Shadow-complexity and trisection genus
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913453823754240 |
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| author | Naoe, Hironobu Ogawa, Masaki |
| author_facet | Naoe, Hironobu Ogawa, Masaki |
| contents | The shadow-complexity is an invariant of closed $4$-manifolds defined by using $2$-dimensional polyhedra called Turaev's shadows, which, roughly speaking, measures how complicated a $2$-skeleton of the $4$-manifold is. In this paper, we define a new version $\mathrm{sc}_{r}$ of shadow-complexity depending on an extra parameter $r\geq0$, and we investigate the relationship between this complexity and the trisection genus $g$. More explicitly, we prove an inequality $g(W) \leq 2+2\mathrm{sc}_{r}(W)$ for any closed $4$-manifold $W$ and any $r\geq1/2$. Moreover, we determine the exact values of $\mathrm{sc}_{1/2}$ for infinitely many $4$-manifolds, and also we classify all the closed $4$-manifolds with $\mathrm{sc}_{1/2}\leq1/2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_21265 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Shadow-complexity and trisection genus Naoe, Hironobu Ogawa, Masaki Geometric Topology 57K41, 57Q15 (Primary) 57R65 (Secondary) The shadow-complexity is an invariant of closed $4$-manifolds defined by using $2$-dimensional polyhedra called Turaev's shadows, which, roughly speaking, measures how complicated a $2$-skeleton of the $4$-manifold is. In this paper, we define a new version $\mathrm{sc}_{r}$ of shadow-complexity depending on an extra parameter $r\geq0$, and we investigate the relationship between this complexity and the trisection genus $g$. More explicitly, we prove an inequality $g(W) \leq 2+2\mathrm{sc}_{r}(W)$ for any closed $4$-manifold $W$ and any $r\geq1/2$. Moreover, we determine the exact values of $\mathrm{sc}_{1/2}$ for infinitely many $4$-manifolds, and also we classify all the closed $4$-manifolds with $\mathrm{sc}_{1/2}\leq1/2$. |
| title | Shadow-complexity and trisection genus |
| topic | Geometric Topology 57K41, 57Q15 (Primary) 57R65 (Secondary) |
| url | https://arxiv.org/abs/2407.21265 |