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

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

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

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