Homological stability for oriented configuration spaces

TL;DR

This paper establishes homological stability for oriented configuration spaces in manifolds with boundary, showing they stabilize more slowly than unordered spaces, and extends the results to certain twisted coefficients.

Contribution

It adapts existing methods to prove homological stability for oriented configuration spaces, revealing a slower stability slope and broadening understanding of stability phenomena.

Findings

Oriented configuration spaces stabilize homologically with a slope of one-third.

Unordered configuration spaces have a stability slope of one-half.

The results extend to certain twisted coefficient systems.

Abstract

We prove homological stability for sequences of "oriented configuration spaces" as the number of points in the configuration goes to infinity. These are spaces of configurations of n points in a connected manifold M of dimension at least 2 which 'admits a boundary', with labels in a path-connected space X, and with an orientation: an ordering of the points up to even permutations. They are double covers of the corresponding unordered configuration spaces, where the points do not have this orientation. To prove our result we adapt methods from a paper of Randal-Williams, which proves homological stability in the unordered case. Interestingly the oriented configuration spaces stabilise more slowly than the unordered ones: the stability slope we obtain is one-third, compared to one-half in the unordered case (these are the best possible slopes in their respective cases). This result can also be interpreted as homological stability for unordered configuration spaces with certain twisted coefficients.