Cohomological Aspects of Magnus Expansions

2005/05/24 by Nariya Kawazumi, Kawazumi, Nariya · 3 citations
Mathematics · #14H10 #20E05 #20F28 #20J05 #57R20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.math/0505497

openalex publication_date 2005/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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 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 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.

Citations

• The stable moduli space of Riemann surfaces: Mumford's conjecture

Cited by

• The crossing matrix and the extended first Johnson homomorphism of a braid group
• Marked fatgraph complexes and surface automorphisms
• Modular representations via the first homology of congruence subgroups of mapping class groups and automorphism groups of free groups

Related

• First cohomology groups of the automorphism group of a free group with coefficients in the abelianization of the IA-automorphism group
• A new filtration of the Magnus kernel of the Torelli group
• Marked fatgraph complexes and surface automorphisms
• On the Andreadakis-Johnson filtration of the automorphism group of a free group
• Canonical lifts of the Johnson homomorphisms to the Torelli groupoid