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

TL;DR

This paper studies torsion elements in the graded modules of the Y-filtration on homology cylinders, revealing that non-trivial torsion at a certain level has order 3, using the LMO functor.

Contribution

It introduces a homomorphism based on the LMO functor to detect torsion and provides a formula for its computation under clasper surgery, advancing understanding of torsion in this context.

Findings

Non-trivial torsion elements in Y_6/ Y_7 have order 3.

A new homomorphism detects torsion in the associated graded modules.

A formula for the homomorphism under clasper surgery is established.

Abstract

Clasper surgery induces the $Y$-filtration $\{Y_n\mathcal{IC}\}_n$ 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 $Y_n\mathcal{IC}/Y_{n+1}$ 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 $Y_6\mathcal{IC}/Y_7$ has order $3$.