Math 8360, Spring 2011

Topics in Topology

Postmortem.

These are some informal thoughts on this course.

The "first sequence in algebraic topology" is starting to approach canon in a number of ways. There are good things about it, in that in gets across important information that's widely useful to a number of fields. There is, however, the perpetual frustration that it doesn't get across a lot of the "enlightenment" that follows from a deeper study of a lot of these topics. It's hard to work the Dold-Thom theorem into the mix in these courses; it's hard to talk about the grander role of simplicial structures in these courses; and as a stable theorist, it's frustrating to not be able to get across almost any of the "derived" philosophy.

So my thought with this class was that I would try to put together something that gave some indication of this material. It's easier to talk about chain complexes, so I wanted to introduce concepts in that context. I wanted to talk about the general yoga of change-of-model that's been effective in homotopy theory and so I thought that I would introduce simplicial sets as an alternative to spaces. I wanted to give something really useful from the land of slightly-advanced homotopy theory, and I thought that rational homotopy theory is probably the best candidate for this. (I still do.)

In order to really talk about a lot of these things in serious detail, you need model categories. It seemed like they would be a fairly strong unifying thread. So I decided that they would form the underlying theme of the course, and the above items would serve as main motivating and illustrating examples.

Things never work out the way that you plan, of course.

• The course started with chain complexes and their homotopy theory. The first thing which was frustrating to me was that I got bogged down in a lot more technical detail than was my original intention during this segment of the course. Moreover, somehow in the course of teaching my first "topics" course I regressed in what I know about pedagogy. Examples are important. I took a long time getting up-and-running with the model structure on chain complexes, and didn't do nearly enough examples.
• Out of a frustration with so many references sticking to the "bounded below" context because it is easier, I thought that I would cover cofibrations of chain complexes in general. By analogy with constructing spaces by cell attachment, I thought that I would work through this in terms of taking iterated attachments of generators. I also left the definitions of "generating cofibrations" and "generating trivial cofibrations" for later. Let me make no bones about this: this was a misguided idea. It bogged the course down in issues about transfinite induction. Showing that it is equivalent to other definitions, and that it gives the "expected" thing for bounded complexes, and that it gives the correct lifting property, took too much work. The definition in Weibel's text is a better one to present and the cell-attachment version should be left as an exercise.
• Too many of the proofs in model categories are not enlightening unless you do them yourself, and involve chasing things from one cofibrant replacement to another. Just like with diagram chasing, perhaps it is best to present, at most, a couple of items in detail and let the references handle the rest. I am still thinking about this.
• For some reason, the small object argument seemed to go more smoothly than I thought.
• Going back and discussing simplicial complexes, Δ-complexes, and simplicial sets, along with their relationships, seems to be an important step.
• I know now why few people go into detail about the Quillen equivalence between simplicial sets and topological spaces. It's almost better to say little other than the definitions.
• Simplicial sets, path composition, and the relationship to category theory actually seemed to go down better than abstract chain-level material. You get a good mix of simplicial methods, simplicial objects in other categories, the Dold-Thom theorem, and the Dold-Kan correspondence. This gives a nice jumping-off point for derived functors, abelian and nonabelian, too.
• Using some of the "for-free" model categories that you get (over- and under-categories) in more detail is something that I wish I'd had more time for. People often want to know about group cohomology with twisted coefficient groups in topological terms, and this is a nice way to do it.
• I talked about spectral sequences from the perspective of exact couples, and to feed into Serre's theory. That was mostly because I wanted to. I don't think it fit with the rest of the course and I'd probably leave it out. Trying to find the "right" journeyman-level exposition of spectral sequences is another job entirely.
• Rational homotopy theory got short shrift, especially for how potent a tool it is and how many examples you can produce. Of course, I should have expected to run out of time, but I ran out far more severely than I would have liked.

So if I were to do things again, I'd probably start in spaces and their relation to simplicial sets, work up to simplicial objects and derived functors, jump over to rational homotopy theory, and then bring in chain complexes closer to the end. I don't like this order, but I feel like it would work better.

I have this memory, from my time as an undergraduate student, of a line from Rotman's "Homological algebra":

This book is my attempt to make Homological Algebra lovable, and I believe that this requires the subject be presented in the context of other mathematics.

(This is from the newer edition, I cannot remember if it is the same as the original.) The role that homological algebra plays in mathematics was changing a lot when the book was published in 1979, and still is changing. But it has not eliminated this need. I essentially came to understand algebraic topology and algebraic geometry through the lens of homological algebra, and sometimes it needs more selling than I think.

I still think that homological algebra is probably one of the best ways to illustrate much of advanced algebraic topology in a hands-on sandbox. I may not have made homological algebra lovable, but I still think that, somewhere out there, there is a course that does what I wanted this course to do.

Should you ever have some experience in a course something like this, I'd really like to hear about it.

(Also, if you really want an accurate postmortem on how a course like this has gone, you need to ask the students instead.)

Tyler Lawson