Topology

Summary

Topology is the mathematical study of the relationships and properties of objects in any number of dimensions, independent of size. Topology plays a crucial role in many branches of mathematics since arguments that can be formulated in topological terms are more general than those that depend on geometrical properties. Topology has also provided models for cosmologists seeking to describe the universe's structure. Predictions about the outcomes of interactions between objects frequently use topological language rather than the exact calculations of classical mechanics. Even biology has found it helpful to use topology to describe the relationships between individual living things and species.

Definition and Basic Principles

Topology is a generalization of geometry, which historically served as a description of the structure of space. In topology, issues of size and measurement are not considered important, and questions about what kinds of overlap two sets of points must have depend on more general properties of the sets. Topology emerged out of the traditional fields of calculus, geometry, and algebra, and there are subdivisions of the field closely related to each. The language of topology looks like that of sets, but there is also room for carrying over ideas from the study of geometrical objects in familiar spaces. The terminology is also based on imagery familiar from geometry.

The applications of topology cut across branches of science and mathematics that can use the language of relationships to describe the objects they study. Topology is abstract enough that it can sometimes seem as though it is not about any particular object, making it even more widely applicable. One of the typical characterizations of topology is “rubber sheet geometry,” where one studies the properties of objects that are preserved under the stretching and squeezing that rubber can undergo. Mathematicians have come up with more abstract ways of describing these properties, which draw on abstract algebra.

Background and History

The word “topology” is a relatively new coinage, but other terms, such as “analysis situs,” have a longer history. In the seventeenth century, the mathematician Gottfried Wilhelm Leibniz proposed the idea of a geometry that would depend on the properties of space more generally than the Euclidean geometry universally studied at the time. In the eighteenth century, the Swiss mathematician Leonhard Euler came up with the first concrete example of an application of topological ideas when he solved a problem about whether one could traverse all the bridges in a certain town without repeating any part of one's path.

The subject remained dormant until the late nineteenth century when the French mathematician Henri Poincaré pursued a couple of lines of study that led to different branches of topology. He looked at the relationship between the algebraic properties of an object and its geometrical properties, which gave rise to geometrical topology. He also studied physical processes in which classical mechanics seemed unable to describe the results, giving rise to the field of nonlinear dynamics, one of the branches that contributed to the rise of chaos theory a hundred years later.

In the twentieth century, the rise of set theory provided a language for talking about collections of points in a more fundamental way than had been possible previously, and an entire school of Polish mathematicians produced fundamental results in what is called point-set topology. As abstract algebra became more sophisticated, its language became crucial for discussing topological spaces, and the discipline of algebraic topology was created to use algebraic techniques to understand topology. Later in the century, the rise of computers gave empirical evidence for how the repetition of functions produced certain kinds of patterns surprisingly independent of the set from which one started and sometimes even the kind of function. Finally, Albert Einstein's general theory of relativity suggested that traditional geometry could not describe the behavior of light, which made physicists appreciate the need to generalize geometry. The rise of various models for the structure of the universe (especially string theory) has put topology at the center of the concern of theoretical astrophysics.

How It Works

Point-Set Topology. The study of sets starts from the notion of a set as a collection of elements. In the simplest cases, one can think of this collection as a list. Still, the mathematically interesting cases usually require an infinite collection, for which the notion of a list is unhelpful. Instead, one uses the idea of a property to define the members of a set as part of a larger collection. The ideas of intersection and union as ways of combining sets help define the notion of a closed and open set. A closed set includes its boundary, while an open set does not include any of its boundary points. For example, a circle with its interior is a closed set, while the interior is an open one. Point-set topology concerns the results of taking intersections and unions of closed and open sets, frequently allowing an infinite number of sets in the operation. Such interactions (when combined with the notion of complementation, taking everything that is not in a given set) give rise to sequences of possible outcomes from repeated processes. The foundations of point-set topology lie in the axioms used for the theory of sets so that different results can come from a selection of different axioms.

The interest of point-set topology outside the area of set theory comes from considering what happens when functions are applied to various kinds of sets of points. In particular, one is frequently interested in continuous functions, which do not allow for jumps in the values of the function. Traditionally, a physical model for a function suggested that it ought to be continuous. Still, the laws of quantum mechanics have made the assumption of continuity a little less useful and have produced the notion of a distribution as a more general function than the continuous case.

The set of points on which functions act in topology is often a manifold. This is a piece of space that is locally like Euclidean space (the familiar version). In other words, one can find some kind of function that connects the manifold's behavior with that of ordinary space. Knowing how the function behaves in Euclidean space and how Euclidean space is related to the manifold provides a description of how the function applies to the manifold. This process of going from the manifold to Euclidean space, operating on Euclidean space, and returning to the manifold is characteristic of mathematics in the twentieth century and beyond.

Algebraic Topology. Algebra became a much more abstract discipline as the nineteenth century progressed. By the arrival of the twentieth century, it was far from its roots in solving polynomial equations using standard arithmetic operations. First, it had been discovered that there were such equations those operations could not solve. Then, there was the challenge of identifying which equations could be solved using old-fashioned means. Finally, there was the question of how to get useful information about the solutions of an equation even when they could not be identified explicitly. The result was the creation of theories of algebraic objects, such as groups, rings, and fields. These had laws corresponding to those for various sets of real numbers, in some cases imitating those of the whole numbers and, in other cases, those of larger sets. In addition, the use of the language of algebra extended beyond sets of numbers to functions and other kinds of mathematical objects.

With all these algebraic structures under consideration, those studying geometrical objects could try to come up with algebraic characteristics for geometry. Just as René Descartes in the seventeenth century had explained how to use coordinate axes to turn questions about geometry into questions that could be handled by the algebra of the time, so the higher-powered algebra of the twentieth century could be brought to bear on identifying and characterizing geometric objects. One of the issues that arose once algebraic characteristics could be attached to geometric objects was whether one could tell that two geometric objects were homeomorphic from enough algebraic information. Different kinds of algebraic objects gave rise to different levels of classification, and the algebraic information offered paths for exploring geometry.

Applications and Products

Graph Theory. One of the areas created by Euler and not followed up on for a century or more was the study of graphs. These are not the objects of algebra that represent the relationship between two variables on a pair of coordinate axes but instead offer a more general way of describing relationships between objects. One uses vertices (points) to stand for the objects themselves and edges (line segments) between the vertices to represent their having a certain kind of property. For example, one can use vertices to stand for numbers and put edges between them if they are within a certain range of one another (less than one unit apart, for example). One can also make the graph directed by putting arrows on the edges if the relationship between the objects represented by the vertices is not symmetric. For example, if the points represent numbers, one could use an edge with an arrow pointing from one point to another if the number corresponding to the first point is a factor of the number corresponding to the second.

Many issues are connected with a graph when it has been used to model the relationship between objects. For example, one can ask what the shortest path is between two points in the graph, where distance is defined not by the usual length from geometry but by the number of edges required to get between the points. Some points cannot be connected by any path, especially when the edges are directed. Results going back to Euler help resolve some of these questions, but others depend on algorithms that take a long time to run.

Applications of graph theory in other sciences are countless. In the middle of the nineteenth century, graphs were already being used in chemistry to represent the processes of transforming one substance into another. In modern times, graphs have been used to represent the proximity of different species to one another in the natural world. Because molecular biology can be used to describe a species, the genetic code is used to characterize the species in mathematical terms. Then, the distance between two codes can be measured by the number of changes required to get from one code to another. For example, the word “lick” can be changed to the word “luck” by altering only one letter, but “lick” requires two changes to get to “risk.” This has become a battleground for those concerned about the theory of evolution because there are questions about whether certain paths would be possible in whatever time is given for the existence of the Earth. It should also be clear that topology does not answer every question in the disciplines to which it can be applied. The question of how likely an alteration in the genetic code is to occur is not part of the domain of graph theory. In general, graph theory has offered a language of representing information and asking questions about it that is independent of geometry. Topology has been invaluable as a means of describing situations, but biologists have not thereby been shown to be dispensable.

Chaos Theory and Fractals. In the nineteenth century, much science fiction was written about the notion of molecules being small worlds with their own inhabitants. The idea was also extended to the possibility the galaxies of which the stars and planets are a part are only particles in a yet larger system. Although these speculations have not been borne out in investigating the universe, the notion that there could be similarity across a difference of scale has been one of the most fruitful ideas in mathematics known as chaos theory. It arose from the empirical evidence that one of the assumptions of classical mathematics was not well-founded: If one takes two numbers that are reasonably close together and applies a continuous function to them, they will stay close together. That is almost the definition of continuity, but problems arose when the application of the function was repeated not just once or twice but thousands or millions of times. What became evident was that points that started as close together as one could imagine would end up unimaginably far apart. What was even more noticeable was that after a sufficiently large number of repetitions of applying a function, the patterns that resulted looked almost as though they had been designed by an artist.

The search for an explanation for these results led to the notion of a fractal. The word is short for “fractional dimension” and refers to the ability of some objects to have a dimension somewhere between two whole numbers. For example, a curve is usually considered a one-dimensional object since it results from applying a function to a straight line. There are, however, curves that can be constructed by an infinite process that are called space-filling because they seem to be able to fill a two-dimensional region (or at least a large part of it). Coming up with a notion of dimension for such objects requires a little more mathematics than the whole numbers. This approach to fractional dimension is useful in studying, for example, the coastline of a country, where magnification frequently leads to a similar shape to the one visible to the naked eye or on a map. There is a fair amount of visual beauty in fractals, which accounts for their prevalence as a subject for books designed to appeal to the general reader. Still, their importance in studying physical phenomena does not depend on their aesthetic characteristic.

The resulting pattern may be stable if a function is applied sufficiently often. The most striking conclusion of chaos theory is the observation that the stability of a configuration is independent of the original configuration from which it arose. There are models for the evolution of a system (such as the game of Life, devised by the British American mathematician John Horton Conway) that offer insights into population changes and dynamics, and the same is true of the study of cellular automata, which are grids with patterns determined by simple rules. Simple rules, if allowed to govern the evolution of a system through enough generations, produce wildly improbable results. Topology offers the best explanation precisely because it works with the smallest initial conditions.

Careers and Course Work

The student interested in pursuing a career in topology would benefit from exposure to as many branches of mathematics as possible. Even though topology is, in some ways, a generalization of geometry, the geometrical foundation for topology does not become lost. Upper-level algebra courses provide part of the language for doing topology. Doctoral work in mathematics or computer science is the best path to a career strictly in topology. However, students with an eye on applications of topology in areas like physics, chemistry, biology, or even business will need to learn about those fields rather than expecting topology to supply all the answers.

Most of those pursuing research in topology are involved in teaching mathematics at the university level. Some topologists could be employed by organizations, such as the National Security Agency, which has a strong interest in cryptanalysis. The same techniques that apply to questions about the genetic code apply to code breaking in general. Although physicists and cosmologists use a good deal of topology, it is generally in an academic setting.

Social Context and Future Prospects

Topology has always suffered from hostility on the part of those who are inclined to see classical mathematics, such as geometry, as sufficient for understanding the universe. It has also been a victim of some of the battles fought within other areas of mathematics. Set theory was involved with challenges due to paradoxes coming out of certain axiomatic foundations. Abstract algebra was viewed as a discipline without content. Catastrophe theory came into existence as a way of applying topology to many disciplines but never lived up to the claims made by its creators. Nevertheless, topology has earned a place within the curriculum at all universities offering a modern picture of mathematics. It was one of the most exciting disciplines in mathematics in the United States for much of the second half of the twentieth century.

Because of its alliance with so many other disciplines, topology should be able to maintain its central role in mathematical education at the university level. Books on topology in chemistry, physics, and other sciences bear witness to its usefulness as a language and a set of ideas. Graduate students no longer need to fear being scorned for studying a discipline like topology. It has survived the occasional overambitious claim and found a lasting role in mathematics and the sciences. Topology continued its utility in interdisciplinary studies as the twenty-first century progressed. Biology, neuroscience, robotics, and computer network technology are just a few examples of the fields using topology. 

Bibliography

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Gilmore, Robert, and Marc Lefranc. The Topology of Chaos. New York: John Wiley & Sons, 2002.

Kim, Minkyung, et al. "Recent Advances in 2D, 3D and Higher-order Topological Photonics." Light: Science & Applications, vol. 9, no. 1, 2020, pp. 1-30, doi.org/10.1038/s41377-020-0331-y. Accessed 3 Jun. 2024.

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