  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 $Y_3\mathcal{IC}_{g,1}/Y_4$ of the $Y$-filtration.
