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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 '11 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 '11 at 9:11
Hi Harry, I meant an example of "little formulas pulled out" ? – Patrick I-Z Jan 4 '11 at 9:26
In my experience, all homotopies are variations of the segment $H(t)=tP+(1-t)Q$. – Fernando Muro May 30 '13 at 10:26
Homotopies are a bit like Turing Machines: one should always give a high-level description because it is almost impossible to figure out what it is doing when written formally. – Benjamin Steinberg May 30 '13 at 17:23
up vote 19 down vote accepted

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 '11 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 '11 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 '11 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 '11 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 '11 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 '11 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 '11 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 '11 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 '11 at 14:30
@Harry: you have a very weird concept of cheating if you think that specifying continuous functions at the point-set level feels like cheating! – Mariano Suárez-Alvarez Jan 4 '11 at 14:39
@Tom: I wonder how common that is. – Harry Gindi Jan 4 '11 at 15:13

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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@Harry Gindi: I had the same problem in the 1960s when writing the first 1968 edition of my book now Topology and Groupoids. Then I was looking at papers of Puppe, and found I had no idea how to construct some of their "diagrammatic" homotopies. with different formulae in each bit. This was part of the motivation for finding the gluing theorem for homotopy equivalences, which is now a standard result in abstract homotopy. My answer to another question illustrates how one gets by the methods there explicit homotopies.

There was the same motivation for using double groupoids in homotopy theory. Using the double groupoid defined here one can enrich the category of Hausdorff spaces over the category of these double groupoids with connection (though this has not been written up in detail) and this can be applied to construct homotopies. For some calculations in double groupoids with connections, see Chapter 6 of Nonabelian Algebraic Topology, particularly on rotations (p.169).

One can also enrich the category of filtered spaces over the symmetric monoidal closed category of crossed complexes, and this again enables calculations of homotopies. As said on p.322 of Nonabelian Algebraic Topology (pdf available), this requires further study.

I am not so sure that the work on model categories helps so directly with the calculation of homotopies, unles the cylinder object is used explicit;ly. The use of homotopies is central to Homological Perturbation Theory, and they are used in Graham Ellis' Homological Algebra Programs, in particular for constructing resolutions by induction with a contracting homotopy.

This paper uses higher homotopy groupoids to discuss Toda brackets.

My own work has largely used cubical methods as a background for conjectures and proofs, because of the ease of describing multiple compositions, and homotopies.

June, 2014 I remember a comment of Raoul Bott which I overheard at the 1958 ICM: "Grothendieck was prepared to work very hard to make things tautological!"

March 2016 The following picture


is part of the argument for proving a formula $\sigma (u +_2 v)= \sigma u +_1 \sigma v$ where $u,v$ are homotopy classes rel vertices of maps $I^2 \to X$ which take the edges to a subspace $A$ and the vertices to a set $C$ of base points, $+_1, +_2$ denote respectively compositions in the vertical and horizontal directions, and $\sigma$ is what is called a "rotation". The diagram is reinterpreted by using the interchange law and rules among the peculiar symbols called "connections". The formula implies the existence of a homotopy and in principal gives it explicitly, but that would be very difficult to write out in terms of the usual type of formulae.

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