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

TL;DR

This paper investigates the embeddability of 2-complexes in four-dimensional space, introduces operations that preserve this property, and characterizes excluded minors within a class of graphs generalizing planarity.

Contribution

It develops new operations that preserve embeddability of 2-complexes in 4-space and identifies the 78 Heawood family graphs as excluded minors for 4-flat graphs.

Findings

Operations preserving embeddability are identified.

A complex is constructed where certain transformations do not preserve embeddability.

All 78 graphs of the Heawood family are excluded minors for 4-flat graphs.

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.