"Homotopy-first" courses in algebraic topology

Question

A first course in algebraic topology, at least the ones I'm familiar with, generally gets students to a point where they can calculate homology right away. Building the theory behind it is generally then left for the bulk of the course, in terms of defining singular homology, proof of the harder Eilenberg-Steenrod axioms, cellular chains, and everything else necessary to show that the result is essentially independent of the definitions. A second course then usually takes up the subject of homotopy theory itself, which is harder to learn and often harder to motivate.

This has some disadvantages, e.g. it leaves a discussion of Eilenberg-Maclane spaces and the corresponding study of cohomology operations far in the distance. However, it gets useful machinery directly to people who are consumers of the theory rather than looking to research it long-term.

Many of the more recent references (e.g. tom Dieck's new text) seem to take the point of view that from a strictly logical standpoint a solid foundation in homotopy theory comes first. I've never seen a course taught this way and I'm not really sure if I know anyone who has, but I've often wondered.

So the question is:

Has anyone taught, or been taught, a graduate course in algebraic topology that studied homotopy theory first? What parts of it have been successful or unsuccessful?

Answer 1

I was a heavily involved TA for such a graduate course in 2006 at UC Berkeley.

We started with a little bit of point-set topology introducing the category of compactly generated spaces. Then we moved into homotopy theory proper. We covered CW-complexes and all the fundamental groups, Van-Kampen's Theorem, etc. From this you can prove some nice classical theorems, like the Fundamental Theorem of Algebra, the Brauwer Fixed Point Theorem, the Borsuk-Ulam Theorem, and that $R^n \neq R^m$ for $n \neq m$. I felt like this part of the course went fairly well and is sufficiently geometric to be suitable for a first level graduate course (you can draw lots of pictures!).

At this point you can take the course in a couple different directions which all seem to have their own disadvantages and problems. The main problem is lack of time. A very natural direction is to discuss obstruction theory, since it is based off of the same ideas and constructions covered so far. However this is not really possible since the students haven't seen homology or cohomology at this point!

Instead, for a bit we discussed the long exact sequences you get from fibrations and cofibrations. You could then try to lead into the definition of cohomology as homotopy classes of maps into a $K(A,n)$. But this definition is fairly abstract and doesn't show one of the main feature of homology/cohomology: It is extremely computable. Still, I could imagine a course trying to develop homology and cohomology from this point of view and leading into CW homology and the Eilenberg-Steenrod axioms.

Another direction you can go is into the theory of fiber bundles (this is what we tried). The part on covering space theory works fairly well and you have all the tools at your disposal. However when you want to do general fiber bundle theory it can be difficult. A natural goal is the construction of classifying spaces and Brown's representability theorem. The problem is that the homotopy invariance of fiber bundles is non-trivial to prove. You should expect to have to spend fair amount of time on this. It is really more suited for a second course on algebraic topology.

The main problem with all of these approaches is that it is difficult to cover the homotopy theory section and still have enough time to cover homology/cohomology properly. You know this has to be the case since it is hard to do the reverse: cover homology and cohomology, and still have enough time to cover homotopy theory properly.

What this means is that you'll be in the slightly distasteful situation of having bunch of students who have taken a first course on algebraic topology, but don't really know about homology or cohomology. This is fine if you know that these students will be taking a second semester of algebraic topology. Then any gaps can be fixed. However, in my experience this is not a realistic expectation. As you well know, you will typically have some students who end up not being interested in algebraic topology and go into analysis or algebraic geometry or some such. Or you might have some students who are second or third year students in other math fields and are taking your course to learn more about homology and cohomology. They would be done a particular disservice by a course focusing on "homotopy first".

Answer 2

There is the Aguilar-Gitler-Prieto book on algebraic topology: Algebraic Topology from a Homotopical Viewpoint. As I recall from browsing it, the book is meant to be a graduate course in algebraic topology, and it introduces both homology and cohomology eventually.

Answer 3

I don't think that "homotopy-first" is a special feature of the recent references. The following classical textbooks begin by introducing the general notions from the homotopy theory:

• Algebraic Topology by E. H. Spanier.
• Algebraic topology - homotopy and homology by R. M. Switzer.

In my opinion, these books provide a basis for a good graduate course.

A Course of Homotopic Topology by D. B. Fuchs and A. T. Fomenko, which is another great textbook, also begins with the homotopy theory.

Answer 4

I was taught algebraic topology from Brayton Gray's "Homotopy Theory" (Academic Press) and the approach was wholly homotopical: Homology and Cohomology are defined using spectra and the constructions are natural and clear. The transition to advanced topics is easy and natural (generalized cohomology theories, for example, including algebraic K-theory).

Answer 5

Novikov (apparently) taught this way: see the 3-volume set Modern Geometry (link to vol.1) with Dubrovin and Fomenko. Volume 2 covers homotopy (among other things) and volume 3 covers homology.

Answer 6

If the "first course" is meant to be taken by all students in pure math, then homotopy theory does not belong there I think; I do not see how learning about homotopy groups of spheres, Eilenberg-MacLane spaces, or obstruction theory could benefit those not interested in topology per se.

If on the other hand, the audience consists of students in geometry/topology, then substantial homotopy theory may (and should) be taught. My personal favorites are texts by Fuchs-Fomenko, and May.

Answer 7

As a graduate student I was taught homotopy first (including higher homotopy groups), then singular homology, and then cohomology. The instructor was quite good, but now I feel that the order of presentation was backwards.

I think starting with homotopy is fine as long as you stay in low dimensions, but degenerates into algebraic nonsense otherwise. I highly recommend Stillwell's book Classical Topology and Combinatorial Group Theory where he takes this approach.

Edit: I am not a topologist. I am probably further from being a topologist than people who have left similar disclaimers.

Answer 8

Disclaimer: I am not a topologist.

I was taught basic homotopy theory (fundamental group, van Kampen, but not sure about higher homotopy groups, that might have been elsewhere) at the end of a point-set topology graduate course based on Munkres's Topology: a first course. As Mikael comments above, $\pi_1$ being so geometric means it can be taught without the need of the standard algebraic topology machinery. Of course, $\pi_1$ is a far cry from homotopy theory, which requires a lot more technology.

Answer 9

After a first course in general topology taught by somebody else, and which ended with introducing $\pi_1$ and covering theory (it has been mentioned above that this is typical in Germany), I recently started a two semester long course on algebraic topology.

Instead of doing full-fledged homotopy theory from the beginning, I introduced higher homotopy groups first, and after proving Blakers-Massey, I got all the typical applications rather quickly (Brouwer fixpoint theorem etc.). Next topics where stable homotopy groups, then homology from an axiomatic viewpoint, later realised as spectral homology. More stuff followed that I don't want to bore you with. The bad thing was that in the end time was too short for a thorough introduction of $K$-theory and bordism, where one could have exploited all the machinery build up to that point.

My experiences? I certainly learned a lot, and some of the students did, too. The way to applications takes a bit longer, but not that much. One has to learn more techniques, but these are "geometric" rather than "algebraic", which I personally like. The motivation for introducing homology and cohomology is different, and maybe not that strong as in other approaches. As a reward, some hard theorems become very easy (the Hurewicz theorem has a two line proof if one knows some basic facts about the Eilenberg-Mac Lane spectrum). Other theorems become really hard, or one needs to restrict to CW complexes as a workaround (Künneth formula, universal coefficients).

I guess I will teach my next topology course (maybe only in some years to come) using the same approach, but a bit more streamlined. I still believe it is easier to learn simplicial singular homology at a later stage (where one is able to appreciate the strength of the simplicial method) than to learn homotopy theory after simplicial theory (with the feeling that one has almost had it all and now needs a lot more technique for just very few additional applications).

Answer 10

This approach is not new. See the book of Gray, Homotopy theory, Academic Press. I am not an expert in topology (My area of research is differential equations and special functions) but I think Gray was a student of Steenrod and the idea of Homotopy first approach originated with Steenrod which motivated Gray to write his book. Or this is what I recall reading.

Gray's book is excellent in his style of presentation and his treatment of the Blakers Massey theorem due to Brodmann int he 1960s. Simpler proofs were given by Dieck, Kamps Puppe in their springer lecture notes in Math. It is in German but very lucidly written. The material now appears in Tom Dieck's book Topologie Walter De Gryuter which I think is simpler to read than his Algebraic Topology book in EMS series. All books of Tom Dieck are superbly clear and writing is extremely thoughtful. These books also take the Homotopy first approach.

Hope these indications help.

Happy Reading Gopala Krishna Srinivasan (IIT Bombay)

Answer 11

This isn't quite what you mean, but I took Igor Frenkel's algebraic topology course as an undergrad. He taught out of Massey's book, A Basic Course in Algebraic Topology. It starts with the classification of 2-manifolds, does the fundamental group and the Seifert-von Kampen theorem, and then does singular homology and cohomology. De Rham cohomology is only there as an appendix. I think the fundamental group is a little bit easier to grasp early on in a first course than singular homology. For cohomology first, you could do something like Bott & Tu, I suppose, but I think this way is a bit more useful because de Rham cohomology is a little too nice for its own good.

Answer 12

As an undergraduate, I took a semester of point-set topology that used Munkres's book Topology, and we studied the fundamental group towards the end of the course. Following that, I took a semester of algebraic topology that used Greenberg and Harper's book Algebraic Topology: A First Course. Greenberg and Harper start off with homotopy theory and introduce higher homotopy groups. However, they don't go very far with homotopy theory before turning their attention to singular homology.

Although there are various things I don't like about Greenberg and Harper's book (for example, I didn't learn about simplicial homology until much later, and I think I would have understood singular homology better if I had first learned simplicial homology), I think that the approach of giving a brief introduction to homotopy groups before proceeding to homology theory works pretty well. It's good to emerge from a one-semester course at least knowing what higher homotopy groups are.

Answer 13

J. P. May's superb book, "A Concise Course in Algebraic Topology," starts with a great deal on homotopy theory, and doesn't really get to homology until nearly half way through. I learned a great deal from this approach, and think that it is the best way to teach algebraic topology. But May's book is probably too difficult for a "first course" in algebraic topology.

Answer 14

Purely homotopic approach to the homology theory was presented by Vladimir Boltyansky during the Second Summer Mathematical School, June-July 1964, and published, pp.3-84, as the first of a volume of three articles. (If there is an interest in this I can go into details). The volume was published by the Mathematical Institute of the Academy of Science of the Ukraine Republic, Kiev (or Kyiv) 1965.

Answer 15

@Tyler Lawson: I just saw this question. Our book published in 2011 and advertised on Nbonabelian Algebraic Topology does exactly that. No (or little) singular homology, no simplicial approximation. It gives many calculations of nonabelian second relative homotopy groups not available by traditional methods. It also gets to the Relative Hurewicz Theorem and the calculation of certain homotopy classes of maps, including the non simply connected case.

It is in a sense a rewrite of algebraic topology on the border between homotopy and homology, using functors defined in terms of homotopy classes of maps, and establishing their main properties directly.

Of course there is a lot of homotopy and homology theory it does not do, for example Poincare duality: I've put that as one of a number problems to solve in the style/techniques of the book!

There are on my preprint page several relevant presentations, and also a recent paper entitled "Modelling and computing homotopy types:I".

Answer 16

I just was passing through-I'd forgotten my responses to it.

The classic book Homotopy Theory: An Introduction to Algebraic Topology by Brayton Gray (official link, Ranicki's archive) also follows the throughline of homotopy as the framing concept of algebraic topology. The down side is that it's a bit more difficult then homology, which has very straightforward algebraic computations to guide it. But homotopy has the advantage of being easier to present geometric implications for.

Either way, a good first course in algebraic topology isn't going to easy for most students, let's face it.