How do facts about the homotopy type of cell complexes shed light on analytic number theory?

Question

I just saw this link text interesting MO question, with a link to this paper, which uses facts from the topology of cell complexes to derive facts of an analytic number theory flavor.

From the perspective of an analytic number theorist, what insight does the topology offer? This approach is capable of proving results whose statements don't involve any topology, for example Theorem 4.1 of the link. Presumably these proofs (and those in the papers cited upon which they depend) could be translated out of the language of algebraic topology and into pure combinatorics and number theory; how big of a mess would this make out of the proofs?

Björner constructs cell complexes for which the Euler characteristic gives the summatory function of the Möbius function -- which is natural, as this is still elementary combinatorics in both cases. However, by the end of the paper he is quoting what appear to be distinctly nontrivial theorems in topology. Is it easy to summarize what these theorems are capable of saying from the number-theoretic point of view?

The proofs of his theorems rely on results from analytic number theory (e.g. Theorem 2.3). To what extent might one hope for results to flow in the other direction?

Answer 1

Björner and others have been very successful in using a mixture of combinatorics and topology to do things such as evaluating alternating sums $\sum (-1)^i a_i $ by first finding a simplicial complex with $a_i$ counting the number of $i$-dimensional faces for each $i$, so that the alternating sum is the Euler characteristic. Topology then can help in that this alternating sum also then equals the alternating sum of ranks of homology groups, which sometimes is a much simpler expression. For instance, in many cases of interest the homology is concentrated in a single degree, e.g. for pure, shellable complexes; in the paper of Björner linked above, the complexes are homotopy equivalent to wedges of spheres. If one wants a purely combinatorial approach, there is the related notion of sign-reversing involution which can be used to cancel pairs of faces, one of which contributes positively to the sum and the other of which contributes negatively; discrete Morse theory is a way to interpret such cancellations topologically in terms of elementary collapses, unifying the combinatorial and topological approaches.

One usually needs a pretty good description of the faces to do the sort of cancellation I mention in an effective manner; to use topology, one needs to know a lot about which faces are incident to each other. If the simplicial complex (or cell complex) comes from number theory, it seems likely one would then need a good bit of number theoretic information as input to this sort of process.

If I understand correctly Vel Nias's recent MO question, that seems to be going after not only numerical data, but also viewing topology of face incidences as a language in which to encode the structure of how different arithmetic progressions of primes may overlap with each other.

Answer 2

My naive opinion is that this question factors into two questions:

1. How do facts from algebraic topology shed light on posets?

2. How do facts about posets shed light on number theory?

The second question is, I think, a little easier to answer. The poset most obviously relevant to number theory is $\mathbb{N}$ under divisibility. In some vague sense the study of this poset is equivalent to the study of Dirichlet series (see for example this MO question). This perspective is, I think, due to Rota.

The first question is not about number theory. Once you realize that you can associate an abstract simplicial complex to a poset in a canonical way then I don't think this idea sounds so strange; it is commonly used by combinatorialists (see for example these notes). There are also some motivational theorems in this area, e.g. every finite CW-complex is weak homotopy equivalent to a finite topological space, and the category of finite $T_0$ spaces is isomorphic to the category of finite posets, but I'm sure an expert could say more here.

Answer 3

There are other places where topology interacts with number theory. The beautiful book by Hirzebruch and Zagier The Atiyah-Singer index theorem and elementary number theory is a good place to see more examples based on rather deep topological results.

The theory of Hilbert modular surfaces is another subject where topology/geometry meets number theory. According to Atiyah, the investigations of these surfaces lead to a remarkable result called the Atiyah-Patodi-Singer index theorem which when applied to Hilbert modular surfaces reveals a connection between the signatures of these varieties and the $L$-functions of the fields of algebraic integers naturally associated to these varieties. Hirzebruch's memoir on this subject is a joy to read.

More than two decades ago it was observed that lattice point counts for certain rational polytopes are intimately related to intersection theory on certain toric varieties.

These are some examples that immediately jumped to my mind.