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arXiv · 2408.12681

On 2-complexes embeddable in 4-space, and the excluded minors of their underlying graphs

Agelos Georgakopoulos·Martin Winter
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Abstract

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 $\Delta\mathrm Y$-transformations. We also construct a CW complex for which $\mathrm Y\Delta$-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.

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Agelos Georgakopoulos, Martin Winter. 2024-08-22. On 2-complexes embeddable in 4-space, and the excluded minors of their underlying graphs. https://arxiv.org/abs/2408.12681

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