Homological stability for automorphism groups

Abstract

Abstract. 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 F I of finite sets and injections relevant to the present context, and define polynomiality for functors in the context of pre-braided monoidal categories.A family of groupsis said to satisfy homological stability if the induced mapsare isomorphisms in a range 0 ≤ i ≤ f (n) increasing with n. In this paper, we prove that homological stability always holds if there is a monoidal category C satisfying a certain hypothesis and a pair of objects A and X in C, such that G n is the group of automorphisms of A ⊕ X ⊕n in C. We show that stability holds not just for constant coefficients, but also for both polynomial and abelian coefficients, without any further assumption on C. The polynomial coefficient systems considered here are functors F : C → Z -Mod satisfying a finite degree condition. They are generalisations of polynomial functors in the sense of functor homology, classically considered in homological stability, and include new examples such as the Burau representation of braid groups. Abelian coefficients are given by functors F : C → ZG ab ∞ -Mod for G ab ∞ the abelianisation of the limit group G ∞ , satisfying the same finite degree condition. These include coefficients such as the sign representation, or determinant-twisted polynomial functors. Such coefficient systems are newer to the subject. One consequence of stability with abelian coefficients is that stability with polynomial coefficients also holds (under the same conditions) for the commutator subgroups G n ≤ G n .Our theorem applies to all the classical examples and gives new stability results with twisted coefficients in particular for symmetric groups, alternating groups, unitary groups, braid groups, mapping class groups of non-orientable surfaces, automorphisms and symmetric automorphisms of free groups, and it proves stability for these groups...

Cohomology of automorphism groups of free groups with twisted coefficients

“…Since that time, in joint work with Nathalie Wahl [25] we proved Conjecture A (and I explain in this paper how a version of Conjecture B follows from it), and in 2014 Søren Galatius explained to me a proof of Conjecture C. Thus these conjectural calculations from my 2010 preprint hold. Recent work of Aurélien Djament [8] and Christine Vespa [28] obtains these calculations by very different means.…”

“…In a 2010 preprint [23], we conjectured (based on the analogy with general linear groups [10] and mapping class groups [17]) that the groups H * (Aut(F n ); V (F n )) should exhibit homological stability in degrees 2 * ≤ n −k −2 when V is a polynomial coefficient system of degree ≤ k. Since then, in joint work with Wahl [25] we have established a quite general homological stability theorem with polynomial coefficients, and using the highly-connected simplicial complexes of [15] it applies in this case. The result obtained is as follows.…”

Twisted homological stability for configuration spaces

“…(They use methods of [SS17] to prove that the category of representations of FI G is Noetherian, and then apply Theorem 4.2 of [PS14] and the known stability of G ≀ Σ n with constant coefficients to deduce twisted stability.) Twisted homological stability with finite degree coefficients is in fact true for G ≀ Σ n (and indeed also G ≀ β n ) for any group G, by Theorem D of [RW17].…”

“…This extends Theorem A in the case (M, X) = (R ∞ , BG), since FI G embeds as a subcategory of B(R ∞ , BG), and precomposition by this embedding preserves degree. Their methods are quite different to those of [RW17] but are in fact more analogous to ours, in that we both proceed by deducing twisted homological stability from untwisted homological stability. (They use methods of [SS17] to prove that the category of representations of FI G is Noetherian, and then apply Theorem 4.2 of [PS14] and the known stability of G ≀ Σ n with constant coefficients to deduce twisted stability.)…”

“…Recently, Krannich [Kra17] has extended the techniques of [RW17] to a topological setting, where one begins with an N-graded E 1 -module over an E 2 -algebra. Considering C(R 2 ) = n C n (R 2 ) as a module over itself, he recovers Theorem D of [RW17] for the braid groups.…”