Homological stability for coloured configuration spaces and symmetric complements

TL;DR

This paper establishes homological stability for specific symmetric space complements, extending conjectures by Vakil and Matchett Wood, through new results on coloured configuration spaces with added points of the same colour.

Contribution

It introduces a homological stability theorem for coloured configuration spaces and symmetric complements, advancing understanding of their topological properties.

Findings

Proves homological stability for symmetric complements of symmetric spaces.

Establishes stability results for coloured configuration spaces with added points.

Extends conjectures by Vakil and Matchett Wood to new classes of subspaces.

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 for subspaces of $\mathrm{Sym}^n(X)$ 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.