Question

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)

Answer 1

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!)

Answer 2

The basic phenomenon is that often the best way to think about "little homotopies" is to use the geometric parts of your brain --- to use primarily your GPU (geometry processing unit), with your arithmetic processing unit, logic processing unit and lexical processing units all in the background, so to speak. However, when writing down a proof, it's customary, and usually easier to transcribe it into symbolic form. This tends to be a one-way process --- it's much harder to start from symbolic formulas and regenerate the geometric intuiton than to start from the geometric intuition and transcribe it into symbolic formulas.

It has become much easier to create reasonable figures illustrating geometric ideas than it used to be (say 20 or 30 years ago), but it's still hard. It's especially hard to directly convey geometric intuition in higher dimensions --- word portraits of geometric ideas can be good, but most mathematical writing neglects them.

I think the best strategy for learning is to avoid reading symbolic definitions of these little homotopies until you have spent some effort thinking about them for yourself, primarily in your head. (Sketches can be good too, but they're often another layer of difficulty. Geometric imagination is not predominantly visual; it's a learned, tricky skill to be able to draw an image on paper that adequately represents a geometric mental model.)

In my experience, the symbolic descriptions often actively interfere with geometric understanding; at first, only use them as hints, for times after you've thought hard and are stuck. It takes time and concentration to build good mental images, but geometric imagination does improve with practice, and it's worth the effort. Eventually, you learn to read the formulas and evoke the geometric images.

Answer 3

Harry, the expression "an explicit representative of the natural homotopy between the identity map and the constant map on a contractible based space" doesn't mean anything to me. Homotopies don't haven't "representatives", and a contractible space doesn't have a "natural" homotopy between the identity and a constant map. I suppose you mean that May could have completed his proof by using the existence of some homotopy, without actually naming a particular one? Or something like that.

I'd say that when I need to make a homotopy, most often I either make it by moving in straight lines or else I make it from another homotopy. How's that for a soft answer to a soft question.

Answer 4

In my experience, the vast majority of homotopies come from some combination of (1) homotopies guaranteed by cofibrations or fibrations and (2) straight-line homotopies.

For example, a standard approach to cellular approximation reduces to the case of a map from a cell to $X \cup D^n$, and then uses the linear structure in the interior of $D^n$ to make sense of straight-line homotopies.

Answer 5

I don't think that there is some point in a typical course where the instructor sits you down and explains how one gets formulas for these things. Some formulas are obvious once you draw a picture, a lot of them seem to involve some sort of linear interpolation. There are certainly times and places where the formulas obscure things and where I have no idea how I might have gotten that particular homotopy if it were up to me.

In what I have been thinking about lately explicit formulas for homotopies are not feasible or even really helpful, you know two things are homotopic for some external reason (read axioms or some such formal thing).

perhaps I will elaborate later, but I suspect that there are others out there with more valuable answers. In fact, I look forward to see ing some of the answers to this question.