Cohomological Aspects of Magnus Expansions

TL;DR

This paper extends Magnus expansions to define Johnson maps on automorphism groups of free groups, revealing cohomological relations and connecting to Morita-Mumford classes and the abelianization of automorphism groups.

Contribution

It introduces a generalized notion of Magnus expansions leading to Johnson maps that satisfy coboundary relations, linking group cohomology with mapping class groups.

Findings

First Johnson map is a twisted 1-cocycle of Aut(F_n).

Cohomology class matches Morita-Mumford classes.

Provides explicit description of abelianization of IA_n.

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 $\operatorname{Aut}(F_{n})$. 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.