Homological stability for symmetric complements

TL;DR

This paper proves a generalized version of a conjecture stating that the rational homology of symmetric complements stabilizes for connected manifolds of dimension at least 2, providing explicit stability ranges.

Contribution

It extends the homological stability conjecture for symmetric complements from complex varieties to connected manifolds of dimension at least 2, with explicit stability bounds.

Findings

Proves homological stability for symmetric complements in higher-dimensional manifolds.

Provides explicit bounds for the stability range.

Generalizes previous conjecture to a broader class of manifolds.

Abstract

Conjecture F from [VW12] states that the complements of closures of certain strata of the symmetric power of a smooth irreducible complex variety exhibit rational homological stability. We prove a generalization of this conjecture to the case of connected manifolds of dimension at least 2 and give an explicit homological stability range.