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

TL;DR

The paper tackles the decidability and structure of embedding 2-dimensional CW complexes into $\\mathbb{R}^4$, focusing on the natural class of 4-flat graphs. It develops a toolkit of embeddability-preserving operations (joining, cloning, collapsing, edge cloning, rerouting, stellification, and $\\Delta\\mathrm{Y}$ transforms) and analyzes their impact on 4-flatness, including a counterexample showing a $\\mathrm{Y}\\Delta$ move can fail to preserve embeddability. A key contribution is the verification that all 78 Heawood graphs are excluded minors for 4-flat graphs, using both computer-assisted and manual case analyses, and the demonstration that the $\\Delta\\mathrm{Y}$ part of van der Holst's conjectures can fail in general. The results yield structural insights into how 4-flatness interacts with variants of full complexes, sliced embeddings, suspensions, and clique sums, advancing the understanding of the embedding problem in 4-space and guiding future work on Ex$(F)$ for this natural minor-closed class.

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 $Δ\mathrm Y$-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.

Figures (15)

• Figure 1: Three graphs from the Heawood family: $K_7$, $K_{3,3,1,1}$ and the Heawood graph. All three are not 4-flat.
• Figure 2: Visualization of $\Delta \mathrm{Y}$- and $\mathrm{Y}\Delta$-trans-for-ma-tions.
• Figure 3: Joining 2-cells across a path $\gamma$.
• Figure 4: Visualization of stellification of a 5-cycle in a 2-complex. The highlighted paths show how an attachment map $\partial c$ gets modified in the process.
• Figure 5: Starting from $K_7$ we first stellify a 4-cycle, then we delete two vertices to obtain a planar graph.

Theorems & Definitions (66)

• Conjecture 1.1: van2006graphs
• Conjecture 1.2: van2006graphs
• Conjecture 1.3: van2006graphs
• Conjecture 1.4: van2006graphs
• Theorem
• Theorem \ref{res:all_2_cells_vw}
• Theorem
• Theorem \ref{res:all_Heawood_are_excluded}
• Example 2.1: $\mathcal{K}_7-\Delta$
• Example 2.2