Treating the differential with discretion

Sophie Bleau

8 November 2024

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The differential is an interesting beast in many respects. It is a map which has been, ironically, integral to the theory of calculus, and has the fascinating property that its square disappears. This is to do with the fact that its partial derivatives always commute, so that they happen to cancel each other out. Last summer I had a go at proving why this happens using a combinatorial version of an existing proof by Michèle Audin, Mihai Damian and Reinie Erné. The goal was to use the subtle force that is Morse theory (sadly no relation to the code of dots and dashes).

Morse theory is all about analysing the topology on a manifold by looking at how differentiable functions act on it. For instance, I can look at the height function on a torus, and see that we have a maximum and a minimum as well as two saddle points. We can see that the type of level sets of the height function changes at each critical point. By level set we just mean a slice where the height function is constant. Any function with this property is called a Morse function, and the rigidity of this constraint means that we get stronger statements about them. For instance, if a Morse function yields exactly two critical points on a manifold then the manifold is homeomorphic to a sphere.

A torus with the level sets of the height function marked, near the bottom of the torus the level sets are a circle, then between the saddle points the level sets are two circles, then finally between the upper saddle point and maximum, the level sets are a circle.

In the title I said we would treat the differential with discretion, by which I mean we translate our manifolds into the discrete setting. To do this we can make the manifolds be simplicial complexes. If you don’t know what a simplicial complex is, the name can look a bit intimidating but it is in reality just a graph, where each ‘vertex’ of the graph is a simplex. A simplex is just a fancy name for a triangle or one of its relatives in different dimensions. An $n$-simplex has $n+1$ vertices all connected to each other, so that a 0-simplex is a vertex, a 1-simplex is an edge, a 2-simplex is a triangle and a 3-simplex is a tetrahedron, and so on.

From left to right: a 0-simplex, 1-simplex, 2-simplex, and 3-simplex.

Simple simplices

A simplicial complex is a gluing together of some simplices. For instance, in the image we have a 3-simplex glued to a 2-simplex, which is glued to two 1-simplices, and a disconnected zero simplex.

If you’re wondering how a simplicial complex can represent a manifold, I can give you an example of the correspondence. A tetrahedron is the simplicial complex version of the sphere, where the discrete Morse function on the tetrahedron corresponds to the height function on the sphere, so that the critical maximum and minimum as we usually know them are represented by the 2-simplex at the top and the 0-simplex at the bottom. This is an example we will come back to in the next section.

So what do functions on these objects look like?

In general, we want it to feel natural to slide down dimension, for instance, from a triangle to one of the edges on its boundary, as in the triangle below. Here I’m saying boundary to mean `at the edge of the simplex’.

We slide down a dimension from a 2-simplex to
an edge on its boundary.

However, despite gravity, occasionally we might wish to jump up a dimension, for instance, from a boundary edge of a triangle to the triangle itself.

We jump up a dimension from a 1-simplex to a
triangle it lives on the boundary of.

We talk about orientations of simplices in everyday Morse discussions, which means defining an ordering on the vertices. On 1-simplices this is very intuitive as we are simply following the flow of the edge, like an arrow. On 2-simplices we take a circular route around the boundary of our triangle, and on 3-simplices we do a sort of `box-step’ dance between the vertices. Conventionally we label vertices by letters, $a, b, c$; edges by pairs of vertices, $ab, bc, ca$; and so on.

The Morse differential, as we will see, is the count of the signs of all the paths from an $(n+1)$-simplex to an $n$-simplex. In particular, if we take a 2-simplex $\alpha$ and drop down two dimensions to a vertex $\gamma$ on its boundary, there are two paths to do this, shown in blue and in pink.

In this diagram the 2-simplex is $\alpha= a bc$, while the vertex $\gamma$ is $a$.

As the direction of the blue path is the opposite of that of the pink, they have opposite signs, so that they cancel each other out. That is, the square of the Morse differential, found by dropping dimension twice, is 0.

A Morse function takes an $n$-simplex either up a dimension or down a dimension, and the rare instances of being able to jump up dimension we call Morse arrows. The distinguishing rule is that there can only ever be at most one Morse arrow associated to any simplex. If a simplex doesn’t have any Morse arrows belonging to it, we call it a critical simplex. I like to think that it is critical of all the other simplices jumping about and being rowdy.

A nice way of displaying the information of possible directed paths is a Hasse diagram.

Below, we have (top left) a simplicial complex consisting of a 2-simplex $abc$ and the simplices in its boundary. On the top right, we show the Hasse diagram, whose arrows correspond to boundary maps.

The Hasse diagram has only critical simplices, since none of the arrows point up. We can choose to make this slightly less trivial by changing the Morse function. This means reversing some of the arrows. Below, we see what effect this has on the simplicial complex, and on the right we see the modified Hasse diagram.

However, this is far too easy so let us glue two of these 2-simplices together and take an arbitrary Morse function on the resulting complex. This gives us the simplicial complex (bottom left) with modified Hasse diagram (bottom right) as shown.

Notice that this satisfies the rule that any individual simplex can only have one Morse arrow (highlighted in yellow) attached. Can you find the critical simplices in this Hasse diagram?

A directed path through a Hasse diagram between two critical simplices two dimensions apart is what we call a flowline, and forms the pivotal concept of this article. To see a flowline, let’s look at a tetrahedron.

We alluded to this earlier: a possible Morse function we can define on the tetrahedron is that representing the height function on a sphere; imagine something like this:

If the top 2-simplex $abc$ is a critical maximum (that is, a source simplex whose arrows all point away from it), and the 0-simplex $d$ is a critical minimum (that is, a target simplex whose arrows all point toward it), the tetrahedron has the Hasse diagram to the right, and here’s an example of a flowline:

You see, we can define an algorithm which changes a flowline bit by bit in a small way to become another flowline, seeking out a flowline with a particular property. You may notice that a flowline travels down and up a dimension almost in alternation, but to get to the level below it must drop down twice through an ‘intermediate simplex’. The intermediate simplex in this flowline, for instance, is the 1-simplex $ab$. We want flowlines whose intermediate simplices are critical. If the flowline has this property, we say it is a critical flowline.

The Hasse diagram for…

…a possible Morse function on two glued triangles

A tetrahedron…

…and the Hasse diagram where $abc$ is a critical maximum and $d$ is a critical minimum

Using this concept of flowline criticality, we define the square differential of a critical $(n+1)$-simplex $\alpha$ in Morse theory to be the signed count of critical flowlines to critical $(n-1)$-simplices. This counts all of the paths that have a critical $(n+1)$-, $n$- and $(n-1)$-simplex. If there are no critical $n$-simplices or $(n-1)$-simplices, then $\partial^2\alpha=0$ since there are no critical flowlines. But usually this is not the case, so we have to do a bit more to definitively prove our case.

So what are the actions we can perform on a flowline? Actually, there are three. We can ‘insert’ a Morse arrow to the intermediate simplex, like so:

In general, insert adds in two Morse arrows either above the intermediate simplex, or below the intermediate simplex, depending on where its Morse arrow is (if it has one at all).

The second action we can perform on a flowline is a `flop’. I showed you earlier that there are two ways down from the 2-simplex to a vertex on its boundary:

The two ways down a triangle.

The two ways down shown on a Hasse diagram

That is, if $ab$ is our intermediate simplex, we can find the other intermediate simplex $ac$ that is a boundary of $abc$ and has $a$ on its boundary. It’s worth noting that if we flop the pink flowline once we obtain the blue flowline, and if we flop the flowline again we will get back to the pink flowline. We can generalise this. If $\beta$ is our intermediate simplex, we can find the other intermediate simplex $\beta’$ that is a boundary of $\alpha$ and has $\gamma$ on its boundary (this $\beta’$ is unique), and flop to this other path, like so:

In general, this flop action is always possible, and self-inverse, since applying it twice gets you back to the same flowline.

The final action is cancel. We can ‘cancel’ out the Morse arrow at an intermediate simplex by removing the redundant path adjacent to the intermediate simplex like so:

In general, we can cancel whenever there is a backwards Morse arrow.

Setting boundaries: know the signs

It’s worth noting that cancel is inverse to insert, so if we devise an algorithm it doesn’t make any sense to apply one after the other. Similarly, going back to our 2-simplex example, we can see that applying flop twice will get us back to the same flowline, so it doesn’t make sense to ever apply two flops in a row.

Ultimately, the algorithm we find looks like this:

The algorithm in action

We sadly can’t apply the algorithm to a critical flowline through the two glued triangles we met earlier, because there aren’t any critical flowlines. In particular, the differential as we define it in discrete Morse theory can only be applied to a critical simplex. In this simplicial complex, since our ingredients are 0-, 1-, and 2-simplices, and if we are dropping two dimensions, we require the existence of a critical 2-simplex in our simplicial complex. One thing we can do, however, is change the Morse function we endow it with in order to remove one of the Morse arrows.

Now our simplicial complex has $abc$, $ac$ and $a$ as critical simplices. The astute reader may have noticed that I’ve highlighted critical simplices. This makes them easier to keep track of for the next step.

Let us start with the easiest critical flowline we can find:

Applying our algorithm to this critical flowline looks like this:

We can see from this that there are an odd number of floperations from start to finish, and that the critical flowlines travel through the simplicial complex in the following way:

The reason that the algorithm terminates at a critical flowline is because the only floperation we can apply to a critical flowline is flop, since there aren’t any Morse arrows to insert or cancel.

Notice that no matter how many times we cycle around each wing of the diagram, we’ll always have an odd number of actions in our algorithm. This is going to be important later when we look at the signs of a flowline. But the punchline of this algorithm is that it terminates at a critical flowline. Not only that, but when we apply the algorithm to a critical flowline, the algorithm is involutive. This means that applying it once to some critical $F’$ gives us a distinct critical flowline $F^{\prime\prime}$, and applying it twice gives us $F’$ again. It is possible to find a flowline through the simplicial complex which isn’t touched by this algorithm, but in this case when we apply the algorithm once and twice to this flowline, we will get a different pair of critical flowlines. In this way, we can partition all flowlines into equivalence classes.

Here, we note three details.

• The algorithm starts with a flowline $F$, and every flowline the algorithm passes through is `equivalent’ to $F$.
• The algorithm terminates when it arrives at a critical flowline.
• There are either 2 or 0 critical flowlines in any given equivalence class.

A consequence of this is that every critical flowline belongs to one equivalence class and there is a unique distinct flowline that is also in that equivalence class.

Another aspect of flowlines that we have only briefly touched on is the concept of a ‘sign’. A flowline can be seen as positive or as negative, and each of the actions of flop, insert and cancel negates the sign of the flowline. We have talked a bit about why this happens for flop, but why do insert and cancel flip the sign of the flowline? Well, the sign of the flowline is calculated in terms of $(-1)$ to the power of half the number of arrows in the flowline. When we insert or cancel, we either increase or decrease the number of arrows by two, meaning that we change the power of $(-1)$ by $\pm 1$.

Applying the algorithm to $F$ gives $F’$…

…then applying the algorithm to $F’$ gives $F^{\prime\prime}$…

To see the space of flowlines, we could, for instance, have a flowline $F$, where applying the algorithm gives $F’$, giving a sequence of flowlines and applying it twice gives $F^{\prime\prime}$, with a possible algorithm step-by-step shown above. In particular, we pass through the same flowlines to get from $F^{\prime\prime}$ to $F’$ as we do to get from the $F’$ to $F^{\prime\prime}$. You may be able to see, now, why the algorithm is involutive.

Now, if $F$ has sign $+$ then we can find the sign of $F’$ and $F^{\prime\prime}$ by looking at the number of actions between them. In particular, we can see that since there is always an odd number of actions (starting with flop and ending with flop) between $F’$ and $F^{\prime\prime}$, they must have opposite signs.

Putting all this information together, we can find an expression for the squared differential in terms of flowlines through a simplicial complex. The squared differential of $\alpha$, some $(n+1)$-simplex, is the signed sum of flowlines $\alpha$ to $\beta$ times the number of flowlines $\beta$ to $\gamma$ for each critical simplex $\gamma$ and each critical $\beta$.

For example, if we imagine a system with one critical $\gamma$ and two critical flowlines $F’$ and $F\prime\prime$ through the system, then
\[\partial^2\alpha=\hspace{-3mm}\sum_{\substack{\text{critical}\\\text{flowlines }F}}\hspace{-3mm} \operatorname{sign}(F)=\operatorname{sign}(F’)+\operatorname{sign}(F^{\prime\prime}).\]
But since $F’$ and $F^{\prime\prime}$ are the only two critical flowlines, they belong to the same equivalence class, and therefore are related by an odd number of operations so that $\operatorname{sign}(F’)=-\operatorname{sign}(F^{\prime\prime})$. Then $\partial^2\alpha=0$.

In general, as critical flowlines come in these pairs, everything reduces to this case, so that $\partial^2\alpha=0$ in any simplicial complex. This is a very satisfying way to see a proof of this geometric identity. In the smooth case, we have better contact with the manifolds themselves, but it’s not so easy to actually compute things. When we see what surfaces look like in this combinatorial abstraction of the problem, we pinpoint the aspects of the space we really care about. And, as we have seen, by abstracting the problem to this discrete analogue, everything cancels out so nicely. So if you’re ever feeling down, just remember that you can always abstract your problems. Discrete mathematics really does come through.

Sophie Bleau

Sophie is a PhD student at the University of Edinburgh. She spends her free time drawing Celtic knots and fangirling over Leonardo da Vinci. She has a cat called Norma, who takes great pleasure disrupting any mathematical progress—usually by sitting on it.

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