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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2005.05381 |
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| _version_ | 1866909464668405760 |
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| author | Conant, James Schneiderman, Rob Teichner, Peter |
| author_facet | Conant, James Schneiderman, Rob Teichner, Peter |
| contents | We show that for each $k\in\mathbb{N}$, a link $L\subset S^3$ bounds a degree $k$ Whitney tower in the 4-ball if and only if it is \emph{$C_k$-concordant} to the unlink. This means that $L$ is obtained from the unlink by a finite sequence of concordances and degree $k$ clasper surgeries. In our construction the trees associated to the Whitney towers coincide with the trees associated to the claspers.
As a corollary to our previous obstruction theory for Whitney towers in the 4-ball, it follows that the $C_k$-concordance filtration of links is classified in terms of Milnor invariants, higher-order Sato-Levine and Arf invariants.
Using a new notion of $k$-repeating twisted Whitney towers, we also classify a natural generalization of the notion of link homotopy, called twisted \emph{self $C_k$-concordance}, in terms of $k$-repeating Milnor invariants and $k$-repeating Arf invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2005_05381 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Clasper Concordance, Whitney towers and repeating Milnor invariants Conant, James Schneiderman, Rob Teichner, Peter Geometric Topology 57M25, 57M27 We show that for each $k\in\mathbb{N}$, a link $L\subset S^3$ bounds a degree $k$ Whitney tower in the 4-ball if and only if it is \emph{$C_k$-concordant} to the unlink. This means that $L$ is obtained from the unlink by a finite sequence of concordances and degree $k$ clasper surgeries. In our construction the trees associated to the Whitney towers coincide with the trees associated to the claspers. As a corollary to our previous obstruction theory for Whitney towers in the 4-ball, it follows that the $C_k$-concordance filtration of links is classified in terms of Milnor invariants, higher-order Sato-Levine and Arf invariants. Using a new notion of $k$-repeating twisted Whitney towers, we also classify a natural generalization of the notion of link homotopy, called twisted \emph{self $C_k$-concordance}, in terms of $k$-repeating Milnor invariants and $k$-repeating Arf invariants. |
| title | Clasper Concordance, Whitney towers and repeating Milnor invariants |
| topic | Geometric Topology 57M25, 57M27 |
| url | https://arxiv.org/abs/2005.05381 |