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Motivated by analogies with basic density theorems in analytic number theory,we introduce a notion (and variations) of the homological density of one spacein another. We use Weil's number field/ function field analogy to predictcoincidences 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, whichseem strange from a purely topological viewpoint, are indeed true.  The obstacle to proving such a theorem with current technology is how to dealwith the combinatorial complexity of all possible "collisions" of points, thisproblem does not arise in the simplest (and classical) case $(m,n)=(1,2)$ ofconfiguration spaces. To overcome this obstacle we develop a method that usesthe Bj\"orner--Wachs theory of lexicographic shellability from algebraiccombinatorics to study such problems. As a consequence we derive newhomological 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 ofclassical results, and also of recent theorems of Church and others.

Coincidences of homological densities, predicted by arithmetic