Coincidences of homological densities, predicted by arithmetic

Abstract

Motivated by analogies with basic density theorems in analytic number theory, we introduce a notion (and variations) of the homological density of one space in another. We use Weil's number field/ function field analogy to predict coincidences for limiting homological densities of various sequences ${\mathcal{Z}}_n^{(d_1,\dots ,d_m)}(X)$ of spaces of $0$-cycles on manifolds $X$. The main theorem in this paper is that these topological predictions, which seem strange from a purely topological viewpoint, are indeed true. The obstacle to proving such a theorem with current technology is how to deal with the combinatorial complexity of all possible "collisions" of points, this problem does not arise in the simplest (and classical) case $(m,n)=(1,2)$ of configuration spaces. To overcome this obstacle we develop a method that uses the Björner--Wachs theory of lexicographic shellability from algebraic combinatorics to study such problems. As a consequence we derive new homological stability theorems for broad classes of $0$-cycles on manifolds. Even in the classical case $(m,n)=(1,2)$ this gives a new, simplified proof of classical results, and also of recent theorems of Church and others.

41 pages. Final Version

References in corpus (2)

• A spectral sequence for stratified spaces and configuration spaces of points
• $E_n$-cell attachments and a local-to-global principle for homological stability

Cited by in corpus (2)

• The homotopy type of spaces of resultants of bounded multiplicity
• Representation stability and arithmetic statistics of spaces of 0-cycles