Weighted distances and topologies on Ran spaces (Sylvain Douteau)Séminaire « Topologie »
Orateur : Sylvain Douteau
Lieu : salle des Séminaires M3
Given a Riemanian manifold M, its Ran space is a stratified space containing all finite configurations of points in M. Its strata, which are configuration spaces of some given cardinality, have been extensively studied, and the space itself is central in factorization homology. Yet, there is some ambiguity on the topology that one should consider on Ran(M). On one hand, there is the weak-topology, obtained by considering Ran(M) as the colimit of its truncation to configurations of bounded cardinal. While extremely natural, this topology is hard to describe. On the other hand, Ran(M) inherits a metric from M, called the Hausdorff distance, which induces another, coarser, topology. In this talk, I will present new topologies on Ran(M), induced from weighted distances which I will introduce. They interpolate between the weak topology and the Hausdorff topology, and collectively allow for a more explicit description of the weak topology.