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

We construct a series of homomorphisms on the $Y$-filtration on the homology cylinders 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 $Y_3\mathcal{C}_{g,1}/Y_4$ of the $Y$-filtration.