Twisted homological stability for configuration spaces

Abstract

Let M be an open, connected manifold. A classical theorem of McDuff and Segal states that the sequence {Cn(M )} of configuration spaces of n unordered, distinct points in M is homologically stable with coefficients in Z -in each degree, the integral homology is eventually independent of n. The purpose of this paper is to prove that this phenomenon also holds for homology with twisted coefficients. We first define an appropriate notion of finite-degree twisted coefficient system for {Cn(M )} and then use a spectral sequence argument to deduce the result from the untwisted homological stability result of McDuff and Segal. The result and the methods are generalisations of those of Betley [Bet02] for the symmetric groups.2010 Mathematics Subject Classification: Primary 55R80; secondary 57N65.

Homological stability for spaces of embedded surfaces

“…Another generalisation of McDuff's stability theorem has been developed by Martin Palmer [Pal12]. He considers the space of all embedded submanifolds of a chosen diffeomorphism type in a background manifold satisfying certain hypotheses, and stabilises by repeatedly adding disjoint copies of a chosen submanifold near the boundary of the background manifold.…”

Generalized Long-Moody functors

“…This framework is generalized by Krannich to a topological setting in [22]. Also, for the case of surface braid groups, similar results are established by Palmer in [28]: in this case, the twisted coefficients are functors B R 2 , * → Z-Mod satisfying a polynomial condition, where B R 2 , * is a certain category with the braid groups as its automorphism groups (in particular, there is a functor Uβ → B R 2 , * which preserves the polynomial degree).…”

$$E_n$$ E n -cell attachments and a local-to-global principle for homological stability

“…Otherwise this proof only gives a (k/2 − 1)-equivalence. We will only do the case (i), noting that the input for case (ii) is Proposition A.2 of [35] or follows from applying Palmer's techniques in [49] to the main result of [22]. This proof is the shortest we are aware of and uses Palmer's result on stability for the homology of configuration spaces with coefficients in certain local systems [49].…”

“…We will only do the case (i), noting that the input for case (ii) is Proposition A.2 of [35] or follows from applying Palmer's techniques in [49] to the main result of [22]. This proof is the shortest we are aware of and uses Palmer's result on stability for the homology of configuration spaces with coefficients in certain local systems [49]. His Corollary 1.6 in particular implies that the map t : H * (C k (M ); H q (F k ; F)) → H * (C k+1 (M ); H q (F k+1 ; F)) induced by a stabilization map, which adds a new point to the configuration with some choice of label in the fiber, is an isomorphism in the range * ≤ k−q 2 where F is some field.…”