On 2-complexes embeddable in 4-space, and the excluded minors of their underlying graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910574697250816 |
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| author | Georgakopoulos, Agelos Winter, Martin |
| author_facet | Georgakopoulos, Agelos Winter, Martin |
| contents | We study the potentially undecidable problem of whether a given 2-dimensional CW complex can be embedded into $\mathbb{R}^4$. We provide operations that preserve embeddability, including joining and cloning of 2-cells, as well as $Δ\mathrm Y$-transformations. We also construct a CW complex for which $\mathrm YΔ$-transformations do not preserve embeddability.
We use these results to study 4-flat graphs, i.e., graphs that embed in $\mathbb{R}^4$ after attaching any number of 2-cells to their cycles; a graph class that naturally generalizes planarity and linklessness. We verify several conjectures of van der Holst; in particular, we prove that each of the 78 graphs of the Heawood family is an excluded minor for the class of 4-flat graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12681 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On 2-complexes embeddable in 4-space, and the excluded minors of their underlying graphs Georgakopoulos, Agelos Winter, Martin Combinatorics Discrete Mathematics Geometric Topology 57Q35, 05C10, 05C83, 05C75 We study the potentially undecidable problem of whether a given 2-dimensional CW complex can be embedded into $\mathbb{R}^4$. We provide operations that preserve embeddability, including joining and cloning of 2-cells, as well as $Δ\mathrm Y$-transformations. We also construct a CW complex for which $\mathrm YΔ$-transformations do not preserve embeddability. We use these results to study 4-flat graphs, i.e., graphs that embed in $\mathbb{R}^4$ after attaching any number of 2-cells to their cycles; a graph class that naturally generalizes planarity and linklessness. We verify several conjectures of van der Holst; in particular, we prove that each of the 78 graphs of the Heawood family is an excluded minor for the class of 4-flat graphs. |
| title | On 2-complexes embeddable in 4-space, and the excluded minors of their underlying graphs |
| topic | Combinatorics Discrete Mathematics Geometric Topology 57Q35, 05C10, 05C83, 05C75 |
| url | https://arxiv.org/abs/2408.12681 |