A High-Dimensional Extension of Wagner's Theorem and the Geometrization of Hypergraphs

This paper introduces a geometric representation of hypergraphs by representing hyperedges as simplices. Building on this framework, we employ homotopy groups to analyze the topological structure of hypergraphs embedded in high-dimensional Euclidean spaces. Leveraging this foundation, we extend Wagner's theorem to Rᵈ. Specifically, we establish that a triangulated d-uniform topological hypergraph embeds into Rᵈ if and only if it contains neither Kd+3ᵈ nor K3,d+1ᵈ as a minor. Here, a triangulated d-uniform topological hypergraph constitutes a geometrized form of a d-uniform hypergraph, while Kd+3ᵈ and K3,d+1ᵈ are the high-dimensional generalizations of the complete graph K₅ and the complete bipartite graph K3,3 in Rᵈ, respectively.