Equivariant concordance of periodic 2-knots in $S^4$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917358365310976 |
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| author | Bohm, Remy |
| author_facet | Bohm, Remy |
| contents | We show that the smooth equivariant concordance group of 2-knots in $S^4$ invariant under a linear $\mathbb{Z}/d\mathbb{Z}$ action is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ for all $d \geq 2$. This is in contrast to the non-equivariant case, in which all 2-knots are slice. We construct a new invariant for these 2-knots, which we call periodic, and show that it fully classifies them up to equivariant concordance. The invariant depends on a variation of the Arf invariant for null-homologous classical knots in arbitrary 3-manifolds with respect to a chosen spin structure. Our proof also shows an identical classification for certain annuli in $S^1 \times B^3$ up to concordance rel. boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16778 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equivariant concordance of periodic 2-knots in $S^4$ Bohm, Remy Geometric Topology We show that the smooth equivariant concordance group of 2-knots in $S^4$ invariant under a linear $\mathbb{Z}/d\mathbb{Z}$ action is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ for all $d \geq 2$. This is in contrast to the non-equivariant case, in which all 2-knots are slice. We construct a new invariant for these 2-knots, which we call periodic, and show that it fully classifies them up to equivariant concordance. The invariant depends on a variation of the Arf invariant for null-homologous classical knots in arbitrary 3-manifolds with respect to a chosen spin structure. Our proof also shows an identical classification for certain annuli in $S^1 \times B^3$ up to concordance rel. boundary. |
| title | Equivariant concordance of periodic 2-knots in $S^4$ |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2508.16778 |