JournalsqtVol. 11, No. 2pp. 379–410

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
Torelli group, Johnson kernel, and invariants of homology spheres cover
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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 d1d_1 associated with the fact that the first MMM\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 d1d_1 is a rational generator of H1(Kg;Z)MgZH^1(\mathcal{K}_g;\mathbb Z)^{\mathcal{M}_g}\cong\mathbb Z where Kg\mathcal{K}_g denotes the Johnson subgroup of the mapping class group Mg\mathcal{M}_g. Hain proved, as a particular case of his fundamental result, that this is the only difference in degree 22. 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 66. We apply this to determine H1(Kg;Q)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 33-spheres of degrees up to 66, including the second and the third Ohtsuki invariants, can be expressed by d1d_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