60. My PhD Thesis: Part 1: Preliminaries - Algebra, Topology, Algebraic Topology

(Epistemic status: A creased, stained map to what were once my favorite hunting grounds. Accessible to anyone who can support substantial abstraction; prior math knowledge is not necessary. In particular, ignorance of calculus is not an obstruction here, but total ignorance of geometry or like, arithmetic or logic, will be. Extremely dense and probably won’t get you there, but at least you’ll ask better questions. Partially dedicated to DG, JM, and PR.) 

The path to my old hunting grounds (as pictured above) is long and twisty and winds through a lot of necessary math along the way. You'll be best served by carefully going through the parentheticals and answering the questions I ask, so as to keep track of the blazes and to keep your footing. You'll still get something out of this if you move more quickly, but you might end up lost further down the line. You might find the pace a little slow if you already know the territory - test yourself by answering those same parentheticals as quickly and off-handedly as I ask them.

Anyhow: let's start down the red path with a first pass at the abstract algebra - we'll need to talk about group theory to set up for everything else along the way. We care about group theory primarily because it models things we care about in the world, like numbers and rotations. It also provides a powerful language for talking about symmetry, structures built on symmetry, and the way that those can resemble each other. This is why my old subfield is called "geometric group theory". We'll get to more of that soon enough; we still need to walk through the basics.

I'm going to provide the classical formal definition of a group first, prepended by more informal glosses of what they really mean. Then I'll point out that you already know many examples of them.

We define a group to be any set \(S\) with a binary operation \(\cdot\) , with the following properties:

• (Closure) Combining any two elements of S puts you back in S. For any \(s, t \in S\), \(s‧t \in S\) as well.
• (Identity element) There's a special element that we call "1" that you can combine with any other element that does nothing to it. There is a special element \(1 \in S\) so that for any \(s \in S\), \(1 \cdot s = s\cdot 1 = s\).
• (Inverse elements) You can always go backwards, uncombining something you had previously mixed in. For any element \(s \in S\), we can always find a \(t \in S\) so that \(s \cdot t = t\cdot s = 1\).
• (Associativity) You don't need to keep track of when you combine what. It doesn't matter how we group pairs for the binary operation: \((s \cdot t) \cdot u = s \cdot (t \cdot u) \), for all \(s, t, u \in S\).

Additionally, we also usually assume that order matters, although sometimes it doesn't have to:

• (Commutativity) What order we combine the things in doesn't matter either. For every \(s, t \in S\), \( s \cdot t = t \cdot s\).

If a group satisfies this last property, we call it a commutative or abelian group, this latter after Niels Henrik Abel.

Better yet, we don't have to worry about where a group came from, just what it does - the word for two groups being "basically the same and we don't care" is "isomorphic"; two groups are isomorphic when there's a way to match up their elements, one to one, where the two corresponding elements do all the same things - combine in the same ways, have inverses that correspond, everything. 

(Extra credit: the identity element is unique, and so are inverses. Can you see why? Select the next line for a hint.)

(Hint: suppose there were two identities... what happens when they combine in either order? What about comparing what happens when multiplying an element by its supposed two inverses, and then multiplying on the left by the same inverse?)

I claim that you already know lots of examples of groups! Pretty much anything you can think of where there's some sensible notion of combining things together (addition, multiplication, composition - that means doing one thing and then doing a second thing) which comes with a way to do nothing and a way to go backwards is a group. (Nonassociative operations are pretty hard to find and generally badly behaved, and non-closed operations can usually be straightforwardly redefined to act on the true set, where we look at everywhere any series of combinations gets you too.)

Here's one: the integers, \(\mathbb{Z}\), which comprise the counting numbers, along with their negatives and zero, with the combining operation being addition. (Check for yourself that this satisfies the group laws, including commutativity.) We usually write this as \((\mathbb{Z}, +)\), and more generally we write down ordered pairs of (group, operation) like this: \((S, \cdot)\), often eliding the operation symbol in practice.

Here's another example: clock arithmetic, be it 12- or 24-hour, and the combining operation is adding times. (Remember that 13 o'clock is really 1 o'clock!) Algebraists would call this \((\mathbb{Z}/12\mathbb{Z}, +)\), or just \((\mathbb{Z}_{12}, +)\), or even just \(\mathbb{Z}_{12}\) with the operation being understood to be the only natural-feeling one that gives \(\mathbb{Z}_{12}\) the structure of a group, unlike (say) multiplication. (As a trailhead to further math we won't get into, because it's off the trail, what's the multiplicative inverse of 6 inside \(\mathbb{Z}_{12}\)? More generally, what happens in groups under multiplication where two elements can combine to 0? The answer requires thinking about rings, not just groups; the search terms you want include "zero divisor" and "principal ideal domain".)

Here's a non-example: the positive counting numbers only, under addition. (Which laws fail? Which laws fail if we think about the odd integers only? Do the even integers form a group?) (Some search terms you might want, if you feel like going down a rabbit hole, are "semigroup" and "monoid". If you do, think about what kinds of physical processes they might correspond to, given which group axioms they lack.)

Here's another nonexample: the integers under multiplication. (Which law fails most obviously?)

Here's another example, closer to our trail: In what ways can you flip a mattress so that it fits back on its frame? The elements here are flips and rotations, and the combining law is "do one operation first, then the other." This gives us our first look at what's called a "group action" - we have some group, and we know how to turn group elements into some kind of other mathematical... thing... which we can apply to some other object. In this case, we have the abstract group called \(D_2\), which consists of just four elements: \(1, a, b, ab\), which is commutative and where every element is its own inverse. We decide that \(a\) means "spin the mattress by a half turn" and that \(b\) means "flip the mattress about its long axis". More generally, whenever we have a set of things we can do to some geometric object that all leave the object unchanged as a shape in its natural space, we can turn that set into a group under composition - "do one thing, and then do a second thing". We call this the symmetry group of that object, e.g. \(D_2\) is the symmetry group of rectangular mattresses and indeed all non-square rectangles.

Speaking of squares, in what ways can you take a square to itself? Given that every square is a rectangle, \(D_2\) is a good place to start looking - half-turn spins should be in the group, as should reflection about any choice of axis - either one passing through the midpoints of opposing sides, or one passing through opposing corners. But we now also get to care about quarter-turns, and our group is actually no longer commutative. (Check this for yourself: fix any axis of reflection. What happens after a quarter-turn rotation followed by that reflection? What about the reflection, followed by the rotation?)

Here's one last example: take a cube - a die, say. The elements are rotations leaving the frame of the cube completely covering up where it started. (What's the natural combining law here? It's one we've seen before.) I won't provide an explicit accounting of the group this time, but I'll point out that it's called the octahedral group for good reason: the symmetry group of rotations on the cube is exactly the same as the symmetry group of rotations on the cube, and likewise if we added in the reflections, too. Geometric duality corresponds to isomorphism of symmetry groups! Put a pin in this whole frame of group actions on geometric objects. We'll think about it more carefully later on.

Two quick concepts to drop before we move on: finite presentation and quotient groups.

First, group presentations and finite presentations in particular. Every group can be written down in the form \( \langle L | R \rangle \), where \(L\) is a set of letters, or generators, and \(R\) is a set of relations - expressions that we either mark as equivalent, or which if all are unmarked, shake out to being the identity element. We call this the presentation of a group; well-behaved groups have only finitely many generators and are called finitely generated. For a few examples from above, we'd write the integers under addition as \( \langle x \rangle \), because it has no nontrivial relations - no number of 1's added together get you a 0. By contrast, the clock group \(\mathbb{Z}_{12}\) gets written as \( \langle x | x^{12} = 1 \rangle \), because when we compose a generator with itself 12 times, we get the identity. For another, we'd write \(D_2\) as \( \langle r, f | r^2 = f^2 = rfrf = 1 \rangle \), and \(D_4\) as \( \langle r, f | r^4 = f^2 = rfrf = 1 \rangle \), where \(r\) is a rotation and \(f\) is a choice of reflection. (We can make lots of different choices of generator - as long as we generate the same group again, we're golden. Maybe think about which choices of elements from \(\mathbb{Z}_{12}\) are generators, and which ones only generate a subgroup.) Finally, you probably would have taken a while to get this on your own, but the orientation-preserving symmetry group of the cube - that is, the one that lacks reflections - is written \( \langle a, b, c, d | a^2 = b^2 = c^2 = d^2 = 1 = aba^{-1}b^{-1} = bdb^{-1}d^{-1}, cac^{-1} = dad, cbc^{-1} = a, dcd = c^{-1}\rangle \). This is kind of a mess, but it turns out to be the exact same presentation that \(S_4\), the group of ways to scramble up four objects, has; the reason we shouldn't be surprised to see it here is something like: "Pick your favorite cube vertex and look at its space-diagonal opposite. Now do that for all 6 of the other vertices, ending up with 4 pairs. Rotations can scramble these pairs arbitrarily, if we don't mind that some of them might get flipped, too.". Once again, peering more deeply into the substructure of the groups unites insights that would be much more annoying to get to the hard way. Especially well-behaved groups - like pretty much all the ones you personally could name right now, probably - also have finitely many relations, and we call those ones finitely presented; it's those that we usually care about, especially in geometric group theory. We can always rewrite the relation expressions to equal out to the identity, but sometimes it's more convenient not to.

What about quotient groups? We've already seen one of them - \(\mathbb{Z}_{12}\) as a quotient group of \(\mathbb{Z}\). What it means for something to be a quotient group of another will become more clear later on, but for now let's focus on the algebraic significance. To construct a quotient group from some group \(G\), pick what's called a normal subgroup \(N \subseteq G\) - a subgroup which the conjugation action by \(G\) fixes. By "conjugation action", I mean that for each \(n \in N\), we look at \(gng^{-1}\) as \(g\) ranges over all of \(G\); the result is called the orbit of \(n\) under this conjugation action. By "fixes", I mean that when we look at where the entirety of \(N\) gets sent under the conjugation action by all of \(G\), it turns out to get sent back to itself.

I've chosen to skip over a lot of the details for why this matters for us, but the quick version is that if we have such an \(N\), we can look at how it carves up \(G\) into \(N\)-sized slices, each of which is called a coset, and which has the form \(gN\) for some \(g \in G\), that is, we left-multiply \(N\) by some element \(g \in G\). We then define a binary operation on \(N\)'s cosets such that \(aN \cdot bN = abN\); this only works because the normality of \(N\) means that any representative element of a coset is as good as any other for the purposes of this operation. Having done this, we move to looking at just the set of cosets instead, with the effect that the entirety of \(N\) gets turned into the identity element of this new group, which we write \(G/N\). For example, this is why we write \(\mathbb{Z}/12\mathbb{Z}\): here, our normal subgroup is \(\langle 12 \rangle\), the set of all multiples of 12, and its cosets are the other residue classes modulo 12.

To end on, here's a challenge problem for the algebra section: for \(\mathbb{Z}_{12}\), we have to be careful about which element we pick for a generator, because not all of them work. For instance, \(\langle 8 \rangle = \{0, 4, 8\} \simeq \mathbb{Z}_3\), and likewise \(\langle 3 \rangle = \{0, 3, 6, 9\} \simeq \mathbb{Z}_4\). But 5 and 7 work just fine, as does (obviously) 1. Why should that be true? And are there groups of this very simple type - cyclic groups, they're called - for which any choice of generating element works just fine, apart from the identity? What's special about those?

 

 

Alright, enough algebra for now - don't worry, though, we'll come back to it later. Let's change gears for a while to follow the green path to talk about basic topology and a little algebraic topology. Topology is the study of continuous space; by this we mean that topology is the study of those aspects of geometry that remain true even after we throw away all information about distance and angle, caring purely about things like connectedness and continuous transformations of space. I won't drop the axioms for what a topology is, because we won't need them and they're a little confusing.

Just like for groups and abstract algebra more generally, while at first glance you might expect topology to have nothing to do with everyday life, it in fact plays a role in all sorts of things in the world around you: everything from the connectivity of roads and telecom networks to playground games. In abstract algebra, we didn't worry about whether two groups """really were""" the same, as long as they did all the same things - recall that we called this property "isomorphism" or "being isomorphic". In the same sort of way, in topology we don't worry about whether two geometric objects """really are""" the same as long as we can continuously deform one into the other - stretching it, bending it, passing it through itself, but never ever cutting or gluing it. If we want to up our demands to make the deformations smooth - that is, differentiable - then we also must rule out creasing our object. The classic joke, then, is that a topologist is someone who can't tell a doughnut from a coffee cup, but can still tell their ass from a hole in the ground; we can continuously warp the doughnut hole into the handle of the cup, letting the rest of the doughnut form the body. Meanwhile, a hole in the ground need not come out anywhere - a digestive tract is very different. 

What remains true when we sculpt our geometric objects out of springy rubber or moldable clay, rather than marble? Could we, for example, turn a sphere into a torus somehow? Thankfully, no. Think about the difference between a doughnut and an orange: you can tie a string around a doughnut, through the hole, so that you can’t remove the string without cutting it or tearing the doughnut, but the same isn’t true of an orange - any loop you tie will just slip right off. That is: you can't leash a sphere, but you can leash a torus, so the two have to be different. (The search terms you want here are "nullhomotopic" and "homotopically nontrivial". Also take a look at the Alexander horned sphere for something really wild!)

 

Above, I've drawn both the inside and outside of a solid sphere and torus. The outside loops would work almost as well if the sphere or torus were hollow, which is still an important distinction for us. We call the hollow sphere - like a basketball - \(S^2\), and the hollow torus - like an inner tube - \(T^2\), where the "2" means that at any point on the surface, it looks a lot like you're on a 2-dimensional plane. Nevertheless, there's no way of leashing a sphere, but there are plenty for a torus. Alright, then... maybe the only thing we need to track is whether or not a given form can be leashed? Turns out, no!

 

 

Here's an example of how this can fail: I've drawn a picture of \(T^2\) next to a picture of \(T^2 \# T^2\), where the "#" stands for "connect-sum". We write the two-holed torus that way because one way of constructing it is the unsurprising one, where we take two copies of the ordinary torus \(T^2\) and attach them together. (Depending on what crafts you like, you might think of this as being like gluing, stitching, welding, or scoring and slipping.)

 

 

So... just saying whether or not there's loops that you can never pull tightly to a point isn't remotely enough to classify geometric objects, even in combination with dimension. Let's poke at this idea some more. Let's go down to inhabit a two-dimensional annular room. (Please imagine a Magic School Bus-style animation.) How many ways are there of winding a rope around in our little world?

 

Alright that's probably not what I want to ask - there's too many of those and we don't really care if there are little wiggles in the rope. How many meaningfully different ways are there, following the rules of topology, where if we can nudge the rope around a bit to turn one wrapping into another, the two wrappings are identical? This is a magic mathematical rope, by the way - it can pass through itself, and you, just fine, and it's infinitely extensible, too, but it has a directionality to it - we'll mark it in our pictures with little arrows. You could also think of our magic mathematical rope as tracing out a path through space, starting at some point, moving through the topological space, and ending up back at its starting point.

 

Well... we can definitely make nullhomotopic loops - loops that can pull tight to a point. We can equally clearly make homotopically nontrivial loops - ones that can't, because they wrap around the middle circle of our tiny world. 

 

Is that all? Nope! We can also wrap the rope around the center circle more than once, and the resulting loop can't be turned into a contractible loop or a single-turn loop, either.

 

For this next part, I'll draw myself as a distinguished (orange) point to save room. What happens if we connect-sum two loops together? Let's take a red loop which wraps around counterclockwise once, and a blue loop that also wraps around counterclockwise once. What do we get?

 

With a little rearrangement, we see that we get back the same loop as above - the one that started off wrapping around the center twice counterclockwise from the start!  

 

Hm... something about this feels familiar. What if we swap the direction that the blue loop wraps in - make it clockwise instead of counterclockwise? (Get your guesses in now!) 

 

As it happens, we get a nullhomotopic loop!

 

 

What exactly might explain that? What mathematical structure have we seen that does something like this? We have a zero-flavored loop - any nullhomotopic loop, which are all the same topologically. We have a way to get a negative-flavored loop, given some loop - just make it wind around the other direction. Stitching two loops together gets us another loop, and it doesn't seem like it should matter what order we do pairwise stitching.

That's right - we have a group! Surprise! We've actually been thinking about the basics of algebraic topology this whole time, skipping right past the horrors of point-set topology. This is why algebraic topology is called that - we use algebraic tools and methods to think about how topological objects work, and to know more about them. What we've done here is construct what’s called a fundamental group: all the ways that you can start somewhere in a shape, move around the shape trailing a string, and come back to the place you started; this lets us examine the holes in a shape. And in fact it's not just any group - we've recovered the integers as the fundamental group of our space. This is how we count the holes in a space: we look at the number of distinct generators - in the form of loops or equivalently letters - required to construct every class of loop in a given topological space. (This is why a straw has but a single hole, as any mathematician will tell you - there is a single type of loop you need: the one that passes all the way through the straw.)

Every topological space has a fundamental group. Some of them are extremely weird. (Your search term here to start off down that rabbit hole is "Hawaiian earring space".) All the same, two things should be immediately evident: that if every loop in a space is contractible, then the fundamental group of that space is the trivial group \(1\), which has only the identity element; and that given how we built up the fundamental group as something about equivalence classes of loops within a space, there should be some strong relationship between the fundamental group of the connect-sum of two spaces, and the fundamental group of the two spaces being connect-summed. But that will have to wait until next time.

Here's a challenge problem to end on, which I promise is related to all this stuff about fundamental groups and algebraic topology: suppose we're hanging up a boulder that we want to be able to drop easily down a hole. We have a big rope attached to it on one end and can attach it securely at the other end, and two pegs on a frame that can hold the boulder's weight. We want to be able to release the boulder by removing either pin. What pattern of winding around the two pins lets us do that? Is there even one?

Happy hunting!