Localization and homological stability of configuration spaces

TL;DR

This paper presents a new proof of rational homological stability for configuration spaces using localization and rational homotopy theory, offering fresh insights into the role of rationals and torsion in homology.

Contribution

It introduces a novel proof technique for homological stability of configuration spaces based on localization and rational homotopy theory, expanding understanding of torsion phenomena.

Findings

New proof of rational homological stability using localization.

Insights into the role of rationals in homological stability.

Results on torsion stability in configuration space homology.

Abstract

In [Chu12], Church used representation stability to prove that the space of configurations of distinct unordered points in a closed manifold exhibit rational homological stability. A second proof was also given by Randal-Williams in [RW11] using transfer maps. We give a third proof of this fact using localization and rational homotopy theory. This gives new insight into the role that the rationals play in homological stability. Our methods also yield new information about stability for torsion in the homology of configuration spaces of points in a closed manifold.