An improvement in the linear stable ranges for ordered configuration spaces

TL;DR

This paper improves the known linear stable ranges for the integral cohomology of ordered configuration spaces on orientable manifolds of dimension at least 3 by leveraging the sparsity of Totaro's spectral sequence.

Contribution

It provides a refined bound on the stable ranges for cohomology, enhancing previous results by exploiting spectral sequence sparsity.

Findings

Improved linear stable range constants for cohomology.

Enhanced understanding of spectral sequence sparsity.

Applicable to orientable manifolds of dimension ≥ 3.

Abstract

Recently, Church-Miller-Nagpal-Reinhold [arXiv:1706.03845] obtained linear stable ranges for the integral cohomology of ordered configuration spaces, in the sense of representation stability. We note that the constants in these linear stable ranges can be further improved for an orientable manifold of dimension at least 3. The idea is to simply exploit the sparsity of Totaro's spectral sequence.