Besides understanding the proof of the fundamental theorem of geometric group theory, figuring out how to classify the homeomorphisms of the torus is one of the first exercises grad students in my field do. It involves lots of matrix multiplication and remembering some facts from linear algebra, which we’ll sorta brush over (mostly because I don’t remember anything from my linear algebra class besides that Stephen Goode always said bee-ta and once told us that if we left his class saying bee-ta then we’ll have learned something. I’d never heard it before so I had no idea what he was talking about until a few years later when I said bee-ta and everyone was like uh you’re not European/Australian… Long story short, Americans say bay-ta for ).
So.. what’s a homeomorphism of the torus? Well we know what the torus is: a donut! Mmmm donuts. But sometime’s it’s hard to do math on a three-dimensional donut (it’s easy to forget where things connect up on the backside) so we often unfold the torus into a square and draw things on the square instead of on the torus. We’ve seen this picture before:
And the idea is that it describes the torus. You glue the red edges together so the arrows line up, and you have a cylinder. Then glue the blue edges (which are now the ends of the cylinder) together so the arrows line up, and you have a torus. Here’s a cartoon:
And here’s some pictures of me doing this in real life:
Visualizing abstractions is one of the hardest skills in math (I think so anyway) but once you’ve started it, it’s difficult to remember that people don’t look at flat squares and see donuts right away. So hopefully this helped you!
Why would we rather use a torus drawn as a square rather than using a 3D model? Well, for one, it’s way easier to draw a square than a torus. But it’s also much easier to mathify things we’re interested in on a square than on a torus: for instance, I can draw a curve on my torus…
Or I can use math to describe it. Notice that the string goes through the hole 3 times (easy to see) and around the hole twice (a little harder to see). So it goes around the skinny side (the longitude) three times and around the flatter side (the latitude) two times. Let’s name this the (3,2)-torus knot. Just like with the curve complex, we don’t care if we wiggle the string a little bit (isotopy).
There’s another cool thing we can do with a square: tile the plane!
I know it doesn’t seem that cool but it is! If I identify every single square with the torus I had before, I can actually draw the (3,2) curve. Here it is drawn on just one square, and then drawn on the big grid:
It might take a little while to understand this picture. On the left, start in the lower left corner. Follow strand 1 and you hit a point about 1/3 the way down the right side. Since the blue arrows are identified, this is the same as the point about 1/3 the way down the left side, which means we go to strand 2. Similarly, the middle of the top and middle of the bottom are identified, so we get to strand 3, and finally strand 4 finishes the loop.
Instead of doing that tracing, we could draw the (3,2)-knot on the tiled plane, so that it goes over 3 times and up 2 times. So we can see this loop as a line with slope 2/3 in the plane, which is way easier to draw than the picture on the left. I added in all the lines on the right to show how the left and right pictures are related.
To summarize so far: a torus is a donut, and we can think of it as embedded in real space (an actual donut) or more abstractly as a square with the sides identified. The plus of the flat picture is that it’s easier to draw, and it can tile the plane, which led to us associating loops of the torus with lines of fractional slope in the plane, just like our bagel loop was the line with slope 2/3.
OK, onto the homeomorphisms part. A morphism is a map that assigns points in $X$ to points in $Y$. For instance, the yellow pages describe a morphism from X=People to Y=Addresses. Yes, I did just refer to the yellow pages. No, I don’t know if they still exist. Deal with it.
A homeomorphism, then, is a special kind of morphism. It goes from one topological object back to itself, so Y=X, and it has an inverse. In our example, there’s no inverse because if I give you an address, you might give me five people back, while if you give me a person, I’ll give you just one person back. To have an inverse, your map has to be one-to-one, which means exactly what it sounds like: each person in X gets exactly one address in Y.
There’s one more property that homeomorphisms have to have: continuity. Roughly, continuity says that if you wiggle around a little bit in $X$, then you only wiggle a little bit in $Y$. The yellow pages map is super not-continuous, because if you go to your alphabetical neighbor in $X$ you might have to cross all of town in $Y$ to find his address. This graph is an example of a function which is continuous:
The topic of this series of posts will be homeomorphisms of the torus: invertible, continuous maps from the donut to itself. Next time we’ll talk about matrices (ooh goody) and how they have to do with these homeomorphisms, and then eventually we’ll classify all homeomorphisms by what kind of matrix they are.