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

Nozaki, Yuta; Sato, Masatoshi; Suzuki, Masaaki

doi:10.2140/gt.2022.26.221

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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 $Y_{3} ℐ 𝒞_{g, 1} ∕ Y_{4}$ of the $Y$ –filtration.

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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

References

Publication

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

Authors

Yuta Nozaki
Organization for the Strategic Coordination of Research and Intellectual Properties Meiji University Tokyo Japan	Graduate School of Advanced Science and Engineering Hiroshima University Hiroshima Japan

Masatoshi Sato
Department of Mathematics Tokyo Denki University Tokyo Japan

Masaaki Suzuki
Department of Frontier Media Science Meiji University Tokyo Japan

Volume 26, issue 1 (2022)