Homological stability for oriented configuration spaces

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.