Resolutions of moduli spaces and homological stability

Randal-Williams, Oscar

doi:10.4171/jems/583

Cited by 44 publications

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“…For constant coefficients, the above theorem is part of Harer's classical stability theorem for mapping class groups [43]. The range obtained here is better than Harer's original range, but is not the best known range, which can be found in [7,68]. For twisted stability, the result is due to Ivanov [49,Thm.…”

Section: 1])mentioning

confidence: 69%

“…For constant coefficients, homological stability for the groups π 0 Diff(F n,b rel ∂ 0 F n,b ) was proved by the second author in [85] and with an improved range by the first author in [68], but for twisted coefficients or for the subgroups T n,b it is new.…”

Section: Oscar Randal-williams and Nathalie Wahlmentioning

confidence: 99%

“…Again, we can characterise the maps X p → A X n in terms of arcs: Note first that there is a unique isotopy class of arcs in M with endpoints on the boundary whose complement in M is a disc (see Figure 7). Such an arc is 1- [85] and [68], a choice was made not to allow genus 0 complement as it did not help-or hinder. As we will see, it does not make a difference here either.)…”

Section: 1])mentioning

confidence: 99%

“…More generally, an embedding of M · · · M in S M n is modelled by a collection of such disjointly embedded 1-sided arcs attached next to one another at the marked interval. By sliding the start points of the arcs along the preceding arcs, one obtains an isomorphism between the (n − 2)-skeleton of S n (A, X) and that of the complex denoted C 0 (S) in [68]. A connectivity bound for C 0 (S) is computed in that paper, and immediately yields the same connectivity for the relevant skeleton: is a non-orientable surface of genus h + n with b boundary components.…”

Section: 1])mentioning

confidence: 99%

“…Let F : U ( n,b π 0 Diff(F n,b rel ∂ 0 F n,b )) → Z -Mod be a coefficient system of degree r at N . Then the map H i (π 0 Diff(F n,b rel ∂ 0 F n,b ); F (F n,b )) −→ H i (π 0 Diff(F n+1,b rel ∂ 0 F n,b ); F (F n+1,b )) is an epimorphism for i ≤ n−3 3 − r and an isomorphism for i ≤ n−6 3 − r. If F is a coefficient system as above then for n ≥ 7 the map H i (T n,b ; F (F n,1 )) −→ H i (T n+1,1 ; F (F n+1,1 )) is an epimorphism for i ≤ n−3 3 − r and an isomorphism for i ≤ n−6 3 − r.For constant coefficients, homological stability for the groups π 0 Diff(F n,b rel ∂ 0 F n,b ) was proved by the second author in [85] and with an improved range by the first author in [68], but for twisted coefficients or for the subgroups T n,b it is new. …”

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Homological stability for automorphism groups

Randal-Williams

Wahl

2017

Advances in Mathematics

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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...

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Section: 1])mentioning

confidence: 69%

Section: Oscar Randal-williams and Nathalie Wahlmentioning

confidence: 99%

Section: 1])mentioning

confidence: 99%

Section: 1])mentioning

confidence: 99%

mentioning

confidence: 98%

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Homological stability for automorphism groups

Randal-Williams

Wahl

2017

Advances in Mathematics

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On the edge of the stable range

Hepworth

2020

Math. Ann.

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We prove a general homological stability theorem for certain families of groups equipped with product maps, followed by two theorems of a new kind that give information about the last two homology groups outside the stable range. (These last two unstable groups are the 'edge' in our title.) Applying our results to automorphism groups of free groups yields a new proof of homological stability with an improved stable range, a description of the last unstable group up to a single ambiguity, and a lower bound on the rank of the penultimate unstable group. We give similar applications to the general linear groups of the integers and of the field of order 2, this time recovering the known stablility range. The results can also be applied to general linear groups of arbitrary principal ideal domains, symmetric groups, and braid groups. Our methods require us to use field coefficients throughout.

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Dynamical and cohomological obstructions to extending group actions

Mann

Nariman

2020

Math. Ann.

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We study cohomological obstructions to extending group actions on the boundary ∂M of a 3-manifold to a C 0 -action on M when ∂M is diffeomorphic to a torus or a sphere. In particular, we show that for a 3-manifold M with torus boundary which is not diffeomorphic to a solid torus, the torus action on the boundary does not extend to a C 0 -action on M . Moreover, we use techniques from 3-manifold topology, homotopy theory, and low-dimensional dynamics to find group actions on a torus and a sphere that are not nullbordant.

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Resolutions of moduli spaces and homological stability

Cited by 44 publications

References 19 publications

Homological stability for automorphism groups

Homological stability for automorphism groups

On the edge of the stable range

Dynamical and cohomological obstructions to extending group actions

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