I'm a beginning graduate student and while my background is primarily in algebraic number theory, I've found myself a bit smitten with the subject of algebraic topology recently after only having read some of Hatcher (Chapters 1 and 2 and I'm currently reading Chapter 3- and this is currently the extent of my topology background).

I'm trying to get an idea of what sorts of problems lay at the intersection of these two fields, for now just so I have a direction of what sort of background I might want to be learning in the next year or so.

I realize this question is rather broad, but does algebraic number theory have any nice applications to topological problems?