Improved homological stability for configuration spaces after inverting 2

TL;DR

This paper extends homological stability results for configuration spaces to Z[1/2] coefficients, improving the stability slope to 1 and clarifying aspects of Segal's original proof for topological manifolds.

Contribution

It proves that the homological stability slope of 1 holds with Z[1/2] coefficients, enhancing previous results that required rational coefficients.

Findings

Stability slope of 1 achieved with Z[1/2] coefficients

Clarification of Segal's proof for topological manifolds

Extension of stability results to broader coefficient systems

Abstract

In Appendix A of his article on rational functions, Segal proved homological stability for configuration spaces with a stability slope of 1/2. This was later improved to a slope of 1 by Church and Randal-Williams if one works with rational coefficients and manifolds of dimension at least $3$. In this note we prove that the stability slope of 1 holds even with Z[1/2] coefficients, and clarify some aspects of Segal's proof for topological manifolds.