Cohomological Aspects of Magnus Expansions

Abstract

We 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 $\mathrm{Aut}(F_n)$. 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 $\mathrm{Aut}(F_n)$ and the mapping class groups for surfaces. The first one means that the first Johnson map is a twisted 1-cocycle of the group $\mathrm{Aut}(F_n)$. 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 $I{A}_n$ of a free group in an explicit way.

Introduction and \S7 are revised

References in corpus (2)

• The stable moduli space of Riemann surfaces: Mumford's conjecture
• Harmonic Magnus Expansion on the Universal Family of Riemann Surfaces

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