## Computing homotopies

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)

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Can you give an example ? – Patrick I-Z Jan 4 2011 at 9:06
Dear Patrick, I think I gave an example. Could you clarify what more you would like to hear? – Harry Gindi Jan 4 2011 at 9:11
Hi Harry, I meant an example of "little formulas pulled out" ? – Patrick I-Z Jan 4 2011 at 9:26
Dear Patrick, look at May's proof of cofiber homotopy equivalence, for instance. – Harry Gindi Jan 4 2011 at 9:30
Harry, for what it's worth I agree that these explicit homotopies are slightly annoying. – Kevin Walker Jan 4 2011 at 16:14

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!)

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@Pete : "Concise" is, I think, a bit more accurate than "Rapid". I don't recall reading the book being a very rapid experience! – Andy Putman Feb 4 2011 at 6:01
Dear Professor May, thanks for your answer! I had skipped the first few chapters, and this was apparently to my detriment. =) – Harry Gindi Feb 4 2011 at 6:25

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.

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I used to think like that - clear and neat geometric constructions sometimes "happen to" have awkward symbolic descriptions. Certainly, whenever I encounter a clumsy symbolic construction in some paper it immediately becomes a challenge for me to recreate it geometrically without the pain of reading the symbols. This said, there's also a deeper look on this issue. If I cannot write up my homotopy in a simple way while geometrically it looks simple, this means I haven't developed an adequate language; and that might prevent me from being able to see how to solve a harder problem. – Sergey Melikhov Jan 10 2011 at 1:35
In fact, I believe the problem is really in that our basic (algebraic and geometric) topology is still missing true combinatorial foundations. By this I mean dealing with simplicial/cubical complexes, or posets, or simplicial sets in a purely combinatorial way without ever resorting to "obvious" homotopies or homeomorphisms on geometric realizations. At first glance this insane idea must seem an overkill (or even making things still worse), if feasible at all. Anyhow, I happen to be working on this, and not out of pedantism but in pursuit of applications that are out of reach otherwise... – Sergey Melikhov Jan 10 2011 at 2:06
@Sergey: I do appreciate simple combinatorial descriptions of homotopies, along with the geometric descriptions. Long ago in college I wrote a senior thesis developing the homotopy theory of finite topologies, and its relationship to cell complexes etc., without being aware of any literature about it. Combinatorial descriptions can be simple and elegant, but they're just different and seem to use different parts of our brain than the geometric descriptions. Translation back and forth is not easy or automatic, although it can be learned. Certainly it's very important for machine computation. – Bill Thurston Jan 10 2011 at 3:09
@Sergey: I should have mentioned (as you must know but others may not) that finite topologies are equivalent to posets; geometric realizations can in fact be thought of as countable posets (like the decimal system). The main point about geometry is that humans have powerful special purpose geometric circuitry, but weak combinatorial circuitry, especially with symbolic data. The situation for computers is reversed; even GPUs are good for general purpose arithmetic and combinatorial computation. – Bill Thurston Jan 10 2011 at 3:17

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.

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Yeah, I realize that was an "abuse of terminology", and you did understand what I was trying to get at. There is a canonical (in a certain sense) "description" of the homotopy by another one of its uses. In general, it's substantially easier to understand what's going on with homotopies if you can give descriptions of them in terms of other maps that you're already working with. That is, I think it's preferable to make homotopies out of other homotopies rather than write them down explicitly. Somehow this seems quite a bit more enlightening. – Harry Gindi Jan 4 2011 at 14:05
Also, specifying continuous functions at the point-set level feels like cheating and doesn't generalize well to other contexts (even simplicial sets). Are you aware of any homotopies appearing in important theorems for which there is only a point-set description? – Harry Gindi Jan 4 2011 at 14:11
A topologist who will remain nameless once proudly told me of writing down a certain homotopy in a certain explicit way in order to deliberately conceal the two-step process by which it had been created. This was in order to prevent a specific other topologist from generalizing the argument. – Tom Goodwillie Jan 4 2011 at 14:30
@Harry: you have a very wierd concept of cheating if you think that specifying continuous functions at the point-set level feels like cheating! – Mariano Suárez-Alvarez Jan 4 2011 at 14:39
@Tom: I wonder how common that is. – Harry Gindi Jan 4 2011 at 15:13
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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.

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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.

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@Sean: I'm pretty comfortable with the model-category-theoretic (damn model theory for taking the name first!) nuts and bolts, but there are often arguments in special cases (say the Hurewicz/Strom model structure on $CGWH_*$ where we can prove the existence of a weak equivalence (that is, a homotopy equivalence that is also a based map but not necessarily with based homotopy inverse) without proving the full existence of a homotopy. – Harry Gindi Jan 4 2011 at 8:55
Rather, a based homotopy. – Harry Gindi Jan 4 2011 at 8:58
I figured you were fine with the formal stuff. My main point was that I am not sure where the stuff comes from either, and I don't feel like I am at a disadvantage because of my lack of knowledge. – Sean Tilson Jan 4 2011 at 18:31