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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?

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Here is a cool application of topology to algebraic number theory, by Ellenberg, Venkatesh, and Westerland: arxiv.org/abs/0912.0325 –  Frank Thorne Jul 27 '12 at 0:03
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There is a lot of interaction between algebraic topology and elliptic curves, for instance. There are not really unrelated parts of math. –  Fernando Muro Jul 27 '12 at 1:03
    
The Adams Conjecture. –  Spice the Bird Jul 27 '12 at 4:16
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Community Wiki? –  David Roberts Jul 27 '12 at 5:56
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@ mark, That is true. –  Spice the Bird Jul 28 '12 at 1:16
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9 Answers 9

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.

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This is a great answer! The only thing missing is a precise definition of the field of study "chromatic homotopy theory". Could you say, for example, exactly which objects are studied there? Or does it refer more to the technique of working a prime at a time? –  Mark Grant Jul 27 '12 at 7:22
    
could you perhaps expand a bit more on (or suggest a reference for) the connections to physics? –  Yosemite Sam Jul 27 '12 at 23:57
    
@Yosemite Sam: I think the connection has to do with the so-called "Witten genus". Physicists - motivated I think by the path integral formulation of quantum field theory - became interested in the geometry and topology of loop spaces of manifolds. The Witten genus is an invariant which is supposed to be the loop space counterpart of the $\hat{A}$ genus in ordinary topology in the sense that it would probably be the index of the Dirac operator on a loop space if such a thing were known to exist... –  Paul Siegel Jul 28 '12 at 1:44
    
... While nobody can make rigorous sense of this interpretation, it is at least known that the Witten genus plays the same role in TMF that the $\hat{A}$ genus plays in K-theory. –  Paul Siegel Jul 28 '12 at 1:48
    
Paul Siegel is correct; the connection originally came through the Witten genus, and the fact that it was was supposed to associate modular forms to certain manifolds. This led into the study of elliptic genera, and their associated lifts to elliptic cohomology theories. –  Tyler Lawson Aug 1 '12 at 2:27
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Take a look at Machlachlan and Reid's book "The Arithmetic of Hyperbolic 3-Manifolds".

Since finite volume hyperbolic structures are unique whenever an $n$-manifold ($n\geq 3$) has them, any invariants of the hyperbolic structure are invariants of the manifold. Hyperbolic manifolds are $K(\pi,1)$-spaces, so they're not just diffeo/homeomorphism invariants, but invariants of the homotopy-type.

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Vandiver's conjecture (about class numbers) can be tackled through algebraic $K$-theory, which is defined via algebraic topology: the conjecture is equivalent to $K_n(\mathbb{Z}) =0$ when $n$ is a multiple of $4$. But that is a really hard problem.

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The field of L-theory, the algebraic K-theory of quadratic forms, lies at the intersection of algebraic topology and of number theory. My impression is that it is an underpopulated discipline partially because it requires background in fields which most graduate students would think of as being disjoint. I think it is both deep and interesting.

A typical problem would be the calculation of a high-dimensional cobordism group (topological problem). You would show this to be isomorphic to a polynomial extension over the integers, and the actual computation would be to calculate the corresponding L-groups for the corresponding polynomial extensions over the rationals (number theory), and then localize to pass to results over the integers.

As a reference, I would recommend any book by Andrew Ranicki (High Dimensional Knot Theory is very nice, for example). See also this book review.

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Name drop: Peter Sarnak analyzes a number theoretic problem which arose in an L-theory context in topology: claymath.org/publications/Gauss_Dirichlet/sarnak.pdf –  Daniel Moskovich Jul 27 '12 at 4:39
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Try "Primes and Knots" - By Toshitake Kohno & Masanori Morishita.

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Is there a particular paper in there that's related to Steven's question? –  Ryan Budney Jul 27 '12 at 1:12
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There is paper titled "Classical Knot Invariants and Elementary Number Theory" written by K. Murasugi. The paper begins with a brief introduction to knot theory and then discusses a few knot invariants and their connection to number theory. It ends with a list of (open?) problems in knot theory related to number theory. –  Henry Zorrilla Jul 27 '12 at 2:17
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Thanks. FYI, that paper of Murasugi is available on Ranicki's webpage: maths.ed.ac.uk/~aar/papers/murasug4.pdf –  Ryan Budney Jul 27 '12 at 2:24
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I think the whole field of anabelian geometry fits the bill, even if it's perhaps more focused on going the other way around (i.e. applying homotopy theory to number theory). Anabelian geometry is a 'program' launched by Grothendieck in his famous Esquisse d'un Programme, and is all about translating arithmetic geometric problems to problems in homotopy theory.

As an example of a specific instance of the anabelian philosophy, we have Grothendieck's celebrated 'section conjecture', which states (in one form) that for a 'nice' curve $X$ over a number field $F$, the rational points are in bijection with the sections of the exact sequence \begin{equation} 1 \rightarrow \pi_1(X_{\bar{F}}) \rightarrow \pi_1(X) \rightarrow G_F \rightarrow 1 \end{equation} where $G_F$ is the absolute Galois group of $F$ and $\pi_1$ is the algebraic (etale) fundamental group. In case the curve is over the complex numbers, the etale $\pi_1$ is the profinite completion of the regular fundamental group, so there is a very close connection to the classical stuff of Hatcher. The conjecture is still a wide open problem, but any proof would mean you could check something of number theoretic interest (existence of rational points on curves) by studying maps between certain generalized homotopy groups!

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This is something I just heard about yesterday. In this article

Nodal quintics in P^4.
B. van Geemen and J. Werner In: Arithmetic of Complex Manifolds; W.-P. Barth, H. Lange Eds. Springer LNM 1399, (1989) 48-59.

(find a copy here: http://users.mat.unimi.it/users/geemen/publ.html).

the authors compute the Betti numbers of certain smooth complex Calabi-Yau manifolds using a combination of techniques from algebraic topology and number theory. If I understand they first reduce the equations to some finite field, then combining the use of etale cohomology, the Lefschetz fixed point theorem and other things they reduce the problem to counting a finite set of points.

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Étale cohomology?

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How is Etale cohomology a problem? –  Ryan Budney Jul 27 '12 at 1:14
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It's the kernel of a good answer though (expand it, "name"!). This does indeed lie at the intersection of algebraic topology and of number theory, and looks like a really nice research field (so it answers the question in the body of the post, but not the title). mathematik.uni-muenchen.de/~morel/ICMfinal1.pdf –  Daniel Moskovich Jul 27 '12 at 4:58
    
My apologies, I was (and am) in a hurry and was hoping to provoke someone more educated than myself to give a better response along these lines. I might fill it out later. –  name Jul 27 '12 at 7:33
    
Also, in response to Daniel's link, motivic homotopy theory is where I arguably have the most experience, and I have also worked through Hatcher's book, and I personally find the two to be completely disjoint. –  name Jul 27 '12 at 7:35
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