Stabilization for mapping class groups of 3-manifolds

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.

v4: improvements in the exposition as well as improvements in the main combinatorial theorem in the paper, concerning complexes built from joins

References in corpus (7)

• Homology stability for outer automorphism groups of free groups
• Homological stability for the mapping class groups of non-orientable surfaces
• The integral cohomology of the group of loops
• Stabilization for the automorphisms of free groups with boundaries
• Erratum to: Homology stability for outer automorphism groups of free groups
• Erratum to ``Stabilization for the automorphisms of free groups with boundaries''
• Configuration spaces of rings and wickets

Cited by in corpus (72)

• Representation stability and finite linear groups
• Homological stability for moduli spaces of high dimensional manifolds. I
• Homological stability for the mapping class groups of non-orientable surfaces
• Homological stability for mapping class groups of surfaces
• Topology of moduli spaces of tropical curves with marked points
• Representations of categories of G-maps
• Assembling homology classes in automorphism groups of free groups
• Stability in the homology of congruence subgroups
• The Torelli group and congruence subgroups of the mapping class group
• Homological stability for automorphism groups
• The homology of the Higman-Thompson groups
• The tropicalization of the moduli space of curves II: Topology and applications
• Central stability for the homology of congruence subgroups and the second homology of Torelli groups
• Higher order representation stability and ordered configuration spaces of manifolds
• Tethers and homology stability for surfaces
• Geometric Cobordism Categories
• Homological stability for families of Coxeter groups
• Homological stability of topological moduli spaces
• Representation stability for the cohomology of the pure string motion groups
• A short exposition of the Madsen-Weiss theorem
• The homology of the Brauer algebras
• A twisted homology fibration criterion and the twisted group-completion theorem
• Homological stability for automorphism groups of RAAGs
• On the edge of the stable range
• On the second homology group of the Torelli subgroup of Aut(F_n)
• Representation stability for the cohomology of the moduli space M_g^n
• Erratum to ``Stabilization for the automorphisms of free groups with boundaries''
• Homological Stability for Diffeomorphism Groups of High Dimensional Handlebodies
• On the cohomology of pure mapping class groups as FI-modules
• Non-integrality of some Steinberg modules
• A topological approach to Neutrino masses by using exotic smoothness
• Linking forms and stabilization of diffeomorphism groups of manifolds of dimension 4n+1
• FI_W-modules and constraints on classical Weyl group characters
• On the top dimensional cohomology groups of congruence subgroups of ${\text{SL}}_n(\mathbb{Z})$
• Twisted homological stability for extensions and automorphism groups of free nilpotent groups
• Stability results for Houghton groups
• The mapping class group of connect sums of $S^2\times S^1$
• Representation stability, secondary stability, and polynomial functors
• Homological stability for moduli spaces of disconnected submanifolds, I
• ${\mathrm{FI}}_G$-modules, orbit configuration spaces, and complex reflection groups
• $E_n$-cell attachments and a local-to-global principle for homological stability
• The homology of the Temperley-Lieb algebras
• A comparison of twisted coefficient systems
• Homological stability for Iwahori-Hecke algebras of type B
• Homological stability for the ribbon Higman--Thompson groups
• Homological stability for subgroups of surface braid groups
• Braiding groups of automorphisms and almost-automorphisms of trees
• The free factor complex and the dualizing module for the automorphism group of a free group
• Vanishing of universal characteristic classes for handlebody groups and boundary bundles
• Homology stability for symmetric diffeomorphism and mapping class groups
• Stability phenomena in the topology of moduli spaces
• Strong Haken via Thin Position
• Braid groups and discrete diffeomorphisms of the punctured disk
• A twisted first homology group of the handlebody mapping class group
• Contractible polyhedra in products of trees and absolute retracts in products of dendrites
• Classical homological stability from the point of view of cells
• A recipe for exotic 2-links in closed 4-manifolds whose components are topological unknots
• Stability for moduli spaces of manifolds with conical singularities
• Finiteness properties for relatives of braided Higman--Thompson groups
• Borel stability for congruence subgroups
• Rational homological stability for groups of partially symmetric automorphisms of free groups
• On the homology of big mapping class groups
• Exotic Dehn twists on sums of two contact 3-manifolds
• Asymptotic mapping class groups of Cantor manifolds and their finiteness properties
• Dynamical and cohomological obstructions to extending group actions
• Disordered arcs and Harer stability
• Homological stability of non-orientable mapping class groups with marked points
• Compact and finite-type support in the homology of big mapping class groups
• Stable cohomology of $Aut(F_n)$ with bivariant twisted coefficients
• Problems on handlebody groups
• Rational homological stability for groups of symmetric automorphisms of free groups
• Cutting and Pasting in the Torelli subgroup of Out($F_n$)