Russ, the requirement is only unreasonable if I restrict my knowledge of the CW complex to the degrees of the attaching maps as indicated in the original question. Thanks for the link to Barmak's paper.

I'm confused. What exactly is inappropriate about applying mathematics to algorithmic trading?

I don't think so: can you prove that only finitely many x-values create cycles? This does not seem obvious at all.

On the other hand, if you just look at the picture, there are tons of cycles...

Josh: presentation groups are 2 dimensional, see (en.wikipedia.org/wiki/Presentation_complex) and you can build your own examples from finitely presented groups. Qiaochu: I don't need an explicit chain map, just isomorphic homology.

Igor: in this case, efficiency is not very important because the input has small size, both in terms of cell count and also in terms of the degrees of attaching maps.

Igor, I am looking at presentation complexes of some small groups and of course the words in many relations contain exponents that are not $\pm 1$. I can add more detail if necessary, but I was trying to keep the question concise.

I like the spirit of this question: visual intuition is often crucial in topology and geometry, and I would certainly benefit from good answers. However, I think it could be much improved if you focus on a question of the type: "are there good visual interpretations of homotopy theoretic constructions such as (blah), for instance by attaching cells?" rather than ask people whether they would personally find such constructions useful.