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Homological stability for oriented configuration spaces

Author: Martin Palmer
Journal: Trans. Amer. Math. Soc. 365 (2013), 3675-3711
MSC (2010): Primary 55R80; Secondary 57N65, 20J06, 57M07
Published electronically: December 4, 2012
MathSciNet review: 3042599
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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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Additional Information

Martin Palmer
Affiliation: Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford, OX1 3LB, United Kingdom

Keywords: Configuration spaces, homology stability, alternating groups
Received by editor(s): June 23, 2011
Received by editor(s) in revised form: November 1, 2011
Published electronically: December 4, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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