2006
DOI: 10.2140/agt.2006.6.603
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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 in [12] and [13], 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 hom… Show more

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Cited by 17 publications
(24 citation statements)
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“…We mention a recent paper [96] of Sakasai for a related work. Also they point out that, although the restriction of σ to M g,1 is injective, σ has a rather big kernel because Ker σ at least contains the group Θ 3 Z = {oriented homology 3-sphere}/homology cobordism.…”
Section: Group Of Homology Cobordism Classes Of Homology Cylindersmentioning
confidence: 99%
“…We mention a recent paper [96] of Sakasai for a related work. Also they point out that, although the restriction of σ to M g,1 is injective, σ has a rather big kernel because Ker σ at least contains the group Θ 3 Z = {oriented homology 3-sphere}/homology cobordism.…”
Section: Group Of Homology Cobordism Classes Of Homology Cylindersmentioning
confidence: 99%
“…The primary significance of Vogel's work arises from its connection to homology cobordism classes of manifolds and embedding theory. (See, for example, [Cha08, CO10, Hec12, LD88, Lev88, Lev89a, Lev89b, Lev94,Sak06]. )…”
Section: Introductionmentioning
confidence: 99%
“…One method for defining r k was already given in our article [28] as an analogue of the extension of the Gassner representation from pure braids to pure string links defined by Le Dimet [17]. More precisely, what we defined in [28] is a noncommutative generalization of the analogue of the Gassner representation.…”
Section: Introductionmentioning
confidence: 99%
“…More precisely, what we defined in [28] is a noncommutative generalization of the analogue of the Gassner representation. (In Section 4, the definition of a noncommutative generalization of the Gassner representation is described for completeness.)…”
Section: Introductionmentioning
confidence: 99%
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