Torsion elements in the associated graded of the $Y$-filtration of the monoid of homology cylinders

Clasper surgery induces the $Y$-filtration $\{Y_n\mathcal{I}\mathcal{C}{\}}_n$ over the monoid of homology cylinders, which serves as a $3$-dimensional analogue of the lower central series of the Torelli group of a surface. In this paper, we investigate the torsion submodules of the associated graded modules of these filtrations. To detect torsion elements, we introduce a homomorphism on $Y_n\mathcal{I}\mathcal{C}\mathrm{/}{Y}_{n+1}$ induced by the degree $n+2$ part of the LMO functor. Additionally, we provide a formula that computes this homomorphism under clasper surgery, and use it to demonstrate that every non-trivial torsion element in $Y_6\mathcal{I}\mathcal{C}\mathrm{/}{Y}_7$ has order $3$.