1. HOMOLOGICAL STABILITY FOR ORIENTED CONFIGURATION SPACES.
- Author
-
PALMER, MARTIN
- Subjects
HOMOLOGY theory ,STABILITY theory ,CONFIGURATION space ,MANIFOLDS (Mathematics) ,DIMENSION theory (Topology) ,MATHEMATICAL sequences ,PERMUTATIONS - 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 Cn(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 ⅓, compared to ½ 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 Z ⊕ Z-coefficients. [ABSTRACT FROM AUTHOR]- Published
- 2013