Abstract. We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the infinite loop space of a Thom spectrum built from the canonical bundle over the Grassmannians of 2-planes in R n+2 . In particular, we show that the stable rational cohomology is a polynomial algebra on generators in degrees 4i-this is the non-oriented analogue of the Mumford conjecture.
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)).
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...
The homology groups of the automorphism group of a free group are known to stabilize as the number of generators of the free group goes to infinity, and this paper relativizes this result to a family of groups that can be defined in terms of homotopy equivalences of a graph fixing a subgraph. This is needed for the second author's recent work on the relationship between the infinite loop structures on the classifying spaces of mapping class groups of surfaces and automorphism groups of free groups, after stabilization and plus-construction. We show more generally that the homology groups of mapping class groups of most compact orientable 3-manifolds, modulo twists along 2 spheres, stabilize under iterated connected sum with the product of a circle and a 2 sphere, and the stable groups are invariant under connected sum with a solid torus or a ball. These results are proved using complexes of disks and spheres in reducible 3 manifolds.
We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any prop with A∞-multiplication-we think of such algebras as A∞-algebras "with extra structure". As applications, we obtain an integral version of the Costello-Kontsevich-Soibelman moduli space action on the Hochschild complex of open TCFTs, the Tradler-Zeinalian action of Sullivan diagrams on the Hochschild complex of strict Frobenius algebras, and give applications to string topology in characteristic zero. Our main tool is a generalization of the Hochschild complex.The Hochschild complex of an associative algebra A admits a degree 1 self-map, Connes-Rinehart's boundary operator B. If A is Frobenius, the (proven) cyclic Deligne conjecture says that B is the ∆-operator of a BV-structure on the Hochschild complex of A. In fact B is part of much richer structure, namely an action by the chain complex of Sullivan diagrams on the Hochschild complex [57,26,28,30]. A weaker version of Frobenius algebras, called here A ∞ -Frobenius algebras, yields instead an action by the chains on the moduli space of Riemann surfaces [11,37,26,28]. Most of these results use a very appealing recipe for constructing such operations introduced by Kontsevich in [38]. Starting from a model for the moduli of curves in terms of the combinatorial data of fatgraphs, the graphs can be used to guide the local-to-global construction of an operation on the Hochschild complex of an A ∞ -Frobenius algebra A -at every vertex of valence n, an n-ary trace is performed.In this paper we develop a general method for constructing explicit operations on the Hochschild complex of A ∞ -algebras "with extra structure", which contains these theorems as special cases. In constrast to the above, our method is global-to-local: we give conditions on a composable collection of operations that ensures that it acts on the Hochschild complex of algebras of a given type; by fiat these operations preserve composition, something that can be hard to verify in the setting of [38]. After constructing the operations globally, we then show how to read-off the action explicitly, so that formulas for individual operations can also be obtained. Doing this we recover the same formuli as in the local-to-global approach. Our construction can be seen as a formalization and extension of the method of [11] which considered the case of A ∞ -Frobenius algebras.Our main result, which we will explain now in more details, gave rise to new computations, including a complete description of the operations on the Hochschild complex of commutative algebras [35], a description of a large complex of operations on the Hochschild complex of commutative Frobenius algebras [34] and a description of the universal operations given any type of algebra [61].An A ∞ -algebra can be described as an enriched symmetric monoidal functor from a certain dg-category A ∞ to Ch, the dg-category of chain complexes over Z. The category A ∞ is what is called a dg-prop, a symmetric monoidal dg-category w...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2025 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.