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?