Homology cylinders and the acyclic closure of a free group

Abstract

We give a Dehn-Nielsen type theorem for the homology cobordism group of homology cylinders by considering its action on the acyclic closure, which was defined by Levine, of a free group. Then we construct an additive invariant of those homology cylinders which act on the acyclic closure trivially. We also describe some tools to study the automorphism group of the acyclic closure of a free group generalizing those for the automorphism group of a free group or the homology cobordism group of homology cylinders.

arXiv commentThis is the version published by Algebraic & Geometric Topology on 7 April 2006

References in corpus (4)

• Claspers and finite type invariants of links
• Cohomological Aspects of Magnus Expansions
• Homology cylinders: an enlargement of the mapping class group
• Bordism Invariants of the Mapping Class Group

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• Hidden torsion, 3-manifolds, and homology cobordism
• Higher-order signature cocycles for subgroups of mapping class groups and homology cylinders
• Generalized Dehn twists on surfaces and homology cylinders
• The Magnus representation and homology cobordism groups of homology cylinders
• Group localization and two problems of Levine