Topology 2: Homeomorphisms

Last time, I talked about how to think of two shapes being the same. If we can deform one into another, they’re homeomorphic! Seems easy enough? Let’s go through some examples of shapes that may not be obviously homeomorphic, or maybe shapes that you thought were the same but are actually different (at least for homeomorphisms)

1. A Triangle and ‘A’

The verdict is in and the answer is nope, they’re not the same shape! “But Osama”, you ask, “can’t you just smush the legs of the A in? You could if it was clay”. This is one of those instances where it’s best to think of rubber and not clay. If you had a rubber band that looks like an A (so a regular rubber band with two arms), you could never stretch or compress it in a way that looks like a triangle, the two legs would always smush out! You could get them really really small, but they’re still there.

Trick: How can you test this in the future to not be fooled? Maybe you think χ will come to your rescue, but try it. Both of their χ’s are 0! The hint is to remove a point. If two shapes are homeomorphic, then removing a point from each should yield homeomorphic shapes! Watch how this fails below

We have two connected pieces in the first one but only one connected piece in the other! This would have happened if I had chosen any point on the triangle! So, there is no way to remove a point on the triangle and not be connected, but there is on the ‘A’! This means they’re not homeomorphic.

Or, you could note that after removing the point, the χ’s are different! The broken ‘A’ has a χ of 2 and the broken triangle has a χ of 1. Sometimes you gotta be sneaky to use χ!

2. An Annulus and a Cylinder

The area between two disks is called an annulus. Turns out it’s homeomorphic to a cylinder! How’s that? Well if you had a rubber cylinder and flattened it inside out, you’d get an annulus! Or if you push the sides of the outer circle in the annulus up, out of the plane, you’d get a cylinder!

3. Cylinder and Mobius Strip

Here’s a new one. A Mobius strip is what you get when you put one twist in a cylinder. How are you gonna tell one of these apart? Here’s an experiment.
Imagine you’re an ant walking on the cylinder. If you walk on the outside of the cylinder and walking in a straight line around the middle, you’ll always stay on the same side of the cylinder. But on a Mobius strip, try the same thing, you end up flipping onto the other side, and then if you take another full walk, you flip back onto the same side. This is weird, how do you even tell the inside from the outside of a Mobius strip?
Answer: You can’t! The Mobius strip only has one side! This is called being non-orientable, there is no inside or outside. Being orientable is an invariant, so homeomorphic objects are either both orientable or both not. After all, you couldn’t ever put a half-twist in a cylinder without cutting and gluing(…)

4. Mobius strip(s)

(…) Okay, I lied. How could the 3 and 1 half-twists be the same if you can only introduce the twists by cutting?

I know I said in the last post that cutting is illegal, it’s non-continuous and you couldn’t blow up or deform an object to cut it? It turns out cutting is legal but in a very very specific sense. You can cut in a small area, modify your object, and glue back as long as you reglue the same way right after. What do I mean?

Let’s try removing the half-twists in any twisted-cylinder. The most “basic” one is the basic cylinder. I cut, there’s nothing to unfurl, and so I reglue with the edges oriented the same way. Great, I’ve shown that a cylinder is the same as a cylinder…
But let’s try the same with the Mobius strip. If you cut and unfurl, your edges are in opposite directions this time. I can’t reglue! So you can’t turn it into a cylinder. What you can do is twist it twice more, putting three half-twists in and then the edges line up, and you can reglue! This means one half twist is the same as three half-twists. And zero is the same as two. So basically, if you have a twisted cylinder, it’s either a real cylinder or a Mobius strip! If the number of half-twists is even, it’s a real cylinder, and if the number of half-twists is odd, it’s a Mobius strip.

5. A blob and all of ℝ² (infinite plane)

Here’s one that may seem like a bit of a stretch (get it)? You can literally infinitely stretch your rubber 2D blob (or clay) and get an infinite plane.

6. A Sphere Minus a Point and an Infinite Plane

Now we look a sphere minus a point, and I’m telling you it’s an infinite plane. How? Imagine you put your fingers insider that hole and stretch the remaining part of the sphere apart, as far as you can. If you burst a balloon (a sphere minus a hole), you basically end up with a blob which you can stretch to infinity. What does this look like? There’s a really neat way called stereographic projection. You take a line through the missing point through any other point, and watch where it intersects the plane. This is where you stretch it to!

7. Knots

In math, knots are loops that begin and end at the same point. So for example, a shoelace knot isn’t really a knot, but if you glued the ends of both laces together, it would be!

It’s surprising then, isn’t it, that all knots are homeomorphic! How could that be? Well, take any knot, cut it anywhere, untangle the whole knot, and then reglue. There’s only “one way” to glue a point to a point, so you don’t have to worry about matching edges.

8. Hopf Link and Disjoint Circles

Okay these are obviously different! One pair of circles is linked and the other isn’t. But nope, they are homeomorphic.
It’s a shame, maybe algebraic topology is too dumb to tell obviously different things apart. But no! You just have to be tricky. Maybe you shouldn’t study the inside of these knots, but what if you studied the outside? Everything in 3D minus the knots. It turns out, these “knot exteriors” are not homeomorphic! It’s hard to visualize and it’s not obvious why these are not homeomorphic, but don’t worry, just because two objects seem homeomorphic doesn’t mean that you don’t have other ways to tell the shapes apart!

I hope I didn’t confuse you too much! Maybe you’re starting to think that homeomorphisms are too weak, so many different objects look the same. But don’t despair, this is far from the truth. You do know what a homeomorphism intuitively is! I’ve just added a new rule. You can

1. Stretch, very very far (infinitely far) [look at example 2, 5, 6]
2. Compress, very very small (infinitely small, but never 0) [look at example 1]
3. Cut, but only if you reglue the same way [3, 4, 7, 8]

And that’s it! No more weird caveats, I promise.

As always, a little Exercise: What are all the letters that are homeomorphic? So, “up to homeomorphism”, how many different shapes of letters are there?