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 with 0-cells, 1-cells and 2-cells, where are integers for which . 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 -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.

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