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!

This post is going to be a short post, and a deviation from what I’ve normally written about; I have been taking a slight break from linear algebra, but expect a post soon regarding vector spaces and lots of stuff we can do with them.

The title of this post comes from a student that I had recently, asking why “squaring both sides” of an equations is legitimate.

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