Integral homology $3$-spheres and the Johnson filtration

The mapping class group of an oriented surface $\Sigma_{g,1}$ of genus $g$ with one boundary component has a natural decreasing filtration $\mathcal{M}_{g,1} \supset \mathcal{M}_{g,1}(1) \supset \mathcal{M}_{g,1}(2) \supset \mathcal{M}_{g,1}(3) \supset \cdots$, where $\mathcal{M}_{g,1}(k)$ is the kernel of the action of $\mathcal{M}_{g,1}$ on the $k^{th}$ nilpotent quotient of $\pi_1(\Sigma_{g,1})$. Using a tree Lie algebra approximating the graded Lie algebra $\bigoplus_{k} \mathcal{M}_{g,1}(k)/\mathcal{M}_{g,1}(k+1)$ we prove that any integral homology sphere of dimension $3$ has for some $g$ a Heegaard decomposition of the form $M = \mathcal{H}_g \coprod_{\iota_g \phi} - \mathcal{H}_g$, where $\phi \in \mathcal{M}_{g,1}(3)$ and $\iota_g$ is such that $\mathcal{H}_g \coprod_{\iota_g} - \mathcal{H}_g= S^3$. This proves a conjecture due to S. Morita and shows that the ``core'' of the Casson invariant is indeed the Casson invariant.