Oftentimes, in the standard algebraic topology books (May, Switzer, Whithead, for instance), there are tricky little proofs that depend on proving that two maps are homotopic. This is comparable to the way we build homotopies, lifts, etc. combinatorially in simplicial homotopy theory, but for some reason I never really acquired the skill-set (maybe the intuition?) to come up with these homotopies in the topological case. I'm just mystified how these little formulas are pulled out of thin air.
Am I missing a key technique that's often taught early-on in an algebraic topology course? Is it tricky even with practice? Have there been any papers that focus on systematic ways of generating these things?
I also noticed that in May's book, he oftentimes writes out explicit formulas for his homotopies, sometimes in a way that obscures the issue at hand (for instance, there is a homotopy that is described by an explicit formula, but it's nothing more than an explicit "representative of the natural homotopy" between the identity map and the constant map on a contractible based space.) How often can these seemingly arbitrary formulas be replaced with more canonical descriptions? (This last question is a soft question to people with experience in topology)
Best Answer
Sometimes easy geometric pictures have awkward seeming algebraic descriptions. On pages 6 and 7 of Concise, I gave examples where I both gave a geometric picture and explicit formulas to make the idea of such translation clear. In other cases, (as in cofiber homotopy equivalence) I just found it quick and easy to write down the homotopies (in terms of other homotopies). Sometimes it is just way too laborious to draw the pictures, other times it is too laborious to write the homotopies out. One should learn to be happily eclectic and absorb all techniques available.
Added by PLC: in the second sentence above, Professor May is referring to his text A Concise Course in Algebraic Topology. (When he taught me the course, the title of the draft copy he handed out to us was A Rapid Course..., but I guess the publishers didn't like that so much!)