Coincidences between 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 Z pd1,...,dmq n pXq 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 pm, nq " p1, 2q of configuration spaces. To overcome this obstacle we apply the Björner-Wachs theory of lexicographic shellability from algebraic combinatorics.

The graph minor theorem in topological combinatorics

“…In this section we study the cohomology of a certain family of hyperplane arrangement complements. In particular, we prove a finite generation result that recovers and expands upon similar results present in [FWW19].…”

“…Observe moreover that the graph G can be seen as an edge with a and b leaves sprouted on its two vertices, respectively. Therefore, Theorem 2.10 implies that our result can be seen as a generalization of the stabilization phenomena observed by [FWW19].…”

“…One observes that the complete bipartite graph K a,b can (essentially) be realized as the line graph complement of some other graph. In particular, the finite generation result of Theorem 1.2 can be seen as a generalization of some results in [FWW19] (see Theorem 2.10).…”

“…is the complete bipartite graph, then one recovers the colored configuration spaces Z D a+b as studied by Farb, Wolfson, and Wood [FWW19]. In this work we will specialize to the family of line graph complements.…”

Stability of the Cohomology of the Space of Complex Irreducible Polynomials in Several Variables

“…The theme of this paper is close to that of Farb-Wolfson-Wood [4], where they proved surprising coincidences in the Poincaré series of certain apparently unrelated spaces, which were predicted by the corresponding point-counting results over finite fields (Theorem 1.2 in [4]). Our Theorem 1.1 and 1.2, as well as the reasoning that leads us to discover them, provide another example where the Grothendieck-Lefschetz trace formula, despite not playing any role in the proofs, can still provide heuristics leading to plausible conjectures, which are then settled by topological methods.…”