Torsion elements in the associated graded of the $Y$-filtration of the monoid of homology cylinders
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| Format: | Preprint |
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2024
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| _version_ | 1866909692213592064 |
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| author | Nozaki, Yuta Sato, Masatoshi Suzuki, Masaaki |
| author_facet | Nozaki, Yuta Sato, Masatoshi Suzuki, Masaaki |
| contents | 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 $3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_10061 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Torsion elements in the associated graded of the $Y$-filtration of the monoid of homology cylinders Nozaki, Yuta Sato, Masatoshi Suzuki, Masaaki Geometric Topology Quantum Algebra 57K16, 57K20 (Primary) 57K31 (Secondary) 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 $3$. |
| title | Torsion elements in the associated graded of the $Y$-filtration of the monoid of homology cylinders |
| topic | Geometric Topology Quantum Algebra 57K16, 57K20 (Primary) 57K31 (Secondary) |
| url | https://arxiv.org/abs/2410.10061 |