Lagrangian traces for the Johnson filtration of the handlebody group

Abstract: 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

S i g m a

. This definition depends on the choice of a Lagrangian of

H

, and we call these operators the emph{Lagrangian traces}. We suppose that

S i g m a

is the boundary of a handlebody with first homology group

H^{'}

, and we show that the Lagrangian traces corresponding to the Lagrangian

o p e r a t o r n a m e K e r (H i g h t a r r o w 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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