See also: Symmetry -- Mathematics and Physics -- Algebraic geometry -- M-theory -- Knot theory -- Noncommutative geometry -- Quantum geometry
Although we can understand a division of mathematics into separate parts dealing with numbers and geometry, it's hard to imagine geometry without numbers. It seems almost contradictory to try to think about measurement and length and angles without using numbers. Yet the classical Greeks (Euclid in particular) established geometry as a field of study in its own right, and at the same time made logic a part of mathematics as well. Thus it became possible to reason about 2-dimensional and 3-dimensional shapes -- with ruler and compass constructions, for example -- quite independently of reducing them to a description consisting only of numerically specified lenghts and angles. (The "ruler" in this case was not assumed to be marked in units of length.)
It wasn't until the time of René Descartes and his "Cartesian" coordinates around 1640, in fact, that geometry was almost completely reduced to a purely numerical form -- what is taught in schools today as "analytic geometry". But ironically, in even more modern times, geometry has been greatly generalized in several directions that eliminate any dependence on measurements and numbers, thereby demonstrating how it really is a very substantial and free-standing part of mathematics on its own.
Perhaps the first steps towards making geometry a truly abstract subject that didn't depend on numbers or measurement were taken beginning in the early 1800s by Carl Friedrich Gauss, Nicolai Ivanovich Lobachevski, Janos Bolyai, and Bernhard Riemann. The issue was Euclid's "parallel postulate", which claimed that given a line and a point not on the line, exactly one other line could be drawn through the point parallel to the original line. Mathematicians ever since Euclid always had the queasy feeling that this axiom was not as intuitively obvious as Euclid's other axioms. No one really doubted that it was true, at least as far as space as we know it is concerned. Yet it was somehow less obvious, and all attempts to prove it from the other axioms failed. It wasn't even possible to find more intuitive axioms from which the parallel postulate could be deduced. In spite of that, it was controversial, almost heresy, to deny the axiom. However, that's what the above-named mathematicians eventually did when they discovered different forms of "geometry" which satisfied Euclid's other axioms, but in which the parallel postulate was plainly false.
The upshot of their work, once it was accepted as legitimate mathematics and not some sort of blasphemy, was that people were then free to think about geometry in a more abstract way. "Geometry" was no longer synonymous with "Euclidean geometry." In particular, notions such as length and angle -- specified as numbers -- no longer needed to be part of the subject. But in that case, then, what was geometry, really?
In one direction, geometry was liberated from number through use of the concept of symmetry. Felix Klein, with his "Erlangen program" in 1872, was the driving force here. The idea was that geometrical notions could be interpreted as properties invariant under specific groups of transformations. This related geometry to the abstract theory of groups, which had been under development since the work of Évariste Galois in 1830. This was a "top down" or "wholistic" view of geometry, in that it did not seek to analyze geometric objects in terms of their constituent parts (such as points or lines).
But even more progress occurred in the opposite direction -- a "bottom up" or "reductionistic" view of geometry. This was in part based on Georg Cantor's set theoretic ideas, which appeared about 1874. Although Cantor's ideas were at first highly controversial, even scandalous (and mistrusted by a few mathematicians even today), the ideas of point set topology eventually emerged from the contributions of a number of mathematicians. The key idea here is to axiomatize the concept of "distance" or "nearness" in terms of the properties of collections of sets -- "open" and "closed" sets.
A little later, however, a different conception of "topology" originated with Henri Poincaré's 1895 book, Analysis situs (which title was, for a time, used as an alternative term for "topology"). The idea here was to deconstruct geometric objects into flat, Euclidean polyhedral pieces (lines, planes, etc.). Using these simpler objects, then, one could compute other things like numbers or abstract groups which were in some sense "characteristic" of the object, such that any other object having the same characteristics could be regarded as essentially the "same" sort of object. Because of this computational emphasis, this form of topology became known as "combinatorial" topology, as opposed to "point set" topology. The field eventually blossomed into what is today called "algebraic" topology.
One of the first results of Poincaré's work was the famous "Poincaré problem" -- the question of whether any 3-dimensional (solid) object having a certain algebraic invariant was essentially the "same" as a 3-sphere (denoted by S^{3}). Of course, the devil is in the details, and the technical description of the algebraic invariants as well as the term "same" need to be explained. We'll go into that later. Let it suffice for now to say that Poincaré's problem is still unresolved, though much progress has been made. The work on the problem during the 20th century has resulted in large parts of the current immense edifice of modern topology. (In the same way that work on Fermat's Last Theorem is responsible for much of algebraic number theory, and even a lot of modern abstract algebra itself.)
It should be noted that Poincaré did not have a very high regard for point set topology, in part because of lingering suspicions about the soundness of Cantorian set theory in general and its lack of "constructiveness" (a technical term) in particular. Nevertheless, the notions of point set topology are essential for giving a rigorous definition of the concept of a "continuous" function. And this concept, in turn, is essential for making rigorous the very idea of what it means for two objects to be topologically "equivalent".
The end result today, after over 100 years of development, is that both point set and combinatorial approaches to topology are essential and complementary parts of the field as a whole. Together they make it possible to rigorously axiomatize topology as a bottom-up theory of geometry that is concerned with the abstract idea of shape, independently of recognizable notions of measurement or "distance". This is why topology is often referred to simplistically as "rubber sheet" geometry. That is, what's important is the relationship between the parts of a geometric object, rather than the numerical values of distances or angles. For (2-dimensional) shapes drawn on a rubber sheet, only the former rather than the latter are invariant.
Interestingly enough, however, Euclid has not been forgotten. In fact, his legacy still remains in the very foundations of the subject. Mathematicians refer to a type of topological space as "Euclidean space". This is simply a topological space that consists of a "product" of 1 or more copies of a straight line. In more familiar terms, what this refers to is nothing other than Descartes' Cartesian coordinates. A plane, for instance, is just a set consisting of points that are identified with pairs of real numbers. These numbers are simply the coordinates of the point relative to two axes which are at right angles to each other (i. e. "orthogonal"). If the set of real numbers is denoted (as usual) by R, then the cartesian plane is R×R. Another way to write that is R^{2}. Within R^{2} all of Euclid's axioms, including the parallel postulate, are true.
Given this, the generalization is impossible to ignore: R^{n} is called n-dimensional Euclidean space. The points of R^{n} are ordered lists of n real numbers. Such lists are also known as n-tuples or "vectors". A very special case is R^{3}, the 3-dimensional space of everyday experience. Since this is the only topological space we know well from direct experience, is it any surprise that it continues to play a key role in geometry? Although it is nearly impossible to visualize spaces of dimension higher than 3, or geometric objects in them, many of the properties of 3-dimensional Euclidean space actually carry over very well to n dimensions, for any larger n.
In fact, Euclidean space is such a fundamental part of our intuitive notions of geometry that it has been given a fundamental role in practically all but the latest generalized geometries that mathematicians still concern themselves with. It's a concept that we refuse to part with. Therefore, the key generalized notion that is central to almost all more abstract forms of geometry -- something called a "manifold" -- incorporates Euclidean space in an essential way, and is really just a slight generalization.
With n-dimensional Euclidean space, R^{n}, just a single coordinate system is needed to describe the position of any point. There are, of course, many coordinate systems possible, because given one, another can be obtained by translation or rotation of the axes. But the point is that it is never necessary to use more than one system at a time to cover the whole space.
With a manifold, on the other hand, this condition is relaxed slightly. It is simply required that there be a Euclidean coordinate system for some "neighborhood" of every point in the space. (A neighborhood of a point is an "open" set that contains the point, where an open set is one of a collection of subsets of the space. This collection must satisfy certain axioms and ultimately defines the topology of the space. As the terminology implies, being in the same neighborhood is the criterion for two points to be "near" each other.) Each coordinate system must have the same dimensionality. That is, each is a copy of R^{n} for the same n. The dimensionality of the manifold is defined as this number. What this all amounts to is that a manifold is a space that is "locally" Euclidean, though (usually) not "globally" Euclidean.
In general, it is possible to select coordinate systems for a given manifold in an unlimited number of ways. However, there is an additional requirement imposed through the technical definition of the coordinate systems. This is a consistency condition that applies whenever coordinate systems overlap. There are different kinds of consistency conditions, and which kind is used determines the "type" of the manifold under consideration. This type, in turn, determines the conditions under which two manifolds are considered to be topologically equivalent.
What we've described so far is a "topological manifold". Additional structures can be defined that specialize the notion as required for various applications. There is, for example, the very important case of the "differential" structure of a manifold, which has to be specified in order to discuss questions of calculus with respect to the manifold -- how to do differentiation and integration on it. This is obviously important for applications in physics. We'll get into some of these additional structures a bit later.
For now, let's conclude this introduction with a review of some pertinent terms which are all part of the big picture of modern geometry, but which are easily confused until you get your bearings. Each of these represents a field of mathematical research -- fields which are all large and have active areas of research at the present time.
The application of geometry to physics is hardly new of course. For instance, the Greeks, and Plato in particular, were fascinated with the fact they discovered that there are only five possible "regular" polyhedra -- solid objects all of whose faces are congruent -- the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Euclid devoted the entire last book of his Elements to an exposition of these figures. Greek philosophers experimented with theories that had all matter constructed out of these five shapes. (Unfortunately, this comported awkwardly with other theories that everything was made of four fundamental elements -- earth, air, fire, and water.) Even as late as the early 1600s, thinkers as sophisticated as Johannes Kepler struggled to formulate a cosmology based on the five "Platonic" solids. In a work called the Mysterium Cosmographicum he theorized that the spacing of the six known planets could be explained by the size ratios of the Platonic solids inscribed within each other.
To be sure, current theories of fundamental physics are somewhat more rigorous. Both directions of abstraction in geometry which were touched on above play a major role. On one hand, we have geometry considered as the study of properties that remain invariant under certain sets of transformations -- think of notions like congruence, similar triangles, etc. -- symmetries, that is. In physics the use of symmetry concepts is now found everywhere, for example:
On the other hand, geometry regarded as a study of the topology of manifolds is also ubiquitous, for instance:
All of these considerations suggest in the strongest possible manner that geometry ultimately holds the explanations for why the universe is the way it is, at a fundamental level.
Einstein was the first major physicist to use geometry in a deep way to not merely express but actually explain physical laws. First, in his special theory of relativity, he showed how the 4-dimensional geometry of spacetime was needed to understand the behavior of light. Then, in his general theory of relativity, he showed how gravity itself is nothing more than an effect of the curvature of spacetime.
Various contributors to quantum theory, such as Herman Weyl and Eugene Wigner, picked up the geometry ball and took it much further. They showed how Lie groups -- representing certain kinds of symmetries and spatial transformations -- allowed a particularly elegant formulation of quantum mechanics. Emmy Noether showed how conservation laws arose from physical equations that are invariant under symmetry operations.
And so the notion of symmetry -- which ultimately has its source in geometrical ideas -- quickly took over physics. From Einstein's symmetry of relatively moving coordinate systems to the (broken) symmetries among different types of forces to the abstract symmetries which classify the properties of elementary particles.
The flood gates were opened. Weyl's investigations ultimately led to the notion of gauge theories, Yang-Mills gauge theories in particular, which allowed for such modern theoretical successes as quantum chromodynamics and the unified electroweak theory. But mathematically such gauge theories had already been studied as connections in certain fiber bundles of a manifold.
And now we find things like:
So there needs to be a new type of geometry -- "quantum geometry" -- that is general enough to provide a mathematical description that is appropriate for both gravitation and quantum mechanics at the same time. Discovering -- or inventing -- this quantum geometry is a major challenge for both mathematics and physics.
After Euclidean space itself, perhaps the most important kind of topological manifold is a sphere. As familiar as the idea of a sphere is, it's necessary to make one point clear. Most of the time when a topologist is talking about a sphere, she is not talking about a "solid" sphere such as the Earth or a baseball. That sort of object is relatively uninteresting, since it is so much like ordinary Euclidean space, except for being of finite extent.
Instead, what the topologist usually has in mind is just the surface of a solid sphere. Of course, when one has to consider spaces of dimension higher than three, the meaning of terms like "solid" and "surface" are not so clear. Therefore, to be precise, a sphere may be defined as the locus of points in a Euclidean space (of some dimension) which are all equidistant from the origin. (We're relying on the fact that a notion of "distance" exists in Euclidean space.)
The common sphere in Euclidean 3-space with which we are most familiar is denoted by S^{2}. This is called, more precisely, a 2-sphere, because as a manifold it is 2-dimensional, even though embedded in 3-space. And the reason it is considered 2-dimensional is that "locally" every point of the sphere can be described by just 2 coordinates. Latitude and longitude will do the trick (except at the poles.) Or indeed, on any patch of the sphere smaller than the whole, rectangular coordinates will do just fine, even though they can't be extended to the entire surface. In other words, a small portion of S^{2} is essentially the same as a small portion of R^{2}. True, it is slightly "curved". However, in order to be able to define curvature, some additional structure is required beyond that of a plain manifold. We'll get to that shortly. As long as a little bending and stretching are allowed, a small piece of S^{2} is topologically equivalent to a small piece of R^{2}. Being a (topological) manifold really isn't anything more than that.
By the definition of a sphere as a set of points equidistant from the origin, it's easy to specify S^{2} as the set of points (x,y,z) in R^{3} that satisfy the equation:
x^{2} + y^{2} + z^{2} = r^{2}where r (a constant) is the radius. (Normally one uses r = 1 in the definition of spheres.)
Notice how the use of an equation to define the object has resulted in the 2-sphere turning out to be a surface instead of a solid object. Had we wanted to define a solid object, we could have done so with an inequality:
x^{2} + y^{2} + z^{2} < r^{2}In fact, topologists do study objects of this sort. The technical term is "ball" rather than "sphere". In this case (3-dimensional) the object is denoted by D^{3}. (Again, it is customary to take r = 1.) However, equations are easier to work with algebraically than inequalities. And plenty of experience has shown that this notion of a "sphere" is more useful topologically.
There's one other subtle point here. Notice that we defined D^{3} using strict inequality. Had we used the non-strict inequality
x^{2} + y^{2} + z^{2} ≤ r^{2}in the definition, the result would be what is called a "manifold with boundary". In this case, the boundary would be precisely S^{2}. Such a set doesn't quite meet the defintion of a manifold, because points on its boundary aren't contained in neighborhoods homeomorphic to neighborhoods in R^{3}. Although manifolds with boundaries can be treated with a little more work, and messiness, in most of the discussion here it will be implicit that we are talking about manifolds without boundaries.
The algebraic definition of a sphere as a locus of points generalizes immediately to any number of dimensions. The 1-sphere S^{1} is simply a circle in R^{2} (the Euclidean plane). The 0-sphere S^{0} consists just of two points (x = ± r). In general, the n-sphere, for nonnegative n, is the set of points (x_{1}, ... , x_{n+1}) in (n+1)-space R^{n+1} such that
x_{1}^{2} + ... + x_{n+1}^{2} = r^{2}Notice how easy it is to describe a general sphere algebraically, even though visualizing S^{n} for n ≥ 3 is impossible for most mortals.
To return to a point hinted at earlier, it is not possible to have a single (Euclidean) coordinate system that works for an entire sphere. On the surface of the Earth, for example, coordinates of latitude and longitude don't work at the North and South poles, because of non-uniqueness -- at 90° North latitude, any value of longitude represents the same point. S^{2} can, however, be exhibited as a manifold using just two coordinate systems.
Of course, spheres are about the simplest examples of manifolds other than Euclidean spaces. For more complicated manifolds, it's quite normal to need more coordinate patches. But as long as more than one is required, we might as well just leave the precise number unspecified.
There are several reasons for spending so much time here talking about spheres. In addition to the fact they make easily understood examples, they play an important role in a part of algebraic topology known as "homotopy theory" by virtue of their simple structure. On the other hand, in spite of their apparent simplicity, spheres present some extremely deep -- and still open -- questions. This is the matter of the Poincaré conjecture, which we will describe later.
Yet another thing about spheres is that they demonstrate how interesting topological objects are defined by algebraic conditions. This leads into the vast, and deep, subject of algebraic geometry.
So far we have been discussing only the most generic sort of manifold, the topological manifold. In practice, most of the interesting questions arise when the manifold has some additional structure. There are various ways in which additional structure can be added -- and a larger number when these additions are combined in different ways. Each different choice tends to have interesting applications to other parts of mathematics or to the "real world".
Recalling the definition of a topological manifold, it was required that it should be possible to provide a Euclidean coordinate system for some neighborhood of every point. In more technical terms, one says that the neighborhood is "homeomorphic" to an open set in the Euclidean space R^{n} (for some n). "Homeomorphism" is an important term. It means a mapping between sets that establishes their topological equivalence. In other words, saying that two topological spaces are "homeomorphic" is a shorthand for saying they are the topologically the same -- that either one can be deformed into the other by bending or stretching, but not cutting or tearing. The key characteristics of a homeomorphism are that it be 1-to-1 and "continuous". It is the "continuous" part which formalizes the notion of not "tearing" the object.
In general, any point of a manifold M is contained within many different neighborhoods. All such neighborhoods overlap to some extent (since they contain at least the original point). If A and B are two such neighborhoods of a point m∈M, there will be two different homeomorphisms to open sets in R^{n} -- call these maps φ_{A} and φ_{B}. What these maps do is simply to provide a coordinate system in each little neighborhood. In other words, they make it possible to associate an n-tuple of real numbers with each point of the manifold. The way things are defined, these maps will be compatible with each other. (Specifically, if the overlap of the sets A and B is denoted by A∩B, then it is also a neighborhood of m, φ_{A}(A∩B) and φ_{B}(A∩B) are open sets in R^{n}, and they are homeomorphic to each other.)
Given all of this, the way that a differentiable structure is defined on the manifold M is simply to require that for some set of neighborhoods of each m∈M and maps like φ_{A}, then there is a compatibility condition analogous to that of the last paragraph between pairs of open sets such as φ_{A}(A∩B) and φ_{B}(A∩B). Specifically, these two open sets should be "diffeomorphic" to each other -- that is, the map between them should be "differentiable", which is defined in terms of the usual notions of calculus in R^{n}.
The need to be so fussy about all this is unfortunate, and the description is still not quite as precise as it could have been. You need not remember all these details. The key point is that whereas plain vanilla topological manifolds are defined in terms of continuous mappings, to define differentiable manifolds we just make the stronger requirement of differentiable mappings instead. Most of the fussiness arises because we can't speak directly of the maps like φ_{A} being differentiable, because we don't have to begin with a definition of what it means for a map defined on the manifold to be differentiable.
In the ordinary calculus of one variable, in order for a function to be differentiable, it must be "smooth" in that it should have no kinks or sharp corners in its graph. So an equivalent term for a differentiable function is a "smooth function". In fact, this is usually understood to mean that derivatives of all orders exist, in which case the function is said to be of class C^{∞}, or "infinitely" differentiable. The same terminology is used for manifolds where the functions relating the coordiniate systems of overlapping neighborhoods have the specified differentiability properties. So differentiable manifolds are also known as smooth manifolds.
There's another, more informal, way to think of all this which might help. You may think of a topological manifold as simply made up of many parts that are small chunks of R^{n} glued together. In this plain vanilla case the "gluing" process is continuous, in that it may warp and bend the parts to fit together, as long as it doesn't tear them. But in the more fastidious differentiable case, the parts must fit together more smoothly, without excessively rough handling. The end result is that you have a nicer manifold that you can safely do calculus on without danger of cutting yourself on sharp edges.
For a function of one variable, y = f(x), we have a good intuitive sense of what is meant by the idea of the tangent to the graph of f(x) at some particular point x_{0}. It is simply a straight line through f(x_{0}) whose slope is numerically the same as the value of the derivative f′(x) at that point, i. e. f′(x_{0}). So a tangent to a curve at some point is simply the straight line which is a best approximattion to the curve at the point. Similarly, one can easily visualize a tangent plane to a 2-dimensional surface at some point as being the best 2-dimensional linear approximation to the surface at that point.
In higher dimensions, our ability to visualize what is going on breaks down, but we can still think of the tangent space of an n-dimensional manifold at a point as being the best linear approximation of the manifold at that point. By analogy with the cases of curves and surfaces, this tangent space should be a "flat" n-dimensional space. But that's exactly what R^{n} is. So we need some way to associate to every point m of a manifold M an n-dimensional linear tangent space.
Just as in the 1-dimensional case of a curve, this is done by using derivatives. Technically, one way to do this is simply to consider all (1-dimensional) curves which lie in the manifold and go through the point m. Each curve through m has a specific direction which can be expressed in terms of derivatives. Given a coordinate system on M around the point m, these derivatives can be expressed in terms of the n partial derivatives with respect to each of the n different coordinate functions. The tangent space of M at the point m, which we can symbolize as T(M,m) is nothing but the set of all possible tangent vectors to all possible curves in M through m.
An element of T(M,m) is, then, a way of specifying both direction and rate of change -- the rate of change in a particular direction. at a particular point. In other words, a directional derivative. This fact about elements of the tangent space T(M,m) is what allows us to define differentiation of functions on M. Suppose f(x) is a function defined on M. That is, f assigns a real number to every point m∈M, so f is a map from M to R. Suppose further that we make a choice of some vector v_{m} in T(M,m) for each m. Such an association that assigns a vector to each point of M is called a "vector field". Then v_{m} gives a recipe for assigning a real number v_{m}(f) corresponding to the function f at each point of m. The technical term for this is that v is a "contravariant 1-form" on M.
T(M,m) has an algebraic structure because any two tangent vectors can be added together to give another tangent vector, and any tangent vector can be multiplied by a "scalar" (i. e. a real number) to give another tangent vector. This kind of algebraic structure is called a "vector space". You can think of the elements of T(M,m) as vectors tangent to M at m. What is a "vector", exactly? Well, in the case of T(M,m) a vector is concretely a directional derivative, which can be expressed in any given coordinate system on M using traditional recipes of calculus. Abstractly, a vector is just any object in an algebraic system that satisfies the axioms for a vector space. It turns out that all finite dimensional n-dimensional vector spaces are essentially the same -- merely copies of R^{n}, consisting of n-tuples of real numbers. The 1-dimensional case is a line, while the 2-dimensional case is a plane, just as expected.
It may seem as though T(M,m) is a stiff dose of abstraction, but it has many uses in manifold theory. Basically, its nice algebraic properties bring a lot of order to the study of the geometry of manifolds. It is a way of "linearizing" an object which may itself be highly curved and nonlinear.
For one thing, as an n-dimensional vector space, T(M,n) has what is called an "inner product" (in fact, many of them). That is a way of "multiplying" two vectors of T(M,m) to produce a scalar. If u and v are elements of T(M,m) then their inner product may be denoted by u⋅v or (u,v)_{m}, to emphasize that everything is specific to the point m of M. When vectors u and v are represented as n-tuples (u_{1}, ..., u_{n}) and (v_{1}, ..., v_{n}), then the inner product is just u_{1}v_{1} + ... + u_{n}v_{n}, or more compactly, Σ_{i} u_{i}v_{i}.
Keep in mind that such an inner product is specific to T(M,m) and can be chosen (somewhat) arbitrarily and separately at each point m of M. In the abstract, this choice of inner product should be made so that it is "smooth" (i. e. differentiable) as one goes from one point to another. When such a choice has been made for all points of the manifold, one says that the manifold has been given a Riemannian structure, after G. F. B. Riemann, who originated so much of this kind of geometry. Given this Riemannian structure, one can then go on to define notions of distance between points of the manifold and curvature of the manifold itself at any particular point. The rest of calculus can be adapted as well -- specifically integration, which makes it possible to define notions of area and volume in the manifold.
And don't forget that all this is possible based on the assumption that M is a differentiable manifold. If M were merely a topological manifold, we would have no way to define a tangent space T(M,m), a Riemannian structure, curvature, or distance. Those concepts simply do not apply in the general case.
Althought abstractly a given differentiable manifold can be endowed with many different Riemannian structures, in practice there is usually one which is favored. In case the manifold M is "embedded" in a Euclidean space -- for instance by some algebraic relationship on coordinates (as S^{2} is embedded in R^{3}) -- then the manifold becomes a submanifold of the Euclidean space and "inherits" a standard inner product from that space. It likewise, then, inherits a specific "natural" notion of length and curvature. (Indeed, it is a fact, known as the Whitney embedding theorem, that a differentiable n-manifold can be embedded in R^{2n+1}.) The theory involving the Riemannian inner product, length, and curvature is known as "Riemannian geometry", which was begun by Riemann and put to such good use by Einstein (for example) in the general theory of relativity.
The tangent space T(M,m) is also a key example of the notion of a "fiber bundle". A fiber bundle is simply a type of topological space which consists of a topological manifold M, and for each m∈M another space such as T(M,m) called the "fiber". Each fiber is the same sort of space (e. g. an n-dimensional vector space in the case of T(M,m)), and appropriate continuity or differentiability conditions are also imposed for the fibers attached to points that are "close" to each other. There is a lot of interesting theory concerned with the topological space that consists of a manifold M and all its tangent spaces -- the collection of which is called the tangent bundle.
The use of fiber bundles is now peravsive in the study of manifolds in most of their many specific forms. Fiber bundles aren't just an exercise in abstraction for its own sake. The applications to physics, in particular, are numerous. But let's defer further discussion in that direction for now, and turn instead to one of the most important open questions in geometry.
Mathematicians (just as practioners of many other scientific fields, like botany or ornithology) are fond of classifying things. Physicists classify elementary particles. Molecular biologists classify protein shapes. Topologists classify manifolds.
The most general way to classify manifolds is in terms of "homeomorphisms". Two manifolds that are homeomorphic to each other are essentially the same. What this means is that one manifold can be deformed into another by bending or stretching, but not by cutting or tearing. Technically, a homeomorphism is a 1-to-1 map between manifolds that is continuous in both directions. The property of being continuous means that open sets of the two manifolds correspond to each other. Since open sets define the topology of an object, this definition guarantees that the topology is the same for homeomorphic manifolds.
Before molecular biology came along and changed the game completely, animals were classified into species, genera, orders, etc. by looking at visible features of an organism which set it apart from related but not identical organisms -- things like bone structure, teeth, hair, and so forth. Biologists have (or had) one advantage over mathematicians. They can declare, by fiat, that some collection of features defines a particular species -- as long as no one comes across specimens having all the required features yet for some reason not being of the same species.
The job is a little harder for topologists. For example, what set of features of a 2-dimensional manifold guarantees it is of the same species (i. e., is homeomorphic to) a 2-sphere, S^{2}? The job is harder, because it is necessary to give a rigorous proof that the asserted list of features of a manifold is necessary and sufficient for being homeomorphic to a sphere.
In this respect we stress the sufficiency aspect, because necessary conditions are relatively simple to come up with. This is because homeomorphism is a strong condition. If a manifold M is homeomorphic to S^{2}, then it must have all of the properties that the sphere has which are preserved by homeomorphism. For example, M must be 2-dimensional. Also, M cannot have a boundary or "edge", because S^{2} is not lacking even a single point, let alone a 1-dimensional edge.
An obvious property, but one that mustn't be overlooked is connectedness. The technical definition behind this intuitive property is that a connected manifold is one that can't be decomposed into two or more parts such that there are no open sets which overlap distinct parts.
Another property that must be present is called "compactness". While there is a technical definition of that property, what it means is that a compact object is finite in extent -- unlike, for example, a 2-dimensional plane such as R^{2}.
Some other properties of a sphere are even more technical in nature -- yet very important. One is that a sphere is "simply connected". This is stronger that merely being connected, because part of the definition of a manifold being simply connected is that the manifold be connected. In addition to that, the definition also requires that every circle (or more precisely, every homemorphic image of S^{1}) can be contracted to a point while staying within the manifold. Among 2-manifolds, S^{2} and R^{2} (which is homeomorphic to S^{2} with one hole) have this property. But S^{2} with more than one hole and R^{2} with any holes do not have the property, since then a circle around just one hole couldn't contract to a point while staying within the manifold.
One is tempted to say that being simply connected means "no holes", but that isn't quite right, because a 2-sphere with a single hole is still simply connected. (However, we can ignore this exception if we consider only manifolds without boundaries.) Another example of a 2-manifold that isn't simply connected is a torus T^{2} (the surface of a donut shape).
It turns out that for 2-manifolds, the properties of being compact, simply connected, and having no boundary are both necessary and sufficient in order for a 2-manifold to be homeomorphic to S^{2}. This was proven in the 1800s. Similar results were proven for T^{2} and other compact 2-manifolds.
The obvious question that stares us in the face is then: what's the situation for 3-manifolds, or manifolds of any finite dimension? What are the necessary and sufficient conditions for an n-manifold to be homeomorphic to S^{n}?
One problem crops up immediately in trying to answer this question. The problem is that simple connectedness is not a strong enough condition. It does not reflect a sufficient amount of the structure of a manifold. Think about a general 3-manifold, or to be more concrete, a copy of ordinary Euclidean 3-space R^{3}. It may contain holes like a Swiss cheese. But unless those holes are infinite in extent, a loop can easily contract to a point yet avoid passing through a hole. The same is true of a compact manifold like the 3-ball D^{3}, as long as internal holes don't reach the surface. Pretty clearly, a copy of D^{3} that contains one or more holes is not homeomorphic to S^{3}.
One suspects that this problem may be easy to fix by considering not only loops (homeomorphic images of S^{1}), but also generalized spheres (homeomorphic images of S^{2}). Examining the question of when such generalized spheres in a 3-manifold can shrink to a point while remaining within the manifold should capture much more of the topology. In particular, for the case of the Swiss cheese (D^{3} with holes in it), any sphere that surrounds a hole can't shrink to a point.
There is a way to formalize this procedure and create an algebraic invariant of any n-manifold M. What you do is consider the set of all continuous images of S^{k} in M, for each k < n. (An additional requirement is that each image pass through some specific point of M, but let's leave that aside, in the interest of brevity.) You then form equivalence classes of such images which can be deformed into each other. A group structure can be defined on the set of these equivalence classes. (When k=1 this is easy to visualize: you "add" two loops by forming the curve that is formed by tracing out the first loop followed immediately by the second.) The resulting group is called the k^{th} "homotopy group" of M, and is denoted by π_{k}(M). If k=1, the group π_{1}(M) is also known as the "fundamental group" of M, and saying M is simply connected is equivalent to saying that the fundamental group is trvial (i. e., consists of just a single element).
Given all this, it is possible to generalize the known result for n=2 and make the following conjecture:
For n ≥ 2, a compact n-manifold M is homeomorphic to S^{n} if and only if π_{k}(M) is equal to π_{k}(S^{n}) for k < n.This is the n-dimensional Poincaré conjecture, or rather, one form of it, since it can be stated in several equivalent forms. Poincaré himself studied this general form of the proposition and claimed, in a 1900 paper, to have a proof for any n ≥ 2. However, he retracted this claim in 1904, and he gave a counterexample to his proof for n=3. He left the question as a problem to be resolved and did nothing further with it.
For almost 60 years mathematicians were stimilated by the challenge of proving the conjecture one way or the other. In the process, a great deal of the machinery of modern topology and manifold theory was created. For most of that time, the case n=3 was expected to be easier to solve than the case for larger n. This turns out to have been a mistake, and may have delayed progress on the general conjecture.
In 1960, less than five years after receiving his PhD, Steve Smale proved the general conjecture for n &ge 5. Undoubtedly, his decision not to bother with cases of smaller n allowed him to succeed with what was in fact the easier part of the problem. (When Smale's success became known, three other mathematicians -- John Stallings, Christopher Zeeman, and Andrew Wallace -- came up with different proofs for slightly different forms of the problem.) It was not until 1981 that the case n=4 was settled, by Michael Freedman. The case n=3 is still an open question. Just about everyone believes it should be true, but no one has found a proof -- despite many erroneous claimed solutions.
Why is the case n=3 so hard? The answer seems to be that in higher dimensions, the study of topology is actually easier because "there is more room to move around". For dimensions 1 and 2, on the other hand (curves and surfaces), there just isn't enough complexity to make the problem hard. Only when n=3 is there enough complexity to make the subject interesting, yet not enough room to maneuver and apply general procedures that work in higher dimensions. The study of 3-manifolds is an active area of research, with many specialized open questions.
We noted that there were other ways to formulate the generalized Poincaré conjecture. Let's say a few words on that. To begin with, note that homotopy groups π_{k}(M) can be defined even when k is greater than or equal to the dimension of M. If M is homeomorphic to S^{n}, then they must have the same homotopy groups for any k. (They are then said to have the same homotopy type.) So that's a necessary condition. Interestingly enough, however, the weaker condition of equality of homotopy groups for k < n is sufficient to ensure homeomorphism.
This is fortunate, since homotopy groups are not in general easy to calculate, even for spheres. It is known that π_{k}(S^{n}) = 0 (i. e., is trivial) for 0 < k < n, while π_{n}(S^{n}) = Z, the additive group of integers. (This is very easy to see for n=1, because the group consists of circles wrapped an integral number of times around S^{1}.) When k > n, the homotopy groups are especially hard to compute and certainly aren't necessarily trivial. π_{3}(S^{2}) = Z, for example. Questions about the homotopy groups of spheres form an active area of research, and involve interesting questions regarding algebra and fiber bundles.
It turns out that there is a similar but different kind of algebraic invariant that can be constructed for manifolds. It is also a group, called a "homology group". There are various ways to define homology groups, though all the definitions tend to be messy and technical. Some definitions are more useful for certain kinds of problems and less useful for others -- which is why there are multiple defnitions. Nevertheless, homology groups are usually easier to compute than homotopy groups. Homology groups are true invariants, because homeomorphic toplogical spaces have the same homology groups.
As with homotopy groups, for a manifold M there are homology groups, denoted by H_{k}(M), for any nonnegative integer k. It turns out that for n &ge 1, and any form of homology theory, H_{k}(S^{n}) = 0 for any nonnegative k except 0 or n, and H_{k}(S^{n}) = Z otherwise. One says that M and S^{n} have the same homology type if all their homology groups are equal. Given all that, an equivalent form of the generalized Poincaré conjecture is:
For n ≥ 2, a compact n-manifold M is homeomorphic to S^{n} if and only if M is simply connected and has the same homology type as S^{n}.
Yet another issue to consider is whether or not M is assumed to be a differentiable manifold. It turns out for the cases that have been resolved, the answer is that it doesn't much matter. One can obtain somewhat sharper results if M is differentiable, but the general situation is the same for general topological manifolds. Of course, for 3-manifolds it conceivably could make a difference. No one knows.
What forms could a resolution of the Poincaré conjecture take? The first possibility is that it could be proven correct for 3-spheres, just as has been done for n-spheres with any other n. This is what most topologists expect to be the case. On the other hand, a counterexample could be found -- some 3-manifold which has the same homotopy or homology as S^{3} but which isn't homeomorphic to it. But how could one tell the supposed counterexample, call it M, isn't homeomorphic to the 3-sphere? That would be the sticky part. There would probably have to be some sort of algebraic invariant of M which can be computed and is different from the same invariant of S^{3}. However, although many other algebraic invariants have been studied, topologists have never found an M that has a different invariant from the 3-sphere, unless the homology or homotopy is different as well.
Interestingly enough, the Poincaré conjecture was not one of the problems in David Hilbert's famous list of 23 outstanding problems of mathematics in 1900. Hilbert and Poincaré were arguably the two leading mathematicians in the world at the time, and there was an element of rivalry between them. But the reason that the problem was omitted from the list was probably that the field of topology was in a state of flux at the time. Indeed, Poincaré had just announced in 1900 (incorrectly) that he'd proven his conjecture for n ≥ 3 -- so as far as Hilbert knew, it was a nonproblem.
Nevertheless, there are hopeful signs.
To begin with, the classification problem for 2-manifolds -- surfaces -- was solved in the 1800s. The answer is that there are only two pieces of information which are required to discriminate between any compact, connected surfaces. One of these is whether or not the surface is "orientable". This means that one can make a consistent definition of "clockwise" on the entire surface. That is, you can take any loop on the surface, decide which direction is clockwise, then continuously move the loop anywhere on the surface, and always preserve the chosen direction. On most surfaces this is possible. On a Möbius band it is not. Of course, a Möbius band isn't, strictly speaking, a 2-manifold, because it has a boundary, consisting of a single circle. However, a Klein bottle, which is harder to visualize because any embedding of it in 3-space necessarily intersects itself, is an example of a nonorientable 2-manifold.
The other piece of information required to classify a surface is related to a number that can be defined for orientable surfaces, the "genus". In that case, roughly speaking, the genus is the number of holes in the surface. A 2-sphere has no holes, so its genus is 0. A torus (donut surface) has one hole, so its genus is 1. And so on. A slightly more exact definition of genus is the number of "handles" that would have to be attached to a sphere in order to yield a surface that is topologically equivalent to the surface in question. There is another number which can be defined by a somewhat more roundabout method for any 2-manifold M, orientable or not, which gives an even better definition of genus. It's called the "Euler characteristic" of the surface, and denoted by χ(M). This number is defined by first replacing the surface by a topologically equivalent one that consists entirely of flat faces, straight edges, and vertices. This can always be done, and the result is said to be "piecewise linear" (PL), for obvious reasons -- all of its parts really are portions of Euclidean space (either R or R^{2}). The technique of using such piecewise linear approximations of surfaces by lines and polygons is very common in topology, and goes back to the 1800s. Given that, one simply counts the total number of faces (F), edges (E), and vertices (V) in the figure. Although V, E, and F will vary depending on the approximating PL surface, it turns out that the number χ(M) = V - E + F is always the same (for homeomorphic manifolds). And furthermore, it satisfies
χ(M) = 2 - 2gwhere g is the genus. This could be taken as the definition of genus for orientable manifolds.
It is tricky to extend the notion of genus to nonorientable 2-manifolds, because attaching handles to spheres always results in an orientable manifold. Nonorientable surfaces are obtained by attaching one or more copies of something called a "cross cap" to a sphere. A cross cap is basically just a Möbius band, and since that has a boundary that is just a circle, it can be "glued" into a circular hole cut in a sphere. For nonorientable surfaces, the genus is defined as the number of attached cross caps. Now, the definition of χ(M) applies equally well for both orientable and nonorientable surfaces. When one figures out what χ(M) is for a nonorientable surface, it turns out that
χ(M) = 2 - gwhere g is the genus in this case (i. e. the number of cross caps).
Given all that, the bottom line is that any compact, connected 2-manifold M is determined uniquely up to homeomorphism by just two pieces of information: the value of χ(M) and whether or not M is orientable. This is the sort of result we would like to have for manifolds of any dimension.
We obviously lack such a classification for 3-manifolds, at present, because it should quickly settle the Poincaré conjecture. All we'd have to do is to take some representative manifold in each class, compute its homology or homotopy, and ask whether that matches the homology/homotopy of S^{3}. (Of course, there will undoubtedly be an infinite number of homeomorphism classes, just as in the 2-dimensional case. So the computation might be easier said than done if the classifying informatation isn't as simple as in the 2-d case.)
The second hopeful sign that classification is possible is that it has been done for 4-manifolds! Michael Freedman's 1981 proof of the Poincaré conjecture for S^{4} was just part of his accomplishement. He in fact provided a classification for 4-manifolds. Just as with 2-manifolds, only two pieces of information are required. The information required is "elementary", but still too involved to explain here. We'll simply note that Freedman's work was mainly algebraic, and did not particularly involve analysis or differential equations, such as would soon appear in subsequent work of Simon Donaldson.
Although we don't need to take into account the metric structure of 2-manifolds for purposes of classification, let's consider it anyhow. It turns out to be true that all 2-manifolds are differentiable, and hence admit a metric structure. Naturally enough, when one is dealing with the metric structure of a manifold, one is said to be working with it "geometrically". That is, the geometry of a manifold deals with the manifold as a more rigid object than a purely topological one. It turns out that a topological 2-manifold can be constructed out of pieces which have one of only three types of geometric structure. Specifically, these pieces can be assumed to have a constant value of curvature. The three types are:
Using the classification of 2-manifolds we already have we note the following:
What Thurston wondered is whether the same thing might happen for 3-manifolds. In other words, could every 3-manifold be homeomorphic to one made up of parts coming exclusively from one of a small number of geometric types? Thurston showed in 1983 that there were just eight possible geometries in three dimensions (as opposed to the three types in two dimensions). The list included Euclidean 3-space, the 3-sphere, and hyperbolic 3-space, plus five other types.
Unfortunately, things didn't go so smoothly from that point on. It turns out that not all 3-manifolds have a geometric structure, although if a manifold does have such a structure, it has to be entirely one of the eight types. The next idea is to break a manifold into multiple parts and try to identify each part as having the same geometric type. That almost worked. In fact, it works for seven of the eight possible types. But it is still unknown whether it can be done when the geometric type is hyperbolic 3-space.
And that is where the geometrization conjecture rests today, as far as 3-manifolds are concerned. What about higher dimensions? At this point the outlook isn't promising. There isn't even a list of possible basic geometries in four or more dimensions. What may come of the geometrization conjecture, or the classification problem in general, is still a very open question.
It's interesting to note that the Poincaré conjecture turned out to be easy in two dimensions, and hard but doable in four or more dimensions, although (so far) uncrackable in three dimensions. And yet the geometrization idea is very easy in two dimensions, still plausible in three dimensions, but offers little hope (at present) in dimensions four or more.
There's obviously quite a bit of strangeness in higher dimensional space, and even in 3-space. Even without taking account of strangeness regarding differentiability we haven't even mentioned yet.
The first sign that the situation wasn't going to remain simple in higher dimensions was when John Milnor found, in 1956, that the 7-sphere S^{7} could have any one of precisely 28 nonequivalent differentiable structures. There was the standard structure inherited from the natural embedding of S^{7} in R^{8} (as the locus of points a unit distance from the origin). But there were 27 other "exotic" structures as well.
Soon thereafter it was found that other spheres of dimension higher than seven also have more than one possible differentiable structure. Perhaps even more surprising, there were manifolds of dimensions greater than four which had no differentiable structure at all, unlike manifolds of dimension three or less. In fact, means were found to distinguish manifolds that had no smooth structure fairly easily from those that did, as long as the dimension was five or more.
All this left the question of what happens with 4-manifolds conspicuously unresolved for awhile: Manifolds of lower dimension always had a unique smooth structure, while manifolds of higher dimensions could at least be classified straightforwardly as to smoothness. But soon, the existence of a nondifferentiable 4-manifold was also proven, as a result of Freedman's work that provided a classification of 4-manifolds and proved the 4-dimensional Poincaré conjecture.
OK, so not all 4-manifolds needed to have a smooth structure. What about uniqueness of the structure if it happens to exist? The answer was that uniqueness wasn't assured. In fact, it turned out that even the most "simple" 4-manifold of all, R^{4}, could have a nonstandard differentiable structure. This surprising result was proven in 1982, the year after Freedman's discoveries, by Simon Donaldson. His methods were interesting and eventually led to even more interesting and powerful results. What was notable about Donaldson's results was the fact they were based on ideas from physics. Specifically, they came out of a study of the Yang-Mills equations of nonabelian gauge theory and a special type of solution called an "instanton".
Although it was surprising that the differentiable structure of R^{4} wasn't unique, it was quite astonishing that there could be infinitely many inequivalent differentiable structures -- uncountably many, in fact. This was discovered a little later by Clifford Taubes. It is fitting that ideas from physics should play a part in the differentiable structure of R^{4}, since that is the space in which physics as we know it (i. e., spacetime) takes place. However, it is rather disconcerting to find that not only is there no unique concept of differentiation in that space, but that there aren't even just finitely many possibilities. This is the case even though the differentiable structure exists and is unique on R^{n} for any n ≠ 4.
What makes 4 so special? No one knows for sure. In some ways, it seems to be an "accident". One clue lies in the way Donaldson built on Freedman's results. In his work it happens to be significant that the group of rotations of R^{4} has a property not shared by rotation groups of R^{n} for any other n -- namely, the group is not "simple", even though all other rotations groups are. In other words, the rotation group of R^{4} is something like a composite number, which is not prime, because it can be expressed as a product of nontrivial factors. Specifically, it is a product of two copies of the rotation group of R^{3}. This, in turn, comes about from the existence of a curious noncommutative 4-dimensional algebra known as the algebra of "quaternions".
Is it possible that the "normal" differentiable structure of R^{4} is not in fact the correct one to use in physics? This doesn't seem likely, since the nonstandard structures are all rather "unnatural" and contrived. On the other hand, quantum mechanics suggests that space at very small distances must have very "unnatural" properties and may not even be smooth at all. Is there some message for us in the bizarre situation with differentiable structures in R^{4}? Again, no one knows.
Let's begin by restating, with a bit more terminology, some things we've already seen. We have considered manifolds from two different points of view. The first point of view is topological, where manifolds are simply taken to be topological spaces with only minimal additional structure. They are defined by the condition that neighborhoods of any point are topologically equivalent -- i. e. "homeomorphic" -- to neighborhoods in R^{n} for some n. Mathematicians use the term "category" for a collection of sets that have some structure and some notion of equivalence that preserves the structure. So the first point of view is concerned with the category of topological manifolds. This is the kind of object that is implied when one refers informally to topology as "rubber sheet geometry", because concepts of rigid shape and distance do not apply.
In the second point of view, one considers topological manifolds that have a differentiable structure. The notion of equivalence requires that the differentiable structure be preserved. The maps establishing equivalence between differentiable manifolds are called diffeomorphisms, and the category is known as the category of differentiable manifolds, or alternatively, smooth manifolds. (Technically, one can also consider manifolds where only a finite degree of differentiability is assumed, whereas "smooth" always implies differentiability of any degree.)
In this terminology, another way of saying that all topological manifolds of dimension three or less have a unique differentiable structure is to say that the topological and smooth categories are essentially the same. Further, the work of Michael Freedman dealt primarily with the category of topological manifolds. His principal result was to show that closed (i. e. compact and without boundary) simply-connected (i. e. with a trivial fundamental group) topological 4-manifolds can be completely classified. This also settled the 4-dimensional Poincaré conjecture as a special case.
Simon Donaldson's results, on the other hand, dealt with the category of smooth manifolds. Not only are there 4-manifolds which have no smooth structure at all, but even the simplest 4-manifold, R^{n}, could have more than one inequivalent smooth structure. So the categories of topological 4-manifolds and smooth 4-manifolds are quite different.
The categories are different in dimensions n ≥ 5 as well, because there are manifolds in those dimensions which either have no smooth structure or multiple smooth structures. However, the situation there is still not so extreme as in 4 dimensions. In proving the Poincaré conjecture for n ≥ 5 Smale developed some powerful techniques, called "surgery" (because it involved constructing new manifolds by stitching together simpler ones) and "cobordism". These techniques work well only when the dimension is 5 or more, because they require a certain freedom to move around without things getting in their own way. Nevertheless, they do make the study of higher dimensional manifolds much easier. In particular, it is possible to be fairly precise about the relationship between topological and smooth manifolds in higher dimensions. Although the categories are not exactly the same, one can determine what sort of smooth structures a topological manifold of dimension greater than 4 can have based on the algebraic topology (e. g. the homology and homotopy groups). The study of the relationships between the possible smooth structures and the topological structure of higher dimensional manifolds is called "smoothing theory". It turns out that although a higher dimensional compact manifold can have multiple smooth structures, it can have only finitely many that are distinct.
All of this simply fails in dimension 4, with the result that even R^{n} can have uncountably many different smooth structures. There is something very special and peculiar about four dimensions.
The methods that Donaldson used were very powerful as well, but quite different from anything used by Smale, Freedman, or other topologists before. These methods are based on a simple but very important idea, which is that partial differential equations -- such as mathematicians had been studying for over 150 years -- could be interpreted as describing certain kinds of geometrical structures associated with differentiable manifolds. These structures can be described generally using the mathematical language of fiber bundles. Although that is a relatively new and highly abstract language, once specialists have grasped it, they have available a means to gain intuitive understanding of the analogies and interplay between partial differential equations and geometrical objects.
The specific case that Donaldson considered was the Yang-Mills type of equations which occur in the theory of nonabelian gauge fields of elementary particle physics. Since this is a theory of physical spacetime, which has four dimensions, it isn't too surprising that it has implications for other 4-manifolds as well. And fortunately so, because as we've seen, the older techniques of algebraic topology work well enough in all dimensions except four, where they seem to be inadequate. (True, they aren't so great in three dimensions either, given the lack of a solution to the Poincaré conjecture.)
Classical algebraic topology provides certain invariants associated with a topological space, most notably the homology and homotopy groups. These obects are "invariant" in the sense that they are the same for all topological spaces that are equivalent (i. e. homeomorphic topological manifolds and diffeomorphic smooth manifolds). But they have a shortcoming, in that they aren't fine-grained enough to distinguish manifolds which are not diffeomorphic even if they are homeomorphic. (In extreme cases, there can even be non-homeomorphic manifolds with the same homology or homotopy. The Poincaré conjecture says this can't happen if one of the manifolds is a sphere. But even this is still an open question for 3-spheres. To keep straight any possible confusion, just remember that "equivalent" manifolds have the same invariants, but having the same invariants doesn't absolutely guarantee equivalence.)
What Donaldson was able to do was to come up with entirely new sorts of invariants that are more discriminating and have a finer "grain". These invariants were derived, ultimately, from objects associated with the Yang-Mills equations -- specifically, "moduli spaces" defined in terms of certain fiber bundles associated with 4-manifolds. Moduli spaces are topological spaces that are not quite manifolds, because they have singularities (such as kinks and creases), so they are not everywhere differentiable, although (as topological spaces) they have a well-defined finite dimension away from the singularties. Also, they are not usually compact.
The solutions of Yang-Mills equations which are physically interesting are called "instantons". They are akin to "solitons", which are wavelike solutions of partial differerential equations that (unlike normal waves) have finite spatial extent. Instantons have finite extent in the time dimension as well as the space dimensions of 4-D spacetime. These instantons can be understood as what is called a "connection" in a certain fiber bundle over a manifold. The moduli spaces referred to above consist of equivalence classes of instantons with respect to equivalence under gauge transformations. Although instantons were first studied for a specific 4-manifold (spacetime), analogous solutions of differential equations turned out to exist for much more general 4-manifolds. And they helped understand the differential topology of the manifolds.
Donaldson found a way to encapsulate these rather abstract ideas in someting quite concrete -- a polynomial, called (of course) a Donaldson polynomial, that is a differential (but not a topological) invariant of a manifold and can often be explicitly calculated. With this tool, he was able to construct topologically equivalent 4-manifolds which had distinct polynomial invariants, so that the manifolds had to be distinct as differentiable manifolds, having inequivalent differentiable structures. This sounds like a great advance, but unfortunately it is very difficult to do computations involving these invariants.
In spite of the seeming innocence of the definition of a complex number, some quite surprising things happen when one generalizes many mathematical concepts from using real numbers to using complex numbers instead. The simplest example involves polynomial equations in one unknown, that is, having the form
x^{n} + c_{n-1}x^{n-1} + ... + c_{1}x + c_{0} = 0where the numbers c_{i} are constants. When one is allowed to consider complex numbers as solutions of such equations, then all conceivable polynomial equations in one variable have n solutions (n is called the degree of the equation), when certain solutions are counted multiple times under appropriate circumstances. This falls out of merely assuming that one particular second degree equation, x^{2} + 1 = 0, has solutions.
Equally surprising things happen in calculus, if you work with functions f(x) which are allowed to take arguments and have values in the complex numbers -- that is, functions whose domain and range lie in C instead of R. For instance, if one assumes merely that such a function has even one derivative (suitably defined), then the function automatically has derivatives of all orders. (Such a function is called "holomorphic" or "analytic".) This property is definitely not true of arbitrary functions of a real variable.
Bernhard Riemann, who made such fundamental contributions to geometry, also molded the theory of complex functions into its present form. These two theories are deeply related, in fact, through the notion of a "Riemann surface" -- a construct which greatly simplified the function theory. The definition of a Riemann surface was essentially a special case of what is now the definition of a manifold -- that is, a set of points which is "locally" Euclidean -- in the 2-dimensional case. (A Riemann surface has 2 dimensions when considered as a real manifold, i. e. a "surface". But it is only 1-dimensional as a complex manifold, because it is locally like C, the "complex plane".)
With the concept of complex numbers in mind, if you look at the defintion of a manifold, the obvious thing to do is simply to replace R with C in the definition. Specifically, M is an n-dimensional complex manifold if it "looks like" C^{n} -- the space of n-tuples of complex numbers -- locally at every point. That is, there is a set of "neighborhoods" that completely cover M (i. e. every point of M belongs to at least one neighborhood) such that each neighborhood maps to a neighborhood of C^{n}. And in addition, neighborhoods that overlap are related by functions that are analytic. (This is exactly analogous to the case of a general manifold where the overlap functions were required to be continuous, and to a differentialble (real) manifold where the overlap functions were required to be (infinitely) differentiable.) These overlap functions are called, collectively, the "complex structure" of the manifold.
Alternatively, one may think of an n-dimensional complex manifold as consisting of chunks that are subsets of C^{n} and are "glued" together with analytic (i. e. complex differentiable) functions.
In view of the other surprising results which crop up when R is replaced with C in a theory, we should not be surprised that very interesting things turn up in the theory of complex manifolds. However, it is a very advanced and difficult theory that has been well developed only in the last few decades, so a number of technical concepts have to be introduced even to describe it. There are big payoffs, however. Among them are deep results (and open questions) in mathematical theories such as algebraic geometry as well as concepts that are central to advanced theories of fundamental physics -- superstring theory, in particular. The primary example of the latter is the notion of a Calabi-Yau manifold.
So let's take a quick tour of complex manifold theory. The theory has quite a lot of structure to it. That is, there are many kinds of objects one can define, and consequently a lot that can be said (or conjectured) about how these objects are related. There are also many different types of complex manifolds that are distinguished by supporting the existence of certain interesting objects.
One of the most important types of complex manifold is the "Kähler" manifold, named after the German mathematician E. Kähler, who originated the concept in the 1930s. These manifolds are distinguished among all complex manifolds by possessing a "Kähler metric". Recall that for (real) differentialble manifolds a metric (i. e., a notion of distance), can be defined using any inner product of vectors in the tangent space of the manifold. Metrics in complex manifolds are obtained in the same way.
In linear algebra, inner products of an n-dimensional space can be defined using n×n matrices. These same matrices can be used to define other sorts of objects called "2-forms", because they are a certain type of function of two arguments. (An inner product, itself, is an example of a 2-form.) As a result, corresponding to a form which yields a metric there is a certain other 2-form called the "fundamental form" of the manifold. This fundamental form encapsulates differential properties of the manifold. When the fundamental form of a complex manifold is of a particularly simple type, the metric it is associated with, as well as the manifold itself, is said to be Kählerian. Many complex manifolds that arise naturally are of this sort. Examples include C^{n} itself and all of its submanifolds, any 1-dimensional complex manifold (such as a Riemann surface), and common objects in algebraic geometry (namely "projective algebraic varieties without singluar points").
Kähler manifolds, then, are important for two reasons: there are many interesting examples, and the theory of them is simplified because of the conditions placed on the fundamental form.
A key property of a manifold that one wants to study is how much it is curved at each point. There are various ways to formalize the notion of curvature. This can also be done using another type of 2-form defined using the differentiable structure of the manifold. The form in question is called the "Ricci curvature tensor".
For particular purposes, forms on a manifold may be regarded as equivalent. For a certain type of equivalence, the resulting equivalence classes are know as "Chern classes", named after Shiing-shen Chern, who developed the theory of "characteristic classes" of manifolds in the 1940s. Chern's theory is of a sort known as "cohomology theory". The mathematician E. Calabi studied such classes in the 1950s and formulated what became known as the "Calabi conjecture", which states that there exists one and only one Kähler form on a compact complex manifold that is in the same Chern class as the Ricci tensor. Calabi himself proved the uniqueness part of his conjecture.
In 1977 Shing-Tung Yau proved the existence part of the Calabi conjecture. Manifolds for which the Ricci curvature is zero are said to be Ricci-flat. Such manifolds have relatively simple geometric properties. Given the proof of the Calabi conjecture, an equivalent condition for Ricci-flatness is that the Chern class be zero. Manifolds that have this property are called "Calabi-Yau manifolds" (or "Calabi-Yau spaces"). These are the manifolds which are so important in superstring theory.
The reason that Calabi-Yau manifolds are so important in superstring theory is that they represent the "compact" part of the 10-dimensional space in which superstrings (are hypothesized to) exist and vibrate. That is, this space is of the form M^{4}×K, where M^{4} is R^{4} with the "Minkowski metric" given by special relativity and K is a 6-dimensional compact Calabi-Yau manifold. As we discuss elsewhere in explaining the theory of superstrings, the topological properties of K determines a lot of the physics of the theory. For instance, the "Euler number" of K determines the number of "generations" of elementary particles that the theory predicts. The problem in superstring theory is to identify the actual manifold K, which is a stumbling block, as Yau himself has determined that there are tens of thousands of possible candidates.
What the physicists found was that they obtained exactly the same physics by using either member of a mirror-symmetric pair of manifolds in a string theory, in spite of the fact that the manifolds were topologically distinct. Mirror symmetry thus became another type of "duality" which had begun to appear in many forms during the early 1990s. Just like other dualities, it was very useful, because computations which were difficult or impossible in one case could (perhaps) be performed easily in the dual theory. But since the physics was the same, it didn't matter which one of the dual theories was used for calculations.
This really got the attention of mathematicians when it turned out that the very same trick could be used to investigate difficult mathematical questions.
Copyright © 2002 by Charles Daney, All Rights Reserved