# "Homotopy-first" courses in algebraic topology

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?

• In Germany it seems to be common to teach the first homotopy ($\pi_1$) before homology. I don't know whether this is good or bad; I am missing any particularly fascinating applications of $\pi_1$. But I guess you don't want $\pi_1$ only. As for $\pi_n$, the way I have been taught it, it is very hard to calculate, and the few things that can be said about it require experience with CW complexes (CW approximation, already necessary to show that small homotopy groups of large spheres are zero) and/or homology (to use the Hurewicz theorem), so it doesn't look like a natural candidate for ... Commented Mar 20, 2010 at 15:51
• I think fundamental group before homology is pretty much standard everywhere. But, as Jose states below, fundamental group just by itself is a far cry from the rest of what is called homotopy theory. Commented Mar 20, 2010 at 17:22
• I have also heard of the homotopy only approach. I once heard that everything (including homology) is just homotopy, but I don't even pretend to understand what this means. Commented Mar 20, 2010 at 17:37
• My first pass through algebraic topology was homotopy-first. They were lecture notes, not from a book. The closest book in the literature would be Peter May's, I suppose. Commented Mar 20, 2010 at 19:20
• My first course in algebraic topology was about both homotopy and homology at the same time! It was taught by two lecturers, one of them focussing on homology, and the other of them (tom Dieck) introducing homotopy. I liked it - it was like two simultaneous storylines meeting eventually. Commented Mar 20, 2010 at 21:12

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".

• Great answer, I feel the same way. I am guessing that people in Russia and Germany have the luxury to follow homotopy-first approach exactly because they are teaching a 2-course-sequence, and maybe the sequence is even mandatory there. Commented Mar 21, 2010 at 22:51
• I have to admit that as a topologist a "homotopy-first" approach is very appealing/tempting. Somehow we understand that the "real meat" of algebraic topology is homotopy theory. If I had students locked into a full year's worth of courses I would absolutely teach homotopy first. I think there are a lot of cool results and ideas that can be expressed from that perspective. I also think that it likely leads to a better understanding of homology/cohomology and how it is just a partial reflection of a deeper and larger world. However I think practical matters usually prohibit this approach. Commented Mar 22, 2010 at 2:28
• Why don't you try to make it a full-year course? Commented Mar 22, 2010 at 7:48
• Just to answer Igor: No, in Germany, topology is not mandatory, except of a basic (and partly rather stupid) course in set-theoretical topology. Commented Mar 22, 2010 at 13:22
• I was in said course; it was indeed a nice course, but it was indeed slightly distasteful to come out of a first semester of algebraic topology without a strong grasp of singular cohomology. Yes, there was also a second semester course, but it was taught by a different professor; it would have been better if it had been taught by the same professor. Commented Mar 29, 2010 at 2:19

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.

• This book introduces homology using the Dold-Thom theorem, which makes it sound like exactly the sort of thing Tyler is thinking about: you build many tools of homology without ever mentioning simplices (singular or otherwise). However, it requires the machinery of quasifibrations. It would be interesting to hear from someone who's used this approach in a first course. Commented Mar 22, 2010 at 5:31
• Sorry for the late comment. My first grad course in algebraic topology was taught from this book. I'd had a more typical undergrad course (using Armstrong), but for some students it was their first (and only) exposure to the field. My peers and I found the course baffling, and it probably dissuaded some students from continuing in algebraic topology. Another bad consequence was never seeing singular homology in a course. It was years before I understood that the (admittedly beautiful) material in that book had a tangible relation to the other approaches to (co)homology that I had learned. Commented Feb 2, 2015 at 2:51

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.

• This is a great answer that I can fully get behind (it's also something that I could never get away with, but that's a different story). Commented Mar 20, 2010 at 19:47
• I would say that Spanier is only partially a homotopy-first textbook. Things like the homotopy extension and lifting properties, fiber bundles, and fibrations are introduced before homology, but higher homotopy groups don't appear until much later. A big problem with the book is of course that it's now quite outdated. For example, CW complexes don't appear until page 400. The book of tom Dieck is much more a homotopy-first textbook, and is certainly much more up to date, having the benefit of 40 years of hindsight. One could say it's Spanier done right. Commented Mar 21, 2010 at 14:36
• Mind if I quote you on that last part in my review next week of tom Dieck for the MAA online,Allen?Had to ask,,,,,,,,, Personally,I consider Joseph Rotman's book to be Spainer done right. Also very beautiful,if you've ever seen them,is Spainer's original lecture notes from Berkeley in the early 1960's. They are much more limited in scope and therefore focused. Commented Mar 22, 2010 at 5:12
• I think Spanier's problem with his book is that he didn't heed Otto von Fredrich's warning to his students which Peter Lax so often quotes:"It's easy to write a book on something if you make the mistake of trying to put everything you know about it into it." Commented Mar 22, 2010 at 5:12
• For better or worse,the geometric/formalistic camps of teaching this subject have hardened thier stances and widened the divide.For the former,your book (and the forthcoming completed versions of the sequels,which hopefully will see the light of day one of these years) will probably be the bibles of the former camp and tom Dieck's and May's books will be those for the latter.tom Dieck's book,in fact,has already been adopted by Clark Barwick at Harvard for his course. Commented Mar 22, 2010 at 5:17

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).

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.

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.

• While I largely agree with you, I think a small amount of homotopy theory belongs in the education of all pure mathematics students. For example, I think it's good for every mathematician to know that $\pi_3(S^2)$ is nontrivial. Commented Sep 7, 2010 at 20:55

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.

• Ronald Brown's TOPOLOGY AND GROUPOIDS is also excellent for this approach,Tony.It has the added benefit of making point-set topology geometric rather then analytic,as it is usually presented. Commented Mar 20, 2010 at 21:24

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.

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).

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)

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.

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.

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.

• It sadly also omits too many important topics,Daniel-like the classification of compact surfaces.But it is definitely a must read for anyone interested in the subject. Commented Sep 7, 2010 at 17:33
• The classification of compact surfaces isn't really part of algebraic topology (and I say that as someone who in some sense specializes in 2-dimensional topology). While May's book has its faults and is probably too brisk for a first pass through the subject (though I took such a class from Peter back when I was a grad student and seemingly turned out ok), I think it covers everything that belongs in a first year grad course in algebraic topology. Commented Sep 7, 2010 at 22:23
• I don't think May wrote the book for beginners. Though I should admit the book is beautifully written. One should know enough examples before reading it. However this book is the most up-to-date algebraic topology textbook and the best book prepared for a future researcher. Commented May 8, 2012 at 17:06

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.

@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".

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.