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Homology stability for outer automorphism groups of free groups
 ALGEBRAIC & GEOMETRIC TOPOLOGY
, 2004
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COHOMOLOGICAL STRUCTURE OF THE MAPPING CLASS GROUP AND BEYOND
, 2005
"... In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer au ..."
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Cited by 24 (3 self)
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In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of free groups, the symplectomorphism group of surfaces. Based on these, we present several conjectures and problems concerning the cohomology of these groups. We are particularly interested in the possible interplays between these cohomology groups rather than merely the structures of individual groups. It turns out that, we have to include, in our considerations, two other groups which contain the mapping class group as their core subgroups and whose structures seem to be deeply related to that of the mapping class group. They are the arithmetic mapping class group and the group of homology cobordism classes of homology cylinders. 1
THE REPRESENTATION OF THE MAPPING CLASS GROUP OF A SURFACE ON ITS FUNDAMENTAL GROUP IN STABLE HOMOLOGY
"... Abstract. The natural action of the mapping class group of an orientable or nonorientable surface on its fundamental group induces a group homomorphism into the automorphism group of a free group. In the light of a recent theorem of Galatius [G2], we determine here the map on stable homology. 1. In ..."
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Cited by 2 (1 self)
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Abstract. The natural action of the mapping class group of an orientable or nonorientable surface on its fundamental group induces a group homomorphism into the automorphism group of a free group. In the light of a recent theorem of Galatius [G2], we determine here the map on stable homology. 1. Introduction and
unknown title
, 2004
"... Homology stability for outer automorphism groups of free groups ..."
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unknown title
, 2004
"... Homology stability for outer automorphism groups of free groups ..."
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unknown title
, 2004
"... Homology stability for outer automorphism groups of free groups ..."
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A Birman exact sequence for Aut(Fn)
, 2012
"... The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the mapping class group, the kernel of the Birman exact sequence i ..."
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The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the mapping class group, the kernel of the Birman exact sequence is a surface braid group. We prove that in the context of the automorphism group of a free group, the natural kernel is finitely generated. However, it is not finitely presentable; indeed, we prove that its second rational homology group has infinite rank by constructing an explicit infinite collection of linearly independent abelian cycles. We also determine the abelianization of our kernel and build a simple infinite presentation for it. The key to many of our proofs are several new generalizations of the Johnson homomorphisms.