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?*



"Homotopy-first" courses in algebraic topology
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". 


