In this post, we’re going to do something that I thought helped me through homology, but that others no doubt think is a huge waste of time: we’re going to explicitly compute the homology groups of trivial or nearly-trivial things explicitly, without appealing to any theorems regarding homeomorphisms or the like.

Recall the things that we’ve done so far.  The steps to find the homology groups in this post (and, more generally, in life) will be as follows:

  1. Find the chain groups and their generators.
  2. Compute explicitly the image and kernel of the boundary maps.
  3. Find a nice way to express the kernel and image of these boundary maps by potentially using different, but equivalent, generators (we’ll get to this step soon; it’ll make more sense then).
  4. Find the quotient \displaystyle \frac{Ker(\delta_{i-1})}{Im(\delta_{i})}, which is equal to the (i-1)-th cellular homology group.
  5. Reflect on this solution: does it make sense?  Is it nice?  Do we love it?

So without further delay, let’s dig right in.

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Two questions usually immediately spring to mind when you are introduced to some mathematical topic: "why should I care?" and, assuming you do care, "how do I use it?"  With homology, we can either have a serious answer or a silly answer; I prefer the latter, so whenever people ask me what I do, I tell them I spent five years studying math so that I can officially say that a donut has one more hole than a sphere.

If you already know about the fundamental group, then you might be saying to yourself, "Alright, I already know how to tell things with holes apart.  We have the fundamental group for that.  And if the fundamental group doesn’t work, then we have higher homotopy groups.  Why do we need homology?"  It turns out that homotopy groups are actually quite difficult to calculate, even for the most simple structures: n-spheres.  In fact, it turns out that even if our sphere is only n dimensional, we can have nontrivial \pi_{r}(S^{n}) for r > n.  That’s kind of crazy!  We’d like it if our topological invariant was a little bit easier to handle.

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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.

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Let me note two things here — one is a mathematical point, one is a technical point. 

First, math: the type of homology I will be introducing here will be cell homology, because I think that it’s the best way for someone to actually get their hands dirty and compute homology groups of spaces.  This is not too much of a loss of generality, since in nice spaces (eg, finite CW complexes) this is the same as most of the other homology theories. 

Now, a technical note.  I am now a beginner user of the bamboo pen tablet, which, so far, is fantastic.  This means that many of my new pictures (whenever possible) will be hand-drawn.  Note that when precision counts, I will continue to use mathematica, but generally drawing things in mathematica is a huge pain past just graphing equations. 

Because my drawing is terrible, in general, if you have any questions about what the pictures mean, please comment and I will try to elaborate.  What seems obvious to me is not necessarily obvious to all of you, so telling me that my drawing of a hexagon looks like a crying cat will help me teach better.

Now onwards to cells!