# Algebraic Topology: First Steps in Cohomology — videos available now!

tl;dr Here’s a free series of 24 lectures on algebraic topology!

In January and February this year, I taught an intensive course for the Australian Mathematical Sciences Institute (AMSI), as part of the annual summer school program. It usually rotates around the country, and students would normally all travel to the host university, and live and study together for four weeks. The course counts for a full semester’s credit towards their honours or masters coursework (I guess in the US system it would be roughly equivalent to a first semester grad course). However, this year things were … different, so the summer school was held online, though still hosted by the University of Adelaide, with lecturers (as usual) from around the country teaching (I ran my lectures on Twitch, inspired by Signum University).

I was flattered to be asked to put in a proposal for a course, and I thought I’d brush off my notes from the last time I taught algebraic topology, and focus more on cohomology, which I felt had to be covered in less detail than I would have liked (more focus was put on covering spaces and fundamental groupoids in that outing). The idea was to work combinatorially as much as possible, using $\Delta$-sets, and talk about topological spaces only to the extent that is needed to define singular cohomology. I also wanted to air an original proof of the long exact sequence (aka zig-zag lemma aka algebraic Mayer–Vietoris) that I had learned from my retired colleague John Rice, that he had once come up with but never really published. In general, I was aiming for general tools that students could then apply to other cohomology theories that might come up in subsequent work. The focus on combinatorial examples and homological algebra was just so that there was something concrete to work with. The content is, I admit, a little non-standard, but I wanted something I would be happy with, not just one more rehash of Hatcher’s freely-available book (and others have mentioned to me something non-Hatcher would be good).

Because of the way AMSI approaches the content of the courses we all taught, I own the rights in the videos and the course content, and so I am releasing the videos of the course under a Creative Commons Attribution (CC-BY) license on YouTube:

I also typed up summary notes, and, because some students wanted it, some brief background notes on modules (both of these are CC-BY licensed as well)

Because of technical issues around the software and my low-spec laptop, the first few lectures do not have great video quality, but it does improve by about lecture 3. Oh, and in one lecture my generic tablet pencil, which I was loaned from my department, had its battery go flat, so I had to improvise somewhat. It only happened once, though!

On the whole I think everything worked reasonably well, aside from the intensity of the workload due to me underestimating quite how prepared I should have been. I ran a Discord server for student discussion (and memes), though the official assessment was ultimately all done in a Canvas instance, for security. The students (some auditing, some taking it for credit) came from a wide variety of backgrounds, from physics, maths, even computational neuroscience. So it was really interesting seeing people engage with the material in different ways, even grapple with concepts they had seen before, but done differently.

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