Cohomological Aspects of Magnus Expansions

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dc.creatorKawazumi, Nariya
dc.date2005-05-24
dc.date2006-09-07
dc.date.accessioned2026-07-07T06:40:01Z
dc.date.available2026-07-07T06:40:01Z
dc.descriptionWe generalize the notion of a Magnus expansion of a free group in order to extend each of the Johnson homomorphisms defined on a decreasing filtration of the Torelli group for a surface with one boundary component to the whole of the automorphism group of a free group $\operatorname{Aut}(F_{n})$. The extended ones are {\it not} homomorphisms, but satisfy an infinite sequence of coboundary relations, so that we call them {\it the Johnson maps}. In this paper we confine ourselves to studying the first and the second relations, which have cohomological consequences about the group $\operatorname{Aut}(F_{n})$ and the mapping class groups for surfaces. The first one means that the first Johnson map is a twisted 1-cocycle of the group $\operatorname{Aut}(F_{n})$. Its cohomology class coincides with ``the unique elementary particle" of all the Morita-Mumford classes on the mapping class group for a surface [Ka1] [KM1]. The second one restricted to the mapping class group is equal to a fundamental relation among twisted Morita-Mumford classes proposed by Garoufalidis and Nakamura [GN] and established by Morita and the author [KM2]. This means we give a simple and coherent proof of the fundamental relation. The first Johnson map gives the abelianization of the induced automorphism group $IA_n$ of a free group in an explicit way.
dc.descriptionIntroduction and \S7 are revised
dc.identifierhttps://arxiv.org/abs/math/0505497
dc.identifierhttp://arxiv.org/abs/math/0505497
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/101294
dc.subjectGeometric Topology
dc.subjectAlgebraic Geometry
dc.subject20J05; 14H10, 20E05, 20F28, 57R20
dc.titleCohomological Aspects of Magnus Expansions
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