I still owe a Part III for the explanation of the linear algebraic classification of homeomorphisms. But let’s take a step away from linear algebra and look at shapes (my favorite!)
We know what homeomorphisms are (continuous functions with continuous inverses), with the famous example in the picture below: a coffee cup turns into a donut and vice-versa.
In fact, this pictures doesn’t just show us the homeomorphism (which says where each point of the coffee cup gets sent to in the torus and vice-versa). It also shows us a homotopy (remember the definition from this post)- essentially, because we can see it traveling through time and back, it’s a homotopy. And in fact, this homotopy is an isotopy– a type of homotopy where at each frozen point in time, the image is homeomorphic to what we started with. An example of a homotopy which is not an isotopy is the map that ends up sending , where x is a real number. Homotopies take place over time, so I would actually write this map as . So when t=0, we have x mapping to x, and when t=1, x maps to -x. One reason this isn’t an isotopy is because when t=1/2, all of the real line gets mapped to the point 0. And mapping everything to 0 isn’t a homeomorphism (what would the continuous inverse be?)
A big part of geometric group theory is using shapes to come up with algebraic theorems, and using algebra to come up with shapes. One thing you can do (which we’re doing RIGHT NOW!) is take a shape, do some algebra, and then make a new shape. To be specific, our first shape is the torus. Our algebra was figuring out the group of homeomorphisms of the torus, also written as Homeo(T)- T for torus. Sometimes you’ll see , to specify that we’re talking about the 2-torus rather than a higher dimension (more on higher dimensional tori later. Isn’t it cool that the plural of torus is tori? Pronounced tor-eye.) Now we’re going to make a new shape from this group of homeomorphisms.
We’ll only consider homeomorphisms isotopic to the identity, written as . Starting in 1962 and finishing in 1965, badass Mary-Elizabeth Hamstrom proved in a series of papers that is contractible (homotopic to a point) if X is a two-manifold with a short list of exceptions [torus, sphere, plane, disk, annulus, disk with a hole in it, plane with a hole in it.]
Let’s look at our current favorite from this list, $Homeo_0(T^2)$. If we start from the identity, which homeomorphisms can we isotope to? Well, I can rotate my torus around its hole-axis, and that ending homeomorphism is definitely isotopic to the identity (the rotation through time is the isotopy; where the points end up is the ending homeomorphism).
Since I can rotate by all the degrees up to 360, which brings me back to the identity, this means that contains a circle- each point on the circle represents rotating the torus by that many degrees.
What else can I do that’ll be isotopic to the identity? I can rotate the torus around its center circle (running through the middle of the donut), like if I was wringing out a towel.
Again, I can do this for 360 degrees before coming back to the identity. So there’s another circle, different from the first one, in .
I can also do any combination of these two: I can rotate 27 degrees around the hole-axis, and then 78 around the center circle. This is true for any numbers between 0 and 360, but then 0 and 360 are the same for both. So far, we have the picture on the left. I colored it to indicate that 0=360 on both axes. Look familiar?
Notice that all four corners are the same homeomorphism: the identity can be had by rotating by 360 degrees in either direction, or in both directions, or by doing nothing. So we’ve shown that actually contains a torus! This is cool because all those other were contractible. In fact, is a torus- we’ve actually described all the homeomorphisms of the torus which are isotopic to the identity- combinations of rotations around the center axis and around the center circle.
Personally I find this much more exciting than classifying the homeomorphisms by trace (yeup that’s happening in Part III, nope Part IV is coming out before Part III and you’re going to like it), probably because it involves shapes rather than numbers.
Update on health: I’m taking antibiotics to help with whooping cough. So that explains why I’ve been sick for a month. My boyfriend walked by and asked what I was doing an hour ago, and I told him I was very busy feeling sorry for myself. Then I wrote this blog post to be less lump-ish.