Intro to Visualizing Topology: Quotient

In topology we often think about what things look like. But fundamentally, these are mathematical objects and the jump to visualization is not trivial.

We often think in topology of gluing two different points together. This concept will become very useful. It means we are taking two points and considering them identical. We pretend there was only ever one point. We visualize this by only drawing one point.

We can glue together different points by using the “quotient”. Anything which has the same remainder under a quotient will be glued together. This idea will hopefully become clearer with examples.

Take the real line \(\mathbb{R}\). What happens if we take the quotient of the real line with the integers? This is written \(\mathbb{R} / \mathbb{Z}\), where \(\mathbb{Z}\) represents the integers. This means we take every real number, and toss out the integer component. We glue together every point which has the same decimal expansion. So we glue \(3.2\) to \(63.2\) and \(\pi\) to \(0.14159\) and \(\pi +4\), etc. The “remainder” of a real number with the integers is this decimal expansion. When you traverse the real number line, you go left and right by adding different numbers. If you can get from one number to another by adding an integer, then they have a special kind of equivalence; they have the same integer remainder.

How can we represent this object? Well, for each infinitely large “class” of real numbers which are glued together, we only draw one point. We can take a representative from each class and build our object like that. It should be pretty clear to see why the interval \([0,1)\) is representative of this object, since every real number will be glued to exactly one point on this interval. But why can’t we choose \(\pi\) to represent the \(0.14159\) class or \(63.2\) to represent the \(0.2\) class? Well, that’s because some of the structure of \(\mathbb{R}\) is preserved. And when we analyze this structure, we observe something very special happens.

The basic structure of \(\mathbb{R}\) is exactly what I touched on before: you traverse it by adding numbers together. Numbers which are close together should stay close together, regardless of how we quotient them. Say we start at \(1.6\). If we want to get to \(1.7\) we just add \(0.1\) and if we want to get to \(3.7\), we have to add \(2.1\). But when we glue everything together, \(1.7\) and \(3.7\) will be represented by the same point, so the distance had better be the same! And this is what we find when we use \([0,1)\). \(1.6\) gets glued to \(0.6\) and \(1.7\) and \(3.7\) both get glued to \(0.7\), and the distance is \(0.1\) in both cases. Notice how \(2.1\) and \(0.1\) are glued together in this model, because they are separated by an integer.

Now, consider \(8.9\) and \(9.1\). You only need to add \(0.2\) to \(8.9\) to get \(9.1\). But when we look at these two points on \([0,1)\), they appear much further. \(8.9\) gets glued to \(0.9\) and \(9.1\) gets glued to \(0.1\). Now to get from \(0.9\) to \(0.1\) we have to add \(-0.8\). Ah! you say, \(-0.8\) is glued to \(0.2\) since they are an integer apart. Absolutely they are, but we see something important occurring. It turns out points towards the ends of \([0,1)\) are actually close together. So the quotient \(\mathbb{R} / \mathbb{Z}\) is not, in fact, the line segment \([0,1)\). It is a circle! We must take the endpoints, \(0\) and \(1\) and glue them together because they are an integer apart!

When you begin to visualize that, you see why we use the word gluing. The idea of plucking out points spread out across the number line and gluing them together might seem a bit bizarre, but it becomes much more concrete when you imagine the line segment folding around to make the endpoints meet into a circle.

As an exercise to the reader, try to imagine what shape we get when you take the entire Cartesian plane \(\mathbb{R}\times\mathbb{R}\) and quotient out \(\mathbb{Z}\times\mathbb{Z}\), meaning we ignore the integer component of every coordinate. How will you fold and glue your surface?

Spoiler: https://www.youtube.com/watch?v=0H5_h-RB0T8

Try to justify what is happening in this animation.

Written by: David Staudinger

Topology & Calculus

If one is familiar with line integrals from calculus, one may have encountered vocabulary used in theorems to describe regions such as open and closed, connected , and simply connected. These all have relatively approachable and easy to understand definitions when considering closed line integrals in \(\mathbb{R}^2\) with interesting topological definitions for other spaces. As any calculus 3 student could tell you, the line integrals for which these definitions are used are force integrals, more specifically the integral along a path of Fᐧdr. A closed line integral is then one for which the path is a complete loop, that is for the curve from \(t = a\) to \(t = b, f(a) = f(b)\). Below is an example of a closed line integral in a force field.
Before translating calculus vocabulary to topological ones, it is necessary to build a foundation in ideas of topology. A given topology is a structure which is defined on an underlying set; consider that \(\mathbb{R}^2\) is the underlying set when discussing the standard, or Euclidean, topology. The topology on the set contains a collection of open sets in which the entire set, the empty set, and unions and intersections of open sets are all open in the topology. The structure of the topology is what dictates what constitutes an open set. The Euclidean topology is the natural topology on anything of the form \(\mathbb{R}^n\), in which open sets are balls around a point where the distance from a point in that ball to the center point is less than some value. A good check for understanding is to verify that open intervals on the real number line are open sets in \(\mathbb{R}\). Finally, a closed set is the complement of an open set; a more visually intuitive definition is that a closed set is a set that contains its limit points, such as closed intervals containing the bounds.
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Now that we can discuss open and closed regions, such as the closed region bounded by a closed line integral, we are ready to analyze these regions. A space, or region, is said to be connected if it cannot be represented as the union of disjoint, non-empty, open subsets. For regions in \(\mathbb{R}^2\), the definition boils down to the region is not clearly numerous distinct regions, as seen below.
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In \(\mathbb{R}^2\), a region is path connected if there exists at least one path between any two points in the region. Topologically, a space is path connected if there is a continuous function, \(f\), from [0,1] to the space where \(f(0)\) is the start point and \(f(1)\) is the end point. The notion of a continuous function is the generalization of drawing a continous path in \(\mathbb{R}^2\).
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Finally, a simply connected region is one where any path between two points can be continuously mapped into any other path between those two points. This can be visualized in \(\mathbb{R}^2\) by a string connected at two points and deforming the path of the string into other paths by only moving the string without any cutting or removing end connections, keeping the string within the space. In \(\mathbb{R}^2\), any region with holes is not simply connected because the path cannot be continuously mapped over that hole because the path would have to “jump” from one side to the other. Your text to link here…
The restrictions of the theorems exist for the “.1%” of cases of vector fields that have strange behavior without these rules. The vortex vector field, $ = $, is an example of when these restrictions are crucial to producing the right answer. The problem with that vector field is that there is a discontinuity at the origin, meaning that regions that should enclose the origin don’t act the same way as they would in other vector fields. This is because the field exists on the punctured plane, \(\mathbb{R}^2\) without the origin, which is a different topology than \(\mathbb{R}^2\) in which the theorems apply. As the reader can, and should, check, the region enclosed by the unit circle in the vortex vector field is a connected, closed, path connected region that is not simply connected. The reader is now encouraged to explore the connectedness of closed line integrals enclosing the origin in other vector fields on the punctured plane. Your text to link here…
Contributor: Joseph Brewster

(Mathematically) Making a 2×2 Rubik’s Cube

Almost everyone has, at some point, played with the deceptively simple puzzle that is a Rubik’s cube. Even before studying pure maths, we had heard about the idea of Group Theory through an interest in Rubik’s Cubes. Rubik’s Cubes are a common example in group theory because of the physical illustration of the mathematical idea of a group. A group is defined as a set equipped with an operation with elements composed into other elements, identity and inverse elements, as well as being associative.Your text to link here… The elements of the group that defines a Rubik’s Cube are all the possible scrambles. The operations are generated by rotating faces of the cube, meaning algorithms that manipulate the cube in more specific ways can always be broken down into a collection of face turns. The identity of the Rubik’s is not moving the cube, which maintains the current scramble. The inverses are just rotating faces in opposite directions or composing those inverses as to undo a collection of moves; this can be thought of as undoing the moves of a scramble to solve the cube again. Finally, associativity can be thought of as the fact that any string of moves can be grouped together into collections of easy to do moves without affecting the algorithm, which is rather important for solving cubes quickly.

The reader may be wondering the exact mathematical construction of a Rubik’s Cube. For example, a clock is used to represent a group called the “cyclic group” because of the repeated cycles through the numbers. Similar to the jump in complexity of movement of the hand of a clock to the movement of pieces of a Rubik’s Cube, the group representing a Rubik’s Cube is too complex for the purpose of this article. For solving a Rubik’s Cube, it is not necessary to know the mechanical workings, but it can help gain a deeper understanding of the cube. The same concept will be applied to the group as the specific construction will not be necessary for following conclusions.Your text to link here…

From a strictly mechanical standpoint, a 2x2 Rubik’s Cube is the same as a 3x3. A 2x2 functions as a 3x3 with smaller edges hidden behind larger corners. The same idea applies to the groups that define each of the different cubes. A quotient group is a group that can be obtained through sending elements of a group to the quotient group through an equivalence relation. For the quotient group Q = G/H, Q is constructed by sending elements of G that differ only by elements of H to the same element in Q. If G were to be the group of a 3x3 and we wanted to make Q a 2x2, how do we construct H. Thinking back to the physical way this is done, we want an operation that “hides” the edge pieces and only gives us corner pieces. The necessary group turns out to be the group of algorithms that move and rotate edge pieces of the Rubik’s Cube. The reader should check for understanding of the following: if all of the possible edge cases are equivalent, the equivalence classes can be represented as different possible corner positions, which is just a 2x2.

The last step to confirm that the results are mathematically sound is to confirm that the group of edge positions and orientations is a normal subgroup of the 3x3. A subgroup means that the edge positions are a subset of the total scrambles and that the moves on a 3x3 are moves on the set of edge positions, which can be checked through an understanding of the Rubik’s Cube itself. To be a “normal” subgroup, the edges must follow the property that the combination of any moves on the 3x3, then moving only edges, and then the inverse of the first moves results in moving edges. Since edges and corners are independent of each other, the corners are unaffected by the middle move and thus scrambled and solved by the first and last move. Only edges can change during this process, meaning that the subgroup of edges is normal.

It has taken groups, quotient groups, equivalence relations, and normal subgroups all to mathematically represent that simply ignoring edges on a 3x3 Rubik’s Cube leaves a 2x2. Armed with this knowledge, the reader may consider what other quotient groups can be created on a 3x3 by making other collections of scrambles equivalent. For example, can the group of moves that only move and rotate corners be treated the same why we treated the group of moves that only move and rotate edges?

Contributors: Joseph Brewster, David Staudinger

Universal Property: How can we define mathematical objects?

In algebra we are interested in thinking about objects. Objects can be many things including numbers, sets, and many more complicated constructions. Category theory is a powerful tool which analyzes these objects and how they are related.

When defining a category we define what the objects are and what the “morphisms” between those objects are. Morphisms are however we choose objects to be related. There are some additional restrictions which we can touch on later.

Let’s look at a simple category: In this category, the objects are positive integers and morphisms compare the value of these numbers. We’ll say the morphism points towards the bigger number. More rigorously, there exists a morphism from \(a\) to \(b\) if and only if \(a\leq b\). We can visualize this with the following diagram:

We can make some obvious statements about morphisms in this category. It is always true that \(a\leq a\). We can also say that if \(a\leq b\) and \(b\leq c\) then \(a\leq c\). These are the additional constrictions on morphisms: every object must have a morphism to itself (called the identity morphism) and that composing morphisms in a row implies the existence of a morphism connecting the ends:

So now we understand a basic category and how it works.

Let’s look at a more interesting category. The objects are still positive integers and now the morphisms are divisibility operators. This means that there is a morphism from \(a\) to \(b\) if and only if \(a\) divides \(b\) (written as \(a | b\)). For example: \(3|9\), \(2|e\) if \(e\) is even.

Let’s take a look at our first Universal Property. Say we have two numbers \(a\) and \(b\) which both divide \(z\). What number \(w\) has the property that no matter what \(z\) is, as long as \(a\) and \(b\) both divide \(w\), it is also true that \(w\) divides \(z\)? Observe the diagram

Take a second to understand the question. For any \(z\) which is divisible by both \(a\) and \(b\), we want to find a \(w\) which is also divisible by both \(a\) and \(b\) which must necessarily divide every choice of \(z\). For every multiple of both \(a\) and \(b\) (called \(z\)), we want to find a multiple of \(a\) and \(b\) (called \(w\)) which always divides it.

Let’s consider some examples. Let \(a=4\), \(b=6\). What are some common multiples of these? There’s \(6\times 4 =24\) but there’s also \(6\times 2 = 4\times 3 = 12\) and \(6\times 100 = 4\times 150 = 600\). These are a few of our candidates for \(w\).

So then, what is a reasonable choice? Can we choose a huge number like \(600\)? No, because \(600\) doesn’t divide \(24\). So, we are looking for the smallest of these values which divides every possible \(z\). A least common multiple, if you will. That would be \(12\). This works for every \(z\) value i.e. every common multiple of \(4\) and \(6\). Any number which is a common multiple, is itself a multiple of the least common multiple. We can choose \(z=12\) or \(z=600\) and it doesn’t matter. \(w=12\) will always work.

Let’s look at what we just did there. We took this diagram, which admittedly we kind of just conjured out of thin air, but we used it to rigorously define the least common multiple. A number is the least common multiple if and only if it satisfies this diagram in this category. This is a rigorous definition. This quality of how the least common multiple interacts with arbitrary numbers is its universal property.

Later, we will discuss at greater length how these kinds of structures can be applied to sets. Even though with the least common multiple there might be simpler ways to define it, we often find in Algebra that the best way to define objects is by their universal property.

As an exercise to the adventurous and curious reader, consider the the following diagram in the same category. What object (number) \(w\) satisfies the universal property:

Remember, solid lines are given. The dotted line is what you are trying to prove exists for any choice of \(z\).

(Hint: \(z\) must satisfy the properties indicated by the arrows to \(a\) and \(b\))

Contributors: David Staudinger, Joseph Brewster