# Torelli group, Johnson kernel, and invariants of homology spheres

### Shigeyuki Morita

The University of Tokyo, Japan### Takuya Sakasai

The University of Tokyo, Japan### Masaaki Suzuki

Meiji University, Tokyo, Japan

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## Abstract

In the late 1980s, 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 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.

## Cite this article

Shigeyuki Morita, Takuya Sakasai, Masaaki Suzuki, Torelli group, Johnson kernel, and invariants of homology spheres. Quantum Topol. 11 (2020), no. 2, pp. 379–410

DOI 10.4171/QT/138