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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

We construct a series of homomorphisms from the Y –filtration on the monoid of homology cylinders to torsion modules via the mod 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 3𝒞g,1Y 4 of the Y –filtration.

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Torelli group, Johnson kernel, homology cylinder, LMO functor, clasper, Jacobi diagram, Johnson homomorphism, Sato–Levine invariant
Mathematical Subject Classification
Primary: 57K16, 57K20
Secondary: 57K31
Received: 9 February 2020
Revised: 24 September 2020
Accepted: 26 October 2020
Published: 5 April 2022
Proposed: Ciprian Manolescu
Seconded: Benson Farb, András I Stipsicz
Yuta Nozaki
Organization for the Strategic Coordination of Research and Intellectual Properties
Meiji University
Graduate School of Advanced Science and Engineering
Hiroshima University
Masatoshi Sato
Department of Mathematics
Tokyo Denki University
Masaaki Suzuki
Department of Frontier Media Science
Meiji University