Topologists are mathematicians who study qualitative questions about geometrical structures. We do not ask: how big is it? but rather: does it have any holes in it? is it all connected together, or can it be separated into parts? A commonly cited example is the London Underground map.
This will not reliably tell you how far it is from Kings Cross to Picadilly, or even the compass direction from one to the other; but it will tell you how the lines connect up between them. In other words, it gives topological rather than geometric information. Again, consider a doughnut and a teacup, both made of BluTack. We can take one of these and transform it into the other by stretching and squeezing, without tearing the BluTack or sticking together bits which were previously separate. It follows that there is no topological difference between the two objects. Consider the problem of building a fusion reactor which confines a plasma by a magnetic field.
Imagine a closed surface surrounding the plasma. At each point on the surface, the component of the magnetic field parallel to the surface must be nonzero, or the plasma will leak out. We are thus led to the following question: given a surface S, is it possible for there to be a field of vectors tangent at each point of S which is nowhere zero? It turns out that this depends solely on the topological nature of S. If S is the surface of a sphere, it is not possible. Magnetic confinement of a ball of plasma just does not work. The only type of surface for which this approach is possible is the inner tube shape, which is of course the solution universally used for such reactors. (I do not claim that engineers needed topologists to point this out; on the contrary, this is a nice example precisely because many people can see for themselves that the claim is true.) Another amusing consequence of the same argument is that at any given time, some point on the Earth's surface is windless.
While these examples are interesting, they are somewhat limited. Topologically speaking there are few possibilities for solid objects; essentially the only variable is the number of holes. Interesting things happen, however, when we work with abstract objects in more than three dimensions. This is not the dry academic game that it might appear at first sight. Consider the ubiquitous use of graphs in science. One often sees graphs of (say) pressure against temperature. Neither variable is naturally a geometric quantity. Nonetheless, they can be represented by distances on a sheet of paper. Moreover, geometric concepts such as slopes of lines and areas of regions can usefully be interpreted in terms of thermodynamics. In principle, none of these geometric ideas are neccessary; all the arguments can be carried through algebraically without any diagrams. Nonetheless, most of us find diagrams helpful and illuminating, because we have a better intuitive feel for geometry than for algebra. Now suppose that instead of two variables (pressure and temperature) we are faced with a problem involving five or six. We might like to draw a six dimensional graph, but we do not have enough space. We can try to imagine what six dimensional geometry might be like, but it is difficult to have confidence in our intuition about such a theoretical construct so far removed from experience. On the other hand, we can try to do the problem by pure algebra, but in doing so we forgo the visual insight which is one of the most powerful faculties of the human mind. In fact, mathematicians have developed a rather strange yet highly fruitful approach to such problems. The questions addressed are officially formulated in terms of logic and algebra rather than geometry, and solutions are required to be expressed in such terms. Where geometrical concepts are used, they must be defined in terms of algebraic ones. All use of such concepts is to be justified purely in terms of such definitions, rather than in terms of geometric intuition. The visual imagination guides the course of the argument, and suggests what one should try to prove, but the argument must stand in its own right. There is an interesting, if limited, analogy with physical science, with rigorous proof playing the role of experiment and geometrical speculation (amongst other things) that of the more mysterious processes by which new theories emerge. Let us consider an example with more than three dimensions. Imagine an industrial robot.
The position and orientation of the hand is determined by deflection of the various joints of the robot arm. Suppose that there are n joints, each of which bends in only one plane (this assumption is not essential). We then have to give n numbers in order to specify the state of the joints. In the case n=2, we could draw a diagram on a sheet of paper with one axis for each joint, so that a point on the diagram would correspond to a possible state of the joints. If n>3, then such a diagram would have to be n-dimensional, so we could not draw it. Nevertheless, we can still reason about a theoretical n-dimensional space (call it X) in which a single point represents a state of the arm. Now consider the position and orientation of the hand. In the usual way, we need three coordinates to specify the position. We next need to specify the direction in which the central axis of the hand points. This needs two numbers: the angle of elevation above the horizontal and the horizontal compass bearing. Note that these two numbers are not entirely independent or well defined. If the elevation is vertical then the compass bearing is meaningless. Also, the physical meaning of the compass bearing is unchanged if we add 360 degrees. We can suppress this by insisting that all angles must be taken between 0 and 360, but then we have a different problem in that the numerical value of the angle could change suddenly (as the arm moves from 1 degree to 359 degrees) without any genuine discontinuity in the behaviour of the robot. There is a topological reason for these problems. If the space of possible orientations of the hand could be simply parametrised by two variables, then it would be topologically a plane. However, it is not a plane. It can instead be thought of as the surface of a sphere, which one imagines centered at a point on the axis of the hand. If the hand held a torch, it would mark a point on the sphere which would determine the orientation. There is one further variable to consider, viz. the twist of the hand about its central axis. This gives a third angle, which again interacts in a complicated way with the other two. All told, we need six parameters to describe the position and orientation of the hand. We can thus consider a six dimensional space Y, each point in which corresponds to a possible state of the hand. We can analyse the structure of Y using the methods of algebraic topology, and learn a number of interesting and nontrivial things about it. Each point in our space X corresponds to a state of the joints of the robot. The state of the joints determines the position and orientation of the hand. Thus, to each point x in the space X we associate a point of Y, which we shall call f(x). However, if we want to achieve a specified position and orientation of the hand, there will in general be many ways to do this. In other words, many different points of X will be associated to the same point of Y. It would be convenient in the design and use of such robots to have a so-called ``inverse mapping''. For each position and orientation of the hand (corresponding to a point y in Y) one would like to choose a state of the joints (corresponding to a point g(y) of X) which puts the hand in that situation (so that f(g(y))=y). Moreover, one would like g(y) to vary continuously with y, so that similar positions of the hand are achieved by similar states of the joints. It turns out, however, that this is topologically impossible. It is thus necessary to devise a more complex control strategy, in which the same hand position may be achieved in different ways at different times. This is typical of the kind of information which can be supplied by topological theory.
Neil Strickland