The last meeting with my 10th grade topology kids this year just finished. We introduced singular homology, calculated the singular homology of a point and discussed homeomorphism invariance.
Next term, we’ll want to show homotopy invariance and that the singular and simplicial homology coincide when applicable. After that, we’ll change directions slightly.
The future after that holds knot theory, was decided today. We’ll want to introduce knots, look at Reidemeister moves and basic knot invariants. I use basic here in a pretty wide sense – we’ll probably do the Jones polynomial and we might even end up doing Khovanov homology if I feel particularly insane late spring.