straightforward reasoning, which we will try to explain. (Quite possibly the mathematicians who do this for a living, and actually prove theorems about such things, have a much better than average capacity for visualizing. But that's not very much needed to understand the general ideas.)
Another thing that most non-mathematicians wonder about, besides just grasping what's going on, is how on Earth it is possible to mathematically prove assertions about higher-dimensional topology -- or even about topology in more "ordinary" spaces of 2 or 3 dimensions. Even for people who've managed to cope with calculus in a 2-dimensional "cartesian" coordinate system, it's daunting to imagine how one ever works with stretchy objects on "rubber sheets".
Here's the trick: You start by making the right definitions of the things you're dealing with. This inevitably requires introducing technical language, and a series of very precise -- some would say painfully fastidious -- definitions.
First, a word of caution. Some of what follows is rather "explicit" and possibly not suitable for folks who have little patience with technical "mumbo jumbo". If that's you... sorry. The aim here is to expose a few technical ideas. If this kind of thing isn't your cup of tea, don't say you weren't warned. If you do have patience, I hope you'll find this educational. On the other hand, if you've studied topology already, you may not find much new here.
The first piece of teminology we need is "manifold". We mentioned that in the previous article. It applies to the particular type of topological objects that we need to deal with. This class of objects includes spheres of all dimensions (such as Sn, for n = 1, 2, 3, ....). It also includes all the objects for which it is even meaningful to ask whether they are topologically "equivalent" to a sphere. And we're not going to get to a more explicit definition of manifold in this article, either, since we need some more basic definitions first.
Defining a "sphere" (or an "n-sphere") is relatively easy, and we actually did it in the previous article for n = 1 or 2. To wit, the 2-sphere S2 is defined as the set of points in 3-space, specified by their coordinates (x,y,z), which satisfy the equation x2 + y2 + z2 = r2. That's a little open-ended, since it actually defines a sphere of radius r for some positive number r. Since we're only concerned with topological properties (which ignore stretching), we may assume r = 1.
In order to state the appropriate definition for n-space, where n is any positive integer, we first have to be more precise about "n-space". When n = 1, what we are taking about is the real number line, denoted by ℝ. The generalization for larger n is denoted by ℝn and called n-space. It consists of points specified by n real numbers in an ordered list called an n-tuple, such as (x1, x2, ..., xn). Very likely you have encountered this called a "vector" or "n-vector", etc. Frequently in mathematics, different terms are used for the same thing. Don't be too distracted by the variations in terminology. Similarly, ℝn is often called, redundantly, Euclidean n-space. Don't let it throw you. (Such Euclidean spaces are critical to defining what a manifold is, but we're not quite ready for that yet.)
Given all that, we can define an n-sphere Sn as the set of points in (n+1)-space ℝn+1 where the (n+1)-tuple (x1, x2, ..., xn+1) satisfies the relationship x12 + ... + xn+12 = 1 among its coordinates. Rigorously, it needs to be proven that this set of points actually constitutes a manifold, but that's not very hard to do, once one has the definition of manifold.
The last major term that's not yet been made more precise is "equivalent". As in the question, "when are two manifolds equivalent?" Now, "equivalent" can have many different meanings in mathematics. The notion of "equivalence" can be formalized in what mathematicians call "category theory". Category theory is a very interesting topic, but saying much more about it here would take us way too far afield. Let it suffice to say here that category theory deals with collections of objects that are all of the same "kind" (or formally, of the same "category") and correspondences between two objects of the same kind. The objects are often (but not necessarily) defined set-theoretically as sets of elements (e. g. the points of a topological object), and the correspondences are called "maps" (or "functions", "arrows", etc.). So as not to get too abstract about this now, in the context of topology one thinks of objects as sets of points and the maps as ordinary functions that specify the correspondence between the points of one object and the points of another.
I suppose most readers are comfortable with the notion of "function", but if not, then in the context of topology you can think of a function that maps one topological object to another as a transformation that twists, stretches, or bends one object into a different shape (the second object). This is how one makes more precise the idea of topology as "rubber sheet geometry".
However, not just any map or function between objects is worthy of consideration, as far as topology is concerned. Colloquially, one says that only maps that do not "rip" or "tear" objects need be considered. A little more formally, what is required is that the admissible functions preserve the essential "structure" of the objects. "Structure" is deliberately vague, because it depends entirely on the "kind" (category) of object under discussion. In particular, when talking about manifolds, preserving their "structure" necessarily entails not "tearing" them. In the context of topological objects (including manifolds) all this can be made more precise by defining additional concepts (e. g. "open" sets, "continuous functions").
But I won't further strain your patience by going into more detail about that now. All that remains to do is explain what "equivalent" means in the topological context. Two topological objects ("spaces" or manifolds) X and Y are said to be "equivalent" if there are admissible functions f:X→Y and g:Y→X such that for any element x of X and y of Y, g(f(x)) = x and f(g(y)) = y. In the notation of function "composition", these conditions can be written as g∘f = IdX and f∘g = IdY. IdX and IdY are the "identity" functions on X and Y respectively. They are the trivial functions which take every point of X or Y to itself. (The function f is the "inverse" of g, and vice versa.)
When such a pair of admissible functions exists for the objects X and Y, topologists say that the objects are "homeomorphic". This is what "equivalent" means in this case. Objects that are homeomorphic to each other are the "same" as far as topology is concerned. Each of the functions in an inverse pair, like f and g, is called a homeomorphism.
Finally, in this terminology, the problem that the Poincaré conjecture addresses is to give sufficient conditions, for a given positive integer n, on an n-dimensional manifold X, so that X is homeomorphic to Sn.
Yes, we really needed all that terminology in order to state precisely what it means to say that X can be twisted, stretched, or bent so that it becomes a sphere, without tearing it. What we gain from going to all this trouble is the technical machinery needed to actually prove or disprove such a statement. There's no free lunch.
In the next installment, we'll describe the additional machinery that's needed to state what conditions must be imposed on a manifold X so that it is equivalent (homeomorphic) to a sphere.
Tags: mathematics, topology, Poincaré conjecture
In the previous post on this subject we looked, from an intuitive point of view, at what the conjecture (now, apparently, a theorem) is about. In a nutshell, it's really quite simple: What properties does an abstract topological object need to have in order to be "equivalent" to a sphere?
We provided a list of properties that have long been known to do the trick in the case of a special kind of sphere, the 2-sphere S2, a 2-dimensional surface (like a soccer ball) in ordinary 3-space.
It's natural, in fact inevitable (at least for a mathematician) to wonder whether a similar result applies in higher dimensions. The next dimension up involves the 3-sphere S3, a 3-dimensional object in "ordinary" 4-space.
At this point, most non-mathematicians start feeling queasy and weak in the knees because of all the traditional hype and mystery often associated with "the 4th dimension". In particular, how on Earth can anyone really understand what's going on with that, when admittedly it's barely possible, at best, to visualize what happens in a space of more than 3 dimensions?
Rest assured, most mathematicians can't really visualize it either. Instead, there are alternative ways to reason about spaces of more than 3 dimensions, and they don't depend much on "visualizing". All it takes is some straightforward reasoning, which we will try to explain. (Quite possibly the mathematicians who do this for a living, and actually prove theorems about such things, have a much better than average capacity for visualizing. But that's not very much needed to understand the general ideas.)
Another thing that most non-mathematicians wonder about, besides just grasping what's going on, is how on Earth it is possible to mathematically prove assertions about higher-dimensional topology -- or even about topology in more "ordinary" spaces of 2 or 3 dimensions. Even for people who've managed to cope with calculus in a 2-dimensional "cartesian" coordinate system, it's daunting to imagine how one ever works with stretchy objects on "rubber sheets".
Here's the trick: You start by making the right definitions of the things you're dealing with. This inevitably requires introducing technical language, and a series of very precise -- some would say painfully fastidious -- definitions.
First, a word of caution. Some of what follows is rather "explicit" and possibly not suitable for folks who have little patience with technical "mumbo jumbo". If that's you... sorry. The aim here is to expose a few technical ideas. If this kind of thing isn't your cup of tea, don't say you weren't warned. If you do have patience, I hope you'll find this educational. On the other hand, if you've studied topology already, you may not find much new here.
The first piece of teminology we need is "manifold". We mentioned that in the previous article. It applies to the particular type of topological objects that we need to deal with. This class of objects includes spheres of all dimensions (such as Sn, for n = 1, 2, 3, ....). It also includes all the objects for which it is even meaningful to ask whether they are topologically "equivalent" to a sphere. And we're not going to get to a more explicit definition of manifold in this article, either, since we need some more basic definitions first.
Defining a "sphere" (or an "n-sphere") is relatively easy, and we actually did it in the previous article for n = 1 or 2. To wit, the 2-sphere S2 is defined as the set of points in 3-space, specified by their coordinates (x,y,z), which satisfy the equation x2 + y2 + z2 = r2. That's a little open-ended, since it actually defines a sphere of radius r for some positive number r. Since we're only concerned with topological properties (which ignore stretching), we may assume r = 1.
In order to state the appropriate definition for n-space, where n is any positive integer, we first have to be more precise about "n-space". When n = 1, what we are taking about is the real number line, denoted by ℝ. The generalization for larger n is denoted by ℝn and called n-space. It consists of points specified by n real numbers in an ordered list called an n-tuple, such as (x1, x2, ..., xn). Very likely you have encountered this called a "vector" or "n-vector", etc. Frequently in mathematics, different terms are used for the same thing. Don't be too distracted by the variations in terminology. Similarly, ℝn is often called, redundantly, Euclidean n-space. Don't let it throw you. (Such Euclidean spaces are critical to defining what a manifold is, but we're not quite ready for that yet.)
Given all that, we can define an n-sphere Sn as the set of points in (n+1)-space ℝn+1 where the (n+1)-tuple (x1, x2, ..., xn+1) satisfies the relationship x12 + ... + xn+12 = 1 among its coordinates. Rigorously, it needs to be proven that this set of points actually constitutes a manifold, but that's not very hard to do, once one has the definition of manifold.
The last major term that's not yet been made more precise is "equivalent". As in the question, "when are two manifolds equivalent?" Now, "equivalent" can have many different meanings in mathematics. The notion of "equivalence" can be formalized in what mathematicians call "category theory". Category theory is a very interesting topic, but saying much more about it here would take us way too far afield. Let it suffice to say here that category theory deals with collections of objects that are all of the same "kind" (or formally, of the same "category") and correspondences between two objects of the same kind. The objects are often (but not necessarily) defined set-theoretically as sets of elements (e. g. the points of a topological object), and the correspondences are called "maps" (or "functions", "arrows", etc.). So as not to get too abstract about this now, in the context of topology one thinks of objects as sets of points and the maps as ordinary functions that specify the correspondence between the points of one object and the points of another.
I suppose most readers are comfortable with the notion of "function", but if not, then in the context of topology you can think of a function that maps one topological object to another as a transformation that twists, stretches, or bends one object into a different shape (the second object). This is how one makes more precise the idea of topology as "rubber sheet geometry".
However, not just any map or function between objects is worthy of consideration, as far as topology is concerned. Colloquially, one says that only maps that do not "rip" or "tear" objects need be considered. A little more formally, what is required is that the admissible functions preserve the essential "structure" of the objects. "Structure" is deliberately vague, because it depends entirely on the "kind" (category) of object under discussion. In particular, when talking about manifolds, preserving their "structure" necessarily entails not "tearing" them. In the context of topological objects (including manifolds) all this can be made more precise by defining additional concepts (e. g. "open" sets, "continuous functions").
But I won't further strain your patience by going into more detail about that now. All that remains to do is explain what "equivalent" means in the topological context. Two topological objects ("spaces" or manifolds) X and Y are said to be "equivalent" if there are admissible functions f:X→Y and g:Y→X such that for any element x of X and y of Y, g(f(x)) = x and f(g(y)) = y. In the notation of function "composition", these conditions can be written as g∘f = IdX and f∘g = IdY. IdX and IdY are the "identity" functions on X and Y respectively. They are the trivial functions which take every point of X or Y to itself. (The function f is the "inverse" of g, and vice versa.)
When such a pair of admissible functions exists for the objects X and Y, topologists say that the objects are "homeomorphic". This is what "equivalent" means in this case. Objects that are homeomorphic to each other are the "same" as far as topology is concerned. Each of the functions in an inverse pair, like f and g, is called a homeomorphism.
Finally, in this terminology, the problem that the Poincaré conjecture addresses is to give sufficient conditions, for a given positive integer n, on an n-dimensional manifold X, so that X is homeomorphic to Sn.
Yes, we really needed all that terminology in order to state precisely what it means to say that X can be twisted, stretched, or bent so that it becomes a sphere, without tearing it. What we gain from going to all this trouble is the technical machinery needed to actually prove or disprove such a statement. There's no free lunch.
In the next installment, we'll describe the additional machinery that's needed to state what conditions must be imposed on a manifold X so that it is equivalent (homeomorphic) to a sphere.
Tags: mathematics, topology, Poincaré conjecture
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