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.

Does the book discuss de Rham cohomology. I was surprised to find that there are now algorithms to compute de Rham cohomology as well.

It might simply be that that was the way I learned it, but I always thought that it was pretty natural to start with simplicial, hit singular, and then finish up with cellular, all with an eye towards the idea that they are computing the same thing. It also mirrors the development of the subject.