Cohomology Intuitively

01 Mar 2021 - Tags: cohomology-intuitively , sage , featured

So I was on mse the other day…

Anyways, somebody asked a question about finding generators in cohomology groups. I think understanding how to compute the generators is important, but it’s equally important to understand what that computation is doing. Regrettably, while there’s some very nice visual intuition¹ for homology classes and what they represent, cohomology groups tend to feel a bit more abstract.

This post is going to be an extension of the answer I gave on the linked question. Cohomology is a big subject, and there were a lot of things that I wanted to include in that answer that I didn’t have space for. A blog post is a much more reasonable setting for something a bit more rambling anyways. That said, everything contained in that answer will also be discussed here, so it’s far from prerequisite reading.

In particular, we’re going to go over Simplicial Cohomology², but we’ll steal language from De Rham Cohomology, and our first example will be a kind of informal cohomology just to get the idea across.

There’s a really nice example that Florian Frick gave when I took his algebraic topology class, and it was such good pedagogy I have to evangelize it. The idea is to study simplicial cohomology for graphs – it turns out to say something which is very down to earth, and we can then view cohomology of more complicated simplicial complexes as a generalization of this.

Graphs are one of very few things that we can really understand, and so using them as a case study for more complex theorems tends to be a good idea. As such, we’ll study what cohomology on graphs is all about, and there will even be some sage code at the bottom so you can check some small cases yourself!

With that out of the way, let’s get started ^_^

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First things first. Let’s give a high level description of what cohomology does for us.

Say you have a geometric object, and you want to define a function on it. This is a very general framework, and “geometric” can mean a lot of different things. Maybe you want to define a continuous function on some topological space. Or perhaps you’re interested in smooth functions on a manifold. The same idea works for defining functions on schemes as well, and the rabbit hole seems to go endlessly deep!

It’s often the case that you know what you want your function to do somewhere (that’s why we want to define it at all!), and then you would like to extend that function to a function defined everywhere.

It is also often the case that the continuity/smothness/etc. constraint means that there’s only one way to define your function locally. So it should be “easy” (for a certain notion of easy) to do the extension in a small neighborhood of anywhere it’s already been defined.

However, sometimes the geometry of your space prevents you from gluing all of these small easy solutions together. You might have all of the pieces lying around to build your function, but the pieces don’t quite fit together right.

Obviously the above argument isn’t entirely rigorous. That said, it does a good job outlining what problem cohomology solves. We had only one choice at every step, and at every step nothing could go wrong. Yet somehow, when we got back where we started, our function was no longer well defined. We thus come to the following obvious question:

It turns out the answer is yes! Our local solutions come from a global solution exactly when the “cohomology class” associated to our function vanishes³.

There’s a bit of a zoo of cohomology theories depending on exactly what kinds of functions you’re trying to define⁴. They all work in a similar way, though: your local solutitions piece together exactly when their cohomology class is $0$. So the nonzero cohomology classes are all of the different “obstructions” to piecing your solutions together. Rather magically, cohomology groups tend to be finite dimensional, and so there are finitely many “basic” obstructions which are responsible for all the ways you might fail to glue your pieces together.

De Rham Cohomology makes this precise by looking at certain differential equations which can be solved locally. Then the cohomology theory tells us which differential equations admit a global solution. In this post, though, we’re going to spend our time thinking about Simplicial Cohomology. Simplicial Cohomology doesn’t have quite as snappy a description, but it’s more combinatorial in nature, which makes it easier to play around with.

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Actually setting up cohomology requires a fair bit of machinery, so before go through the formalities I want to take a second to detail the problem we’ll solve.

Take your favorite graph, but make sure you label the vertices. My favorite graph (at least for the purposes of explanation) is this one:

Notice the edges are always oriented from the smaller vertex to the bigger one. This is not an accident, and keeping a consistent choice of orientation is important for what follows. The simplest approach is to order your vertices, then follow the convention of $\text{small} \to \text{large}$

Now our problem will be to “integrate” a function defined on the edges to one defined on the vertices. What do I mean by this? Let’s see some concrete examples:

Here we see a function defined on the edges. Indeed, we could write this more formally as

\[\begin{aligned} f(e_{01}) &= 5 \\ f(e_{02}) &= 5 \\ f(e_{12}) &= 0 \end{aligned}\]

The goal now is to find a function on the vertices whose difference along each edge agrees with our function. This is what I mean when I say we’re “integrating” this edge function to the vertices. It’s not hard to see that the following works:

Again, if you like symbols, we can write this as

\[\begin{aligned} F(v_0) &= 3 \\ F(v_1) &= 8 \\ F(v_2) &= 8 \\ \end{aligned}\]

Then we see for each edge $f(e_{ij}) = F(v_j) - F(v_i)$. This may seem like a weird problem to try and solve, but at least we solved it! Notice we pick up an arbitrary constant when we do this – We can set $v_0 = C$ for any $C$ we want as long as $v_1 = v_2 = C+5$. This is one parallel with integration, and helps justify our language.

As some more justification, notice this obeys a kind of “fundamental theorem of calculus”: If you want to know the total edge values along some path, \(\displaystyle \sum_{v_{k_1} \to v_{k_2} \to \ldots \to v_{k_n}} f(e_{k_i, k_{i+1}})\), that turns out to be exactly $F(k_n) - F(k_1)$ for some “antiderivative” $F$ of $f$.

Let’s look at a different function now:

You can quickly convince yourself that no matter how hard you try, you can’t integrate this function. There is no antiderivative in the sense that no function on the vertices can possibly be compatible with our function on the edges.

After all, say we assign $v_0$ the value $C$. Then $v_1$ is forced into the value $C+5$ in order to agree with $e_{01}$. But then because of $e_{12}$ we must set $v_2 = C+5$ as well, and uh oh! Our hand was forced at every step, but looking at $e_{02}$ we see were out of luck.

This should feel at least superficially similar to the $\sqrt{\cdot}$ example from earlier. At each step along the way, it’s easy to solve our problem: If you know what $F(v_i)$ is, and you see an edge $v_i \to v_j$, just assign $F(v_j)$ to $F(v_i) + f(e_{ij})$. The problem comes from making these choices consistently, which turns out to not always be possible!

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Ok, so now we know what problem we’re trying to solve. When can we find an antiderivative for one of these edge functions? The machinery ends up being a bit complicated, but that’s in part because we’re working with graphs, which are one dimensional simplicial complexes. This exact same setup works for spaces of arbitrary dimension, so it makes sense that it would feel a bit overpowered for our comparatively tiny example.

First things first, we look at the free abelian groups generated by our $n$-dimensional cells. For us, we only have $0$-dimensional vertices and $1$-dimensional edges. So we have to consider two groups:

\[\mathbb{Z}E \text{ and } \mathbb{Z}V\]

For the example from before, that means

\[\mathbb{Z} \{ e_{01}, e_{12}, e_{02} \} \text{ and } \mathbb{Z} \{ v_0, v_1, v_2 \}\]

which are both isomorphic to $\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$.

Second things second. We want to connect these two groups together in a way that reflects the combinatorial structure. We do this with the Boundary Map $\partial : \mathbb{Z}E \to \mathbb{Z}V$. This map takes an edge to its “boundary”, so $\partial e_{01} = v_1 - v_0$. Since we have a basis floating around anyways, it’s convenient to represent this map by a matrix:

\[\partial = \begin{pmatrix} -1 & 0 & -1 \\ 1 & -1 & 0 \\ 0 & 1 & 1 \end{pmatrix}\]

So for instance,

\[\partial e_{01} = \begin{pmatrix} -1 & 0 & -1 \\ 1 & -1 & 0 \\ 0 & 1 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} -1 \\ 1 \\ 0 \end{pmatrix} = v_1 - v_0\]

Now our groups assemble into a Chain Complex:

\[\cdots \to 0 \to 0 \to \mathbb{Z}E \overset{\partial}{\longrightarrow} \mathbb{Z}V\]

The extra groups $0$ correspond to higher dimensional simplices that aren’t present for us. If we filled in our cycle with a $2$-dimensional triangular face, then we would have an extra group $\mathbb{Z}F$ and an extra map (which is also called $\partial$, rather abusively) from $\mathbb{Z}F \to \mathbb{Z}E$ which takes a face to its boundary (this might also help explain the term “boundary”). Then if we had a “cycle” of faces, we could fill them in with a (solid) tetrahedron. So we would have a new group $\mathbb{Z}T$, equipped with a map $\partial : \mathbb{Z}T \to \mathbb{Z}F$ taking each tetrahedron to its boundary of faces. Of course, this goes on and on into higher and higher dimensions⁵.

There’s actually a technical condition to be a chain complex that’s automatically satisfied for us because our chain only has one interesting “link”. Given an $n+2$-dimensional simplex $\sigma$, we need to know that $\partial \partial \sigma = 0$. I won’t say much more about it now, but I might write a blog post giving examples of higher-dimensional simplicial cohomology at some point. When that happens, we’ll have no choice but to go into more detail.

So we know that elements of $\mathbb{Z}E$ (resp. $\mathbb{Z}V$) represent (linear combinations of) edges (resp. vertices). Of course, we want to look at functions defined on the edges and vertices. So our next step is to dualize this chain:

\[\cdots \leftarrow \text{Hom}(0, \mathbb{R}) \leftarrow \text{Hom}(0, \mathbb{R}) \leftarrow \text{Hom}(\mathbb{Z}E, \mathbb{R}) \overset{\partial^T}{\longleftarrow} \text{Hom}(\mathbb{Z}V, \mathbb{R})\]

We’re now looking at all (linear) functions from $\mathbb{Z}E \to \mathbb{R}$ (resp. $\mathbb{Z}V \to \mathbb{R}$). By the universal property of free abelian groups, though, we know that the functions $\mathbb{Z}E \to \mathbb{R}$ correspond exactly to the functions $E \to \mathbb{R}$ (extended linearly).

Moreover, our boundary operator $\partial$ has become a coboundary operator $\partial^T$ that points the other direction⁶. Here if $F : V \to \mathbb{R}$ then we define $\partial^T f : E \to \mathbb{R}$ by

\[(\partial^T F) (e) = F(\partial e)\]

Moreover, our notation $\partial^T$ is not misleading. $\text{Hom}(\mathbb{Z}V, \mathbb{R})$ has a basis of characteristic functions

\[\{ \chi_{v_0}, \chi_{v_1}, \chi_{v_2} \}\]

where

\[\chi_{v_i}(v_j) = \begin{cases} 1 & i=j \\ 0 & i \neq j \end{cases}\]

Similarly $\text{Hom}(\mathbb{Z}E, \mathbb{R})$ has a basis of characteristic functions, and it turns out that, with respect to these “dual bases”, the map $\partial^T$ is actually represented by the transpose of $\partial$!

We’re in the home stretch! The second half of that box alludes to something important: A function $f$ can be integrated exactly when it is in the image of $\partial^T$. With this in mind, we’re finally led to the definition of the cohomlogy group of our graph:

Since the only map $0 \to \mathbb{R}$ is the trivial one, we can rewrite our complex as:

\[\cdots \overset{0}{\leftarrow} 0 \overset{0}{\leftarrow} 0 \overset{0}{\leftarrow} \text{Hom}(\mathbb{Z}E, \mathbb{R}) \overset{\partial^T}{\longleftarrow} \text{Hom}(\mathbb{Z}j, \mathbb{R})\]

Then we define⁷

\[H^1 = \frac { \text{Ker}\big ( \partial^T : \text{Hom}(\mathbb{Z}E, \mathbb{R}) \to 0 \big ) }{ \text{Im} \big ( \partial^T : \text{Hom}(\mathbb{Z}V, \mathbb{R}) \to \text{Hom}(\mathbb{Z}E, \mathbb{R}) \big ) }\]

Since there are no two dimensional faces, $\partial^T : \mathbb{Z}E \to 0$ is the trivial map, and so its kernel is everything. In light of this, we see a slightly simpler definition of $H^1$:

\[H^1 = \frac { \text{Hom}(\mathbb{Z}E, \mathbb{R}) }{ \text{Im} \big ( \partial^T : \text{Hom}(\mathbb{Z}V, \mathbb{R}) \to \text{Hom}(\mathbb{Z}E, \mathbb{R}) \big ) }\]

This says the elements of $H^1$ are exactly the functions on edges, but we’ve quotiented out by all the functions that we can integrate to the vertices! So if we want to check if a function can be integrated, we just compute its cohomology class and check if it’s $0$.

Moreover, the basis of $H^1$ as an $\mathbb{R}$-vector space give us a collection of “basic” non-integrable functions. Then every function on the edges can be written as an integrable one, plus some linear combination of the basic nonintegrable ones. This dramatically reduces the number of things we have to think about! From the point of view of integration, we only need to worry about the “good” functions (which admit antiderivatives) and (typically finitely many) “bad” functions which we can handle on a case-by-case basis.

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We’ve spent some time now talking about what cohomology is. But again, part of its power comes from how computable it is. Without the exposition, you can see it’s really a three step process:

1. Turn your combinatorial data into a chain complex by taking free abelian groups and writing down boundary matrices $\partial$.

2. Dualize by hitting each group with $\text{Hom}(\cdot, \mathbb{R})$

3. Compute the kernels and images of $\partial^T$, then take quotients.

Steps $1$ and $2$ should feel very good, and hopefully you’re aware that taking kernels and images of a linear map should be easy (even if I know I’m pretty rusty). It turns out computing a vector space quotient is also easy, though it’s much less widely taught. That doesn’t matter, though, since sage absolutely remmebers how to do it.

Since it’s so computable, and the best way to gain intuition for things is to work through examples, I’ve included some code to do just that!

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