Anton, of course that makes sense. Thank you. Yemon, I have made some changes that I hope will clarify matters somewhat. Bill: this sounds perfect for my needs. Could you please provide a reference?

John, what are $\lambda^p$ and $\Lambda^p$?

Dear Liviu, thank you for this comprehensive answer, I will take a look at the references. Is the link to Fu's notes broken?

Dear quid, thanks for your general advice. However, (1.) I was not belittling anyone either implicitly or otherwise, and (2.) as a general advice, there is only one 'e' in belittling.

I would love to see answers to this, but I fear that any minute now the army of "serious" users will try to close this question. I would like to pre-emptively request: please guys, can we leave this one alone?

Igor: I don't think this works since the inverse of an integer matrix need not be an integer matrix. Can you elaborate on your solution?

Igor: I was pointed to this paper by the reviewer: mrzv.org/publications/zzph-mmt/socg11 but I should warn you that I haven't yet had time to go through it and verify the claim

Dan, thanks. Regarding your projection map: you do have a homotopy equivalence because there is a homotopy version of the Vietoris-Begle theorem due to Smale (search: A Vietoris Mapping Theorem for homotopy). Essentially, if you require the fibers of your map to be contractible (instead of just acyclic), then you have a homotopy (instead of just homology) equivalence.