Torelli group, Johnson kernel and invariants of homology spheres

In the late 1980's, it was shown that the Casson invariant appears in the difference between the two filtrations of the Torelli group: the lower central series and the Johnson filtration, and that its core part was identified with the secondary characteristic class $d_1$ associated with the fact that the first $\mathrm{MMM}$ class vanishes on the Torelli group (however it turned out that Johnson proved the former part highly likely prior to the above, see Remark 1.1). This secondary class $d_1$ is a rational generator of $H^1(\mathcal{K}_g;\mathbb{Z})^{\mathcal{M}_g}\cong\mathbb{Z}$ where $\mathcal{K}_g$ denotes the Johnson subgroup of the mapping class group $\mathcal{M}_g$. Hain proved, as a particular case of his fundamental result, that this is the only difference in degree $2$. In this paper, we prove that no other invariant than the above gives rise to new rational difference between the two filtrations up to degree $6$. We apply this to determine $H_1(\mathcal{K}_g;\mathbb{Q})$ explicitly by computing the description given by Dimca, Hain and Papadima. We also show that any finite type rational invariant of homology $3$-spheres of degrees up to $6$, including the second and the third Ohtsuki invariants, can be expressed by $d_1$ and lifts of Johnson homomorphisms.