This winter I am planning on teaching a small seminar (20 lectures 45 minutes each) for high school students. I was was given the freedom to choose the topic of the seminar, but it is supposed to be about some "advanced" mathematics in an elementary exposition.
I was thinking about lecturing on algebraic topology. I wanted to focus on algebraic techniques. My idea was to introduce some basic objects (spaces like surfaces, graphs, knots) and then try to explain to students how to study those objects using algebra (some basic knot invariants, Euler characteristics of graphs and surfaces, maybe even fundamental groups).
My main problem is: references. I have started learning topology already knowing a decent amount of analysis and algebra, so I don't know many elementary topology books. The only book I know is the Prasolov's book "Intuitive topology". It is a very nice introduction to topology! But unfortunately, it does not talk much about the algerbaic side of topology, just a bit about invariants of knots. Other than that, I don't know any good reference for basic algebraic topology aimed at advanced high school students.
To summarize, my questions are:
- Do you know a good reference (books or notes) for basic algebraic topology accessible for advanced high school students?
- Maybe you can suggest some other nice topics in topology I could make a course based on? In this case, I would be very happy if you also suggest an appropriate reference!
Thank you very much for your help!
Best Answer
Take a look at A Combinatorial Introduction to Topology by Michael Henle. The point-set topology is done gently, he uses combinatorial methods to bypass some otherwise complicated proofs (to get the Brouwer fixed point theorem, for example), and he uses what is essentially mod 2 homology (if I remember right) to prove some other results, like the Jordan curve theorem.
This doesn't quite do what you want, but parts of it might be useful.