Five tori in $S^4$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917987129229312 |
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| author | Martelli, Bruno |
| author_facet | Martelli, Bruno |
| contents | Ivansic proved that there is a link $L$ of five tori in $S^4$ with hyperbolic complement. We describe $L$ explicitly with pictures, study its properties, and discover that $L$ is in many aspects similar to the Borromean rings in $S^3$. In particular the following hold: (1) Any two tori in $L$ are unlinked, but three are not; (2) The complement $M = S^4 \setminus L$ is integral arithmetic hyperbolic; (3) The symmetry group of $L$ acts $k$-transitively on its components for all $k$; (4) The double branched covering over $L$ has geometry $\mathbb H^2 \times \mathbb H^2$; (5) The fundamental group of $M$ has a nice presentation via commutators; (6) The Alexander ideal has an explicit simple description; (7) Every class $x \in H^1(M,Z) = Z^5$ with non-zero xi is represented by a perfect circle-valued Morse function; (8) By longitudinal Dehn surgery along $L$ we get a closed 4-manifold with fundamental group $Z^5$; (9) The link $L$ can be put in perfect position. This leads also to the first descriptions of a cusped hyperbolic 4-manifold as a complement of tori in $\mathbb{RP}^4$ and as a complement of some explicit Lagrangian tori in the product of two surfaces of genus two. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03460 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Five tori in $S^4$ Martelli, Bruno Geometric Topology Differential Geometry Ivansic proved that there is a link $L$ of five tori in $S^4$ with hyperbolic complement. We describe $L$ explicitly with pictures, study its properties, and discover that $L$ is in many aspects similar to the Borromean rings in $S^3$. In particular the following hold: (1) Any two tori in $L$ are unlinked, but three are not; (2) The complement $M = S^4 \setminus L$ is integral arithmetic hyperbolic; (3) The symmetry group of $L$ acts $k$-transitively on its components for all $k$; (4) The double branched covering over $L$ has geometry $\mathbb H^2 \times \mathbb H^2$; (5) The fundamental group of $M$ has a nice presentation via commutators; (6) The Alexander ideal has an explicit simple description; (7) Every class $x \in H^1(M,Z) = Z^5$ with non-zero xi is represented by a perfect circle-valued Morse function; (8) By longitudinal Dehn surgery along $L$ we get a closed 4-manifold with fundamental group $Z^5$; (9) The link $L$ can be put in perfect position. This leads also to the first descriptions of a cusped hyperbolic 4-manifold as a complement of tori in $\mathbb{RP}^4$ and as a complement of some explicit Lagrangian tori in the product of two surfaces of genus two. |
| title | Five tori in $S^4$ |
| topic | Geometric Topology Differential Geometry |
| url | https://arxiv.org/abs/2401.03460 |