Decomposition Theorem for Homology Groups of Symmetric Groups

Nakaoka, Minoru

doi:10.2307/1969878

Cited by 90 publications

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“…Previously known results were: Nakaoka [60] (Σ n with constant coefficients), Betley [6] (Σ n with more restrictive polynomial coefficients) and Hausmann [48] (A n with constant coefficients), Arnold [1] (β n with constant coefficients) and Church-Farb [19] (β n more restrictive polynomial coefficients), Segal [74] (β S n with constant coefficients), Frenkel and Callegaro [34,10] (complete computation for β n with constant coefficients), and Hatcher-Wahl [47] (G Σ n and G β S n with constant coefficients). Automorphism groups of free groups.…”

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confidence: 95%

“…A consequence of Theorem A is the following: Corollary B. Let C, A, X and F : C → Z -Mod be as in Theorem A and assume thatThe assumption on the stability slope k ≥ 2 in the first part of Theorem A and k ≥ 3 in the two other statements is known to be best possible in this level of generality as shown for example by the computations [60] in the case of symmetric groups with constant coefficients and [48] in the case of their commutator subgroups, the alternating groups, also with constant coefficients.In Section 3.2, we briefly explain how the method of group completion can be used to compute the stable homology in the case of constant and constant abelian coefficients.To apply the above result to a given family of groups G 1 → G 2 → · · · , we must first identify the family as a family of automorphism groups in a monoidal groupoid. If the groupoid admits a braiding and satisfies cancellation, we can apply our Theorem 1.10 to obtain an associated homogeneous category and a family of semi-simplicial sets W n (A, X) • .…”

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confidence: 99%

“…The assumption on the stability slope k ≥ 2 in the first part of Theorem A and k ≥ 3 in the two other statements is known to be best possible in this level of generality as shown for example by the computations [60] in the case of symmetric groups with constant coefficients and [48] in the case of their commutator subgroups, the alternating groups, also with constant coefficients.…”

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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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Homological stability for automorphism groups

Randal-Williams

Wahl

2017

Advances in Mathematics

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“…each homology group stabilizes with P . We also prove an analogous result for boundary connected sum, and a version for the quotient group of the mapping class group by twists along spheres and disks, a group closely related to the automorphism group of the fundamental group of the manifold.Homological stability theorems were first found in the sixties for symmetric groups by Nakaoka [36] and linear groups by Quillen, and now form the foundation of modern algebraic K-theory (see for example [28, Part IV] and [42]). Stability theorems for mapping class groups of surfaces were obtained in the eighties by Harer and Ivanov [14,25] and recently turned out to be a key ingredient to a solution of the Mumford conjecture about the homology of the Riemann moduli space [30].…”

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confidence: 99%

Stabilization for mapping class groups of 3-manifolds

Hatcher

Wahl²

2010

Duke Math. J.

122

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Abstract. We prove that the homology of the mapping class group of any 3-manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3-manifold when both manifolds are compact and orientable. The stabilization also holds for the quotient group by twists along spheres and disks, and includes as particular cases homological stability for symmetric automorphisms of free groups, automorphisms of certain free products, and handlebody mapping class groups. Our methods also apply to manifolds of other dimensions in the case of stabilization by punctures.The main result of this paper is a homological stability theorem for mapping class groups of 3-manifolds, where the stabilization is by connected sum with an arbitrary 3-manifold. More precisely, we show that given any two compact, connected, oriented 3-manifolds N and P with ∂N = ∅, the homology groupis independent of the number n of copies of P in the connected sum, as long as n ≥ 2i + 2, i.e. each homology group stabilizes with P . We also prove an analogous result for boundary connected sum, and a version for the quotient group of the mapping class group by twists along spheres and disks, a group closely related to the automorphism group of the fundamental group of the manifold.Homological stability theorems were first found in the sixties for symmetric groups by Nakaoka [36] and linear groups by Quillen, and now form the foundation of modern algebraic K-theory (see for example [28, Part IV] and [42]). Stability theorems for mapping class groups of surfaces were obtained in the eighties by Harer and Ivanov [14,25] and recently turned out to be a key ingredient to a solution of the Mumford conjecture about the homology of the Riemann moduli space [30]. The other main examples of families of groups for which stability has been known are braid groups [1] and automorphism groups of free groups [17,18].The present paper extends significantly the class of groups for which homological stability is known to hold. It suggests that it is a widespread phenomenon among families of groups containing enough 'symmetries'. In addition to the already mentioned stability theorems for mapping class groups of 3-manifolds, corollaries of our main result include stability for handlebody subgroups of surface mapping class groups, symmetric automorphism groups of free groups, and automorphism groups of free products * n G for many groups G. Using similar techniques we obtain stability results also for mapping class groups π 0 Diff(M −{n points} rel ∂M ) for M any m-dimensional manifold with boundary, m ≥ 2 (even the case m = 2 is new here), as well as for the fundamental group π 1 Conf(M, n) of the configuration space of n unordered points in M . Our paper thus also unifies previous known results as we recover stability for braid groups (as π 1 Conf(D 2 , n)), symmetric groups (as π 1 Conf(D 3 , n)) and automorphism groups of free groups (as Aut( * n Z)).

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“…For the symmetric group this was proved by Nakaoka ([Nak60]) and for Aut(F n ) by Hatcher and Vogtmann ([HV98a], [HV04]). The homology groups H k (Σ n ) are completely known.…”

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confidence: 87%