It's been many days now since Afra Zamorodian announced his new monograph 'Topology and Computing'. Now that I finally got a chance to read an excerpt, I am encouraged ! Finally, I am hoping that there is a way to understand Betti numbers without needing an entire year's worth of study :).
Great job, Afra ! Topology is an area that has many intriguing connections with geometry and algorithms in general (Kneser's conjecture, the Kahn-Saks-Sturtevant partial resolution of the evasiveness conjecture), and we need books that can bridge the gap. Another "geometric' text is the wonderful book by Matousek on the Borsuk-Ulam theorem, and a more 'combinatoric' text in this regard is the survey 'Topological Methods' by Björner in the Handbook of Combinatorics.