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We prove that all homology 3-spheres are $J_4$-equivalent, i.e. that anyhomology 3-sphere can be obtained from one another by twisting one of itsHeegaard splittings by an element of the mapping class group acting triviallyon the fourth nilpotent quotient of the fundamental group of the gluingsurface. We do so by exhibiting an element of $J_4$, the fourth term of theJohnson filtration of the mapping class group, on which (the core of) theCasson invariant takes the value $1$. In particular, this provides an explicitexample of an element of $J_4$ that is not a commutator of length $2$ in theTorelli group.

Triviality of the $J_4$-equivalence among homology 3-spheres