Homological stability for coloured configuration spaces and symmetric complements

Abstract

We prove a homological stability theorem for certain complements of symmetric spaces. This is a variant of a conjecture by Vakil and Matchett Wood [VMW12, Conjecture F] for subspaces of Sym n X where X is an open manifold admitting a boundary. To do this we prove a homological stability result for a type of "coloured" configuration space by adding points of the same colour.

Plant complexes and homological stability for Hurwitz spaces

“…Is there also a general theorem for the stabilization in the direction of a fixed conjugacy class, i.e., sequences {ξ +ne i }, where e i is a unit vector? This is motivated by the corresponding result for colored configuration spaces in [Tra13]. -Does the homological stabilization carry over to base spaces of higher genus?…”