# Algebraic topology in high school?

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!

• This seems really interesting to do. The only thing is that I'm not sure how one would present algebraic objects to high school students and have time to do interesting topology. I've heard of a book (haven't read it) called "Geometry and the Imagination". A professor at my undergrad taught a course with the same title, so I'll look for any notes on that and let you know if I find any. Oct 8, 2014 at 0:38
• Ah, the book is by Hilbert, and available here: amazon.com/Geometry-Imagination-CHEL-Chelsea-Publishing/dp/… and notes by Rob Kusner are available here: gang.umass.edu/~kusner/class/462hw Oct 8, 2014 at 0:41
• In 2001 (and possibly in other years), the Stanford University Mathematics Camp covered this topic for high school students, much as you're describing. I think their focus was on surfaces. If you contacted somebody involved with the program, they might have notes or references, or at least useful suggestions. (I somehow doubt there is much online) Oct 8, 2014 at 1:03

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.

Have a look at the articles, including "Making a mathematical Exhibition", on my Popularisation and Teaching web page. The actual exhibition on knots is part of this web page.

The intention was mainly to use the idea of knots to present the methods of mathematics to a general audience. See also this article on knots.

I have given masterclasses to 13 year olds on "Spherical geometry" and on "higher dimensions".

Just to add some light relief, I have added on my Popularisation page a link to cartoons of David Piggins (with the agreement of his family).

At one time, if not now, Chinn/Steenrod's First Concepts of Topology (freely available here) was probably the standard book for what you're looking for. You might also want to look at Mitch Struble's Stretching a Point and Donovan Johnson's Topology. The Rubber-Sheet Geometry, although I think the connections to algebraic topology are less in these books than is the case with Chinn/Steenrod's book.

I suggest William S. Massey, A Basic Course in Algebraic Topology, and the Brouwer fixed-point theorem in dimension 2, if it hasn't to be a very original topic.

Apart from suggestions given above, you can have a look at $"From \ Geometry\ to \ Topology"$ by H.Graham Flegg ( http://www.amazon.com/Geometry-Topology-Dover-Books-Mathematics/dp/0486419614 ) ... it contains a lot of topics like Euler Characteristic, idea of Simply Connected Spaces, Homotopy, Colouring of Graphs, Jordan Curve Theorem etc. and that too without much technical dificulties.

I am surprised nobody has mentioned The Shape of Space (Jeff Weeks) yet. This is a brilliant introductory text, more for topology and geometry in general rather than specifically algebraic topology. It is suitable for high school students, and introduces basic properties of surfaces, topological manifolds (with a big focus on $$3$$-manifolds), a little bit of point-set topology, orientability, and the Euler Characteristic. The exercises are the right level of difficulty for able high school students.