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In a recent paper we defined a new filtration of the mapping class group--the"Lagrangian" filtration. We here determine the successive quotients of thisfiltration, up to finite index. As an application we show that, for anyadditive invariant of finite-type (e.g. the Casson invariant), and any level ofthe Lagrangian filtration, there is a homology 3-sphere which has a Heegaarddecomposition whose gluing diffeomorphism lies at that level, on which thisinvariant is non-zero. In a final section we examine the relationship betweenthe Johnson and Lagrangian filtrations.

The Lagrangian filtration of the mapping class group and finite-type invariants of homology spheres