Homological stability for automorphism groups

By Nathalie Wahl, O. Randal-Williams. 2014

166 citations recorded by the source corpus.

Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, homological stability for the family of groups. We show that stability also holds with both polynomial and abelian twisted coefficients, with no further assumptions. This new construction of a family of spaces from a family of groups recovers known spaces in the classical examples of stable families of groups, such as the symmetric groups, general linear groups and mapping class groups. By making systematic the proofs of classical stability results, we show that they all hold with the same type of coefficient systems, obtaining in particular without any further work new stability theorems with twisted coefficients for the symmetric groups, braid groups, automorphisms of free groups, unitary groups, mapping class groups of non-orientable surfaces and mapping class groups of 3-manifolds. Our construction can also be applied to families of groups not considered before in the context of homological stability. As a byproduct of our work, we construct the braided analogue of the category FI of finite sets and injections relevant to the present context, and define polynomiality for functors in the context of pre-braided monoidal categories.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/95bf3182e9c49843c0dec15ec25296483bf024a7

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Citations in the corpus

FI-modules and the cohomology of modular representations of symmetric groups (2015, background): “include: cohomology of free groups via polynomial functors [DPV]; rational cohomology of classical groups via polynomial functors [To], [Ku]; homology groups Ht(GLn(R), T(Adn(R)) where T : Z-Mod → Z-Mod is a functor of finite degree [Dw]; homology groups Ht(Sn,Vn) where V is a finitely generated FI-”
Representation stability and finite linear groups (2014, methodology): “ted category (A,⊛) whose monoidal structure is symmetric (more precisely, which is equipped with a symmetry). Remark 1.9. After a version of this paper was distributed, Nathalie Wahl posted her paper [Wah], which uses an axiomatization similar to complemented categories to prove homological stabilit”