UC Irvine

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 Zn(d1,…,dm)(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. One obstacle to proving such a theorem is 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 apply the Björner–Wachs theory of lexicographic shellability from algebraic combinatorics.