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.
G. Bergman. (2016)cite arxiv:1602.00034Comment: Comments: 50 pages. Main change from version 2: Lemma 26 strengthened to include contractibility statement which in previous version was noted as "likely" in paragraph following that result. Several small typoes also corrected.