The new Bulletin of the AMS is out. It has a review of Computational Homology, a book that I have not read, but was very tempted by at the bookstore. Sadly, my library doesn’t have it. Homology provides an interesting pedagogical challenge. If you just wanted to convey the idea of it, you would probably start with simplicial or cubical homology (I think this is the approach Rotman takes in his book), but if you wanted to train future researchers in the subject, you’d be tempted to skip that and go straight to singular or cellular homology. Most graduate courses probably opt for the latter, but perhaps we’ll begin to see applied courses that take the former route.