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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?

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3 Answers 3

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.

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I don't see how that poset is related to Dirchlet series. –  Will Sawin Aug 6 '12 at 23:15
    
@Will: the poset from Björner's paper or $\mathbb{N}$ under divisibility? (The latter is explained in the link, or see also Doubilet, Rota, and Stanley's On the foundations of combinatorial theory IV: the idea of generating function (dedekind.mit.edu/~rstan/pubs/pubfiles/10.pdf)). –  Qiaochu Yuan Aug 6 '12 at 23:32
    
Qiaochu: as long as you are talking about a simplicial complex or even a regular CW complex, as in the other MO question, it seems safe to factor through posets, but if you have a nonregular CW complex, then you may lose information at this step -- since two nonhomeomorphic finite CW complexes can have the same closure poset. –  Patricia Hersh Aug 7 '12 at 0:37
    
Very cool, thank you. I think (2) has a lot of depth also -- for example Brun's sieve could be interpreted in terms of posets, and gives you good bounds (for example) on the number of twin primes, and good estimates on the number of twin almost-primes. –  Frank Thorne Aug 10 '12 at 18:45

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.

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n the context of general cell complexes over a coefficient ring R, one should note that discrete Morse theory becomes severely limited: for the simple homotopy equivalence to hold, the discrete Vector field is only allowed to pair adjacent cells $\sigma < \tau$ if the degree of the attaching map from the boundary of $\tau$ onto $\sigma$ is a unit as well as a central element of $R$. This is made clear in the work of Welker etc. here in a purely algebraic setting: www.maths.ed.ac.uk/~aar/papers/jollwelk.pdf –  Vidit Nanda Aug 7 '12 at 3:17
    
Good point, though one doesn't have to worry about this when dealing with simplicial complexes or regular CW complexes. I like this reference of Jollenbeck-Welker that you mention -- I've looked at it too. –  Patricia Hersh Aug 7 '12 at 4:04
    
Thanks, quid! I knew how to make an umlaut in latex, but don't know how to do it in MathOverflow. –  Patricia Hersh Aug 7 '12 at 15:34
    
You are welcome! The way I type them is unfortunately not portable (my keyboard happens to support them). I think there is no convenient/specific way on MO. Just general html ways I think. Or, what I often find convenient is copy-pasting from some other page (wikipedia is a good place to find any character). For the html way I think &ouml; should do. (Added: it is slightly strange that the &ouml; actually stays as is, but I think and via preview on the test site tried this out that in answer or question it does work and I have vague recollection that comments behave differntlty reg html) –  quid Aug 7 '12 at 23:20
    
Thanks! I agree that it works in an answer -- I just tested that too. Now let me do a comment experiment here, leaving the result for anyone who is interested to see: Anders Bj&ouml;rner –  Patricia Hersh Aug 10 '12 at 1:02

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.

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