Homology Primer 1: Building with Cells.
December 19, 2010
When I began doing actual calculations for the homology groups, I was surprised by how simple the idea was given the right structures: in the finite case it reduces to finding the kernel and image of a few matrices. This is in direct contrast to how I originally learned the subject: axiomatically. What I’ve described below is how to begin building CW-complexes. These structures are a nice way to begin learning how to find homology groups (in this case, we call them “cellular homology groups”, but they turn out to be equivalent to most of the other homologies you’ve probably heard of in the case of CW-complexes) and are relatively simple and intuitive to describe. Let’s begin by talking about the basic building blocks of CW-complexes: cells.
You may already know what simplices (sometimes spelled “simplexes”) are: they’re things which look like triangles and higher-dimensional triangles: a point, a line, a triangle, a tetrahedron, and so on. If not, think about building a gingerbread house: you can piece together various shapes (usually squares, which are made up of two triangles) to build a nice shape. In much the same way, we can build up many structures using just simplices. We need only require that we paste them nicely and that no more than finitely many attach to anywhere.
Cells are similar to simplices in that they’re essentially disks (filled in circles instead of simplices being filled in triangles) and higher dimensional versions of disks, but using cell structure has the added advantage of being able to piece together these cells in a more flexible way: we no longer need to think about the strict requirements of triangulations, and we need only worry about a less rigid policy. The price we pay for this simplicity is that in addition to cells, we also need to keep track of attaching maps, which tells us how each cell is glued into the structure — and this map is not always so obvious! But it’s this attaching map which allows us to actually compute the homology of certain structures. We’ll talk about these structures (called CW-complexes) after we’ve defined cells.
Definition: An n-cell, for a general , is homeomorphic to the closed -ball . In we can think of this as
In other words, all the points which are less than or equal to a certain distance (in the unit disk’s case, that distance is 1) away from a certain point (like the origin, in this case). Let’s do the first few cases of cells, and then an example of how to glue them together.
First, we have 0-cells. These are just points. There’s not much to say about these. They look like this.
Next, 1-cells are just line segments. Also, not much to say about them.
After that, we have 2-cells, which, by definition, are the two-dimensional closed disk, denoted . This looks like a “filled in” circle:
The last one that I can draw is the 3-cell, which is just a “filled in” sphere, the 3-disk, denoted .
Good. Now, the cells themselves aren’t really all that interesting; the interesting part is what we can build with them. In particular, we can start gluing them together to make structures called CW-complexes.
The first question that should come to your mind is, “What does the CW stand for?” The answer is simple: “Civil War.” Naw, it means “Closure-finite” + “Weak topology”, but, really, that’s kind of dull. This latter (“real”) name should give us some hints as to what our complex should actually be like, but let’s detail this explicitly.
Definition: A CW-Complex is a (hausdorff) space which has a partition of open cells (meaning the interior of the cells we’ve described above; so a line without endpoints, a disk without its outermost ring, a sphere without its outermost shell, and so on) such that the following hold:
- For each cell -cell, usually denoted which is read “the -th -cell”, we have a continuous map from the closed -disk into such that restricting to the interior of the domain -disk gives a homeomorphism onto our cell (as in, it takes the interior of the domain disk onto the cell, which should be homeomorphic to that disk anyway) and the image of the boundary of the domain under is contained in a union of finitely many cells with dimension less than (this last part will be explained in a picture below).
- A subset of is closed if and only if the (potentially empty) intersection with each cell is closed.
This second condition is the same thing as saying we give the CW-complex the weak topology. In this primer, we will not focus too much energy on the weak topology aspect of our spaces, since we will only be working with finite spaces.
So, this definition may not seem so intuitive at first reading, but let’s draw a picture or two to show what’s actually going on here. For the sake of simplicity, I’m going to make a CW-complex which is homeomorphic to a disk and then one which is homeomorphic to a sphere.
Example 1: Homeomorphic to .
Suppose someone gave you a single 0-cell, a single 1-cell, and a single 2-cell. How can you glue them together to make a two dimensional disk? Here’s a good way:
Here, I’ve tried to use some color to denote which cells I’m using. The red point is my 0-cell, the blue line is my 1-cell, and the greenish disk is my 2-cell. How am I attaching all of this together? Well, let’s do it step by step. First, we glue the 0-cell to one side of the 1-cell.
In this picture, the arrows are trying to denote approximately where the 2-cell is going. Now we just have to just check the definition of CW-complex to make sure that this is actually a CW-complex; note that our “attaching maps” are actual maps. We didn’t explicitly say what they were in this case, but we usually know approximately what they do; they’re the maps that tell us to “glue” the line ends to the point, and the edges of the disk to the line.
The first part of the definition of CW-complex says that we need an attaching map for each of the cells we’re gluing into the space.
For the 0-cell, call it , this map is trivial. We usually don’t talk too much about this map.
For the 1-cell, call it , we have . Note that , so this is really a map from the unit interval into our 1-cell. First, let’s restrict this map to the interior of . Note that since is a line segment, the interior is homeomorphic to . So, what did we do with ? Well, we wrapped it around and glued both ends to the 0-cell. So, what does this function do? It sends the interior of the line to every point on the circle except the ones attached to the 0-cell. You can think of this as taking the open interval and bending it into making a circle minus a little point at the top. Next, we need that the map also maps the boundary of our 1-cell into lower dimensional cells. Where does it map the boundary? The boundary of our 1-cell (the endpoints of the line) are both mapped to the 0-cell, which is a cell of lower dimension. Good.
For the 2-cell, call it , we have . Let’s restrict the map to the interior of which is just the filled in circle without a boundary. What happens to this? We’ve glued it inside of that circle that the 0- and 1-cell made, but the interior never touches it: it’s only the boundary that is “attached” to the 0- and 1-cell. Thus, we just map the interior of into what we put inside the circle made by the 0- and 1-cell. Note that, in my picture above, this looks a lot like the interior of the 2-cell, but we could have, say, bent the sides to make the edges formed by the 0- and 1-cell look like a square. Then we’d need to map the interior of into the interior of the square. Last, where does the boundary of go under ? Since it’s glued to the 1-cell and (at a point) the 0-cell, it is in the (finite) union , which are two simplices of lower dimension. Good.
Thus, our cells along with their attaching maps give us a structure homeomorphic to the disk as shown above.
Example 2. Homeomorphic to .
Let’s do one more, and we can even use the previous example to help us out here. We won’t do this one in such detail, but we will show the construction. There are a number of good ways to make (we will show a few later!) but can you think of one where you use a single 0-cell, a single 1-cell, and two 2-cells?
I call this construction “the hamburger” for what I hope are obvious reasons.
We make a circle out of the 0- and 1-cell as we did before, but instead of attaching a single 2-cell along the boundary, we sort of make the two 2-cells into bowls and then attach them along the boundary. Note that the 2-cells don’t touch anywhere except their boundaries where they meet at the circle made up of the 0- and 1-cell.
Proving that this is a CW complex is exactly the same as above, except we have one extra 2-cell attaching map (for the extra 2-cell). But this new map is essentially the same thing as the first 2-cell map, except we have to be slightly careful and just note that the two 2-cells don’t touch anywhere except the boundary. This gives us a nice CW-complex which is homeomorphic to the sphere.
But What About the Torus?
I’ll mention the construction of the torus because we’ll use essentially the same construction to construct a number of spaces to take the homology of in the next part of the primer.
We can actually make CW-complexes out of quotient spaces — this is much less scary than it sounds! For example, recall that we can construct the torus out of the following quotient space:
Where we have a filled in square and associate the edges with arrows together if they have the same arrows, and we “twist” if the arrows are pointing in the opposite direction. Here, there is no twisting. Notice that, in all, we have one vertex (ie, one 0-cell), two edges (ie, two 1-cells), and one “filled in square” (ie, one 2-cell) once we identify both sets of edges together to make the torus. This shows that there is a construction of the torus which uses one 0-cell, two 1-cells, and one 2-cell; the attaching maps are a bit trickier, but they turn out to be dead easy (in this case, at least!) once we give the edges names. We’ll do this when we explicitly compute the homology groups of the torus. For now, this is enough to get us started; next post will be diving into what the homology group is, why we should care about it, and how to compute it for a (very) simple space.
Do the following things. Note that there is more than one solution for each problem. The hints guide towards the “standard” model.
- Construct a CW-complex homeomorphic to a Mobius strip. How many 0-cells do you need? How many 1-cells do you need? How many 2-cells do you need? Do you need any other cells? (Hint: use a square like the one above. You only need to glue a pair of edges, though.)
- Construct a CW-complex homeomorphic to the Klein bottle. (Hint: look up what the quotient space of a Klein bottle looks like, or go onto the next part of the primer to see how we’ve made it.) How many 0-cells do you need? How many 1-cells do you need? How many 2-cells do you need? Do you need any other cells?
- Can you make a CW-complex homeomorphic to the sphere another way than we did above? (Hint: Can you make one using just a 0-cell and a 2-cell? What would you have to attach the boundary of the 2-cell to?)
- Look up the Projective Plane (usually called or just ) and look for its quotient space (it’s going to be a square with identified edges; we’ll do this in the next part of the primer if you can’t find it). Make a CW-complex homeomorphic to the projective plane. How many 0-cells do you need? How many 1-cells do you need? How many 2-cells do you need? Do you need any more cells? (Note: this is the first example of a complex of something we can’t actually “visualize” in three dimensions; nonetheless, we can construct it!)