Johnson homomorphisms (1980-1985) and their extensions by Morita (1989-1995) are fundamental tools in the study of the Mapping Class Groups. They allow one to investigate these groups from a topological point of view. For example, the fi rst Johnson homomorphism is related to the Casson invariant of homology 3-spheres.

Garoufalidis and Levine (1997-2005) reinforced this relation, bringing fi nite type invariants to the study of Johnson filtration subgroups of the Mapping Class Group. More recently, the universal fi nite type invariant of closed 3-dimensional manifolds introduced by Le, Murakami, and Ohtsuki in 1998 was extended to a functor, and from the latter a Jacobi-diagrammatic homomorphism (called the LMO homomorphism) was derived. The degree truncations of its tree-reduction are essentially the Johnson-Morita homomorphisms.

This relation was completely clari fied by Massuyeau (2009), who used Magnus expansions (in the sense of Kawazumi), and introduced the notion of symplectic Magnus expansion to show that the LMO functor defines a specifi c symplectic expansion , whose associated infinitesimal Dehn-Nielsen representation (which defi nes a total Johnson map ) coincides with up to Levine's isomorphism .

Moreover, the tree reduction of the LMO homomorphism (defi ned on the monoid of homology cylinders) factorizes to the homology cobordism group of homology cylinders, in which the Torelli group embeds.

After surveying the above constructions and relationships, in particular x-ing notations, we will focus on two applications. Firstly, we can identify some relationships between the topology of integral homology 3-spheres and properties of the groups of the Johnson filtration. Secondly, via weight systems one can connect to properties of the Weyl algebras of metrized Lie algebras , with the goal to recover by this procedure the representations of the Mapping Class Group obtained (via skein) from quantum Reshetikhin-Turaev invariants of 3-dimensional manifolds.