12,599 research outputs found
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
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