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

Abstract Clasper surgery induces the ‐filtration 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 induced by the degree part of the LMO functor. Additionally, we provide a formula that computes this homomorphism under clasper surgery, and use it to demonstrate that every nontrivial torsion element in has order 3.