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

Yuta Nozaki; Masatoshi Sato; Masaaki Suzuki

doi:10.1112/topo.70028

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arXiv · 2410.10061

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

Yuta Nozaki·Masatoshi Sato·Masaaki Suzuki

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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 $3$.

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Yuta Nozaki, Masatoshi Sato, Masaaki Suzuki. 2024-10-14. Torsion elements in the associated graded of the $Y$-filtration of the monoid of homology cylinders. https://doi.org/10.1112/topo.70028

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