Abstract
In this paper, we define trace-like operators on a subspace of the space of derivations of the free Lie algebra generated by the first homology group H of a surface Σ. This definition depends on the choice of a Lagrangian of H, and we call these operators the Lagrangian traces. We suppose that Σ is the boundary of a handlebody with first homology group H′, and we show that, in degree greater than 2, the Lagrangian traces corresponding to the Lagrangian Ker(H→H′) vanish on the image by the Johnson homomorphisms of the elements of the Johnson filtration that extend to the handlebody.
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