Applications of topology to computer science
Personally, I think the most interesting application of topology was the work done by Herlihy and Shavit. They used algebraic topology to characterize asynchronous distributed computation and gave new proofs of important known results and knocked out a number of long-standing open problems. They won the 2004 Godel prize for that work.

"The Topological Structure of Asynchronous Computation"
by Maurice Herlihy and Nir Shavit,
Journal of the ACM, Vol. 46 (1999), 858-923,