Given an oriented surface bounding a handlebody, we study the subgroup of its mapping class group defined as the intersection of the handlebody group and the second term of the Johnson filtration: A ∩ J 2 . We introduce two trace-like operators, inspired by Morita's trace, and show that their kernels coincide with the images by the second Johnson homomorphism τ 2 of J 2 and A ∩ J 2 , respectively. In particular, we answer by the negative to a question asked by Levine about an algebraic description of τ 2 (A ∩ J 2 ). By the same techniques, and for a Heegaard surface in S 3 , we also compute the image by τ 2 of the intersection of the Goeritz group G with J 2 .

We prove that all homology 3-spheres are J 4 J_4 -equivalent, i.e. that any homology 3-sphere can be obtained from one another by twisting one of its Heegaard splittings by an element of the mapping class group acting trivially on the fourth nilpotent quotient of the fundamental group of the gluing surface. We do so by exhibiting an element of J 4 J_4 , the fourth term of the Johnson filtration of the mapping class group, on which (the core of) the Casson invariant takes the value 1 1 . In particular, this provides an explicit example of an element of J 4 J_4 that is not a commutator of length 2 2 in the Torelli group.

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