Nathalie Wahl

Stabilization for mapping class groups of 3-manifolds

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.

Homological stability for the mapping class groups of non-orientable surfaces

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 ℝn+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.

AUTOMORPHISMS OF FREE GROUPS WITH BOUNDARIES

The automorphisms of free groups with boundaries form a fam- ily of groups An,k closely related to mapping class groups, with the stan- dard automorphisms of free groups as An,0 and (essentially) the symmetric automorphisms of free groups as A0,k. We construct a contractible space Ln,k on which An,k acts with finite stabilizers and finite quotient space and deduce a range for the virtual cohomological dimension of An,k. We also give a presentation of the groups and calculate their first homology group. AMS Classification 20F65, 20F28; 20F05

Erratum to: Homology stability for outer automorphism groups of free groups

In Hatcher‐Vogtmann [2] a proof was presented that the homology of certain groups An;s is independent of both n and s for n sufficiently large. The groups An;s include Aut.Fn/ (sD 1) and Out.Fn/ (sD 0). In August of 2005 Nathalie Wahl discovered an error in the proof, and the purpose of this note is to fix that error. We assume the reader is familiar with [2], whose notation and conventions we will use here without further comment. The error occurs in the first part of the proof of Theorem 5, showing that the map W Hi.An;sC2/! Hi.AnC1;s/ is injective for n i and s 1. The argument used a diagram chase in the following diagram: Hi.An;sC2;An 1;sC4/ ! Hi 1.An 1;sC4/ ? y ? y Hi.AnC1;s;An;sC2/ ! Hi 1.An;sC2/ It was asserted that the top horizontal and right vertical arrows were successive maps in the long exact sequence of the pair .An;sC2;An 1;sC4/, and hence their composition was the zero map, but in fact the group An;sC2 in the lower right corner of the diagram is a different subgroup of AnC1;s from the An;sC2 in the upper left corner, so that is not induced by the inclusion map of the pair. It is in fact true that the composition is the zero map for n sufficiently large, but a proof seems to require the results proved in this correction. We correct the problem by giving a completely new proof of stability with respect to s for s 1, complementing the earlier proof of stability with respect to n. The new proof entirely avoids the diagram displayed above and instead focuses on the map W An;s!An;sC1 . In the range where is an isomorphism, the relation D 2

Stabilization for the automorphisms of free groups with boundaries

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 authors 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.

Infinite loop space structure(s) on the stable mapping class group

Abstract Tillmann introduced two infinite loop space structures on the plus construction of the classifying space of the stable mapping class group, each with different computational advantages (Invent. Math. 130 (1997) 257; Math. Ann. 317 (2000) 613). The first one uses disjoint union on a suitable cobordism category, whereas the second uses an operad which extends the pair of pants multiplication (i.e. the double loop space structure introduced by Miller, J. Differential Geom. 24 (1986) 1). She conjectured that these two infinite loop space structures were equivalent, and managed to prove that the first delooping are the same. In this paper, we resolve the conjecture by proving that the two structures are indeed equivalent, exhibiting an explicit geometric map.

From Mapping Class Groups to Automorphism Groups of Free Groups

It is shown that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups induces an infinite loop map on the classifying spaces of the stable groups after plus construction. The proof uses automorphisms of free groups with boundaries which play the role of mapping class groups of surfaces with several boundary components.

Homological stability for automorphism groups of RAAGs

We show that the homology of the automorphism group of a right-angled Artin group stabilizes under taking products with any right-angled Artin group.

Antimatroids of finite character

Abstract.We give an axiomatisation of antimatroids of finite character through compatible orderings. P.H. Edeleman and R.E. Jamison show that, in the finite case, the building blocks of antimatroids are in some sense their compatible orderings. We propose a definition of antimatroids for the general case justified by an extension of their result to the infinite case. This implies a study of maximal chains in antimatroids. The extension is not straightforward as certain properties of the maximal chains are lost in the infinite case.