Homological stability for oriented configuration spaces

Palmer, Martin

doi:10.1090/S0002-9947-2012-05743-6

Homological stability for oriented configuration spaces
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Trans. Amer. Math. Soc. 365 (2013), 3675-3711 Request permission

Abstract:

In this paper we prove (integral) homological stability for the sequences of spaces $C_n^+ (M,X)$. These are the spaces of configurations of $n$ points in a connected manifold 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 unordered configuration spaces $C_n (M,X)$, and indeed 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 $\frac 13$, compared to $\frac 12$ in the unordered case (and these are the best possible slopes in their respective cases).

This result can also be interpreted as homological stability for the unordered configuration spaces with certain twisted $\mathbb {Z} \oplus \mathbb {Z}$-coefficients.

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