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?
Best Answer
Chromatic homotopy theory is one such point of interaction between the two subjects.
The story of chromatic homotopy theory is a long one, but a version of the history might go as follows. (This is highly abbreviated, revisionist, insufficiently referenced, and overlooks many aspects and contributions of many people.)
Ordinary (co)homology theory is developed and turns out to be a useful tool.
Later, certain "generalized" cohomology theories are developed, such as K-theory and bordism. These are, in some sense, built out of ordinary cohomology, but some of them seem quite capable of sifting out interesting information, telling us geometric facts, or reassembling some nasty torsion information into something more accessible.
Flipping roles, generalized cohomology theories can be studied in their own right. They come from a category called the stable homotopy category (which is much like a derived category of chain complexes), and each of them can be determined by a certain amount of data involving cohomology operations. Much of this data can be recovered by looking at how the generalized cohomology theory behaves on certain spaces (projective spaces and classifying spaces being the canonical examples).
After a lot of hard work (with some of the bigger names including Adams, Milnor, and Quillen, though I am leaving a lot of important names out) it is discovered, starting from almost pure calculation, that the stable homotopy category has a connection to the category of 1-dimensional formal groups, via the study of characteristic classes.
Further study affirms this connection. Each generalized cohomology theory determines some amount of formal group data. Certain theories that were particularly interesting turn out to have particularly interesting formal group data. Certain computational tools have interpretations in terms of formal groups.
Then - ! - making use of this interpretation systematically, via things like BP-theory and the Adams-Novikov spectral sequence, leads to better qualitative understanding of the stable homotopy category, new guesses about what phenomena can occur (e.g. the Ravenel conjectures), new techniques which are computationally useful, and new theorems (e.g. the solution of most of the Ravenel conjectures).
Later, these things also find connections with mathematical physics, via a track through mathematical physics, string manifolds, modular forms, elliptic curves, and formal group laws. This leads to the development of elliptic cohomology theories and topological modular forms.
However, we still have very little understanding of why this connection arose in the first place, and most of the ways of showing that it exists at all are still through pure computation. Constructive tools are still missing.
Here is a link to Lurie's recent course notes on the subject; Mike Hopkins has an ICM address on this topic which is quite nice; there are many other references.