Algebraic topology in high school?

Solution 1:

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.

Solution 2:

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).