Torelli group, Johnson kernel and invariants of homology spheres

TL;DR

This paper investigates the relationship between the Torelli group filtrations and invariants of homology spheres, showing that up to degree 6, no new invariants arise beyond the known secondary class $d_1$ and Johnson homomorphisms.

Contribution

It proves that no other rational invariants distinguish the filtrations beyond degree 6 and explicitly computes the first homology of the Johnson subgroup.

Findings

No new rational invariants up to degree 6 beyond $d_1$ and Johnson homomorphisms.

Explicit computation of $H_1( ext{Johnson subgroup}; extbf{Q})$.

All finite type rational invariants of homology 3-spheres up to degree 6 are generated by $d_1$ and Johnson lifts.

Abstract

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.