TITLE: What are some interesting problems in the intersection of Algebraic Number Theory and Algebraic Topology?
QUESTION [30 upvotes]: 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?

REPLY [6 votes]: There are applications of the theory of cyclotomic fields to free actions of finite groups on $S^n$. The existence of such actions is tied to the class groups of certain cyclotomic fields $\mathbb{Q}(\mu_N)$ and their maximal real subfields $\mathbb{Q}(\mu_N)^+$.
You can find a brief introduction to this concept on p265 in Lang's Units and Class Groups in Number Theory and Algebraic Geometry: http://projecteuclid.org/euclid.bams/1183548780
Here are some of the relevant references he cites:
J. MILGRAM, Odd index subgroups of units in cyclotomic fields and applica-
tions, Springer Lecture Notes no. 854 (1981).
J. MILGRAM, Patching techniques in surgery and the solution of the compact
space form problem.
D. KUBERT, The 2-primary component of the ideal class group in cyclotomic
fields.
C. T. C. WALL, Classification of hermitian forms VI, Ann. of Math. 103 (1976)
pp. 1-80.