Whitney tower concordance and knots in homology spheres
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914091593891840 |
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| author | Davis, Christopher William |
| author_facet | Davis, Christopher William |
| contents | In a groundbreaking work A. Levine proved the surprising result that there exist knots in homology spheres which are not smoothly concordant to any knot in $S^3$, even if one allows for concordances in homology cobordisms. Since then subsequent works due to Hom-Levine-Lidman and Zhou have strengthened this result showing that there are many knots in homology spheres which are not smoothly concordant to knots in $S^3$. In this paper we present evidence that the opposite is true topologically. We study the Whitney tower filtration of concordance due to Cochran-Orr-Teichner and prove that modulo any term in this filtration every knot (or link) in a homology sphere is equivalent to a knot (or link) in $S^3$. As an application we recover the main result of [Davis2019], namely that the solvable filtration similarly fails to distinguish links in homology spheres from links in $S^3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_14509 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Whitney tower concordance and knots in homology spheres Davis, Christopher William Geometric Topology 57K10, 57N70 In a groundbreaking work A. Levine proved the surprising result that there exist knots in homology spheres which are not smoothly concordant to any knot in $S^3$, even if one allows for concordances in homology cobordisms. Since then subsequent works due to Hom-Levine-Lidman and Zhou have strengthened this result showing that there are many knots in homology spheres which are not smoothly concordant to knots in $S^3$. In this paper we present evidence that the opposite is true topologically. We study the Whitney tower filtration of concordance due to Cochran-Orr-Teichner and prove that modulo any term in this filtration every knot (or link) in a homology sphere is equivalent to a knot (or link) in $S^3$. As an application we recover the main result of [Davis2019], namely that the solvable filtration similarly fails to distinguish links in homology spheres from links in $S^3$. |
| title | Whitney tower concordance and knots in homology spheres |
| topic | Geometric Topology 57K10, 57N70 |
| url | https://arxiv.org/abs/2303.14509 |