TITLE: How do facts about the homotopy type of cell complexes shed light on analytic number theory?
QUESTION [13 upvotes]: 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?

REPLY [5 votes]: 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.