Professor Jeremy Brazas

Topology is one of the major areas of modern mathematics. Despite its foundational importance in the math world, if you ask a random person on the street what topology is, they may first think you said "topography" and then after realizing that you're describing an area of mathematics will admit to never hearing of it before. The reason for this is probably because topology is largely an abstract subject that is usually only studied by graduate students or undergraduate students taking senior level mathematics or physics courses.

From my perspective, Topology is a beautiful subject that is used by nearly all other areas of mathematics. If you are unfamiliar with topology but are curious about it, here are a few simple ways to understand what it is all about.

Topology is the abstract study of space and shape

A space is a set of points and some extra information that tells you how the points are related to each other. For instance, a metric space is a set X of points where a positive real number distance d(x,y) is assigned to each pair of points x and y in X and the distance assignment has to follow some rules. By adding a notion of "closeness" between points in X, you've given your set some kind of "shape." It is no longer just a formless collection of points. Adding a topology to a set is doing the same sort of thing - a topology gives a formless set some kind of shape. Topology is a theoretical field of mathematics, which is the study of space and shape and how to tell when two spaces are equivalent or not. 

In classical geometry, you use numerical values like distance, angle, area, volume, etc. as characteristics that describe some property of a figure. Objects are considered to be rigid. Two objects are only geometrically equivalent if there is a rigid transformation (like a rotation, reflection, or translation) taking one to the other. A large circle is different than a small circle simply because the radii are different. A square and a circle are different because a circle is smooth and contains no line segments. In topology, you loosen the rigidity that is inherent to classical geometry. You become less specific, imagining that your objects are made out of some flexible material, and you begin to look at general characteristics of the object - a small circle and large circle are each just one-dimensional closed curves so they topologically equivalent. Similarly, a square (without interior) and a circle are topologically the same because what really matters is that you have a one-dimensional curve with a single hole in it. 

To a topologist, a donut (solid torus) and a coffee cup are equivalent because each is a 3-dimensional solid with a single hole (the donut hole and the hole in the mug handle). If you see a topologist struggling at breakfast, this may explain the issue. Kindly remind them their breakfast should be understood geometrically and not topologically.

Why is a circle different from a solid disk? That is, why are they non-equivalent topological spaces? Easy, one has a hole in it and the other does not. This may seem like a trivial observation but to actually have mathematics which rigorously detects the difference between the two is not so simple. It's important to remember that intuition is not the same thing as a mathematical argument and that sometimes our intuition is wrong. Topology is the area of mathematics which helps us make this kind of reasoning into logically rigorous mathematics.

What does a topologist actually do? Think of it this way. When biologists study organisms, they keep track of the observed differences between the organisms. The general differences can be used to place organisms into broad classification groups such as the taxonomic rank: species, genus, family, order, class, phylum, kingdom. If the DNA sequence of an organism could be fully decoded, we could use it to uniquely characterize that individual. In a similar way, topologists want to know how to classify spaces. We want to understand what kinds of spaces and shapes exist (even abstractly) and how to distinguish between them. By doing this, we're unraveling the limits of the human ability to apply logic to understand our universe. Topologists are like shape-biologists; we answer questions about topological spaces so that we may formally distinguish between them and characterize them at various levels of classification - from very general characteristics to decoding the "DNA of the space" (called the homeomorphism type of the space). Occasionally, we find beautiful results that completely and perfectly characterize a space with a single sentence.

Topology is the study of continuity

The idea of continuity is pervasive in mathematics. A function from one space X to another Y is continuous if small differences of input lead to small differences of output, that is, if you don't tear apart your domain X in any way as you map it into Y. Anywhere that continuity is used, even in applied mathematics, topology is involved in some form or another. It may be hiding in the background (e.g. in the form of convergence), but  some bit of topology is probably being taken advantage of. For example, the proof of the Extreme Value Theorem (every continuous function on a closed interval has a maximum and minimum) in Calculus is often glossed over in undergraduate classes. The reason the EVT and its higher dimensional analogues are true in the first place hinges on the topological concepts of compactness and linearly ordered spaces. 

The objects studied in topology are called topological spaces. Each space can be thought of as its own world, with it's own set of points and data about how close the points are to each other. Some of these worlds may seem quite strange to us. The way in which we study how topological spaces are related is by using the notion of a continuous function. Often, we can think of a continuous function from one space X to another Y as a way of mapping the first into the second in a way that does not violate the closeness rules of the spaces - nearby points in X must be sent to nearby points in Y.

With this in mind, two topological spaces are equivalent if one of the spaces can be continuously deformed (by a bijection) into the other and if the deformation can be inverted continuously as well. If you had a donut made out of playdough you could mold it (without tearing or puncturing the dough) into a coffee cup - I've done it... although the coffee cup looked a little rough. You could also start with the playdough coffee cup and turn it back into the donut. So the two are equivalent. 

Now imagine smashing your playdough donut into a solid ball. That operation is continuous because you didn't have to tear it at all - just smash it into a ball. On the other hand, try to start with a ball and turn it into a donut. To do this, you must puncture it at some point to get the hole in the middle. When you do this, some points that were very close to each other in the ball are now far apart on opposite sides of the hole. This violates continuity. So it is not possible to continuously turn a ball into a donut - hence the two shapes are different as topological spaces.

A topology is a thing

The word "topology" is used to describe the entire field of study, however,  a "topology" is a thing, a mathematical object that we can study.

Start with a set of points X. This could be a finite set or an infinite set. 

We want to think of X as a space - as it's own universe whose points (or elements) are the elements of the set X. But a set is an unstructured thing. It's just an abstract, unordered collection of points with no structure. We need to add more structure to the set to know how close points are to each other within this universe. Different "closeness structures" may describe different universes!

A topology on X is this data of "closeness". It contains the information that tells us how points are related in X. In particular, a topology T on X is a set of subsets of X - these subsets, elements of the topology T, are called open sets. These open subsets in T need to follow some rules: the intersection of two open sets must be open and an arbitrary union of open sets must be open. For logical purposes, we also insist that the empty set and X itself are open. ,

A topological space is an ordered pair (X,T) consisting of X with some topology T on X. We can now think of a topological space as a universe where two points x and y may or may not be close to each other depending on how the topology T tells us they are related.

This set-theoretic definition may seem weird and abstract at first, but after studying topology for a while you will find that it is, in fact, very well-motivated.

There may be many different topologies on a given space X. You could have two topologies on the same set X and the two resulting spaces may be completely different. The question in topology is then: how can we formally decide when spaces are the same or different?

The word "Algebra" is commonly understood as the algebra that you learn in middle school and high school: the manipulation of symbols and equations that follow very specific rules. In "abstract algebra" sometimes called "modern algebra," you focus on these rules and forget that the things you were adding or multiply were numbers. After all, numbers are just symbols that abstract quantities we experience in the real world.  For instance, in group theory, you start with a set of things (not necessarily numbers) and some operation that allows you to take two elements x and y and form a new element z from them. This operation may not have anything to do with the notion of multiplication that we're used to but there is no harm in calling this operation "multiplication" and writing x*y=z since what matters about the operation is the rule it follows and not the name we give it. This operation needs to satisfy some rules that come from high school algebra: associativity (x*y)*z=x*(y*z), there needs to be an element like 1 called the identity for which 1*x=x*1=x, and each x needs to come with an inverse y which satisfies x*y=y*x=1. 

Algebraic Topology is the study of topological spaces using tools from abstract algebra. Moving from geometry to topology already requires us already to loosen things up and imagine that spaces become more flexible. But in topology you can't collapse structures because this would violate bijectivity. Moving from topology to algebraic topology takes this loosening up even further. Now, two spaces become the same if I can continuously deform one into the other in finite time. For instance, a solid disk and a single point are obviously different in topology simply because a disk has more than one point. However, I can imagine shrinking the disk in time through smaller and smaller disks until at the end of a finite time period I've morphed it into the single center point. Functionally, I can undo this procedure as well. So in algebraic topology (specifically homotopy theory), a solid disk is equivalent to a single point!

It may seem like we've taken things too far - everything becomes too fuzzy and vague. But actually, taking this coarse viewpoint allows us to focus on the even more generic structures of spaces. Things that should have been easy in topology but were actually very hard (like telling the difference between two and three-dimensional real space) become easy precisely because we've broadened our viewpoint. To actually formalize this idea and turn it into mathematics, we look for algebraic structures (like groups) that describe certain properties of our spaces that we want to remember. For instance, homotopy and homology groups detect, in some sense, the existence of high-dimensional "holes" and "twists" in our spaces.

 

Though I have interests in many areas of mathematics and my background is in traditional homotopy theory, my expertise is in the algebraic topology of locally complicated topological spaces. Sometimes this is called "wild algebraic topology" or just "wild topology." I usually only use the term “wild” colloquially with other experts in the field because this term can be misleading. Without knowing better, it is easy to think that fundamental groups and higher homotopy groups of spaces with infinite shrinking sequences of holes, twists, and other deformations are unfathomably complicated. However, the opposite is often true. Despite their cardinality, their algebraic structure is often quite rigid, coming with practical ways to study and apply it. Like any progressive field of mathematics, the theory can get very technical on some days but, in many ways, the objects studied are better understood and far more accessible to the mathematical public than many aspects of traditional algebraic topology. 

Part of, but not all, of wild topology could be considered "infinitary algebraic topology" because the binding concept that both appears nearly everywhere and distinguishes it from traditional content is that the invariants involved typically come with natural infinite product operations akin to infinite sums in the real line or complex plane. In contrast with binary, trinary, and other finitary operations studied in many algebraic fields, the term ``infinitary" refers to partially-defined operations with infinitely many inputs that extend some familiar binary operation. In this setting, the interactions between the algebraic and topological structures are richer than in classical algebraic topology. 

In the past two decades, leaps in progress and understanding have started to mold this work into a systematic theory. Much of this progress, including remarkable methods for analyzing and classifying homotopy types and fundamental groups of one-dimensional spaces like the earring space, Sierpinski carpet, and Menger cube, has been led by the innovative work of Katsuya Eda. Many advances have been motivated by techniques from geometric group theory and topological group theory. On the other hand, infinitary algebraic topology provides an increasingly useful structural theory for locally complicated spaces routinely studied in various areas of mathematics. For example, topological spaces associated with (complex and topological) dynamical systems, boundaries of groups, and inverse systems often have nontrivial local geometric features and cannot be adequately studied or classified using traditional techniques in algebraic topology.

My blog Wild Topology is written to provide a readable and intuitive approach to this content. Other useful things to read about if you're interested in this kind of topology: generalized covering space theories, topologized fundamental groups, and shape theory.

Of course, many open problems remain in this challenging and beautiful area of mathematics!

Click Here for my Open Problem List.

This is my own list of open problems in infinitary/wild algebraic topology. Some of these problems are very hard and fairly well-known. Some of them are less well known but still hard and some may be answerable with some careful effort. This list does not speak for “the field” or really anyone but myself. However, if you have an interesting problem that you think should be on it, feel free to send it to me.