Topology 4: Homology
Hi all! It’s been a while since I’ve written. Truthfully, I’ve been wanting to introduce some “advanced” concepts while keeping the material readable. Today we’re going to talk about homology! Homology is a tool we use to count holes in a space. It’s a very smart way to count holes, because as we’ll see, defining a hole is kind of hard. Unfortunately, the “output” of homology are groups or vector spaces. But we’ll deal with this by only looking at the “rank” of the group or vector space, which is a number. This will tell us how many holes we have!
So what is a hole?
It’s kind of hard to define what a hole is. How do you define the absence of something? What is even absent? The space? Let’s try the simplest holes — just removing a point.
On the left, we removed a point from the plane (R² \ (0,0)), and we have an obvious hole
On the right, we removed a point from the circle, and note that you can unravel it and get all of the real number line! In other words, (S¹\point) is homeomorphic to R. Removing a point changes the shape of something (obviously). But removing a point did not cause a hole! R is connected so there’s no holes there.
Weird side fact: You would probably think of the circle as something finite. But once you remove a point, you can get something homeomorphic to something very infinite. What happened? How did removing a point make something bigger? [The topological idea related to this weird notion is called compactness].
So, removing points does not always create a hole. But what about the opposite question? Are all holes created by just removing a point?
Sadly, no. Think about the hole inside a ball (i.e. the hole inside S²). This wasn’t made by removing a point. Now what?
Cycles and Boundaries
Let’s return to R²\(0,0). Here’s one way to consider the difference between R² and R²\(0,0).
Consider both spaces, R² and R²\(0,0). If I put a circle in each (green), then note that I can fill in the space on the right, but I can’t really fill in the space on the left. You might be asking “what do you mean, Osama? I can fill it in the same way”. Sure, but you can see there’s something weird in the way you would fill on the left if you wanted, it doesn’t make a “nice” shape. I know this is subjective. Technically, what I want is the “filling” in to be possible using (solid/filled-in) triangles. “But Osama, the circle is curved, you can’t fill it with triangles”. Okay you got me, but something homeomorphic to a triangle (so disks, squares etc.) works too. So for example, I could fill the circle like this:
Here are two ways of fitting something homeomorphic to a triangle in a disk! It may be hard to describe why, but intuitively, you note that you couldn’t fit the triangles in the R²\(0,0) picture.
Why do we use triangles? We actually use something called simplices. These are the “simplest” shapes in any dimension. A 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a filled-in triangle, a 3-simplex is a solid tetrahedron, and so on and so forth. An n-simplex has n+1 points and all possible edges, sides, faces etc are filled in. Note that no simplices have holes (we fill everything possible in!). We denote the n-simplex by a triangle and n (see below).
Now, let me rephrase. We know that R² \ (0,0) has a hole because I cannot fill a circle with a bunch of 2-simplices. Any attempt will accidentally “cover” the (0,0)!
So, the big idea to catch a hole is: take closed things (like circles) and see if you can fill them in a nice way. If you can’t, there’s a hole there!
These closed objects are called cycles. If a cycle can be filled, it’s called a boundary, clearly because it’s the boundary of the filled in shape. So to measure a hole we look at cycles that are not boundaries. This is the big idea (and almost the definition) of homology!
More Details on Cycles and Boundaries
The definition I’ve given is pretty general and doesn’t mean anything right now. Let’s try again. Let us define an n-chain as the sum of n-simplices. What do I mean by sum? I mean that you just put them together. For example, below I have a 1-chain given by taking four 1-simplices and putting them together, this is a + b + c + d. I also have a 2-chain given by taking two 2-simplices and putting them together, i.e. this is A + B.
If a chain has no boundary, we call it a cycle. For example, the left has no boundary so it is a cycle. The right is not a cycle. The boundary of that filled in thing are the sides, i.e. the left 1-chain! In other words, the left side is not only a cycle, it’s also a boundary! This is basically the same 1-chain and 2-chain as in the R² figure, since a square is homeomorphic to a circle and the filled in square is homeomorphic to a disk.
Note that if you take the boundary twice you get nothing! The 2-chain on the right, if you take the boundary, you get the 1-chain on the left. If you take the boundary again, it’s empty!
If you’re getting confused, just keep the circle and filling it in idea in mind! This is the idea of homology anyway.
Looking for Some More Cycles and Boundaries
Let’s considers some cycles and boundaries on the torus now.
I’ve drawn three cycles on the Torus here in red, blue and green. They’re all cycles because they’re circles so have no boundary. Note that you can’t fill in the red and blue circles (this is only the surface of a torus, there is no inside), but you can fill in the green one, so the dark green cycle is a boundary. This means the red and blue cycles actually represent holes but the green one doesn’t. In fact, these red and blue cycles are pretty much the only type of holes on the torus.
Defining Homology (For Real)
Here is the part I’ve been shying away from. Given a space X, we have a series of vector spaces (think R²) for each dimension.
Do not worry about the “division”. We only care about the idea, which is the following. You take all possible p-cycles inside the space X, see which ones are boundaries, and get rid of them. What you are left with is called the p’th homology group of X.
We associate one copy of R to each hole. So for example, for the Torus, we found 2 holes using 1-cycles, so H¹(T²) = R². We are saying that there are 2 cycles that are not boundaries.
This may seem very abstract right now, and also, who is drawing triangles and stuff? I’ll give you a feel of what H⁰, H¹ and H² mean, since these have easier meanings.
Also, note that since there are no simplices higher than the dimension of the space, you only have to consider up to the max dimension of the space. A sphere is a surface so dimension 2, so you only have to look at H⁰, H¹ and H².
What is H⁰(X)?
p = 0 means 0-simplices i.e. points.
The green point is a 0-simplex. It is a cycle because it’s boundary is well…empty, so it is automatically 0. But a point can’t be a boundary of a 1-chain. Note that every 1-chain either closes up on itself (makes a cycle) or is just a squiggly line so has two points in its boundary. So great! Green point is a cycle that is not a boundary. Are there any other kinds? No. If you had two points, it’s a boundary. If you had three points, you can join two and you have one point which is a boundary, same one as above, and so on and so forth. So there is really only one cycle that is not a boundary. Great! Now I’m tempted to say that H⁰(X) = R¹ for any space! But not quite…
What if your space had two pieces? Like two different disks. Each point by itself is a cycle, same reason as above. But here we have that the sum of the two is also a cycle, and no longer a boundary! There’s no path that goes from one to the other. So we really have 2 cycles that are not boundaries! This leads to our first big result.
H⁰(X) = R^n when there are n different connected pieces. Each copy of R corresponds to one piece.
Restated, H⁰ tells you the number of connected pieces of your space!
What about H¹(X)?
H¹(X) should feel natural at this point! It’s the only example I’ve been using. H¹(X) measures loops that cannot be filled in by a disk. Here’s an example calculation of H¹(X) for a new space.
On S², any loop can be filled in! There are no cycles that are not also boundaries. So H¹(S²) = 0. In fact, this shows that the torus is not homeomorphic to the sphere, since H¹(T²) = R².
Last one — H²(X)
This one is maybe the easiest. 2-cycles are sums of triangles with no boundary. An easy way to get this is to glue triangles together and form a closed shape, like a sphere or a torus!
But wait, aren’t these the boundary of the inside? Yes and no! These aren’t the boundary of the inside because there is no inside in this shape! If I looked at a solid sphere or solid torus, yes it is a boundary. But here I am talking about just the surface. So in fact, the whole surface is a 2-cycle that is not a boundary!
In essence, H²(X) measures the number of voids you can fill in a space by blowing air into it. These voids are enclosed by these 2-cycles! Both the sphere and torus have one void, so H²(S²) = H²(T²) = R.
Again, H²(X) just measures how many “insides” there are.
“Trivial” Exercise
What are all the homology groups of R²? R³? R⁴? Generalize. Recall, you’re looking for cycles that are not boundaries (but can’t you always fill anything in on sheet of paper, in space and higher? And intuitively are there any holes?)
Less trivial exercise
What is H¹(S¹)? What about H²(S²) [we calculated this above]. Do you have any guesses for H^n(S^n)? [you don’t have the tools to calculate this]
Closing Notes
Here’s a quick summary of what the first 3 homology groups mean again.
The big takeaway is the k’th homology measures k-dimensional holes by looking at k-cycles that are not boundaries.
Where to go next? And cool connections!
- H¹(X) is really special, also easy to visualize. It’s all about loops! In fact, there’s a different invariant that calculates 1-dimensional holes for you too, called the fundamental group, denoted 𝜋_1(X). They offer a lot of the same information. That may be a post for another time, because that’s a little easier to grasp than homology.
- If you do H⁰(X)-H¹(X)+H²(X)-H³(X)…you actually get the Euler characteristic (Topology 1) again! That’s another way to calculate this.
- I accidentally used superscripts instead of subscripts throughout this article. Superscripts denote cohomology, and for all the spaces we’ve looked at today, they’re equal, so don’t fret.
- Homology has some very strong connections with vector calculus. If you remember exact vector fields, conservative vector fields, potentials, path independence and so on, they’re all actually just (co)homology.
- In the same spirit, gradient, divergence and curl are are cohomological concepts. In fact, Green’s theorems, Stokes’ theorem and the divergence theorem are all just the following topological statement (which I know doesn’t mean anything now, but is just a very pure and clean topological statement)
