Extensions of Johnson's and Morita's homomorphisms that map to finitely generated abelian groups

TL;DR

This paper extends Johnson's and Morita's homomorphisms to crossed homomorphisms mapping into finitely generated abelian groups, using group cohomology and nilpotent geometry techniques.

Contribution

It introduces new extensions of Johnson and Morita homomorphisms as crossed homomorphisms into finitely generated abelian groups, enhancing previous results.

Findings

Extended Johnson homomorphisms as crossed homomorphisms

Extended Morita homomorphisms as crossed homomorphisms

Developed polynomial straightening in nilpotent spaces

Abstract

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improves on the author's previous results [Algebr. Geom. Topol. 7 (2007):1297-1326]. To prove the first result, we express the higher Johnson homomorphisms as coboundary maps in group cohomology. Our methods involve the topology of nilpotent homogeneous spaces and lattices in nilpotent Lie algebras. In particular, we develop a notion of the "polynomial straightening" of a singular homology chain in a nilpotent homogeneous space.