Recently in between studying for my finals I’ve been learning about a whole load of different things in algebraic geometry, a main one being cohomology in its various forms (sheaf, Čech, de Rahm, … even a bit of étale cohomology using sites).

The general idea of cohomology that I have at the moment is this:  we could say that, at least in part,

modern geometry  studies “spaces” by understanding functions on those spaces

where “space” could mean “variety”, “scheme”, “manifold”, “site”, and many other things. Cohomology is a way of understanding the functions on spaces in a pretty high-level way. It’s a way of going from local information about functions defined on neighbourhoods of points to global information about how we can extend those functions to global functions. The geometry of the “space” dictates whether functions can be “extended”, and cohomology is a way for us to unpack this information. Therefore we could say

cohomology is the study of global properties of a space by looking at local-to-global obstructions.

The words “local-to-global” should immediately bring to mind sheaves, and indeed cohomology basically seems to be the measure of losing exactness of sequences of sheaves

$0\to F\to G\to H\to 0$

after taking global sections:

$0\to F(X)\to G(X)\to H(X)$

since the map on the right might not always be surjective. Sheaf cohomology seems to be the most general way (so far, to my understanding) of measuring these obstructions – and on suitably nice spaces, sheaf cohomology starts to agree with its more easily-calculable mates.

Anyway, I don’t want to talk so much about cohomology right now. Instead I wanted to briefly point out something cool on the Čech cohomology Wikipedia page. They have a picture there of a Penrose triangle:

By Tobias R. – Metoc – Own work, CC BY-SA 2.5, https://commons.wikimedia.org/w/index.php?curid=2520370

Obviously this is the famous example of an “impossible object”: one that cannot exist in our 3D space. Things like this have delighted loads of people over the ages, and arguably great artists like M.C. Escher made careers out of exploiting these weird objects.

After spending ages working on cohomology in the abstract, I was amazed to hear that the “impossibility” of this Penrose triangle can be described by cohomology! This nice article by Penrose himself explains how the “impossibility” of the figure arises in the following way: we could chop up the figure into three pieces, each containing a corner of the triangle. By doing this we get three absolutely standard “corner pieces” $C_1, C_2, C_3$ that have no paradoxical qualities. The issue arises in gluing these pieces together, because we don’t have any information about the distance each piece is from the observer.

To explain what happens in the article, let’s fix two points $P_{i, j} \in C_i$ such that when the corners are laid out as if to be glued, e.g. $P_{1,2}$ is superimposed over $P_{2,1}$: in general $P_{i,j}$ and $P_{j,i}$ are the same to the observer except for the fact that they may be a different distance away. Let’s let $O$ denote the observer and

$d_{i j} = \frac{\text{distance}(O, P_{i,j})}{\text{distance}(O, P_{j,i})}$.

The ratios $d_{i,j}$ don’t depend on the actual distance the points are from the observer – only their ratios of the points with each other. By moving a one of the corners $C_i$ towards/away from the observer by a certain distance, we change the ratios $d_{i,j}$ and $d_{i,k}$ by the same factor $\lambda >0$. Therefore if the Penrose triangle could be realised as a 3D object it would be possible to simultaneously reduce each $d_{i,j}$ to 1 by transformations of this type. That would mean the cocyle

$(d_{1,2}, d_{2,3}, d_{3,1})$

would be a coboundary, and would therefore be trivial in the first cohomology group of the annulus that surrounds the figure. But this is a nontrivial group, so not all cocycles are coboundaries, and it turns out this one is nontrivial.

So the nontriviality of the first cohomology group with real coefficients (distances from the observer) means that there are certain shapes, like the Penrose triangle, that cannot actually exist! The problem was that we had a shape (not the Penrose triangle, but the annulus surrounding it) and we had three open sets on it, each with a “corner” of the Penrose triangle drawn on it. Each of these open sets also had a function “distance from the observer” defined on it. The topology of the shape prevented us from being able to glue these functions together to get a globally consistent distance function. Which is precisely what causes the paradox with the Penrose triangle – at any point it is locally a consistent shape, but our brains cannot “glue” the distance information together to get a global shape that actually exists.