CW-complexes

So as I spend my days progressing through the very dense, very slow-moving genius that is Hatcher’s book Algebraic Topology, I’m constantly reminded of things I’m not very good at.

And believe me: There are lots of things in that book I”m not very good at.

One thing I’ve always struggled with were the technical details of CW-complexes. I’ve mentioned that before. As a result, I’ve spent the better part of an afternoon gathering online resources, etc., that would shed insight onto the things I’m not clear about. While I wouldn’t bet money on this, I feel confident that I’m now better-equipped to recognize and understand the theoretical construction and properties of CW-complexes…

…and yet, I still find that I can’t do many (as in, I can do almost none) of the problems in Hatcher.

I’m pretty sure I understand the gist for problem 0.14 (for example), which asks you to put a CW-cell structure onto $S^2$ with $v$ 0-cells, $e$ 1-cells and $f$ 2-cells, where $v,e,f$ are integers for which $v-e+f=2$. The gist seems simple: Add on a cell in a precise, regimented way, and define a characteristic map which correctly “incorporates” the additional construction into the given construction. I get that. But where do I start?

I just don’t seem to know enough to transition from sticking my toes in the water to jumping in and taking a swim. I’m pretty sure jumping in at this juncture means drowning.

As I’m sometimes inclined to do, I dug up some other solutions to see if they could shed insight. I feel like Tarun’s solution is overly complicated, while I feel like Dr. Robbin’s solution assumes way way more knowledge than I have at my disposal.

This leaves me feeling a little lost on the algebraic topology front, meaning I’m going to have to dig through some of the resources I have at home and try to figure something out.

Gah.

For completeness, it should be noted that I’ve scrounged up a couple of resources online: Find them here and here.

Maybe in the meantime, I’ll dig up a diversion or two: I’m thinking some differential geometry or maybe even some $D$-module theory. Maybe I’ll be adding some more around these parts later.

In the mean time, check out these easy-to-read, casual expositions on the proofs of the ABC conjecture and the Bounded Gap Conjecture for Primes. I find the first story particularly intriguing.