You know the routine. You come across a topological space X and you need to find its fundamental group. Unfortunately, X is an unfamiliar space and it's too difficult to look at explicit loops and relations. So what do you do? You look for another space Y that is homotopy equivalent to X and whose fundamental group is much easier to compute. And voila! Since X and Y are homotopy equivalent, you know that the fundamental group of X is isomorphic to the fundamental group of Y. Mission accomplished. Below is a list of some homotopy equivalences which I think are pretty clever and useful to keep in your back pocket for, say, a qualifying exam or some other pressing topological occasion.
A. Radulescu-Banu. (2006)cite http://arxiv.org/abs/math/0610009arxiv:math/0610009Comment: Ams-latex, 158 pages. Corrections to Thm. 6.4.1 and Def. 7.2.5.
B. Toen, and G. Vezzosi. (2004)cite http://arxiv.org/abs/math/0404373arxiv:math.AG/0404373Comment: 228 pages; final version to appear in Memoirs of the AMS.
K. Hess. (2006)cite http://arxiv.org/abs/math/0604626arxiv:math.AT/0604626Comment: A slight revision (some minor errors corrected) of lecture notes from a minicourse given in the summer school "Interactions between Homotopy Theory and Algebra," August 2004. (28 pages).