Homological stability for moduli spaces of high dimensional manifolds. II

Abstract. By proving that several new complexes of embedded disks are highly connected, we obtain several new homological stability results. Our main result is homological stability for topological chiral homology on an open manifold with coefficients in certain partial framed En-algebras. Using this, we prove a special case of a conjecture of Vakil and Wood on homological stability for complements of closures of particular strata in the symmetric powers of an open manifold and we prove that the bounded symmetric powers of closed manifolds satisfy homological stability rationally.

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“…The following is part of Theorem 2.4 of [GRW12]. Its proof is a generalization of the Lemma 3.1 of [HW10].…”

Section: Highly Connected Bounded Charge Complexesmentioning

confidence: 92%

Homological stability for topological chiral homology of completions

Kupers

Miller

2016

Advances in Mathematics

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“…The proofs of the stability theorems given here are generalisations of work of Charney [15], Dwyer [31], van der Kallen [80], and Galatius and the first author [36].…”

mentioning

confidence: 83%

“…Following the terminology of [36,47], we call a simplicial complex X weakly Cohen-Macaulay of dimension n if (i) X is (n − 1)-connected, and (ii) for every p ≥ 0, the link of each p-simplex in X is (n − p − 2)-connected. The complex X is called locally weakly Cohen-Macaulay of dimension n if it just satisfies property (ii).…”

Section: 1mentioning

confidence: 99%

“…If k ≤ n − 1, there is a map g : D k+1 → |Y | such that g restricts tof on S k . Again, after changing g by a homotopy we can assume that it is simplicial with respect to a PL triangulation of D k+1 extending the triangulation of S k , and in fact, by [36,Thm. 2.4], the map g can be chosen such that for each vertex v in D k+1 which does not lie in S k…”

Section: 1mentioning

confidence: 99%

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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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“…In the case when d > 2, the fact is a consequence of two theorems of Galatius and Randal-Williams [10,9]. For d = 1, the range of degrees can be improved to * ≤ 2g/3.…”

Section: Manifolds With a Fixed Disk And Homological Stabilitymentioning

confidence: 99%

Relations among characteristic classes of manifold bundles

Grigoriev

2017

Geom. Topol.

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We study relations among characteristic classes of smooth manifold bundles with highly-connected fibers. For bundles with fiber the connected sum of g copies of a product of spheres S d × S d and odd d , we find numerous algebraic relations among so-called "generalized Miller-Morita-Mumford classes". For all g > 1, we show that these infinitely many classes are algebraically generated by a finite subset.Our results contrast with the fact that there are no algebraic relations among these classes in a range of cohomological degrees that grows linearly with g, according to recent homological stability results. In the case of surface bundles (d = 1), our approach recovers some previously known results about the structure of the classical "tautological ring", as introduced by Mumford, using only the tools of algebraic topology. 55R40; 57R22, 55T10 IntroductionLet M be a 2d -dimensional closed oriented smooth manifold. We denote by Diff M the topological group of orientation-preserving diffeomorphisms of M . The bar construction can be used to construct the space BDiff(M) that classifies bundles with fiber M . For any characteristic class of vector bundles p ∈ H * +2d (BSO 2d ; Q), we will define a generalized Miller-Morita-Mumford class (or just kappa class) κ p ∈ H * (BDiff(M); Q). These are the simplest examples of characteristic classes of bundles 1 with fiber M and structure group Diff M .We are mainly interested in the case where the fiber is #g S d × S d , the connected sum of g copies of S d × S d . More generally, we let the fiber to be a highly-connected manifold (see Definition 2.5) of genus g and dimension 2d , denoted M 2d g or M g . Recall that H * (BSO 2d ; Q) = Q[p 1 , . . . , p d−1 , e], where p i is the Pontryagin class of degree 4i and e is the Euler class of degree 2d . Let S ⊂ H * (BSO 2d ; Q) consist of the monomials 1 A geometric example of such a bundle is a proper submersion f : E → B of smooth, oriented manifolds that has M as its fiber.arXiv:1310.6804v3 [math.AT]

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Homological stability for moduli spaces of high dimensional manifolds. II

Cited by 61 publications

References 26 publications

Homological stability for topological chiral homology of completions

Homological stability for topological chiral homology of completions

Homological stability for automorphism groups

Relations among characteristic classes of manifold bundles

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