## Struggles with the Continuum (Part 2)

9 September, 2016

Last time we saw that that nobody yet knows if Newtonian gravity, applied to point particles, truly succeeds in predicting the future. To be precise: for four or more particles, nobody has proved that almost all initial conditions give a well-defined solution for all times!

The problem is related to the continuum nature of space: as particles get arbitrarily close to other, an infinite amount of potential energy can be converted to kinetic energy in a finite amount of time.

I left off by asking if this problem is solve by more sophisticated theories. For example, does the ‘speed limit’ imposed by special relativity help the situation? Or might quantum mechanics help, since it describes particles as ‘probability clouds’, and puts limits on how accurately we can simultaneously know both their position and momentum?

We begin with quantum mechanics, which indeed does help.

### The quantum mechanics of charged particles

Few people spend much time thinking about ‘quantum celestial mechanics’—that is, quantum particles obeying Schrödinger’s equation, that attract each other gravitationally, obeying an inverse-square force law. But Newtonian gravity is a lot like the electrostatic force between charged particles. The main difference is a minus sign, which makes like masses attract, while like charges repel. In chemistry, people spend a lot of time thinking about charged particles obeying Schrödinger’s equation, attracting or repelling each other electrostatically. This approximation neglects magnetic fields, spin, and indeed anything related to the finiteness of the speed of light, but it’s good enough explain quite a bit about atoms and molecules.

In this approximation, a collection of charged particles is described by a wavefunction $\psi,$ which is a complex-valued function of all the particles’ positions and also of time. The basic idea is that $\psi$ obeys Schrödinger’s equation

$\displaystyle{ \frac{d \psi}{dt} = - i H \psi}$

where $H$ is an operator called the Hamiltonian, and I’m working in units where $\hbar = 1.$

Does this equation succeeding in predicting $\psi$ at a later time given $\psi$ at time zero? To answer this, we must first decide what kind of function $\psi$ should be, what concept of derivative applies to such funtions, and so on. These issues were worked out by von Neumann and others starting in the late 1920s. It required a lot of new mathematics. Skimming the surface, we can say this.

At any time, we want $\psi$ to lie in the Hilbert space consisting of square-integrable functions of all the particle’s positions. We can then formally solve Schrödinger’s equation as

$\psi(t) = \exp(-i t H) \psi(0)$

where $\psi(t)$ is the solution at time $t.$ But for this to really work, we need $H$ to be a self-adjoint operator on the chosen Hilbert space. The correct definition of ‘self-adjoint’ is a bit subtler than what most physicists learn in a first course on quantum mechanics. In particular, an operator can be superficially self-adjoint—the actual term for this is ‘symmetric’—but not truly self-adjoint.

In 1951, based on earlier work of Rellich, Kato proved that $H$ is indeed self-adjoint for a collection of nonrelativistic quantum particles interacting via inverse-square forces. So, this simple model of chemistry works fine. We can also conclude that ‘celestial quantum mechanics’ would dodge the nasty problems that we saw in Newtonian gravity.

The reason, simply put, is the uncertainty principle.

In the classical case, bad things happen because the energy is not bounded below. A pair of classical particles attracting each other with an inverse square force law can have arbitrarily large negative energy, simply by being very close to each other. Since energy is conserved, if you have a way to make some particles get an arbitrarily large negative energy, you can balance the books by letting others get an arbitrarily large positive energy and shoot to infinity in a finite amount of time!

When we switch to quantum mechanics, the energy of any collection of particles becomes bounded below. The reason is that to make the potential energy of two particles large and negative, they must be very close. Thus, their difference in position must be very small. In particular, this difference must be accurately known! Thus, by the uncertainty principle, their difference in momentum must be very poorly known: at least one of its components must have a large standard deviation. This in turn means that the expected value of the kinetic energy must be large.

This must all be made quantitative, to prove that as particles get close, the uncertainty principle provides enough positive kinetic energy to counterbalance the negative potential energy. The Kato–Lax–Milgram–Nelson theorem, a refinement of the original Kato–Rellich theorem, is the key to understanding this issue. The Hamiltonian $H$ for a collection of particles interacting by inverse square forces can be written as

$H = K + V$

where $K$ is an operator for the kinetic energy and $V$ is an operator for the potential energy. With some clever work one can prove that for any $\epsilon > 0,$ there exists $c > 0$ such that if $\psi$ is a smooth normalized wavefunction that vanishes at infinity and at points where particles collide, then

$| \langle \psi , V \psi \rangle | \le \epsilon \langle \psi, K\psi \rangle + c.$

Remember that $\langle \psi , V \psi \rangle$ is the expected value of the potential energy, while $\langle \psi, K \psi \rangle$ is the expected value of the kinetic energy. Thus, this inequality is a precise way of saying how kinetic energy triumphs over potential energy.

By taking $\epsilon = 1,$ it follows that the Hamiltonian is bounded below on such
states $\psi$:

$\langle \psi , H \psi \rangle \ge -c$

But the fact that the inequality holds even for smaller values of $\epsilon$ is the key to showing $H$ is ‘essentially self-adjoint’. This means that while $H$ is not self-adjoint when defined only on smooth wavefunctions that vanish at infinity and at points where particles collide, it has a unique self-adjoint extension to some larger domain. Thus, we can unambiguously take this extension to be the true Hamiltonian for this problem.

To understand what a great triumph this is, one needs to see what could have gone wrong! Suppose space had an extra dimension. In 3-dimensional space, Newtonian gravity obeys an inverse square force law because the area of a sphere is proportional to its radius squared. In 4-dimensional space, the force obeys an inverse cube law:

$\displaystyle{ F = -\frac{Gm_1 m_2}{r^3} }$

Using a cube instead of a square here makes the force stronger at short distances, with dramatic effects. For example, even for the classical 2-body problem, the equations of motion no longer ‘almost always’ have a well-defined solution for all times. For an open set of initial conditions, the particles spiral into each other in a finite amount of time!

Hyperbolic spiral – a fairly common orbit in an inverse cube force.

The quantum version of this theory is also problematic. The uncertainty principle is not enough to save the day. The inequalities above no longer hold: kinetic energy does not triumph over potential energy. The Hamiltonian is no longer essentially self-adjoint on the set of wavefunctions that I described.

In fact, this Hamiltonian has infinitely many self-adjoint extensions! Each one describes different physics: namely, a different choice of what happens when particles collide. Moreover, when $G$ exceeds a certain critical value, the energy is no longer bounded below.

The same problems afflict quantum particles interacting by the electrostatic force in 4d space, as long as some of the particles have opposite charges. So, chemistry would be quite problematic in a world with four dimensions of space.

With more dimensions of space, the situation becomes even worse. In fact, this is part of a general pattern in mathematical physics: our struggles with the continuum tend to become worse in higher dimensions. String theory and M-theory may provide exceptions.

Next time we’ll look at what happens to point particles interacting electromagnetically when we take special relativity into account. After that, we’ll try to put special relativity and quantum mechanics together!

### For more

For more on the inverse cube force law, see:

• John Baez, The inverse cube force law, Azimuth, 30 August 2015.

It turns out Newton made some fascinating discoveries about this law in his Principia; it has remarkable properties both classically and in quantum mechanics.

The hyperbolic spiral is one of 3 kinds of orbits possible in an inverse cube force; for the others see:

Cotes’s spiral, Wikipedia.

The picture of a hyperbolic spiral was drawn by Anarkman and Pbroks13 and placed on Wikicommons under a Creative Commons Attribution-Share Alike 3.0 Unported license.

## Struggles with the Continuum (Part 1)

8 September, 2016

Is spacetime really a continuum? That is, can points of spacetime really be described—at least locally—by lists of four real numbers $(t,x,y,z)$? Or is this description, though immensely successful so far, just an approximation that breaks down at short distances?

Rather than trying to answer this hard question, let’s look back at the struggles with the continuum that mathematicians and physicists have had so far.

The worries go back at least to Zeno. Among other things, he argued that that an arrow can never reach its target:

That which is in locomotion must arrive at the half-way stage before it arrives at the goal.Aristotle summarizing Zeno

and Achilles can never catch up with a tortoise:

In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.Aristotle summarizing Zeno

These paradoxes can now be dismissed using our theory of real numbers. An interval of finite length can contain infinitely many points. In particular, a sum of infinitely many terms can still converge to a finite answer.

But the theory of real numbers is far from trivial. It became fully rigorous only considerably after the rise of Newtonian physics. At first, the practical tools of calculus seemed to require infinitesimals, which seemed logically suspect. Thanks to the work of Dedekind, Cauchy, Weierstrass, Cantor and others, a beautiful formalism was developed to handle real numbers, limits, and the concept of infinity in a precise axiomatic manner.

However, the logical problems are not gone. Gödel’s theorems hang like a dark cloud over the axioms of mathematics, assuring us that any consistent theory as strong as Peano arithmetic, or stronger, cannot prove itself consistent. Worse, it will leave some questions unsettled.

For example: how many real numbers are there? The continuum hypothesis proposes a conservative answer, but the usual axioms of set theory leaves this question open: there could vastly more real numbers than most people think. And the superficially plausible axiom of choice—which amounts to saying that the product of any collection of nonempty sets is nonempty—has scary consequences, like the existence of non-measurable subsets of the real line. This in turn leads to results like that of Banach and Tarski: one can partition a ball of unit radius into six disjoint subsets, and by rigid motions reassemble these subsets into two disjoint balls of unit radius. (Later it was shown that one can do the job with five, but no fewer.)

However, most mathematicians and physicists are inured to these logical problems. Few of us bother to learn about attempts to tackle them head-on, such as:

nonstandard analysis and synthetic differential geometry, which let us work consistently with infinitesimals,

constructivism, which avoids proof by contradiction: for example, one must ‘construct’ a mathematical object to prove that it exists,

finitism (which avoids infinities altogether),

ultrafinitism, which even denies the existence of very large numbers.

This sort of foundational work proceeds slowly, and is now deeply unfashionable. One reason is that it rarely seems to intrude in ‘real life’ (whatever that is). For example, it seems that no question about the experimental consequences of physical theories has an answer that depends on whether or not we assume the continuum hypothesis or the axiom of choice.

But even if we take a hard-headed practical attitude and leave logic to the logicians, our struggles with the continuum are not over. In fact, the infinitely divisible nature of the real line—the existence of arbitrarily small real numbers—is a serious challenge to almost all of the most widely used theories of physics.

Indeed, we have been unable to rigorously prove that most of these theories make sensible predictions in all circumstances, thanks to problems involving the continuum.

One might hope that a radical approach to the foundations of mathematics—such as those listed above—would allow avoid some of the problems I’ll be discussing. However, I know of no progress along these lines that would interest most physicists. Some of the ideas of constructivism have been embraced by topos theory, which also provides a foundation for calculus with infinitesimals using synthetic differential geometry. Topos theory and especially higher topos theory are becoming important in mathematical physics. They’re great! But as far as I know, they have not been used to solve the problems I want to discuss here.

Today I’ll talk about one of the first theories to use calculus: Newton’s theory of gravity.

### Newtonian Gravity

In its simplest form, Newtonian gravity describes ideal point particles attracting each other with a force inversely proportional to the square of their distance. It is one of the early triumphs of modern physics. But what happens when these particles collide? Apparently the force between them becomes infinite. What does Newtonian gravity predict then?

Of course real planets are not points: when two planets come too close together, this idealization breaks down. Yet if we wish to study Newtonian gravity as a mathematical theory, we should consider this case. Part of working with a continuum is successfully dealing with such issues.

In fact, there is a well-defined ‘best way’ to continue the motion of two point masses through a collision. Their velocity becomes infinite at the moment of collision but is finite before and after. The total energy, momentum and angular momentum are unchanged by this event. So, a 2-body collision is not a serious problem. But what about a simultaneous collision of 3 or more bodies? This seems more difficult.

Worse than that, Xia proved in 1992 that with 5 or more particles, there are solutions where particles shoot off to infinity in a finite amount of time!

This sounds crazy at first, but it works like this: a pair of heavy particles orbit each other, another pair of heavy particles orbit each other, and these pairs toss a lighter particle back and forth. Xia and Saari’s nice expository article has a picture of the setup:

Each time the lighter particle gets thrown back and forth, the pairs move further apart from each other, while the two particles within each pair get closer together. And each time they toss the lighter particle back and forth, the two pairs move away from each other faster!

As the time $t$ approaches a certain value $t_0,$ the speed of these pairs approaches infinity, so they shoot off to infinity in opposite directions in a finite amount of time, and the lighter particle bounces back and forth an infinite number of times!

Of course this crazy behavior isn’t possible in the real world, but Newtonian physics has no ‘speed limit’, and we’re idealizing the particles as points. So, if two or more of them get arbitrarily close to each other, the potential energy they liberate can give some particles enough kinetic energy to zip off to infinity in a finite amount of time! After that time, the solution is undefined.

You can think of this as a modern reincarnation of Zeno’s paradox. Suppose you take a coin and put it heads up. Flip it over after 1/2 a second, then flip it over after 1/4 of a second, and so on. After one second, which side will be up? There is no well-defined answer. That may not bother us, since this is a contrived scenario that seems physically impossible. It’s a bit more bothersome that Newtonian gravity doesn’t tell us what happens to our particles when $t = t_0.$

Your might argue that collisions and these more exotic ‘noncollision singularities’ occur with probability zero, because they require finely tuned initial conditions. If so, perhaps we can safely ignore them!

This is a nice fallback position. But to a mathematician, this argument demands proof.

A bit more precisely, we would like to prove that the set of initial conditions for which two or more particles come arbitrarily close to each other within a finite time has ‘measure zero’. This would mean that ‘almost all’ solutions are well-defined for all times, in a very precise sense.

In 1977, Saari proved that this is true for 4 or fewer particles. However, to the best of my knowledge, the problem remains open for 5 or more particles. Thanks to previous work by Saari, we know that the set of initial conditions that lead to collisions has measure zero, regardless of the number of particles. So, the remaining problem is to prove that noncollision singularities occur with probability zero.

It is remarkable that even Newtonian gravity, often considered a prime example of determinism in physics, has not been proved to make definite predictions, not even ‘almost always’! In 1840, Laplace wrote:

We ought to regard the present state of the universe as the effect of its antecedent state and as the cause of the state that is to follow. An intelligence knowing all the forces acting in nature at a given instant, as well as the momentary positions of all things in the universe, would be able to comprehend in one single formula the motions of the largest bodies as well as the lightest atoms in the world, provided that its intellect were sufficiently powerful to subject all data to analysis; to it nothing would be uncertain, the future as well as the past would be present to its eyes. The perfection that the human mind has been able to give to astronomy affords but a feeble outline of such an intelligence.Laplace

However, this dream has not yet been realized for Newtonian gravity.

I expect that noncollision singularities will be proved to occur with probability zero. If so, the remaining question would why it takes so much work to prove this, and thus prove that Newtonian gravity makes definite predictions in almost all cases. Is this is a weakness in the theory, or just the way things go? Clearly it has something to do with three idealizations:

• point particles whose distance can be arbitrarily small,

• potential energies that can be arbitrariy large and negative,

• velocities that can be arbitrarily large.

These are connected: as the distance between point particles approaches zero, their potential energy approaches $-\infty,$ and conservation of energy dictates that some velocities approach $+\infty.$

Does the situation improve when we go to more sophisticated theories? For example, does the ‘speed limit’ imposed by special relativity help the situation? Or might quantum mechanics help, since it describes particles as ‘probability clouds’, and puts limits on how accurately we can simultaneously know both their position and momentum?

Next time I’ll talk about quantum mechanics, which indeed does help.

2 September, 2016

I’m now going to try to announce all my new writings in one place: on Twitter.

Why? Well, someone I respect said he’s been following my online writings, off and on, ever since the old days of This Week’s Finds. He wishes it were easier to find my new stuff all in one place. Right now it’s spread out over several locations:

Azimuth: serious posts on environmental issues and applied mathematics, fairly serious popularizations of diverse scientific subjects.

Google+: short posts of all kinds, mainly light popularizations of math, physics, and astronomy.

The n-Category Café: posts on mathematics, leaning toward category theory and other forms of pure mathematics that seem too intimidating for the above forums.

Visual Insight: beautiful pictures of mathematical objects, together with explanations.

Diary: more personal stuff, and polished versions of the more interesting Google+ posts, just so I have them on my own website.

It’s absurd to expect anyone to look at all these locations to see what I’m writing. Even more absurdly, I claimed I was going to quit posting on Google+, but then didn’t. So, I’ll try to make it possible to reach everything via Twitter.

Unlike Facebook, you don’t need to join Twitter to see what people put there. Furthermore, you can see it while blocking cookies. So, I feel okay about this approach to broadcasting my stuff to a larger audience. (Some of my best friends are very concerned with privacy. In fact, I lost touch with one when he said he would only communicate with me in encrypted emails.)

I currently have 4 followers.

## Topological Crystals (Part 4)

28 August, 2016

Okay, let’s look at some examples of topological crystals. These are what got me excited in the first place. We’ll get some highly symmetrical crystals, often in higher-dimensional Euclidean spaces. The ‘triamond’, above, is a 3d example.

### Review

First let me remind you how it works. We start with a connected graph $X.$ This has a space $C_0(X,\mathbb{R})$ of 0-chains, which are formal linear combinations of vertices, and a space $C_1(X,\mathbb{R})$ of 1-chains, which are formal linear combinations of edges.

We choose a vertex in $X.$ Each path $\gamma$ in $X$ starting at this vertex determines a 1-chain $c_\gamma,$ namely the sum of its edges. These 1-chains give some points in $C_1(X,\mathbb{R}).$ These points are the vertices of a graph $\overline{X}$ called the maximal abelian cover of $X.$ The maximal abelian cover has an edge from $c_\gamma$ to $c_{\gamma'}$ whenever the path $\gamma'$ is obtained by adding an extra edge to $\gamma.$ We can think of this edge as a straight line segment from $c_\gamma$ to $c_{\gamma'}.$

So, we get a graph $\overline{X}$ sitting inside $C_1(X,\mathbb{R}).$ But this is a high-dimensional space. To get something nicer we’ll project down to a lower-dimensional space.

There’s boundary operator

$\partial : C_1(X,\mathbb{R}) \to C_0(X,\mathbb{R})$

sending any edge to the difference of its two endpoints. The kernel of this operator is the space of 1-cycles, $Z_1(X,\mathbb{R}).$ There’s an inner product on the space of 1-chains such that edges form an orthonormal basis, so we get an orthogonal projection

$\pi : C_1(X,\mathbb{R}) \to Z_1(X,\mathbb{R})$

We can use this to take the maximal abelian cover $\overline{X}$ and project it down to the space of 1-cycles. The hard part is checking that $\pi$ is one-to-one on $\overline{X}.$ But that’s what I explained last time! It’s true whenever our original graph $X$ has no bridges: that is, edges whose removal would disconnect our graph, like this:

So, when $X$ is a bridgeless graph, we get a copy of the maximal abelian cover embedded in $Z_1(X,\mathbb{R}).$ This is our topological crystal.

Let’s do some examples.

### Graphene

I showed you this one before, but it’s a good place to start. Let $X$ be this graph:

Since this graph has 3 edges, its space of 1-chains is 3-dimensional. Since this graph has 2 holes, its 1-cycles form a plane in that 3d space. If we take paths $\gamma$ in $X$ starting at the red vertex, form the 1-chains $c_\gamma,$ and project them down to this plane, we get this:

Here the 1-chains $c_\gamma$ are the white and red dots. They’re the vertices of the maximal abelian cover $\overline{X},$ while the line segments between them are the edges of $\overline{X}.$ Projecting these vertices and edges onto the plane of 1-cycles, we get our topological crystal:

This is the pattern of graphene, a miraculous 2-dimensional form of carbon. The more familiar 3d crystal called graphite is made of parallel layers of graphene connected with some other bonds.

Puzzle 1. Classify bridgeless connected graphs with 2 holes (or more precisely, a 2-dimensional space of 1-cycles). What are the resulting 2d topological crystals?

### Diamond

Now let’s try this graph:

Since it has 3 holes, it gives a 3d crystal:

This crystal structure is famous! It’s the pattern used by a diamond. Up to translation it has two kinds of atoms, corresponding to the two vertices of the original graph.

### Triamond

Now let’s try this graph:

Since it has 3 holes, it gives another 3d crystal:

This is also famous: it’s sometimes called a ‘triamond’. If you’re a bug crawling around on this crystal, locally you experience the same topology as if you were crawling around on a wire-frame model of a tetrahedron. But you’re actually on the maximal abelian cover!

Up to translation the triamond has 4 kinds of atoms, corresponding to the 4 vertices of the tetrahedron. Each atom has 3 equally distant neighbors lying in a plane at 120° angles from each other. These planes lie in 4 families, each parallel to one face of a regular tetrahedron. This structure was discovered by the crystallographer Laves, and it was dubbed the Laves graph by Coxeter. Later Sunada called it the ‘$\mathrm{K}_4$ lattice’ and studied its energy minimization properties. Theoretically it seems to be a stable form of carbon. Crystals in this pattern have not yet been seen, but this pattern plays a role in the structure of certain butterfly wings.

Puzzle 2. Classify bridgeless connected graphs with 3 holes (or more precisely, a 3d space of 1-cycles). What are the resulting 3d topological crystals?

### Lonsdaleite and hyperquartz

There’s a crystal form of carbon called lonsdaleite that looks like this:

It forms in meteor impacts. It does not arise as 3-dimensional topological crystal.

Puzzle 3. Show that this graph gives a 5-dimensional topological crystal which can be projected down to give lonsdaleite in 3d space:

Puzzle 4. Classify bridgeless connected graphs with 4 holes (or more precisely, a 4d space of 1-cycles). What are the resulting 4d topological crystals? A crystallographer with the wonderful name of Eon calls this one hyperquartz, because it’s a 4-dimensional analogue of quartz:

All these classification problems are quite manageable if you notice there are certain ‘boring’, easily understood ways to get new bridgeless connected graphs with $n$ holes from old ones.

### Platonic crystals

For any connected graph $X,$ there is a covering map

$q : \overline{X} \to X$

The vertices of $\overline{X}$ come in different kinds, or ‘colors’, depending on which vertex of $X$ they map to. It’s interesting to look at the group of ‘covering symmetries’, $\mathrm{Cov}(X),$ consisting of all symmetries of $\overline{X}$ that map vertices of same color to vertices of the same color. Greg Egan and I showed that if $X$ has no bridges, $\mathrm{Cov}(X)$ also acts as symmetries of the topological crystal associated to $X.$ This group fits into a short exact sequence:

$1 \longrightarrow H_1(X,\mathbb{Z}) \longrightarrow \mathrm{Cov}(X) \longrightarrow \mathrm{Aut}(X) \longrightarrow 1$

where $\mathrm{Aut}(X)$ is the group of all symmetries of $X.$ Thus, every symmetry of $X$ is covered by some symmetry of its topological crystal, while $H_1(X,\mathbb{Z})$ acts as translations of the crystal, in a way that preserves the color of every vertex.

For example consider the triamond, which comes from the tetrahedron. The symmetry group of the tetrahedron is this Coxeter group:

$\mathrm{A}_3 = \langle s_1, s_2, s_3 \;| \; (s_1s_2)^3 = (s_2s_3)^3 = s_1^2 = s_2^2 = s_3^2 = 1\rangle$

Thus, the group of covering symmetries of the triamond is an extension of $\mathrm{A}_3$ by $\mathbb{Z}^3.$ Beware the notation here: this is not the alternating group on the 3 letters. In fact it’s the permutation group on 4 letters, namely the vertices of the tetrahedron!

We can also look at other ‘Platonic crystals’. The symmetry group of the cube and octahedron is the Coxeter group

$\mathrm{B}_3 = \langle s_1, s_2, s_3 \;| \; (s_1s_2)^3 = (s_2s_3)^4 = s_1^2 = s_2^2 = s_3^2 = 1\rangle$

Since the cube has 6 faces, the graph formed by its vertices and edges a 5d space of 1-cycles. The corresponding topological crystal is thus 5-dimensional, and its group of covering symmetries is an extension of $\mathrm{B}_3$ by $\mathbb{Z}^5.$ Similarly, the octahedron gives a 7-dimensional topological crystal whose group of covering symmetries, an extension of $\mathrm{B}_3$ by $\mathbb{Z}^7.$

The symmetry group of the dodecahedron and icosahedron is

$\mathrm{H}_3 = \langle s_1, s_2, s_3 \;| \; (s_1s_2)^3 = (s_2s_3)^5= s_1^2 = s_2^2 = s_3^2 = 1\rangle$

and these solids give crystals of dimensions 11 and 19. If you’re a bug crawling around on the the second of these, locally you experience the same topology as if you were crawling around on a wire-frame model of a icosahedron. But you’re actually in 19-dimensional space, crawling around on the maximal abelian cover!

There is also an infinite family of degenerate Platonic solids called ‘hosohedra’ with two vertices, $n$ edges and $n$ faces. These faces cannot be made flat, since each face has just 2 edges, but that is not relevant to our construction: the vertices and edges still give a graph. For example, when $n = 6,$ we have the ‘hexagonal hosohedron’:

The corresponding crystal has dimension $n-1,$ and its group of covering symmetries is an extension of $\mathrm{S}_n \times \mathbb{Z}/2$ by $\mathbb{Z}^{n-1}.$ The case $n = 3$ gives the graphene crystal, while $n = 4$ gives the diamond.

### Exotic crystals

We can also get crystals from more exotic highly symmetrical graphs. For example, take the Petersen graph:

Its symmetry group is the symmetric group $\mathrm{S}_5.$ It has 10 vertices and 15 edges, so its Euler characteristic is $-5,$ which implies that its space of 1-cycles is 6-dimensional. It thus gives a 6-dimensional crystal whose group of covering symmetries is an extension of $\mathrm{S}_5$ by $\mathbb{Z}^6.$

Two more nice examples come from Klein’s quartic curve, a Riemann surface of genus three on which the 336-element group $\mathrm{PGL}(2,\mathbb{F}_7)$ acts as isometries. These isometries preserve a tiling of Klein’s quartic curve by 56 triangles, with 7 meeting at each vertex. This picture is topologically correct, though not geometrically:

From this tiling we obtain a graph $X$ embedded in Klein’s quartic curve. This graph has $56 \times 3 / 2 = 84$ edges and $56 \times 3 / 7 = 24$ vertices, so it has Euler characteristic $-60.$ It thus gives a 61-dimensional topological crystal whose group of covering symmetries is extension of $\mathrm{PGL}(2,\mathbb{F}_7)$ by $\mathbb{Z}^{61}.$

There is also a dual tiling of Klein’s curve by 24 heptagons, 3 meeting at each vertex. This gives a graph with 84 edges and 56 vertices, hence Euler characteristic $-28.$ From this we obtain a 29-dimensional topological crystal whose group of covering symmetries is an extension of $\mathrm{PGL}(2,\mathbb{F}_7)$ by $\mathbb{Z}^{29}.$

### The packing fraction

Now that we’ve got a supply of highly symmetrical crystals in higher dimensions, we can try to study their structure. We’ve only made a bit of progress on this.

One interesting property of a topological crystal is its ‘packing fraction’. I like to call the vertices of a topological crystal atoms, for the obvious reason. The set $A$ of atoms is periodic. It’s usually not a lattice. But it’s contained in the lattice $L$ obtained by projecting the integral 1-chains down to the space of 1-cycles:

$L = \{ \pi(c) : \; c \in C_1(X,\mathbb{Z}) \}$

We can ask what fraction of the points in this lattice are actually atoms. Let’s call this the packing fraction. Since $Z_1(X,\mathbb{Z})$ acts as translations on both $A$ and $L,$ we can define it to be

$\displaystyle{ \frac{|A/Z_1(X,\mathbb{Z})|}{|L/Z_1(X,\mathbb{Z})|} }$

For example, suppose $X$ is the graph that gives graphene:

Then the packing fraction is 2/3, as can be seen here:

For any bridgeless connected graph $X,$ it turns out that the packing fraction equals

$\displaystyle{ \frac{|V|}{|T|} }$

where $V$ is the set of vertices and $T$ is the set of spanning trees. The main tool used to prove this is Bacher, de la Harpe and Nagnibeda’s work on integral cycles and integral cuts, which in turn relies on Kirchhoff’s matrix tree theorem.

Greg Egan used Mathematica to count the spanning trees in the examples discussed above, and this let us work out their packing fractions. They tend to be very low! For example, the maximal abelian cover of the dodecahedron gives an 11-dimensional crystal with packing fraction 1/27,648, while the heptagonal tiling of Klein’s quartic gives a 29-dimensional crystal with packing fraction 1/688,257,368,064,000,000.

So, we have some very delicate, wispy crystals in high-dimensional spaces, built from two simple ideas in topology: the maximal abelian cover of a graph, and the natural inner product on 1-chains. They have intriguing connections to tropical geometry, but they are just beginning to be understood in detail. Have fun with them!

For more, see:

• John Baez, Topological crystals.

Part 1 – the basic idea.

Part 2 – the maximal abelian cover of a graph.

Part 3 – constructing topological crystals.

Part 4 – examples of topological crystals.

## Topological Crystals (Part 3)

6 August, 2016

Last time I explained how to create the ‘maximal abelian cover’ of a connected graph. Now I’ll say more about a systematic procedure for embedding this into a vector space. That will give us a topological crystal, like the one above.

Some remarkably symmetrical patterns arise this way! For example, starting from this graph:

we get this:

Nature uses this pattern for crystals of graphene.

Starting from this graph:

we get this:

Nature uses this for crystals of diamond! Since the construction depends only on the topology of the graph we start with, we call this embedded copy of its maximal abelian cover a topological crystal.

Today I’ll remind you how this construction works. I’ll also outline a proof that it gives an embedding of the maximal abelian cover if and only if the graph has no bridges: that is, edges that disconnect the graph when removed. I’ll skip all the hard steps of the proof, but they can be found here:

• John Baez, Topological crystals.

### The homology of graphs

I’ll start with some standard stuff that’s good to know. Let $X$ be a graph. Remember from last time that we’re working in a setup where every edge $e$ goes from a vertex called its source $s(e)$ to a vertex called its target $t(e)$. We write $e: x \to y$ to indicate that $e$ is going from $x$ to $y$. You can think of the edge as having an arrow on it, and if you turn the arrow around you get the inverse edge, $e^{-1}: y \to x$. Also, $e^{-1} \ne e$.

The group of integral 0-chains on $X$, $C_0(X,\mathbb{Z})$, is the free abelian group on the set of vertices of $X$. The group of integral 1-chains on $X$, $C_1(X,\mathbb{Z})$, is the quotient of the free abelian group on the set of edges of $X$ by relations $e^{-1} = -e$ for every edge $e$. The boundary map is the homomorphism

$\partial : C_1(X,\mathbb{Z}) \to C_0(X,\mathbb{Z})$

such that

$\partial e = t(e) - s(e)$

for each edge $e$, and

$Z_1(X,\mathbb{Z}) = \ker \partial$

is the group of integral 1-cycles on $X$.

Remember, a path in a graph is a sequence of edges, the target of each one being the source of the next. Any path $\gamma = e_1 \cdots e_n$ in $X$ determines an integral 1-chain:

$c_\gamma = e_1 + \cdots + e_n$

For any path $\gamma$ we have

$c_{\gamma^{-1}} = -c_{\gamma},$

and if $\gamma$ and $\delta$ are composable then

$c_{\gamma \delta} = c_\gamma + c_\delta$

Last time I explained what it means for two paths to be ‘homologous’. Here’s the quick way to say it. There’s groupoid called the fundamental groupoid of $X$, where the objects are the vertices of $X$ and the morphisms are freely generated by the edges except for relations saying that the inverse of $e: x \to y$ really is $e^{-1}: y \to x$. We can abelianize the fundamental groupoid by imposing relations saying that $\gamma \delta = \delta \gamma$ whenever this equation makes sense. Each path $\gamma : x \to y$ gives a morphism which I’ll call $[[\gamma]] : x \to y$ in the abelianized fundamental groupoid. We say two paths $\gamma, \gamma' : x \to y$ are homologous if $[[\gamma]] = [[\gamma']]$.

Here’s a nice thing:

Lemma A. Let $X$ be a graph. Two paths $\gamma, \delta : x \to y$ in $X$ are homologous if and only if they give the same 1-chain: $c_\gamma = c_\delta$.

Proof. See the paper. You could say they give ‘homologous’ 1-chains, too, but for graphs that’s the same as being equal.   █

We define vector spaces of 0-chains and 1-chains by

$C_0(X,\mathbb{R}) = C_0(X,\mathbb{Z}) \otimes \mathbb{R}, \qquad C_1(X,\mathbb{R}) = C_1(X,\mathbb{Z}) \otimes \mathbb{R},$

respectively. We extend the boundary map to a linear map

$\partial : C_1(X,\mathbb{R}) \to C_0(X,\mathbb{R})$

We let $Z_1(X,\mathbb{R})$ be the kernel of this linear map, or equivalently,

$Z_1(X,\mathbb{R}) = Z_0(X,\mathbb{Z}) \otimes \mathbb{R} ,$

and we call elements of this vector space 1-cycles. Since $Z_1(X,\mathbb{Z})$ is a free abelian group, it forms a lattice in the space of 1-cycles. Any edge of $X$ can be seen as a 1-chain, and there is a unique inner product on $C_1(X,\mathbb{R})$ such that edges form an orthonormal basis (with each edge $e^{-1}$ counting as the negative of $e$.) There is thus an orthogonal projection

$\pi : C_1(X,\mathbb{R}) \to Z_1(X,\mathbb{R}) .$

This is the key to building topological crystals!

### The embedding of atoms

We now come to the main construction, first introduced by Kotani and Sunada. To build a topological crystal, we start with a connected graph $X$ with a chosen basepoint $x_0$. We define an atom to be a homology class of paths starting at the basepoint, like

$[[\alpha]] : x_0 \to x$

Last time I showed that these atoms are the vertices of the maximal abelian cover of $X$. Now let’s embed these atoms in a vector space!

Definition. Let $X$ be a connected graph with a chosen basepoint. Let $A$ be its set of atoms. Define the map

$i : A \to Z_1(X,\mathbb{R})$

by

$i([[ \alpha ]]) = \pi(c_\alpha) .$

That $i$ is well-defined follows from Lemma A. The interesting part is this:

Theorem A. The following are equivalent:

(1) The graph $X$ has no bridges.

(2) The map $i : A \to Z_1(X,\mathbb{R})$ is one-to-one.

Proof. The map $i$ is one-to-one if and only if for any atoms $[[ \alpha ]]$ and $[[ \beta ]]$, $i([[ \alpha ]]) = i([[ \beta ]])$ implies $[[ \alpha ]]= [[ \beta ]]$. Note that $\gamma = \beta^{-1} \alpha$ is a path in $X$ with $c_\gamma = c_{\alpha} - c_\beta$, so

$\pi(c_\gamma) = \pi(c_{\alpha} - c_\beta) = i([[ \alpha ]]) - i([[ \beta ]])$

Since $\pi(c_\gamma)$ vanishes if and only if $c_\gamma$ is orthogonal to every 1-cycle, we have

$c_{\gamma} \textrm{ is orthogonal to every 1-cycle} \; \iff \; i([[ \alpha ]]) = i([[ \beta ]])$

On the other hand, Lemma A says

$c_\gamma = 0 \; \iff \; [[ \alpha ]]= [[ \beta ]].$

Thus, to prove (1)$\iff$(2), it suffices to that show that $X$ has no bridges if and only if every 1-chain $c_\gamma$ orthogonal to every 1-cycle has $c_\gamma =0$. This is Lemma D below.   █

The following lemmas are the key to the theorem above — and also a deeper one saying that if $X$ has no bridges, we can extend $i : A \to Z_1(X,\mathbb{R})$ to an embedding of the whole maximal abelian cover of $X$.

For now, we just need to show that any nonzero 1-chain coming from a path in a bridgeless graph has nonzero inner product with some 1-cycle. The following lemmas, inspired by an idea of Ilya Bogdanov, yield an algorithm for actually constructing such a 1-cycle. This 1-cycle also has other desirable properties, which will come in handy later.

To state these, let a simple path be one in which each vertex appears at most once. Let a simple loop be a loop $\gamma : x \to x$ in which each vertex except $x$ appears at most once, while $x$ appears exactly twice, as the starting point and ending point. Let the support of a 1-chain $c$, denoted $\mathrm{supp}(c)$, be the set of edges $e$ such that $\langle c, e\rangle> 0$. This excludes edges with $\langle c, e \rangle= 0$, but also those with $\langle c , e \rangle < 0$, which are inverses of edges in the support. Note that

$c = \sum_{e \in \mathrm{supp}(c)} \langle c, e \rangle .$

Thus, $\mathrm{supp}(c)$ is the smallest set of edges such that $c$ can be written as a positive linear combination of edges in this set.

Okay, here are the lemmas!

Lemma B. Let $X$ be any graph and let $c$ be an integral 1-cycle on $X$. Then for some $n$ we can write

$c = c_{\sigma_1} + \cdots + c_{\sigma_n}$

where $\sigma_i$ are simple loops with $\mathrm{supp}(c_{\sigma_i}) \subseteq \mathrm{supp}(c)$.

Proof. See the paper. The proof is an algorithm that builds a simple loop $\sigma_1$ with$\mathrm{supp}(c_{\sigma_1}) \subseteq \mathrm{supp}(c)$. We subtract this from $c$, and if the result isn’t zero we repeat the algorithm, continuing to subtract off 1-cycles $c_{\sigma_i}$ until there’s nothing left.   █

Lemma C. Let $\gamma: x \to y$ be a path in a graph $X$. Then for some $n \ge 0$ we can write

$c_\gamma = c_\delta + c_{\sigma_1} + \cdots + c_{\sigma_n}$

where $\delta: x \to y$ is a simple path and $\sigma_i$ are simple loops with $\mathrm{supp}(c_\delta), \mathrm{supp}(c_{\sigma_i}) \subseteq \mathrm{supp}(c_\gamma)$.

Proof. This relies on the previous lemma, and the proof is similar — but when we can’t subtract off any more $c_{\sigma_i}$’s we show what’s left is $c_\delta$ for a simple path $\delta: x \to y$.   █

Lemma D. Let $X$ be a graph. Then the following are equivalent:

(1) $X$ has no bridges.

(2) For any path $\gamma$ in $X$, if $c_\gamma$ is orthogonal to every 1-cycle then $c_\gamma = 0$.

Proof. It’s easy to show a bridge $e$ gives a nonzero 1-chain $c_e$ that’s orthogonal to all 1-cycles, so the hard part is showing that for a bridgeless graph, if $c_\gamma$ is orthogonal to every 1-cycle then $c_\gamma = 0$. The idea is to start with a path for which $c_\gamma \ne 0$. We hit this path with Lemma C, which lets us replace $\gamma$ by a simple path $\delta$. The point is that a simple path is a lot easier to deal with than a general path: a general path could wind around crazily, passing over every edge of our graph multiple times.

Then, assuming $X$ has no bridges, we use Ilya Bogdanov’s idea to build a 1-cycle that’s not orthogonal to $c_\delta$. The basic idea is to take the path $\delta : x \to y$ and write it out as $\delta = e_1 \cdots e_n$. Since the last edge $e_n$ is not a bridge, there must be a path from $y$ back to $x$ that does not use the edge $e_n$ or its inverse. Combining this path with $\delta$ we can construct a loop, which gives a cycle having nonzero inner product with $c_\delta$ and thus with $c_\gamma$.

I’m deliberately glossing over some difficulties that can arise, so see the paper for details!   █

### Embedding the whole crystal

Okay: so far, we’ve taken a connected bridgeless graph $X$ and embedded its atoms into the space of 1-cycles via a map

$i : A \to Z_1(X,\mathbb{R}) .$

These atoms are the vertices of the maximal abelian cover $\overline{X}$. Now we’ll extend $i$ to an embedding of the whole graph $\overline{X}$ — or to be precise, its geometric realization $|\overline{X}|$. Remember, for us a graph is an abstract combinatorial gadget; its geometric realization is a topological space where the edges become closed intervals.

The idea is that just as $i$ maps each atom to a point in the vector space $Z_1(X,\mathbb{R})$, $j$ maps each edge of $|\overline{X}|$ to a straight line segment between such points. These line segments serve as the ‘bonds’ of a topological crystal. The only challenge is to show that these bonds do not cross each other.

Theorem B. If $X$ is a connected graph with basepoint, the map $i : A \to Z_1(X,\mathbb{R})$ extends to a continuous map

$j : |\overline{X}| \to Z_1(X,\mathbb{R})$

sending each edge of $|\overline{X}|$ to a straight line segment in $Z_1(X,\mathbb{R})$. If $X$ has no bridges, then $j$ is one-to-one.

Proof. The first part is easy; the second part takes real work! The problem is to show the edges don’t cross. Greg Egan and I couldn’t do it using just Lemma D above. However, there’s a nice argument that goes back and uses Lemma C — read the paper for details.

As usual, history is different than what you read in math papers: David Speyer gave us a nice proof of Lemma D, and that was good enough to prove that atoms are mapped into the space of 1-cycles in a one-to-one way, but we only came up with Lemma C after weeks of struggling to prove the edges don’t cross.   █

### Connections to tropical geometry

Tropical geometry sets up a nice analogy between Riemann surfaces and graphs. The Abel–Jacobi map embeds any Riemann surface $\Sigma$ in its Jacobian, which is the torus $H_1(\Sigma,\mathbb{R})/H_1(\Sigma,\mathbb{Z})$. We can similarly define the Jacobian of a graph $X$ to be $H_1(X,\mathbb{R})/H_1(X,\mathbb{Z})$. Theorem B yields a way to embed a graph, or more precisely its geometric realization $|X|$, into its Jacobian. This is the analogue, for graphs, of the Abel–Jacobi map.

After I put this paper on the arXiv, I got an email from Matt Baker saying that he had already proved Theorem A — or to be precise, something that’s clearly equivalent. It’s Theorem 1.8 here:

• Matthew Baker and Serguei Norine, Riemann–Roch and Abel–Jacobi theory on a finite graph.

This says that the vertices of a bridgeless graph $X$ are embedded in its Jacobian by means of the graph-theoretic analogue of the Abel–Jacobi map.

What I really want to know is whether someone’s written up a proof that this map embeds the whole graph, not just its vertices, into its Jacobian in a one-to-one way. That would imply Theorem B. For more on this, try my conversation with David Speyer.

Anyway, there’s a nice connection between topological crystallography and tropical geometry, and not enough communication between the two communities. Once I figure out what the tropical folks have proved, I will revise my paper to take that into account.

Next time I’ll talk about more examples of topological crystals!

Part 1 – the basic idea.

Part 2 – the maximal abelian cover of a graph.

Part 3 – constructing topological crystals.

Part 4 – examples of topological crystals.

## Renewable Energy News

1 August, 2016

Some good news:

• Ed Crooks, Balance of power tilts from fossil fuels to renewable energy, Financial Times, 26 July 2016.

These are strange days in the energy business. Startling headlines are emerging from the sector that would have seemed impossible just a few years ago.

The Dubai Electricity and Water Authority said in May it had received bids to develop solar power projects that would deliver electricity costing less than three cents per kilowatt hour. This established a new worldwide low for the contracted cost of delivering solar power to the grid—and is priced well below the benchmark of what the emirate and other countries typically pay for electricity from coal-fired stations.

In the UK, renowned for its miserable overcast weather, solar panels contributed more power to the grid than coal plants for the month of May.

In energy-hungry Los Angeles, the electricity company AES is installing the world’s largest battery, with capacity to power hundreds of thousands of homes at times of high demand, replacing gas-fired plants which are often used at short notice to increase supply to the grid.

Trina Solar, the Chinese company that is the world’s largest solar panel manufacturer, said it had started selling in 20 new markets last year, from Poland to Mauritius and Nepal to Uruguay.

[…]

Some new energy technologies, meanwhile, are not making much progress, such as the development of power plants that capture and store the carbon dioxide they produce. It is commonly assumed among policymakers that carbon capture has become essential if humankind is to enjoy the benefits of fossil fuels while avoiding their polluting effects.

It is clear, too, that the growth of renewables and other low-carbon energy sources will not follow a straight line. Investment in “clean” energy has been faltering this year after hitting a record in 2015, according to Bloomberg New Energy Finance. For the first half of 2016, it is down 23 per cent from the equivalent period last year.

Even so, the elements are being put in place for what could be a quite sudden and far-reaching energy transition, which could be triggered by an unexpected and sustained surge in oil prices. If China or India were to make large-scale policy commitments to electric vehicles, they would have a dramatic impact on the outlook for oil demand.

I’m also interested in Elon Musk’s Gigafactory: a lithium-ion battery factory in Nevada with a projected capacity of 50 gigawatt-hours/year of battery packs in 2018, ultimately ramping up to 150 GWh/yr. These battery packs are mainly designed for Musk’s electric car company, Tesla.

So far, Tesla is having trouble making lots of cars: its Fremont, California plant theoretically has the capacity to make 500,000 cars per year, but last year it only built 50,000. For some of the reasons, see this:

• Matthew Debord, Tesla has to overcome a major problem for its massive new Gigafactory to succeed, Singapore Business Insider, 1 August 2016.

Basically, it’s hard to make cars as efficiently as traditional auto companies have learned to do, and as long as people don’t buy many electric cars, it’s hard to get better quickly.

Still, Musk has big dreams for his Gigafactory, which I can only applaud. Here’s what it should look like when it’s done:

## Topological Crystals (Part 2)

27 July, 2016

We’re building crystals, like diamonds, purely from topology. Last time I said how: you take a graph $X$ and embed its maximal abelian cover into the vector space $H_1(X,\mathbb{R}).$ Now let me say a bit more about the maximal abelian cover. It’s not nearly as famous as the universal cover, but it’s very nice.

First I’ll whiz though the basic idea, and then I’ll give the details.

### The basic idea

By ‘space’ let me mean a connected topological space that’s locally nice. The basic idea is that if $X$ is some space, its universal cover $\widetilde{X}$ is a covering space of $X$ that covers all other covering spaces of $X.$ The maximal abelian cover $\overline{X}$ has a similar universal property—but it’s abelian, and it covers all abelian connected covers. A cover is abelian if its group of deck transformations is abelian.

The cool part is that universal covers are to homotopy theory as maximal abelian covers are to homology theory.

What do I mean by that? For starters, points in $\widetilde{X}$ are just homotopy classes of paths in $X$ starting at some chosen basepoint. And the points in $\overline{X}$ are just ‘homology classes’ of paths starting at the basepoint.

But people don’t talk so much about ‘homology classes’ of paths. So what do I mean by that? Here a bit of category theory comes in handy. Homotopy classes of paths in $X$ are morphisms in the fundamental groupoid of $X.$ Homology classes of paths are morphisms in the abelianized version of the fundamental groupoid!

But wait a minute — what does that mean? Well, we can abelianize any groupoid by imposing the relations

$f g = g f$

whenever it makes sense to do so. It makes sense to do so when you can compose the morphisms $f : x \to y$ and $g : x' \to y'$ in either order, and the resulting morphisms $f g$ and $g f$ have the same source and the same target. And if you work out what that means, you’ll see it means

$x = y = x' = y'$

But now let me say it all much more slowly, for people who want a more relaxed treatment.

### The details

There are lots of slightly different things called ‘graphs’ in mathematics, but in topological crystallography it’s convenient to work with one that you’ve probably never seen before. This kind of graph has two copies of each edge, one pointing in each direction.

So, we’ll say a graph $X = (E,V,s,t,i)$ has a set $V$ of vertices, a set $E$ of edges, maps $s,t : E \to V$ assigning to each edge its source and target, and a map $i : E \to E$ sending each edge to its inverse, obeying

$s(i(e)) = t(e), \quad t(i(e)) = s(e) , \qquad i(i(e)) = e$

and

$i(e) \ne e$

for all $e \in E.$

That inequality at the end will make category theorists gag: definitions should say what’s true, not what’s not true. But category theorists should be able to see what’s really going on here, so I leave that as a puzzle.

For ordinary folks, let me repeat the definition using more words. If $s(e) = v$ and $t(e) = w$ we write $e : v \to w,$ and draw $e$ as an interval with an arrow on it pointing from $v$ to $w.$ We write $i(e)$ as $e^{-1},$ and draw $e^{-1}$ as the same interval as $e,$ but with its arrow reversed. The equations obeyed by $i$ say that taking the inverse of $e : v \to w$ gives an edge $e^{-1} : w \to v$ and that $(e^{-1})^{-1} = e.$ No edge can be its own inverse.

A map of graphs, say $f : X \to X',$ is a pair of functions, one sending vertices to vertices and one sending edges to edges, that preserve the source, target and inverse maps. By abuse of notation we call both of these functions $f.$

I started out talking about topology; now I’m treating graphs very combinatorially, but we can bring the topology back in. From a graph $X$ we can build a topological space $|X|$ called its geometric realization. We do this by taking one point for each vertex and gluing on one copy of $[0,1]$ for each edge $e : v \to w,$ gluing the point $0$ to $v$ and the point $1$ to $w,$ and then identifying the interval for each edge $e$ with the interval for its inverse by means of the map $t \mapsto 1 - t.$

Any map of graphs gives rise to a continuous map between their geometric realizations, and we say a map of graphs is a cover if this continuous map is a covering map. For simplicity we denote the fundamental group of $|X|$ by $\pi_1(X),$ and similarly for other topological invariants of $|X|.$ However, sometimes I’ll need to distinguish between a graph $X$ and its geometric realization $|X|.$

Any connected graph $X$ has a universal cover, meaning a connected cover

$p : \widetilde{X} \to X$

that covers every other connected cover. The geometric realization of $\widetilde{X}$ is connected and simply connected. The fundamental group $\pi_1(X)$ acts as deck transformations of $\widetilde{X},$ meaning invertible maps $g : \widetilde{X} \to \widetilde{X}$ such that $p \circ g = p.$ We can take the quotient of $\widetilde{X}$ by the action of any subgroup $G \subseteq \pi_1(X)$ and get a cover $q : \widetilde{X}/G \to X.$

In particular, if we take $G$ to be the commutator subgroup of $\pi_1(X),$ we call the graph $\widetilde{X}/G$ the maximal abelian cover of the graph $X,$ and denote it by $\overline{X}.$ We obtain a cover

$q : \overline{X} \to X$

whose group of deck transformations is the abelianization of $\pi_1(X).$ This is just the first homology group $H_1(X,\mathbb{Z}).$ In particular, if the space corresponding to $X$ has $n$ holes, this is the free abelian group on
$n$ generators.

I want a concrete description of the maximal abelian cover! I’ll build it starting with the universal cover, but first we need some preliminaries on paths in graphs.

Given vertices $x,y$ in $X,$ define a path from $x$ to $y$ to be a word of edges $\gamma = e_1 \cdots e_\ell$ with $e_i : v_{i-1} \to v_i$ for some vertices $v_0, \dots, v_\ell$ with $v_0 = x$ and $v_\ell = y.$ We allow the word to be empty if and only if $x = y$; this gives the trivial path from $x$ to itself.

Given a path $\gamma$ from $x$ to $y$ we write $\gamma : x \to y,$ and we write the trivial path from $x$ to itself as $1_x : x \to x.$ We define the composite of paths $\gamma : x \to y$ and $\delta : y \to z$ via concatenation of words, obtaining a path we call $\gamma \delta : x \to z.$ We call a path from a vertex $x$ to itself a loop based at $x.$

We say two paths from $x$ to $y$ are homotopic if one can be obtained from the other by repeatedly introducing or deleting subwords of the form $e_i e_{i+1}$ where $e_{i+1} = e_i^{-1}.$ If $[\gamma]$ is a homotopy class of paths from $x$ to $y,$ we write $[\gamma] : x \to y.$ We can compose homotopy classes $[\gamma] : x \to y$ and $[\delta] : y \to z$ by setting $[\gamma] [\delta] = [\gamma \delta].$

If $X$ is a connected graph, we can describe the universal cover $\widetilde{X}$ as follows. Fix a vertex $x_0$ of $X,$ which we call the basepoint. The vertices of $\widetilde{X}$ are defined to be the homotopy classes of paths $[\gamma] : x_0 \to x$ where $x$ is arbitrary. The edges in $\widetilde{X}$ from the vertex $[\gamma] : x_0 \to x$ to the vertex $[\delta] : x_0 \to y$ are defined to be the edges $e \in E$ with $[\gamma e] = [\delta].$ In fact, there is always at most one such edge. There is an obvious map of graphs

$p : \widetilde{X} \to X$

sending each vertex $[\gamma] : x_0 \to x$ of $\widetilde{X}$ to the vertex
$x$ of $X.$ This map is a cover.

Now we are ready to construct the maximal abelian cover $\overline{X}.$ For this, we impose a further equivalence relation on paths, which is designed to make composition commutative whenever possible. However, we need to be careful. If $\gamma : x \to y$ and $\delta : x' \to y' ,$ the composites $\gamma \delta$ and $\delta \gamma$ are both well-defined if and only if $x' = y$ and $y' = x.$ In this case, $\gamma \delta$ and $\delta \gamma$ share the same starting point and share the same ending point if and only if $x = x'$ and $y = y'.$ If all four of these equations hold, both $\gamma$ and $\delta$ are loops based at $x.$ So, we shall impose the relation $\gamma \delta = \delta \gamma$ only in this case.

We say two paths are homologous if one can be obtained from another by:

• repeatedly introducing or deleting subwords $e_i e_{i+1}$ where
$e_{i+1} = e_i^{-1},$ and/or

• repeatedly replacing subwords of the form

$e_i \cdots e_j e_{j+1} \cdots e_k$

by those of the form

$e_{j+1} \cdots e_k e_i \cdots e_j$

where $e_i \cdots e_j$ and $e_{j+1} \cdots e_k$ are loops based at the same vertex.

My use of the term ‘homologous’ is a bit nonstandard here!

We denote the homology class of a path $\gamma$ by $[[ \gamma ]].$ Note that if two paths $\gamma : x \to y,$ $\delta : x' \to y'$ are homologous then $x = x'$ and $y = y'.$ Thus, the starting and ending points of a homology class of paths are well-defined, and given any path $\gamma : x \to y$ we write $[[ \gamma ]] : x \to y .$ The composite of homology classes is also well-defined if we set

$[[ \gamma ]] [[ \delta ]] = [[ \gamma \delta ]]$

We construct the maximal abelian cover of a connected graph $X$ just as we constructed its universal cover, but using homology classes rather than homotopy classes of paths. And now I’ll introduce some jargon that should make you start thinking about crystals!

Fix a basepoint $x_0$ for $X.$ The vertices of $\overline{X},$ or atoms, are defined to be the homology classes of paths $[[\gamma]] : x_0 \to x$ where $x$ is arbitrary. Any edge of $\overline{X},$ or bond, goes from some atom $[[ \gamma]] : x_0 \to x$ to the some atom $[[ \delta ]] : x_0 \to y.$ The bonds from $[[ \gamma]]$ to $[[ \delta ]]$ are defined to be the edges $e \in E$ with $[[ \gamma e ]] = [[ \delta ]].$ There is at most one bond between any two atoms. Again we have a covering map

$q : \overline{X} \to X .$

The homotopy classes of loops based at $x_0$ form a group, with composition as the group operation. This is the fundamental group $\pi_1(X)$ of the graph $X.$ This is isomorphic as the fundamental group of the space associated to $X.$ By our construction of the universal cover, $\pi_1(X)$ is also the set of vertices of $\widetilde{X}$ that are mapped to $x_0$ by $p.$ Furthermore, any element $[\gamma] \in \pi_1(X)$ defines a deck transformation of $\widetilde{X}$ that sends each vertex $[\delta] : x_0 \to x$ to the vertex $[\gamma] [\delta] : x_0 \to x.$

Similarly, the homology classes of loops based at $x_0$ form a group with composition as the group operation. Since the additional relation used to define homology classes is precisely that needed to make composition of homology classes of loops commutative, this group is the abelianization of $\pi_1(X).$ It is therefore isomorphic to the first homology group $H_1(X,\mathbb{Z})$ of the geometric realization of $X.$

By our construction of the maximal abelian cover, $H_1(X,\mathbb{Z})$ is also the set of vertices of $\overline{X}$ that are mapped to $x_0$ by $q.$ Furthermore, any element $[[\gamma]] \in H_1(X,\mathbb{Z})$ defines a deck transformation of $\overline{X}$ that sends each vertex $[[\delta]] : x_0 \to x$ to the vertex $[[\gamma]] [[\delta]] : x_0 \to x.$

So, it all works out! The fundamental group $\pi_1(X)$ acts as deck transformations of the universal cover, while the first homology group $H_1(X,\mathbb{Z})$ acts as deck transformations of the maximal abelian cover.

Puzzle for experts: what does this remind you of in Galois theory?

We’ll get back to crystals next time.