Volume 26, issue 1 (2022)

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Abelian quotients of the $Y$–filtration on the homology cylinders via the LMO functor

Yuta Nozaki, Masatoshi Sato and Masaaki Suzuki

Geometry & Topology 26 (2022) 221–282
DOI: 10.2140/gt.2022.26.221
Abstract

We construct a series of homomorphisms from the $Y$–filtration on the monoid of homology cylinders to torsion modules via the mod $ℤ$ reduction of the LMO functor. The restrictions of our homomorphisms to the lower central series of the Torelli group do not factor through Morita’s refinement of the Johnson homomorphism. We use it to show that the abelianization of the Johnson kernel of a closed surface has torsion elements. We also determine the third graded quotient ${Y}_{3}\mathsc{ℐ}{\mathsc{𝒞}}_{g,1}∕{Y}_{4}$ of the $Y$–filtration.

Keywords
Torelli group, Johnson kernel, homology cylinder, LMO functor, clasper, Jacobi diagram, Johnson homomorphism, Sato–Levine invariant
Mathematical Subject Classification
Primary: 57K16, 57K20
Secondary: 57K31