Coincidences of homological densities, predicted by arithmetic

TL;DR

This paper predicts and proves coincidences in homological densities of spaces of 0-cycles on manifolds using analogies from number theory and combinatorial topology, introducing new stability results and simplified proofs.

Contribution

It introduces a novel notion of homological density inspired by number theory, and develops a combinatorial method to prove topological predictions about these densities.

Findings

Proves that predicted coincidences of homological densities are true.

Develops a new method using lexicographic shellability for complex combinatorial problems.

Provides new homological stability theorems for 0-cycles on manifolds.

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 $\mathcal{Z}^{(d_1,\ldots,d_m)}_n(X)$ 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 Bj\"orner--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.