Stabilization for mapping class groups of 3-manifolds

Abstract

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)).

Homological stability for automorphism groups

“…3 − r and an isomorphism for i ≤ n−5 3 − r. For G n with constant coefficients, this was first proved in [47]. An example of a group G n in the theorem is for M = D 3 and N = S 1 × D 2 , where G n is the handlebody mapping class group of a surface of genus n.…”

“…The first part of the theorem was conjectured by the first author for Aut(F n ) in [65]. Previously known results were: Hatcher-Vogtmann [44] (Aut(F n ) with constant coefficients) and HatcherWahl [47] (Σ Aut(F n ) with constant coefficients). A generalisation of the above theorem to automorphism groups of certain free products is given in Theorem 5.7.…”

“…A generalisation of the above theorem to automorphism groups of certain free products is given in Theorem 5.7. A further generalisation to certain families of subgroups as described in [47,Cor. 1.3] can likewise be carried out.…”

“…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.…”

Extending triangulations of the 2-sphere to the 3-disk preserving a 4-coloring

“…I also want to thank Nathalie Wahl for telling me about the relation between her result in [Hatcher and Wahl 2010] and Conjecture 4.…”