A short exposition of the Madsen-Weiss theorem

This is an exposition of a proof of the Madsen-Weiss Theorem, which asserts that the homology of mapping class groups of surfaces, in a stable dimension range, is isomorphic to the homology of a certain infinite loopspace that arises naturally when one applies the "scanning method". The proof given here utilizes simplifications introduced by Galatius and Randal-Williams.Comment: Version 2 adds three appendices containing background material: (1) Gramain's proof of the Earle-Eells theorem on contractibility of the components of diffeomorphism groups of surfaces, (2) the calculation of the stable rational homology, and (3) a proof of the Group Completion Theorem following an argument of Galatius. The exposition of the paper has also been reorganized significantl

Homology stability for outer automorphism groups of free groups

We prove that the quotient map from Aut(F_n) to Out(F_n) induces an isomorphism on homology in dimension i for n at least 2i+4. This corrects an earlier proof by the first author and significantly improves the stability range. In the course of the proof, we also prove homology stability for a sequence of groups which are natural analogs of mapping class groups of surfaces with punctures. In particular, this leads to a slight improvement on the known stability range for Aut(F_n), showing that its i-th homology is independent of n for n at least 2i+2.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-54.abs.htm

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

We correct the proof of Theorem 5 of the paper Homology stability for outer automorphism groups of free groups, by the first two authors.Comment: 7 pages, 3 figure

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

Configuration spaces of rings and wickets

The main result in this paper is that the space of all smooth links in Euclidean 3-space isotopic to the trivial link of n components has the same homotopy type as its finite-dimensional subspace consisting of configurations of n unlinked Euclidean circles (the "rings" in the title). There is also an analogous result for spaces of arcs in upper half-space, with circles replaced by semicircles (the "wickets" in the title). A key part of the proofs is a procedure for greatly reducing the complexity of tangled configurations of rings and wickets. This leads to simple methods for computing presentations for the fundamental groups of these spaces of rings and wickets as well as various interesting subspaces. The wicket spaces are also shown to be K(G,1)'s.Comment: 28 pages. Some revisions in the expositio

Finiteness of classifying spaces of relative diffeomorphism groups of 3-manifolds

The main theorem shows that if M is an irreducible compact connected orientable 3-manifold with non-empty boundary, then the classifying space BDiff(M rel dM) of the space of diffeomorphisms of M which restrict to the identity map on boundary(M) has the homotopy type of a finite aspherical CW-complex. This answers, for this class of manifolds, a question posed by M Kontsevich. The main theorem follows from a more precise result, which asserts that for these manifolds the mapping class group H(M rel dM) is built up as a sequence of extensions of free abelian groups and subgroups of finite index in relative mapping class groups of compact connected surfaces.Comment: 19 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol1/paper7.abs.htm