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

Abstract

We construct a series of homomorphisms from the $Y$-filtration on the monoid of homology cylinders to torsion modules via the mod $\mathbb{Z}$ reduction of the LMO functor. The restriction of our homomorphism to the lower central series of the Torelli group does 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 $Y_3{\mathcal{I}\mathcal{C}}_{g,1}/Y_4$ of the $Y$-filtration.

arXiv comment49 pages, 7 figures

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Cited by in corpus (4)

• Triviality of the $J_4$-equivalence among homology 3-spheres
• On the kernel of the surgery map restricted to the 1-loop part
• A non-commutative Reidemeister-Turaev torsion of homology cylinders
• Torsion elements in the associated graded of the $Y$-filtration of the monoid of homology cylinders